%------------------------------------------------------------------------------
% File     : ITP246^3 : TPTP v8.2.0. Released v8.1.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem VEBT_Succ 00795_053196
% 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_00795_053196 [Des22]

% Status   : Theorem
% Rating   : 0.50 v8.2.0, 0.69 v8.1.0
% Syntax   : Number of formulae    : 11234 (6275 unt; 977 typ;   0 def)
%            Number of atoms       : 26809 (11987 equ;   0 cnn)
%            Maximal formula atoms :   71 (   2 avg)
%            Number of connectives : 104965 (2569   ~; 512   |;1523   &;91781   @)
%                                         (   0 <=>;8580  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   39 (   5 avg)
%            Number of types       :   85 (  84 usr)
%            Number of type conns  : 3106 (3106   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  896 ( 893 usr;  69 con; 0-8 aty)
%            Number of variables   : 23057 (1889   ^;20505   !; 663   ?;23057   :)
% SPC      : TH0_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 01:26:34.367
%------------------------------------------------------------------------------
% Could-be-implicit typings (84)
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thf(ty_n_t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Extended____Nat__Oenat_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J,type,
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thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J,type,
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thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_M_Eo_J_J,type,
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thf(ty_n_t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__Nat__Onat_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_M_Eo_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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thf(ty_n_t__Set__Oset_It__Extended____Nat__Oenat_J,type,
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thf(ty_n_t__List__Olist_It__Complex__Ocomplex_J,type,
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thf(ty_n_t__Set__Oset_It__Complex__Ocomplex_J,type,
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thf(ty_n_t__Filter__Ofilter_It__Real__Oreal_J,type,
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thf(ty_n_t__Option__Ooption_It__Num__Onum_J,type,
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thf(ty_n_t__Option__Ooption_It__Nat__Onat_J,type,
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thf(ty_n_t__Filter__Ofilter_It__Nat__Onat_J,type,
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thf(ty_n_t__Set__Oset_It__String__Ochar_J,type,
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thf(ty_n_t__List__Olist_It__Real__Oreal_J,type,
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thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
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thf(ty_n_t__List__Olist_It__Num__Onum_J,type,
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thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
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thf(ty_n_t__List__Olist_It__Int__Oint_J,type,
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thf(ty_n_t__VEBT____Definitions__OVEBT,type,
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thf(ty_n_t__Set__Oset_It__Rat__Orat_J,type,
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thf(ty_n_t__Set__Oset_It__Num__Onum_J,type,
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thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
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thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
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thf(ty_n_t__Code____Numeral__Ointeger,type,
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thf(ty_n_t__Product____Type__Ounit,type,
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thf(ty_n_t__Extended____Nat__Oenat,type,
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thf(ty_n_t__List__Olist_I_Eo_J,type,
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thf(ty_n_t__Complex__Ocomplex,type,
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thf(ty_n_t__String__Oliteral,type,
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thf(ty_n_t__Set__Oset_I_Eo_J,type,
    set_o: $tType ).

thf(ty_n_t__String__Ochar,type,
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thf(ty_n_t__Real__Oreal,type,
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thf(ty_n_t__Rat__Orat,type,
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thf(ty_n_t__Num__Onum,type,
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thf(ty_n_t__Nat__Onat,type,
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thf(ty_n_t__Int__Oint,type,
    int: $tType ).

% Explicit typings (893)
thf(sy_c_Archimedean__Field_Oceiling_001t__Rat__Orat,type,
    archim2889992004027027881ng_rat: rat > int ).

thf(sy_c_Archimedean__Field_Oceiling_001t__Real__Oreal,type,
    archim7802044766580827645g_real: real > int ).

thf(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_001t__Rat__Orat,type,
    archim3151403230148437115or_rat: rat > int ).

thf(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_001t__Real__Oreal,type,
    archim6058952711729229775r_real: real > int ).

thf(sy_c_Archimedean__Field_Ofrac_001t__Rat__Orat,type,
    archimedean_frac_rat: rat > rat ).

thf(sy_c_Archimedean__Field_Ofrac_001t__Real__Oreal,type,
    archim2898591450579166408c_real: real > real ).

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

thf(sy_c_Archimedean__Field_Oround_001t__Real__Oreal,type,
    archim8280529875227126926d_real: real > int ).

thf(sy_c_BNF__Def_Orel__fun_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001_062_I_062_It__Nat__Onat_Mt__Rat__Orat_J_M_062_It__Nat__Onat_Mt__Rat__Orat_J_J_001_062_I_062_It__Nat__Onat_Mt__Rat__Orat_J_M_062_It__Nat__Onat_Mt__Rat__Orat_J_J,type,
    bNF_re1962705104956426057at_rat: ( ( nat > rat ) > ( nat > rat ) > $o ) > ( ( ( nat > rat ) > nat > rat ) > ( ( nat > rat ) > nat > rat ) > $o ) > ( ( nat > rat ) > ( nat > rat ) > nat > rat ) > ( ( nat > rat ) > ( nat > rat ) > nat > rat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001_062_It__Nat__Onat_Mt__Rat__Orat_J,type,
    bNF_re895249473297799549at_rat: ( ( nat > rat ) > ( nat > rat ) > $o ) > ( ( nat > rat ) > ( nat > rat ) > $o ) > ( ( nat > rat ) > nat > rat ) > ( ( nat > rat ) > nat > rat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001_Eo_001_Eo,type,
    bNF_re728719798268516973at_o_o: ( ( nat > rat ) > ( nat > rat ) > $o ) > ( $o > $o > $o ) > ( ( nat > rat ) > $o ) > ( ( nat > rat ) > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001t__Real__Oreal_001_062_I_062_It__Nat__Onat_Mt__Rat__Orat_J_M_062_It__Nat__Onat_Mt__Rat__Orat_J_J_001_062_It__Real__Oreal_Mt__Real__Oreal_J,type,
    bNF_re4695409256820837752l_real: ( ( nat > rat ) > real > $o ) > ( ( ( nat > rat ) > nat > rat ) > ( real > real ) > $o ) > ( ( nat > rat ) > ( nat > rat ) > nat > rat ) > ( real > real > real ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001t__Real__Oreal_001_062_I_062_It__Nat__Onat_Mt__Rat__Orat_J_M_Eo_J_001_062_It__Real__Oreal_M_Eo_J,type,
    bNF_re4521903465945308077real_o: ( ( nat > rat ) > real > $o ) > ( ( ( nat > rat ) > $o ) > ( real > $o ) > $o ) > ( ( nat > rat ) > ( nat > rat ) > $o ) > ( real > real > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001t__Real__Oreal_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001t__Real__Oreal,type,
    bNF_re3023117138289059399t_real: ( ( nat > rat ) > real > $o ) > ( ( nat > rat ) > real > $o ) > ( ( nat > rat ) > nat > rat ) > ( real > real ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001t__Real__Oreal_001_Eo_001_Eo,type,
    bNF_re4297313714947099218al_o_o: ( ( nat > rat ) > real > $o ) > ( $o > $o > $o ) > ( ( nat > rat ) > $o ) > ( real > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001_062_It__Int__Oint_M_Eo_J_001_062_It__Int__Oint_M_Eo_J,type,
    bNF_re3403563459893282935_int_o: ( int > int > $o ) > ( ( int > $o ) > ( int > $o ) > $o ) > ( int > int > $o ) > ( int > int > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001_062_It__Int__Oint_Mt__Int__Oint_J_001_062_It__Int__Oint_Mt__Int__Oint_J,type,
    bNF_re711492959462206631nt_int: ( int > int > $o ) > ( ( int > int ) > ( int > int ) > $o ) > ( int > int > int ) > ( int > int > int ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001_Eo_001_Eo,type,
    bNF_re5089333283451836215nt_o_o: ( int > int > $o ) > ( $o > $o > $o ) > ( int > $o ) > ( int > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint,type,
    bNF_re4712519889275205905nt_int: ( int > int > $o ) > ( int > int > $o ) > ( int > int ) > ( int > int ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001_062_It__Nat__Onat_M_Eo_J_001_062_It__Nat__Onat_M_Eo_J,type,
    bNF_re578469030762574527_nat_o: ( nat > nat > $o ) > ( ( nat > $o ) > ( nat > $o ) > $o ) > ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
    bNF_re1345281282404953727at_nat: ( nat > nat > $o ) > ( ( nat > nat ) > ( nat > nat ) > $o ) > ( nat > nat > nat ) > ( nat > nat > nat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001_Eo_001_Eo,type,
    bNF_re4705727531993890431at_o_o: ( nat > nat > $o ) > ( $o > $o > $o ) > ( nat > $o ) > ( nat > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
    bNF_re5653821019739307937at_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > ( nat > nat ) > ( nat > nat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Num__Onum_001t__Num__Onum_001_062_It__Num__Onum_Mt__Int__Oint_J_001_062_It__Num__Onum_Mt__Int__Oint_J,type,
    bNF_re8402795839162346335um_int: ( num > num > $o ) > ( ( num > int ) > ( num > int ) > $o ) > ( num > num > int ) > ( num > num > int ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Num__Onum_001t__Num__Onum_001t__Int__Oint_001t__Int__Oint,type,
    bNF_re1822329894187522285nt_int: ( num > num > $o ) > ( int > int > $o ) > ( num > int ) > ( num > int ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    bNF_re5228765855967844073nt_int: ( product_prod_int_int > product_prod_int_int > $o ) > ( ( product_prod_int_int > product_prod_int_int ) > ( product_prod_int_int > product_prod_int_int ) > $o ) > ( product_prod_int_int > product_prod_int_int > product_prod_int_int ) > ( product_prod_int_int > product_prod_int_int > product_prod_int_int ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001_Eo_001_Eo,type,
    bNF_re8699439704749558557nt_o_o: ( product_prod_int_int > product_prod_int_int > $o ) > ( $o > $o > $o ) > ( product_prod_int_int > $o ) > ( product_prod_int_int > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    bNF_re7145576690424134365nt_int: ( product_prod_int_int > product_prod_int_int > $o ) > ( product_prod_int_int > product_prod_int_int > $o ) > ( product_prod_int_int > product_prod_int_int ) > ( product_prod_int_int > product_prod_int_int ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat_001_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001_062_It__Rat__Orat_Mt__Rat__Orat_J,type,
    bNF_re7627151682743391978at_rat: ( product_prod_int_int > rat > $o ) > ( ( product_prod_int_int > product_prod_int_int ) > ( rat > rat ) > $o ) > ( product_prod_int_int > product_prod_int_int > product_prod_int_int ) > ( rat > rat > rat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat_001_Eo_001_Eo,type,
    bNF_re1494630372529172596at_o_o: ( product_prod_int_int > rat > $o ) > ( $o > $o > $o ) > ( product_prod_int_int > $o ) > ( rat > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat,type,
    bNF_re8279943556446156061nt_rat: ( product_prod_int_int > rat > $o ) > ( product_prod_int_int > rat > $o ) > ( product_prod_int_int > product_prod_int_int ) > ( rat > rat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_001_062_It__Int__Oint_M_Eo_J,type,
    bNF_re717283939379294677_int_o: ( product_prod_nat_nat > int > $o ) > ( ( product_prod_nat_nat > $o ) > ( int > $o ) > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( int > int > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001_062_It__Int__Oint_Mt__Int__Oint_J,type,
    bNF_re7408651293131936558nt_int: ( product_prod_nat_nat > int > $o ) > ( ( product_prod_nat_nat > product_prod_nat_nat ) > ( int > int ) > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > ( int > int > int ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint_001_Eo_001_Eo,type,
    bNF_re6644619430987730960nt_o_o: ( product_prod_nat_nat > int > $o ) > ( $o > $o > $o ) > ( product_prod_nat_nat > $o ) > ( int > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint_001t__Nat__Onat_001t__Nat__Onat,type,
    bNF_re4555766996558763186at_nat: ( product_prod_nat_nat > int > $o ) > ( nat > nat > $o ) > ( product_prod_nat_nat > nat ) > ( int > nat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint,type,
    bNF_re7400052026677387805at_int: ( product_prod_nat_nat > int > $o ) > ( product_prod_nat_nat > int > $o ) > ( product_prod_nat_nat > product_prod_nat_nat ) > ( int > int ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
    bNF_re4202695980764964119_nat_o: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( ( product_prod_nat_nat > $o ) > ( product_prod_nat_nat > $o ) > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    bNF_re3099431351363272937at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( ( product_prod_nat_nat > product_prod_nat_nat ) > ( product_prod_nat_nat > product_prod_nat_nat ) > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_Eo_001_Eo,type,
    bNF_re3666534408544137501at_o_o: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( $o > $o > $o ) > ( product_prod_nat_nat > $o ) > ( product_prod_nat_nat > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat_001t__Nat__Onat,type,
    bNF_re8246922863344978751at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( nat > nat > $o ) > ( product_prod_nat_nat > nat ) > ( product_prod_nat_nat > nat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    bNF_re2241393799969408733at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( product_prod_nat_nat > product_prod_nat_nat ) > ( product_prod_nat_nat > product_prod_nat_nat ) > $o ).

thf(sy_c_Binomial_Obinomial,type,
    binomial: nat > nat > nat ).

thf(sy_c_Binomial_Ogbinomial_001t__Complex__Ocomplex,type,
    gbinomial_complex: complex > nat > complex ).

thf(sy_c_Binomial_Ogbinomial_001t__Int__Oint,type,
    gbinomial_int: int > nat > int ).

thf(sy_c_Binomial_Ogbinomial_001t__Nat__Onat,type,
    gbinomial_nat: nat > nat > nat ).

thf(sy_c_Binomial_Ogbinomial_001t__Rat__Orat,type,
    gbinomial_rat: rat > nat > rat ).

thf(sy_c_Binomial_Ogbinomial_001t__Real__Oreal,type,
    gbinomial_real: real > nat > real ).

thf(sy_c_Bit__Operations_Oand__int__rel,type,
    bit_and_int_rel: product_prod_int_int > product_prod_int_int > $o ).

thf(sy_c_Bit__Operations_Oand__not__num,type,
    bit_and_not_num: num > num > option_num ).

thf(sy_c_Bit__Operations_Oand__not__num__rel,type,
    bit_and_not_num_rel: product_prod_num_num > product_prod_num_num > $o ).

thf(sy_c_Bit__Operations_Oconcat__bit,type,
    bit_concat_bit: nat > int > int > int ).

thf(sy_c_Bit__Operations_Oor__not__num__neg,type,
    bit_or_not_num_neg: num > num > num ).

thf(sy_c_Bit__Operations_Oor__not__num__neg__rel,type,
    bit_or3848514188828904588eg_rel: product_prod_num_num > product_prod_num_num > $o ).

thf(sy_c_Bit__Operations_Oring__bit__operations__class_Onot_001t__Code____Numeral__Ointeger,type,
    bit_ri7632146776885996613nteger: code_integer > code_integer ).

thf(sy_c_Bit__Operations_Oring__bit__operations__class_Onot_001t__Int__Oint,type,
    bit_ri7919022796975470100ot_int: int > int ).

thf(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit_001t__Code____Numeral__Ointeger,type,
    bit_ri6519982836138164636nteger: nat > code_integer > code_integer ).

thf(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit_001t__Int__Oint,type,
    bit_ri631733984087533419it_int: nat > int > int ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand_001t__Code____Numeral__Ointeger,type,
    bit_se3949692690581998587nteger: code_integer > code_integer > code_integer ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand_001t__Int__Oint,type,
    bit_se725231765392027082nd_int: int > int > int ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand_001t__Nat__Onat,type,
    bit_se727722235901077358nd_nat: nat > nat > nat ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit_001t__Int__Oint,type,
    bit_se8568078237143864401it_int: nat > int > int ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit_001t__Nat__Onat,type,
    bit_se8570568707652914677it_nat: nat > nat > nat ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit_001t__Code____Numeral__Ointeger,type,
    bit_se1345352211410354436nteger: nat > code_integer > code_integer ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit_001t__Int__Oint,type,
    bit_se2159334234014336723it_int: nat > int > int ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit_001t__Nat__Onat,type,
    bit_se2161824704523386999it_nat: nat > nat > nat ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask_001t__Code____Numeral__Ointeger,type,
    bit_se2119862282449309892nteger: nat > code_integer ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask_001t__Int__Oint,type,
    bit_se2000444600071755411sk_int: nat > int ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask_001t__Nat__Onat,type,
    bit_se2002935070580805687sk_nat: nat > nat ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor_001t__Int__Oint,type,
    bit_se1409905431419307370or_int: int > int > int ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor_001t__Nat__Onat,type,
    bit_se1412395901928357646or_nat: nat > nat > nat ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit_001t__Code____Numeral__Ointeger,type,
    bit_se7788150548672797655nteger: nat > code_integer > code_integer ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit_001t__Int__Oint,type,
    bit_se545348938243370406it_int: nat > int > int ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit_001t__Nat__Onat,type,
    bit_se547839408752420682it_nat: nat > nat > nat ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__Code____Numeral__Ointeger,type,
    bit_se2793503036327961859nteger: nat > code_integer > code_integer ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__Int__Oint,type,
    bit_se7879613467334960850it_int: nat > int > int ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__Nat__Onat,type,
    bit_se7882103937844011126it_nat: nat > nat > nat ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit_001t__Code____Numeral__Ointeger,type,
    bit_se1745604003318907178nteger: nat > code_integer > code_integer ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit_001t__Int__Oint,type,
    bit_se2923211474154528505it_int: nat > int > int ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit_001t__Nat__Onat,type,
    bit_se2925701944663578781it_nat: nat > nat > nat ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit_001t__Code____Numeral__Ointeger,type,
    bit_se8260200283734997820nteger: nat > code_integer > code_integer ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit_001t__Int__Oint,type,
    bit_se4203085406695923979it_int: nat > int > int ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit_001t__Nat__Onat,type,
    bit_se4205575877204974255it_nat: nat > nat > nat ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor_001t__Code____Numeral__Ointeger,type,
    bit_se3222712562003087583nteger: code_integer > code_integer > code_integer ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor_001t__Int__Oint,type,
    bit_se6526347334894502574or_int: int > int > int ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor_001t__Nat__Onat,type,
    bit_se6528837805403552850or_nat: nat > nat > nat ).

thf(sy_c_Bit__Operations_Osemiring__bits__class_Obit_001t__Code____Numeral__Ointeger,type,
    bit_se9216721137139052372nteger: code_integer > nat > $o ).

thf(sy_c_Bit__Operations_Osemiring__bits__class_Obit_001t__Int__Oint,type,
    bit_se1146084159140164899it_int: int > nat > $o ).

thf(sy_c_Bit__Operations_Osemiring__bits__class_Obit_001t__Nat__Onat,type,
    bit_se1148574629649215175it_nat: nat > nat > $o ).

thf(sy_c_Bit__Operations_Otake__bit__num,type,
    bit_take_bit_num: nat > num > option_num ).

thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_Oand__num,type,
    bit_un1837492267222099188nd_num: num > num > option_num ).

thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_Oand__num__rel,type,
    bit_un5425074673868309765um_rel: product_prod_num_num > product_prod_num_num > $o ).

thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_Oxor__num,type,
    bit_un6178654185764691216or_num: num > num > option_num ).

thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_Oxor__num__rel,type,
    bit_un3595099601533988841um_rel: product_prod_num_num > product_prod_num_num > $o ).

thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oand__num,type,
    bit_un7362597486090784418nd_num: num > num > option_num ).

thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oand__num__rel,type,
    bit_un4731106466462545111um_rel: product_prod_num_num > product_prod_num_num > $o ).

thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oxor__num,type,
    bit_un2480387367778600638or_num: num > num > option_num ).

thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oxor__num__rel,type,
    bit_un2901131394128224187um_rel: product_prod_num_num > product_prod_num_num > $o ).

thf(sy_c_Code__Numeral_Obit__cut__integer,type,
    code_bit_cut_integer: code_integer > produc6271795597528267376eger_o ).

thf(sy_c_Code__Numeral_Odivmod__abs,type,
    code_divmod_abs: code_integer > code_integer > produc8923325533196201883nteger ).

thf(sy_c_Code__Numeral_Odivmod__integer,type,
    code_divmod_integer: code_integer > code_integer > produc8923325533196201883nteger ).

thf(sy_c_Code__Numeral_Ointeger_Oint__of__integer,type,
    code_int_of_integer: code_integer > int ).

thf(sy_c_Code__Numeral_Ointeger_Ointeger__of__int,type,
    code_integer_of_int: int > code_integer ).

thf(sy_c_Code__Numeral_Ointeger__of__num,type,
    code_integer_of_num: num > code_integer ).

thf(sy_c_Code__Numeral_Onat__of__integer,type,
    code_nat_of_integer: code_integer > nat ).

thf(sy_c_Code__Numeral_Onum__of__integer,type,
    code_num_of_integer: code_integer > num ).

thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Extended____Nat__Oenat,type,
    comple2295165028678016749d_enat: set_Extended_enat > extended_enat ).

thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Filter__Ofilter_It__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_J,type,
    comple8358262395181532106omplex: set_fi4554929511873752355omplex > filter6041513312241820739omplex ).

thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Filter__Ofilter_It__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
    comple2936214249959783750l_real: set_fi7789364187291644575l_real > filter2146258269922977983l_real ).

thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Real__Oreal,type,
    comple4887499456419720421f_real: set_real > real ).

thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Set__Oset_It__Nat__Onat_J,type,
    comple7806235888213564991et_nat: set_set_nat > set_nat ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Extended____Nat__Oenat,type,
    comple4398354569131411667d_enat: set_Extended_enat > extended_enat ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Real__Oreal,type,
    comple1385675409528146559p_real: set_real > real ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Nat__Onat_J,type,
    comple7399068483239264473et_nat: set_set_nat > set_nat ).

thf(sy_c_Complex_OArg,type,
    arg: complex > real ).

thf(sy_c_Complex_Ocis,type,
    cis: real > complex ).

thf(sy_c_Complex_Ocnj,type,
    cnj: complex > complex ).

thf(sy_c_Complex_Ocomplex_OComplex,type,
    complex2: real > real > complex ).

thf(sy_c_Complex_Ocomplex_OIm,type,
    im: complex > real ).

thf(sy_c_Complex_Ocomplex_ORe,type,
    re: complex > real ).

thf(sy_c_Complex_Ocsqrt,type,
    csqrt: complex > complex ).

thf(sy_c_Complex_Oimaginary__unit,type,
    imaginary_unit: complex ).

thf(sy_c_Complex_Orcis,type,
    rcis: real > real > complex ).

thf(sy_c_Deriv_Odifferentiable_001t__Real__Oreal_001t__Real__Oreal,type,
    differ6690327859849518006l_real: ( real > real ) > filter_real > $o ).

thf(sy_c_Deriv_Ohas__derivative_001t__Real__Oreal_001t__Real__Oreal,type,
    has_de1759254742604945161l_real: ( real > real ) > ( real > real ) > filter_real > $o ).

thf(sy_c_Deriv_Ohas__field__derivative_001t__Real__Oreal,type,
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thf(sy_c_Deriv_Ohas__vector__derivative_001t__Real__Oreal,type,
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thf(sy_c_Divides_Oadjust__div,type,
    adjust_div: product_prod_int_int > int ).

thf(sy_c_Divides_Oadjust__mod,type,
    adjust_mod: int > int > int ).

thf(sy_c_Divides_Odivmod__nat,type,
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thf(sy_c_Divides_Oeucl__rel__int,type,
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thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux_001t__Int__Oint,type,
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thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux_001t__Nat__Onat,type,
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thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod_001t__Int__Oint,type,
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thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod_001t__Nat__Onat,type,
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thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step_001t__Int__Oint,type,
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thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step_001t__Nat__Onat,type,
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thf(sy_c_Euclidean__Division_Oeuclidean__semiring__class_Oeuclidean__size_001t__Int__Oint,type,
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thf(sy_c_Euclidean__Division_Oeuclidean__semiring__class_Oeuclidean__size_001t__Nat__Onat,type,
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    euclid3395696857347342551nt_int: int > int ).

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thf(sy_c_Extended__Nat_OeSuc,type,
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thf(sy_c_Extended__Nat_Oenat,type,
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thf(sy_c_Extended__Nat_Oenat_Ocase__enat_001_Eo,type,
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thf(sy_c_Extended__Nat_Oenat_Ocase__enat_001t__Extended____Nat__Oenat,type,
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thf(sy_c_Extended__Nat_Othe__enat,type,
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thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Rat__Orat,type,
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thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Real__Oreal,type,
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thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Rat__Orat,type,
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thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Real__Oreal,type,
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thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Rat__Orat,type,
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thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Real__Oreal,type,
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thf(sy_c_Filter_Oat__bot_001t__Real__Oreal,type,
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thf(sy_c_Filter_Oat__top_001t__Nat__Onat,type,
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thf(sy_c_Filter_Oat__top_001t__Real__Oreal,type,
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thf(sy_c_Filter_Oeventually_001t__Nat__Onat,type,
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thf(sy_c_Filter_Oeventually_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_Filter_Oeventually_001t__Real__Oreal,type,
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thf(sy_c_Filter_Ofilterlim_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_Filter_Ofilterlim_001t__Nat__Onat_001t__Real__Oreal,type,
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thf(sy_c_Filter_Ofilterlim_001t__Real__Oreal_001t__Real__Oreal,type,
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thf(sy_c_Filter_Ofiltermap_001t__Real__Oreal_001t__Real__Oreal,type,
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thf(sy_c_Filter_Oprincipal_001t__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J,type,
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thf(sy_c_Filter_Oprod__filter_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_Finite__Set_Ocard_001_Eo,type,
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thf(sy_c_Finite__Set_Ocard_001t__Complex__Ocomplex,type,
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thf(sy_c_Finite__Set_Ocard_001t__Int__Oint,type,
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thf(sy_c_Finite__Set_Ocard_001t__List__Olist_It__Nat__Onat_J,type,
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thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
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thf(sy_c_Finite__Set_Ofinite_001t__Extended____Nat__Oenat,type,
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thf(sy_c_Finite__Set_Ofinite_001t__Int__Oint,type,
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thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
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thf(sy_c_Fun_Obij__betw_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
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thf(sy_c_Fun_Obij__betw_001t__Nat__Onat_001t__Complex__Ocomplex,type,
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thf(sy_c_Fun_Obij__betw_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_Fun_Ocomp_001_062_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Fun_Ocomp_001t__Int__Oint_001t__Nat__Onat_001t__Int__Oint,type,
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thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Real__Oreal_001t__Nat__Onat,type,
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thf(sy_c_Fun_Oid_001_Eo,type,
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thf(sy_c_Fun_Oid_001t__Nat__Onat,type,
    id_nat: nat > nat ).

thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__String__Ochar,type,
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thf(sy_c_Fun_Oinj__on_001t__Real__Oreal_001t__Real__Oreal,type,
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thf(sy_c_Fun_Omap__fun_001t__Int__Oint_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_001_062_It__Int__Oint_M_Eo_J,type,
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thf(sy_c_Fun_Omap__fun_001t__Int__Oint_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat_001t__Nat__Onat,type,
    map_fu2345160673673942751at_nat: ( int > product_prod_nat_nat ) > ( nat > nat ) > ( product_prod_nat_nat > nat ) > int > nat ).

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thf(sy_c_Fun_Omap__fun_001t__Rat__Orat_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001_062_It__Rat__Orat_Mt__Rat__Orat_J,type,
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thf(sy_c_Fun_Omap__fun_001t__Rat__Orat_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat,type,
    map_fu5673905371560938248nt_rat: ( rat > product_prod_int_int ) > ( product_prod_int_int > rat ) > ( product_prod_int_int > product_prod_int_int ) > rat > rat ).

thf(sy_c_Fun_Omap__fun_001t__Real__Oreal_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001_062_I_062_It__Nat__Onat_Mt__Rat__Orat_J_M_062_It__Nat__Onat_Mt__Rat__Orat_J_J_001_062_It__Real__Oreal_Mt__Real__Oreal_J,type,
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thf(sy_c_Fun_Omap__fun_001t__Real__Oreal_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001t__Real__Oreal,type,
    map_fu7146612038024189824t_real: ( real > nat > rat ) > ( ( nat > rat ) > real ) > ( ( nat > rat ) > nat > rat ) > real > real ).

thf(sy_c_Fun_Omap__fun_001t__Real__Oreal_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001_Eo_001_Eo,type,
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thf(sy_c_Fun_Ostrict__mono__on_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_Fun_Othe__inv__into_001t__Real__Oreal_001t__Real__Oreal,type,
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thf(sy_c_Fun__Def_Opair__leq,type,
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thf(sy_c_Fun__Def_Opair__less,type,
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thf(sy_c_GCD_OGcd__class_OGcd_001t__Nat__Onat,type,
    gcd_Gcd_nat: set_nat > nat ).

thf(sy_c_GCD_Obezw,type,
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thf(sy_c_GCD_Obezw__rel,type,
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thf(sy_c_GCD_Ogcd__class_Ogcd_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_GCD_Ogcd__class_Ogcd_001t__Int__Oint,type,
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thf(sy_c_GCD_Ogcd__class_Ogcd_001t__Nat__Onat,type,
    gcd_gcd_nat: nat > nat > nat ).

thf(sy_c_GCD_Ogcd__class_Olcm_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_GCD_Ogcd__class_Olcm_001t__Int__Oint,type,
    gcd_lcm_int: int > int > int ).

thf(sy_c_GCD_Ogcd__class_Olcm_001t__Nat__Onat,type,
    gcd_lcm_nat: nat > nat > nat ).

thf(sy_c_GCD_Ogcd__nat__rel,type,
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thf(sy_c_Groups_Oabs__class_Oabs_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Groups_Oabs__class_Oabs_001t__Complex__Ocomplex,type,
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thf(sy_c_Groups_Oabs__class_Oabs_001t__Int__Oint,type,
    abs_abs_int: int > int ).

thf(sy_c_Groups_Oabs__class_Oabs_001t__Rat__Orat,type,
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thf(sy_c_Groups_Oabs__class_Oabs_001t__Real__Oreal,type,
    abs_abs_real: real > real ).

thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Complex__Ocomplex_M_Eo_J,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Int__Oint_M_Eo_J,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__List__Olist_It__Nat__Onat_J_M_Eo_J,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Nat__Onat_M_Eo_J,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Real__Oreal_M_Eo_J,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Complex__Ocomplex,type,
    minus_minus_complex: complex > complex > complex ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Extended____Nat__Oenat,type,
    minus_3235023915231533773d_enat: extended_enat > extended_enat > extended_enat ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
    minus_minus_int: int > int > int ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
    minus_minus_nat: nat > nat > nat ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Rat__Orat,type,
    minus_minus_rat: rat > rat > rat ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Real__Oreal,type,
    minus_minus_real: real > real > real ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Complex__Ocomplex_J,type,
    minus_811609699411566653omplex: set_complex > set_complex > set_complex ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Extended____Nat__Oenat_J,type,
    minus_925952699566721837d_enat: set_Extended_enat > set_Extended_enat > set_Extended_enat ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Int__Oint_J,type,
    minus_minus_set_int: set_int > set_int > set_int ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
    minus_7954133019191499631st_nat: set_list_nat > set_list_nat > set_list_nat ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
    minus_minus_set_nat: set_nat > set_nat > set_nat ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Real__Oreal_J,type,
    minus_minus_set_real: set_real > set_real > set_real ).

thf(sy_c_Groups_Oone__class_Oone_001t__Code____Numeral__Ointeger,type,
    one_one_Code_integer: code_integer ).

thf(sy_c_Groups_Oone__class_Oone_001t__Complex__Ocomplex,type,
    one_one_complex: complex ).

thf(sy_c_Groups_Oone__class_Oone_001t__Extended____Nat__Oenat,type,
    one_on7984719198319812577d_enat: extended_enat ).

thf(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
    one_one_int: int ).

thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
    one_one_nat: nat ).

thf(sy_c_Groups_Oone__class_Oone_001t__Rat__Orat,type,
    one_one_rat: rat ).

thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
    one_one_real: real ).

thf(sy_c_Groups_Oplus__class_Oplus_001t__Code____Numeral__Ointeger,type,
    plus_p5714425477246183910nteger: code_integer > code_integer > code_integer ).

thf(sy_c_Groups_Oplus__class_Oplus_001t__Complex__Ocomplex,type,
    plus_plus_complex: complex > complex > complex ).

thf(sy_c_Groups_Oplus__class_Oplus_001t__Extended____Nat__Oenat,type,
    plus_p3455044024723400733d_enat: extended_enat > extended_enat > extended_enat ).

thf(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
    plus_plus_int: int > int > int ).

thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
    plus_plus_nat: nat > nat > nat ).

thf(sy_c_Groups_Oplus__class_Oplus_001t__Num__Onum,type,
    plus_plus_num: num > num > num ).

thf(sy_c_Groups_Oplus__class_Oplus_001t__Rat__Orat,type,
    plus_plus_rat: rat > rat > rat ).

thf(sy_c_Groups_Oplus__class_Oplus_001t__Real__Oreal,type,
    plus_plus_real: real > real > real ).

thf(sy_c_Groups_Oplus__class_Oplus_001t__String__Oliteral,type,
    plus_plus_literal: literal > literal > literal ).

thf(sy_c_Groups_Osgn__class_Osgn_001t__Code____Numeral__Ointeger,type,
    sgn_sgn_Code_integer: code_integer > code_integer ).

thf(sy_c_Groups_Osgn__class_Osgn_001t__Complex__Ocomplex,type,
    sgn_sgn_complex: complex > complex ).

thf(sy_c_Groups_Osgn__class_Osgn_001t__Int__Oint,type,
    sgn_sgn_int: int > int ).

thf(sy_c_Groups_Osgn__class_Osgn_001t__Rat__Orat,type,
    sgn_sgn_rat: rat > rat ).

thf(sy_c_Groups_Osgn__class_Osgn_001t__Real__Oreal,type,
    sgn_sgn_real: real > real ).

thf(sy_c_Groups_Otimes__class_Otimes_001t__Code____Numeral__Ointeger,type,
    times_3573771949741848930nteger: code_integer > code_integer > code_integer ).

thf(sy_c_Groups_Otimes__class_Otimes_001t__Complex__Ocomplex,type,
    times_times_complex: complex > complex > complex ).

thf(sy_c_Groups_Otimes__class_Otimes_001t__Extended____Nat__Oenat,type,
    times_7803423173614009249d_enat: extended_enat > extended_enat > extended_enat ).

thf(sy_c_Groups_Otimes__class_Otimes_001t__Int__Oint,type,
    times_times_int: int > int > int ).

thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
    times_times_nat: nat > nat > nat ).

thf(sy_c_Groups_Otimes__class_Otimes_001t__Num__Onum,type,
    times_times_num: num > num > num ).

thf(sy_c_Groups_Otimes__class_Otimes_001t__Rat__Orat,type,
    times_times_rat: rat > rat > rat ).

thf(sy_c_Groups_Otimes__class_Otimes_001t__Real__Oreal,type,
    times_times_real: real > real > real ).

thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Code____Numeral__Ointeger,type,
    uminus1351360451143612070nteger: code_integer > code_integer ).

thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Complex__Ocomplex,type,
    uminus1482373934393186551omplex: complex > complex ).

thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Int__Oint,type,
    uminus_uminus_int: int > int ).

thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Rat__Orat,type,
    uminus_uminus_rat: rat > rat ).

thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Real__Oreal,type,
    uminus_uminus_real: real > real ).

thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Nat__Onat_J,type,
    uminus5710092332889474511et_nat: set_nat > set_nat ).

thf(sy_c_Groups_Ozero__class_Ozero_001t__Code____Numeral__Ointeger,type,
    zero_z3403309356797280102nteger: code_integer ).

thf(sy_c_Groups_Ozero__class_Ozero_001t__Complex__Ocomplex,type,
    zero_zero_complex: complex ).

thf(sy_c_Groups_Ozero__class_Ozero_001t__Extended____Nat__Oenat,type,
    zero_z5237406670263579293d_enat: extended_enat ).

thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
    zero_zero_int: int ).

thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
    zero_zero_nat: nat ).

thf(sy_c_Groups_Ozero__class_Ozero_001t__Rat__Orat,type,
    zero_zero_rat: rat ).

thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
    zero_zero_real: real ).

thf(sy_c_Groups_Ozero__class_Ozero_001t__String__Oliteral,type,
    zero_zero_literal: literal ).

thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
    groups7754918857620584856omplex: ( complex > complex ) > set_complex > complex ).

thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Int__Oint_001t__Int__Oint,type,
    groups4538972089207619220nt_int: ( int > int ) > set_int > int ).

thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Nat__Onat,type,
    groups3542108847815614940at_nat: ( nat > nat ) > set_nat > nat ).

thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Real__Oreal,type,
    groups6591440286371151544t_real: ( nat > real ) > set_nat > real ).

thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
    groups3708469109370488835omplex: ( complex > complex ) > set_complex > complex ).

thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Complex__Ocomplex_001t__Rat__Orat,type,
    groups225925009352817453ex_rat: ( complex > rat ) > set_complex > rat ).

thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Complex__Ocomplex_001t__Real__Oreal,type,
    groups766887009212190081x_real: ( complex > real ) > set_complex > real ).

thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Extended____Nat__Oenat_001t__Complex__Ocomplex,type,
    groups4622424608036095791omplex: ( extended_enat > complex ) > set_Extended_enat > complex ).

thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Extended____Nat__Oenat_001t__Int__Oint,type,
    groups2878480467620962989at_int: ( extended_enat > int ) > set_Extended_enat > int ).

thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Extended____Nat__Oenat_001t__Nat__Onat,type,
    groups2880970938130013265at_nat: ( extended_enat > nat ) > set_Extended_enat > nat ).

thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Extended____Nat__Oenat_001t__Rat__Orat,type,
    groups2245840878043517529at_rat: ( extended_enat > rat ) > set_Extended_enat > rat ).

thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Extended____Nat__Oenat_001t__Real__Oreal,type,
    groups97031904164794029t_real: ( extended_enat > real ) > set_Extended_enat > real ).

thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Int__Oint_001t__Complex__Ocomplex,type,
    groups7440179247065528705omplex: ( int > complex ) > set_int > complex ).

thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Int__Oint_001t__Int__Oint,type,
    groups1705073143266064639nt_int: ( int > int ) > set_int > int ).

thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Int__Oint_001t__Rat__Orat,type,
    groups1072433553688619179nt_rat: ( int > rat ) > set_int > rat ).

thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Int__Oint_001t__Real__Oreal,type,
    groups2316167850115554303t_real: ( int > real ) > set_int > real ).

thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Nat__Onat_001t__Complex__Ocomplex,type,
    groups6464643781859351333omplex: ( nat > complex ) > set_nat > complex ).

thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Nat__Onat_001t__Int__Oint,type,
    groups705719431365010083at_int: ( nat > int ) > set_nat > int ).

thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Nat__Onat_001t__Nat__Onat,type,
    groups708209901874060359at_nat: ( nat > nat ) > set_nat > nat ).

thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Nat__Onat_001t__Rat__Orat,type,
    groups73079841787564623at_rat: ( nat > rat ) > set_nat > rat ).

thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Nat__Onat_001t__Real__Oreal,type,
    groups129246275422532515t_real: ( nat > real ) > set_nat > real ).

thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Real__Oreal_001t__Complex__Ocomplex,type,
    groups713298508707869441omplex: ( real > complex ) > set_real > complex ).

thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Real__Oreal_001t__Int__Oint,type,
    groups4694064378042380927al_int: ( real > int ) > set_real > int ).

thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Real__Oreal_001t__Nat__Onat,type,
    groups4696554848551431203al_nat: ( real > nat ) > set_real > nat ).

thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Real__Oreal_001t__Rat__Orat,type,
    groups4061424788464935467al_rat: ( real > rat ) > set_real > rat ).

thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Real__Oreal_001t__Real__Oreal,type,
    groups1681761925125756287l_real: ( real > real ) > set_real > real ).

thf(sy_c_Groups__List_Ocomm__semiring__0__class_Ohorner__sum_001_Eo_001t__Int__Oint,type,
    groups9116527308978886569_o_int: ( $o > int ) > int > list_o > int ).

thf(sy_c_Groups__List_Omonoid__add__class_Osum__list_001t__Nat__Onat,type,
    groups4561878855575611511st_nat: list_nat > nat ).

thf(sy_c_HOL_OThe_001t__Int__Oint,type,
    the_int: ( int > $o ) > int ).

thf(sy_c_HOL_OThe_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    the_Pr4378521158711661632nt_int: ( product_prod_int_int > $o ) > product_prod_int_int ).

thf(sy_c_HOL_OThe_001t__Real__Oreal,type,
    the_real: ( real > $o ) > real ).

thf(sy_c_If_001_062_It__Int__Oint_Mt__Int__Oint_J,type,
    if_int_int: $o > ( int > int ) > ( int > int ) > int > int ).

thf(sy_c_If_001_062_It__Nat__Onat_M_062_It__Int__Oint_Mt__Int__Oint_J_J,type,
    if_nat_int_int: $o > ( nat > int > int ) > ( nat > int > int ) > nat > int > int ).

thf(sy_c_If_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    if_nat_nat_nat: $o > ( nat > nat > nat ) > ( nat > nat > nat ) > nat > nat > nat ).

thf(sy_c_If_001_062_It__Nat__Onat_Mt__Rat__Orat_J,type,
    if_nat_rat: $o > ( nat > rat ) > ( nat > rat ) > nat > rat ).

thf(sy_c_If_001t__Code____Numeral__Ointeger,type,
    if_Code_integer: $o > code_integer > code_integer > code_integer ).

thf(sy_c_If_001t__Complex__Ocomplex,type,
    if_complex: $o > complex > complex > complex ).

thf(sy_c_If_001t__Extended____Nat__Oenat,type,
    if_Extended_enat: $o > extended_enat > extended_enat > extended_enat ).

thf(sy_c_If_001t__Int__Oint,type,
    if_int: $o > int > int > int ).

thf(sy_c_If_001t__List__Olist_It__Int__Oint_J,type,
    if_list_int: $o > list_int > list_int > list_int ).

thf(sy_c_If_001t__List__Olist_It__Nat__Onat_J,type,
    if_list_nat: $o > list_nat > list_nat > list_nat ).

thf(sy_c_If_001t__Nat__Onat,type,
    if_nat: $o > nat > nat > nat ).

thf(sy_c_If_001t__Num__Onum,type,
    if_num: $o > num > num > num ).

thf(sy_c_If_001t__Option__Ooption_It__Nat__Onat_J,type,
    if_option_nat: $o > option_nat > option_nat > option_nat ).

thf(sy_c_If_001t__Option__Ooption_It__Num__Onum_J,type,
    if_option_num: $o > option_num > option_num > option_num ).

thf(sy_c_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J,type,
    if_Pro5737122678794959658eger_o: $o > produc6271795597528267376eger_o > produc6271795597528267376eger_o > produc6271795597528267376eger_o ).

thf(sy_c_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    if_Pro6119634080678213985nteger: $o > produc8923325533196201883nteger > produc8923325533196201883nteger > produc8923325533196201883nteger ).

thf(sy_c_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    if_Pro3027730157355071871nt_int: $o > product_prod_int_int > product_prod_int_int > product_prod_int_int ).

thf(sy_c_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    if_Pro6206227464963214023at_nat: $o > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ).

thf(sy_c_If_001t__Rat__Orat,type,
    if_rat: $o > rat > rat > rat ).

thf(sy_c_If_001t__Real__Oreal,type,
    if_real: $o > real > real > real ).

thf(sy_c_If_001t__Set__Oset_It__Int__Oint_J,type,
    if_set_int: $o > set_int > set_int > set_int ).

thf(sy_c_If_001t__VEBT____Definitions__OVEBT,type,
    if_VEBT_VEBT: $o > vEBT_VEBT > vEBT_VEBT > vEBT_VEBT ).

thf(sy_c_Infinite__Set_Owellorder__class_Oenumerate_001t__Nat__Onat,type,
    infini8530281810654367211te_nat: set_nat > nat > nat ).

thf(sy_c_Int_OAbs__Integ,type,
    abs_Integ: product_prod_nat_nat > int ).

thf(sy_c_Int_ORep__Integ,type,
    rep_Integ: int > product_prod_nat_nat ).

thf(sy_c_Int_Oint__ge__less__than,type,
    int_ge_less_than: int > set_Pr958786334691620121nt_int ).

thf(sy_c_Int_Oint__ge__less__than2,type,
    int_ge_less_than2: int > set_Pr958786334691620121nt_int ).

thf(sy_c_Int_Ointrel,type,
    intrel: product_prod_nat_nat > product_prod_nat_nat > $o ).

thf(sy_c_Int_Onat,type,
    nat2: int > nat ).

thf(sy_c_Int_Opcr__int,type,
    pcr_int: product_prod_nat_nat > int > $o ).

thf(sy_c_Int_Opower__int_001t__Real__Oreal,type,
    power_int_real: real > int > real ).

thf(sy_c_Int_Oring__1__class_OInts_001t__Code____Numeral__Ointeger,type,
    ring_11222124179247155820nteger: set_Code_integer ).

thf(sy_c_Int_Oring__1__class_OInts_001t__Complex__Ocomplex,type,
    ring_1_Ints_complex: set_complex ).

thf(sy_c_Int_Oring__1__class_OInts_001t__Int__Oint,type,
    ring_1_Ints_int: set_int ).

thf(sy_c_Int_Oring__1__class_OInts_001t__Rat__Orat,type,
    ring_1_Ints_rat: set_rat ).

thf(sy_c_Int_Oring__1__class_OInts_001t__Real__Oreal,type,
    ring_1_Ints_real: set_real ).

thf(sy_c_Int_Oring__1__class_Oof__int_001t__Code____Numeral__Ointeger,type,
    ring_18347121197199848620nteger: int > code_integer ).

thf(sy_c_Int_Oring__1__class_Oof__int_001t__Complex__Ocomplex,type,
    ring_17405671764205052669omplex: int > complex ).

thf(sy_c_Int_Oring__1__class_Oof__int_001t__Int__Oint,type,
    ring_1_of_int_int: int > int ).

thf(sy_c_Int_Oring__1__class_Oof__int_001t__Rat__Orat,type,
    ring_1_of_int_rat: int > rat ).

thf(sy_c_Int_Oring__1__class_Oof__int_001t__Real__Oreal,type,
    ring_1_of_int_real: int > real ).

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Extended____Nat__Oenat,type,
    inf_in1870772243966228564d_enat: extended_enat > extended_enat > extended_enat ).

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Nat__Onat,type,
    inf_inf_nat: nat > nat > nat ).

thf(sy_c_Lattices_Osemilattice__neutr__order_001t__Nat__Onat,type,
    semila1623282765462674594er_nat: ( nat > nat > nat ) > nat > ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Extended____Nat__Oenat,type,
    sup_su3973961784419623482d_enat: extended_enat > extended_enat > extended_enat ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Nat__Onat,type,
    sup_sup_nat: nat > nat > nat ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Nat__Onat_J,type,
    sup_sup_set_nat: set_nat > set_nat > set_nat ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    sup_su6327502436637775413at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).

thf(sy_c_Lattices__Big_Olinorder__class_OMax_001t__Extended____Nat__Oenat,type,
    lattic921264341876707157d_enat: set_Extended_enat > extended_enat ).

thf(sy_c_Lattices__Big_Olinorder__class_OMax_001t__Nat__Onat,type,
    lattic8265883725875713057ax_nat: set_nat > nat ).

thf(sy_c_Lifting_OQuotient_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001t__Real__Oreal,type,
    quotie3684837364556693515t_real: ( ( nat > rat ) > ( nat > rat ) > $o ) > ( ( nat > rat ) > real ) > ( real > nat > rat ) > ( ( nat > rat ) > real > $o ) > $o ).

thf(sy_c_Limits_OBfun_001t__Nat__Onat_001t__Real__Oreal,type,
    bfun_nat_real: ( nat > real ) > filter_nat > $o ).

thf(sy_c_List_Oappend_001t__Int__Oint,type,
    append_int: list_int > list_int > list_int ).

thf(sy_c_List_Oappend_001t__Nat__Onat,type,
    append_nat: list_nat > list_nat > list_nat ).

thf(sy_c_List_Odrop_001t__Nat__Onat,type,
    drop_nat: nat > list_nat > list_nat ).

thf(sy_c_List_Olast_001t__Nat__Onat,type,
    last_nat: list_nat > nat ).

thf(sy_c_List_Olinorder__class_Osorted__list__of__set_001t__Nat__Onat,type,
    linord2614967742042102400et_nat: set_nat > list_nat ).

thf(sy_c_List_Olist_OCons_001t__Int__Oint,type,
    cons_int: int > list_int > list_int ).

thf(sy_c_List_Olist_OCons_001t__Nat__Onat,type,
    cons_nat: nat > list_nat > list_nat ).

thf(sy_c_List_Olist_ONil_001t__Int__Oint,type,
    nil_int: list_int ).

thf(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
    nil_nat: list_nat ).

thf(sy_c_List_Olist_Ohd_001t__Nat__Onat,type,
    hd_nat: list_nat > nat ).

thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Nat__Onat,type,
    map_nat_nat: ( nat > nat ) > list_nat > list_nat ).

thf(sy_c_List_Olist_Omap_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
    map_VE8901447254227204932T_VEBT: ( vEBT_VEBT > vEBT_VEBT ) > list_VEBT_VEBT > list_VEBT_VEBT ).

thf(sy_c_List_Olist_Oset_001_Eo,type,
    set_o2: list_o > set_o ).

thf(sy_c_List_Olist_Oset_001t__Complex__Ocomplex,type,
    set_complex2: list_complex > set_complex ).

thf(sy_c_List_Olist_Oset_001t__Extended____Nat__Oenat,type,
    set_Extended_enat2: list_Extended_enat > set_Extended_enat ).

thf(sy_c_List_Olist_Oset_001t__Int__Oint,type,
    set_int2: list_int > set_int ).

thf(sy_c_List_Olist_Oset_001t__List__Olist_It__Nat__Onat_J,type,
    set_list_nat2: list_list_nat > set_list_nat ).

thf(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
    set_nat2: list_nat > set_nat ).

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

thf(sy_c_List_Olist_Oset_001t__VEBT____Definitions__OVEBT,type,
    set_VEBT_VEBT2: list_VEBT_VEBT > set_VEBT_VEBT ).

thf(sy_c_List_Olist_Osize__list_001t__VEBT____Definitions__OVEBT,type,
    size_list_VEBT_VEBT: ( vEBT_VEBT > nat ) > list_VEBT_VEBT > nat ).

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

thf(sy_c_List_Olist__update_001t__VEBT____Definitions__OVEBT,type,
    list_u1324408373059187874T_VEBT: list_VEBT_VEBT > nat > vEBT_VEBT > list_VEBT_VEBT ).

thf(sy_c_List_Onth_001_Eo,type,
    nth_o: list_o > nat > $o ).

thf(sy_c_List_Onth_001t__Complex__Ocomplex,type,
    nth_complex: list_complex > nat > complex ).

thf(sy_c_List_Onth_001t__Extended____Nat__Oenat,type,
    nth_Extended_enat: list_Extended_enat > nat > extended_enat ).

thf(sy_c_List_Onth_001t__Int__Oint,type,
    nth_int: list_int > nat > int ).

thf(sy_c_List_Onth_001t__List__Olist_It__Nat__Onat_J,type,
    nth_list_nat: list_list_nat > nat > list_nat ).

thf(sy_c_List_Onth_001t__Nat__Onat,type,
    nth_nat: list_nat > nat > nat ).

thf(sy_c_List_Onth_001t__Num__Onum,type,
    nth_num: list_num > nat > num ).

thf(sy_c_List_Onth_001t__Product____Type__Oprod_I_Eo_M_Eo_J,type,
    nth_Product_prod_o_o: list_P4002435161011370285od_o_o > nat > product_prod_o_o ).

thf(sy_c_List_Onth_001t__Product____Type__Oprod_I_Eo_Mt__Int__Oint_J,type,
    nth_Pr1649062631805364268_o_int: list_P3795440434834930179_o_int > nat > product_prod_o_int ).

thf(sy_c_List_Onth_001t__Product____Type__Oprod_I_Eo_Mt__Nat__Onat_J,type,
    nth_Pr5826913651314560976_o_nat: list_P6285523579766656935_o_nat > nat > product_prod_o_nat ).

thf(sy_c_List_Onth_001t__Product____Type__Oprod_I_Eo_Mt__VEBT____Definitions__OVEBT_J,type,
    nth_Pr6777367263587873994T_VEBT: list_P7495141550334521929T_VEBT > nat > produc2504756804600209347T_VEBT ).

thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J,type,
    nth_Pr8326237132889035090at_num: list_P1726324292696863441at_num > nat > product_prod_nat_num ).

thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J,type,
    nth_Pr6456567536196504476um_num: list_P3744719386663036955um_num > nat > product_prod_num_num ).

thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_M_Eo_J,type,
    nth_Pr4606735188037164562VEBT_o: list_P3126845725202233233VEBT_o > nat > produc334124729049499915VEBT_o ).

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

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

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

thf(sy_c_List_Onth_001t__Real__Oreal,type,
    nth_real: list_real > nat > real ).

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

thf(sy_c_List_Oproduct_001_Eo_001_Eo,type,
    product_o_o: list_o > list_o > list_P4002435161011370285od_o_o ).

thf(sy_c_List_Oproduct_001_Eo_001t__Int__Oint,type,
    product_o_int: list_o > list_int > list_P3795440434834930179_o_int ).

thf(sy_c_List_Oproduct_001_Eo_001t__Nat__Onat,type,
    product_o_nat: list_o > list_nat > list_P6285523579766656935_o_nat ).

thf(sy_c_List_Oproduct_001_Eo_001t__VEBT____Definitions__OVEBT,type,
    product_o_VEBT_VEBT: list_o > list_VEBT_VEBT > list_P7495141550334521929T_VEBT ).

thf(sy_c_List_Oproduct_001t__Nat__Onat_001_Eo,type,
    product_nat_o: list_nat > list_o > list_P7333126701944960589_nat_o ).

thf(sy_c_List_Oproduct_001t__Nat__Onat_001t__Num__Onum,type,
    product_nat_num: list_nat > list_num > list_P1726324292696863441at_num ).

thf(sy_c_List_Oproduct_001t__Nat__Onat_001t__VEBT____Definitions__OVEBT,type,
    produc7156399406898700509T_VEBT: list_nat > list_VEBT_VEBT > list_P5647936690300460905T_VEBT ).

thf(sy_c_List_Oproduct_001t__Num__Onum_001t__Num__Onum,type,
    product_num_num: list_num > list_num > list_P3744719386663036955um_num ).

thf(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001_Eo,type,
    product_VEBT_VEBT_o: list_VEBT_VEBT > list_o > list_P3126845725202233233VEBT_o ).

thf(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
    produc7292646706713671643BT_int: list_VEBT_VEBT > list_int > list_P4547456442757143711BT_int ).

thf(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
    produc7295137177222721919BT_nat: list_VEBT_VEBT > list_nat > list_P7037539587688870467BT_nat ).

thf(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
    produc4743750530478302277T_VEBT: list_VEBT_VEBT > list_VEBT_VEBT > list_P7413028617227757229T_VEBT ).

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

thf(sy_c_List_Osorted__wrt_001t__Int__Oint,type,
    sorted_wrt_int: ( int > int > $o ) > list_int > $o ).

thf(sy_c_List_Osorted__wrt_001t__Nat__Onat,type,
    sorted_wrt_nat: ( nat > nat > $o ) > list_nat > $o ).

thf(sy_c_List_Otake_001t__Nat__Onat,type,
    take_nat: nat > list_nat > list_nat ).

thf(sy_c_List_Otake_001t__VEBT____Definitions__OVEBT,type,
    take_VEBT_VEBT: nat > list_VEBT_VEBT > list_VEBT_VEBT ).

thf(sy_c_List_Oupt,type,
    upt: nat > nat > list_nat ).

thf(sy_c_List_Oupto,type,
    upto: int > int > list_int ).

thf(sy_c_List_Oupto__aux,type,
    upto_aux: int > int > list_int > list_int ).

thf(sy_c_List_Oupto__rel,type,
    upto_rel: product_prod_int_int > product_prod_int_int > $o ).

thf(sy_c_Nat_OSuc,type,
    suc: nat > nat ).

thf(sy_c_Nat_Ocompow_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
    compow_nat_nat: nat > ( nat > nat ) > nat > nat ).

thf(sy_c_Nat_Onat_Ocase__nat_001_Eo,type,
    case_nat_o: $o > ( nat > $o ) > nat > $o ).

thf(sy_c_Nat_Onat_Ocase__nat_001t__Nat__Onat,type,
    case_nat_nat: nat > ( nat > nat ) > nat > nat ).

thf(sy_c_Nat_Onat_Ocase__nat_001t__Option__Ooption_It__Num__Onum_J,type,
    case_nat_option_num: option_num > ( nat > option_num ) > nat > option_num ).

thf(sy_c_Nat_Onat_Opred,type,
    pred: nat > nat ).

thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Code____Numeral__Ointeger,type,
    semiri4939895301339042750nteger: nat > code_integer ).

thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Complex__Ocomplex,type,
    semiri8010041392384452111omplex: nat > complex ).

thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Extended____Nat__Oenat,type,
    semiri4216267220026989637d_enat: nat > extended_enat ).

thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
    semiri1314217659103216013at_int: nat > int ).

thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
    semiri1316708129612266289at_nat: nat > nat ).

thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Rat__Orat,type,
    semiri681578069525770553at_rat: nat > rat ).

thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
    semiri5074537144036343181t_real: nat > real ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_I_Eo_J,type,
    size_size_list_o: list_o > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Complex__Ocomplex_J,type,
    size_s3451745648224563538omplex: list_complex > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Extended____Nat__Oenat_J,type,
    size_s3941691890525107288d_enat: list_Extended_enat > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Int__Oint_J,type,
    size_size_list_int: list_int > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
    size_s3023201423986296836st_nat: list_list_nat > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
    size_size_list_nat: list_nat > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Num__Onum_J,type,
    size_size_list_num: list_num > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_I_Eo_M_Eo_J_J,type,
    size_s1515746228057227161od_o_o: list_P4002435161011370285od_o_o > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__Int__Oint_J_J,type,
    size_s2953683556165314199_o_int: list_P3795440434834930179_o_int > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__Nat__Onat_J_J,type,
    size_s5443766701097040955_o_nat: list_P6285523579766656935_o_nat > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__VEBT____Definitions__OVEBT_J_J,type,
    size_s4313452262239582901T_VEBT: list_P7495141550334521929T_VEBT > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_M_Eo_J_J,type,
    size_s6491369823275344609_nat_o: list_P7333126701944960589_nat_o > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__VEBT____Definitions__OVEBT_J_J,type,
    size_s4762443039079500285T_VEBT: list_P5647936690300460905T_VEBT > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_M_Eo_J_J,type,
    size_s9168528473962070013VEBT_o: list_P3126845725202233233VEBT_o > nat ).

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

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J_J,type,
    size_s6152045936467909847BT_nat: list_P7037539587688870467BT_nat > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J_J,type,
    size_s7466405169056248089T_VEBT: list_P7413028617227757229T_VEBT > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Real__Oreal_J,type,
    size_size_list_real: list_real > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
    size_s6755466524823107622T_VEBT: list_VEBT_VEBT > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__Num__Onum,type,
    size_size_num: num > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__VEBT____Definitions__OVEBT,type,
    size_size_VEBT_VEBT: vEBT_VEBT > nat ).

thf(sy_c_Nat__Bijection_Olist__decode,type,
    nat_list_decode: nat > list_nat ).

thf(sy_c_Nat__Bijection_Olist__decode__rel,type,
    nat_list_decode_rel: nat > nat > $o ).

thf(sy_c_Nat__Bijection_Olist__encode,type,
    nat_list_encode: list_nat > nat ).

thf(sy_c_Nat__Bijection_Olist__encode__rel,type,
    nat_list_encode_rel: list_nat > list_nat > $o ).

thf(sy_c_Nat__Bijection_Oprod__decode,type,
    nat_prod_decode: nat > product_prod_nat_nat ).

thf(sy_c_Nat__Bijection_Oprod__decode__aux,type,
    nat_prod_decode_aux: nat > nat > product_prod_nat_nat ).

thf(sy_c_Nat__Bijection_Oprod__decode__aux__rel,type,
    nat_pr5047031295181774490ux_rel: product_prod_nat_nat > product_prod_nat_nat > $o ).

thf(sy_c_Nat__Bijection_Oprod__encode,type,
    nat_prod_encode: product_prod_nat_nat > nat ).

thf(sy_c_Nat__Bijection_Oset__decode,type,
    nat_set_decode: nat > set_nat ).

thf(sy_c_Nat__Bijection_Oset__encode,type,
    nat_set_encode: set_nat > nat ).

thf(sy_c_Nat__Bijection_Otriangle,type,
    nat_triangle: nat > nat ).

thf(sy_c_NthRoot_Oroot,type,
    root: nat > real > real ).

thf(sy_c_NthRoot_Osqrt,type,
    sqrt: real > real ).

thf(sy_c_Num_OBitM,type,
    bitM: num > num ).

thf(sy_c_Num_Oinc,type,
    inc: num > num ).

thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Code____Numeral__Ointeger,type,
    neg_nu8804712462038260780nteger: code_integer > code_integer ).

thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Complex__Ocomplex,type,
    neg_nu7009210354673126013omplex: complex > complex ).

thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Int__Oint,type,
    neg_numeral_dbl_int: int > int ).

thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Rat__Orat,type,
    neg_numeral_dbl_rat: rat > rat ).

thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Real__Oreal,type,
    neg_numeral_dbl_real: real > real ).

thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Code____Numeral__Ointeger,type,
    neg_nu7757733837767384882nteger: code_integer > code_integer ).

thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Complex__Ocomplex,type,
    neg_nu6511756317524482435omplex: complex > complex ).

thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Int__Oint,type,
    neg_nu3811975205180677377ec_int: int > int ).

thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Rat__Orat,type,
    neg_nu3179335615603231917ec_rat: rat > rat ).

thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Real__Oreal,type,
    neg_nu6075765906172075777c_real: real > real ).

thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Code____Numeral__Ointeger,type,
    neg_nu5831290666863070958nteger: code_integer > code_integer ).

thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Complex__Ocomplex,type,
    neg_nu8557863876264182079omplex: complex > complex ).

thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Int__Oint,type,
    neg_nu5851722552734809277nc_int: int > int ).

thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Rat__Orat,type,
    neg_nu5219082963157363817nc_rat: rat > rat ).

thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Real__Oreal,type,
    neg_nu8295874005876285629c_real: real > real ).

thf(sy_c_Num_Oneg__numeral__class_Osub_001t__Int__Oint,type,
    neg_numeral_sub_int: num > num > int ).

thf(sy_c_Num_Onum_OBit0,type,
    bit0: num > num ).

thf(sy_c_Num_Onum_OBit1,type,
    bit1: num > num ).

thf(sy_c_Num_Onum_OOne,type,
    one: num ).

thf(sy_c_Num_Onum_Ocase__num_001t__Option__Ooption_It__Num__Onum_J,type,
    case_num_option_num: option_num > ( num > option_num ) > ( num > option_num ) > num > option_num ).

thf(sy_c_Num_Onum_Osize__num,type,
    size_num: num > nat ).

thf(sy_c_Num_Onum__of__nat,type,
    num_of_nat: nat > num ).

thf(sy_c_Num_Onumeral__class_Onumeral_001t__Code____Numeral__Ointeger,type,
    numera6620942414471956472nteger: num > code_integer ).

thf(sy_c_Num_Onumeral__class_Onumeral_001t__Complex__Ocomplex,type,
    numera6690914467698888265omplex: num > complex ).

thf(sy_c_Num_Onumeral__class_Onumeral_001t__Extended____Nat__Oenat,type,
    numera1916890842035813515d_enat: num > extended_enat ).

thf(sy_c_Num_Onumeral__class_Onumeral_001t__Int__Oint,type,
    numeral_numeral_int: num > int ).

thf(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
    numeral_numeral_nat: num > nat ).

thf(sy_c_Num_Onumeral__class_Onumeral_001t__Rat__Orat,type,
    numeral_numeral_rat: num > rat ).

thf(sy_c_Num_Onumeral__class_Onumeral_001t__Real__Oreal,type,
    numeral_numeral_real: num > real ).

thf(sy_c_Num_Opow,type,
    pow: num > num > num ).

thf(sy_c_Num_Opred__numeral,type,
    pred_numeral: num > nat ).

thf(sy_c_Num_Osqr,type,
    sqr: num > num ).

thf(sy_c_Option_Ooption_ONone_001t__Nat__Onat,type,
    none_nat: option_nat ).

thf(sy_c_Option_Ooption_ONone_001t__Num__Onum,type,
    none_num: option_num ).

thf(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    none_P5556105721700978146at_nat: option4927543243414619207at_nat ).

thf(sy_c_Option_Ooption_OSome_001t__Nat__Onat,type,
    some_nat: nat > option_nat ).

thf(sy_c_Option_Ooption_OSome_001t__Num__Onum,type,
    some_num: num > option_num ).

thf(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    some_P7363390416028606310at_nat: product_prod_nat_nat > option4927543243414619207at_nat ).

thf(sy_c_Option_Ooption_Ocase__option_001_Eo_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    case_o184042715313410164at_nat: $o > ( product_prod_nat_nat > $o ) > option4927543243414619207at_nat > $o ).

thf(sy_c_Option_Ooption_Ocase__option_001t__Int__Oint_001t__Num__Onum,type,
    case_option_int_num: int > ( num > int ) > option_num > int ).

thf(sy_c_Option_Ooption_Ocase__option_001t__Num__Onum_001t__Num__Onum,type,
    case_option_num_num: num > ( num > num ) > option_num > num ).

thf(sy_c_Option_Ooption_Ocase__option_001t__Option__Ooption_It__Num__Onum_J_001t__Num__Onum,type,
    case_o6005452278849405969um_num: option_num > ( num > option_num ) > option_num > option_num ).

thf(sy_c_Option_Ooption_Omap__option_001t__Num__Onum_001t__Num__Onum,type,
    map_option_num_num: ( num > num ) > option_num > option_num ).

thf(sy_c_Option_Ooption_Othe_001t__Nat__Onat,type,
    the_nat: option_nat > nat ).

thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Nat__Onat_M_Eo_J,type,
    bot_bot_nat_o: nat > $o ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Extended____Nat__Oenat,type,
    bot_bo4199563552545308370d_enat: extended_enat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
    bot_bot_nat: nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Extended____Nat__Oenat_J,type,
    bot_bo7653980558646680370d_enat: set_Extended_enat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Int__Oint_J,type,
    bot_bot_set_int: set_int ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
    bot_bot_set_nat: set_nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Num__Onum_J,type,
    bot_bot_set_num: set_num ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Rat__Orat_J,type,
    bot_bot_set_rat: set_rat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Real__Oreal_J,type,
    bot_bot_set_real: set_real ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    bot_bot_set_set_nat: set_set_nat ).

thf(sy_c_Orderings_Oord__class_OLeast_001t__Extended____Nat__Oenat,type,
    ord_Le1955565732374568822d_enat: ( extended_enat > $o ) > extended_enat ).

thf(sy_c_Orderings_Oord__class_OLeast_001t__Nat__Onat,type,
    ord_Least_nat: ( nat > $o ) > nat ).

thf(sy_c_Orderings_Oord__class_OLeast_001t__Real__Oreal,type,
    ord_Least_real: ( real > $o ) > real ).

thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Complex__Ocomplex_M_Eo_J,type,
    ord_less_complex_o: ( complex > $o ) > ( complex > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Extended____Nat__Oenat_M_Eo_J,type,
    ord_le8499522857272258027enat_o: ( extended_enat > $o ) > ( extended_enat > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Int__Oint_M_Eo_J,type,
    ord_less_int_o: ( int > $o ) > ( int > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Nat__Onat_M_Eo_J,type,
    ord_less_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Real__Oreal_M_Eo_J,type,
    ord_less_real_o: ( real > $o ) > ( real > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Code____Numeral__Ointeger,type,
    ord_le6747313008572928689nteger: code_integer > code_integer > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Extended____Nat__Oenat,type,
    ord_le72135733267957522d_enat: extended_enat > extended_enat > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
    ord_less_int: int > int > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
    ord_less_nat: nat > nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Num__Onum,type,
    ord_less_num: num > num > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Rat__Orat,type,
    ord_less_rat: rat > rat > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
    ord_less_real: real > real > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Code____Numeral__Ointeger_J,type,
    ord_le1307284697595431911nteger: set_Code_integer > set_Code_integer > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Complex__Ocomplex_J,type,
    ord_less_set_complex: set_complex > set_complex > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Extended____Nat__Oenat_J,type,
    ord_le2529575680413868914d_enat: set_Extended_enat > set_Extended_enat > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Int__Oint_J,type,
    ord_less_set_int: set_int > set_int > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
    ord_less_set_nat: set_nat > set_nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Num__Onum_J,type,
    ord_less_set_num: set_num > set_num > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Rat__Orat_J,type,
    ord_less_set_rat: set_rat > set_rat > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Real__Oreal_J,type,
    ord_less_set_real: set_real > set_real > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    ord_less_set_set_nat: set_set_nat > set_set_nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Code____Numeral__Ointeger,type,
    ord_le3102999989581377725nteger: code_integer > code_integer > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Extended____Nat__Oenat,type,
    ord_le2932123472753598470d_enat: extended_enat > extended_enat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Filter__Ofilter_It__Nat__Onat_J,type,
    ord_le2510731241096832064er_nat: filter_nat > filter_nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
    ord_less_eq_int: int > int > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
    ord_less_eq_nat: nat > nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Num__Onum,type,
    ord_less_eq_num: num > num > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Rat__Orat,type,
    ord_less_eq_rat: rat > rat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
    ord_less_eq_real: real > real > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Code____Numeral__Ointeger_J,type,
    ord_le7084787975880047091nteger: set_Code_integer > set_Code_integer > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Complex__Ocomplex_J,type,
    ord_le211207098394363844omplex: set_complex > set_complex > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Extended____Nat__Oenat_J,type,
    ord_le7203529160286727270d_enat: set_Extended_enat > set_Extended_enat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Int__Oint_J,type,
    ord_less_eq_set_int: set_int > set_int > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
    ord_less_eq_set_nat: set_nat > set_nat > $o ).

thf(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 ).

thf(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 ).

thf(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 ).

thf(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 ).

thf(sy_c_Orderings_Oord__class_Omax_001t__Extended____Nat__Oenat,type,
    ord_ma741700101516333627d_enat: extended_enat > extended_enat > extended_enat ).

thf(sy_c_Orderings_Oord__class_Omax_001t__Int__Oint,type,
    ord_max_int: int > int > int ).

thf(sy_c_Orderings_Oord__class_Omax_001t__Nat__Onat,type,
    ord_max_nat: nat > nat > nat ).

thf(sy_c_Orderings_Oord__class_Omin_001t__Extended____Nat__Oenat,type,
    ord_mi8085742599997312461d_enat: extended_enat > extended_enat > extended_enat ).

thf(sy_c_Orderings_Oord__class_Omin_001t__Nat__Onat,type,
    ord_min_nat: nat > nat > nat ).

thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Nat__Onat,type,
    order_Greatest_nat: ( nat > $o ) > nat ).

thf(sy_c_Orderings_Oorder__class_Oantimono_001t__Nat__Onat_001t__Real__Oreal,type,
    order_9091379641038594480t_real: ( nat > real ) > $o ).

thf(sy_c_Orderings_Oorder__class_Omono_001t__Extended____Nat__Oenat_001t__Extended____Nat__Oenat,type,
    order_4130057895858720880d_enat: ( extended_enat > extended_enat ) > $o ).

thf(sy_c_Orderings_Oorder__class_Omono_001t__Nat__Onat_001t__Nat__Onat,type,
    order_mono_nat_nat: ( nat > nat ) > $o ).

thf(sy_c_Orderings_Oorder__class_Omono_001t__Nat__Onat_001t__Real__Oreal,type,
    order_mono_nat_real: ( nat > real ) > $o ).

thf(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Nat__Onat_001t__Nat__Onat,type,
    order_5726023648592871131at_nat: ( nat > nat ) > $o ).

thf(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Real__Oreal_001t__Real__Oreal,type,
    order_7092887310737990675l_real: ( real > real ) > $o ).

thf(sy_c_Orderings_Oordering__top_001t__Nat__Onat,type,
    ordering_top_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > nat > $o ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Extended____Nat__Oenat,type,
    top_to3028658606643905974d_enat: extended_enat ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_Eo_J,type,
    top_top_set_o: set_o ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
    top_top_set_nat: set_nat ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Ounit_J,type,
    top_to1996260823553986621t_unit: set_Product_unit ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Real__Oreal_J,type,
    top_top_set_real: set_real ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__String__Ochar_J,type,
    top_top_set_char: set_char ).

thf(sy_c_Power_Opower__class_Opower_001t__Code____Numeral__Ointeger,type,
    power_8256067586552552935nteger: code_integer > nat > code_integer ).

thf(sy_c_Power_Opower__class_Opower_001t__Complex__Ocomplex,type,
    power_power_complex: complex > nat > complex ).

thf(sy_c_Power_Opower__class_Opower_001t__Int__Oint,type,
    power_power_int: int > nat > int ).

thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
    power_power_nat: nat > nat > nat ).

thf(sy_c_Power_Opower__class_Opower_001t__Rat__Orat,type,
    power_power_rat: rat > nat > rat ).

thf(sy_c_Power_Opower__class_Opower_001t__Real__Oreal,type,
    power_power_real: real > nat > real ).

thf(sy_c_Product__Type_OPair_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    produc3209952032786966637at_nat: ( nat > nat > nat ) > produc7248412053542808358at_nat > produc4471711990508489141at_nat ).

thf(sy_c_Product__Type_OPair_001_062_It__Nat__Onat_M_062_It__Num__Onum_Mt__Num__Onum_J_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J_J,type,
    produc851828971589881931at_num: ( nat > num > num ) > produc2963631642982155120at_num > produc3368934014287244435at_num ).

thf(sy_c_Product__Type_OPair_001_Eo_001_Eo,type,
    product_Pair_o_o: $o > $o > product_prod_o_o ).

thf(sy_c_Product__Type_OPair_001_Eo_001t__Int__Oint,type,
    product_Pair_o_int: $o > int > product_prod_o_int ).

thf(sy_c_Product__Type_OPair_001_Eo_001t__Nat__Onat,type,
    product_Pair_o_nat: $o > nat > product_prod_o_nat ).

thf(sy_c_Product__Type_OPair_001_Eo_001t__VEBT____Definitions__OVEBT,type,
    produc2982872950893828659T_VEBT: $o > vEBT_VEBT > produc2504756804600209347T_VEBT ).

thf(sy_c_Product__Type_OPair_001t__Code____Numeral__Ointeger_001_Eo,type,
    produc6677183202524767010eger_o: code_integer > $o > produc6271795597528267376eger_o ).

thf(sy_c_Product__Type_OPair_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
    produc1086072967326762835nteger: code_integer > code_integer > produc8923325533196201883nteger ).

thf(sy_c_Product__Type_OPair_001t__Int__Oint_001t__Int__Oint,type,
    product_Pair_int_int: int > int > product_prod_int_int ).

thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Nat__Onat,type,
    product_Pair_nat_nat: nat > nat > product_prod_nat_nat ).

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

thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    produc487386426758144856at_nat: nat > product_prod_nat_nat > produc7248412053542808358at_nat ).

thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J,type,
    produc1195630363706982562at_num: nat > product_prod_nat_num > produc2963631642982155120at_num ).

thf(sy_c_Product__Type_OPair_001t__Num__Onum_001t__Num__Onum,type,
    product_Pair_num_num: num > num > product_prod_num_num ).

thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    produc6161850002892822231at_nat: product_prod_nat_nat > product_prod_nat_nat > produc859450856879609959at_nat ).

thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001_Eo,type,
    produc8721562602347293563VEBT_o: vEBT_VEBT > $o > produc334124729049499915VEBT_o ).

thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__Extended____Nat__Oenat,type,
    produc581526299967858633d_enat: vEBT_VEBT > extended_enat > produc7272778201969148633d_enat ).

thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
    produc736041933913180425BT_int: vEBT_VEBT > int > produc4894624898956917775BT_int ).

thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
    produc738532404422230701BT_nat: vEBT_VEBT > nat > produc9072475918466114483BT_nat ).

thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
    produc537772716801021591T_VEBT: vEBT_VEBT > vEBT_VEBT > produc8243902056947475879T_VEBT ).

thf(sy_c_Product__Type_OSigma_001t__Nat__Onat_001t__Nat__Onat,type,
    produc457027306803732586at_nat: set_nat > ( nat > set_nat ) > set_Pr1261947904930325089at_nat ).

thf(sy_c_Product__Type_Oapsnd_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
    produc6499014454317279255nteger: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Int__Oint,type,
    produc1553301316500091796er_int: ( code_integer > code_integer > int ) > produc8923325533196201883nteger > int ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Nat__Onat,type,
    produc1555791787009142072er_nat: ( code_integer > code_integer > nat ) > produc8923325533196201883nteger > nat ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Num__Onum,type,
    produc7336495610019696514er_num: ( code_integer > code_integer > num ) > produc8923325533196201883nteger > num ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J,type,
    produc9125791028180074456eger_o: ( code_integer > code_integer > produc6271795597528267376eger_o ) > produc8923325533196201883nteger > produc6271795597528267376eger_o ).

thf(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: ( code_integer > code_integer > produc8923325533196201883nteger ) > produc8923325533196201883nteger > produc8923325533196201883nteger ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Complex__Ocomplex_001t__Complex__Ocomplex_001_Eo,type,
    produc6771430404735790350plex_o: ( complex > complex > $o ) > produc4411394909380815293omplex > $o ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001_Eo,type,
    produc4947309494688390418_int_o: ( int > int > $o ) > product_prod_int_int > $o ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint,type,
    produc8211389475949308722nt_int: ( int > int > int ) > product_prod_int_int > int ).

thf(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: ( int > int > product_prod_int_int ) > product_prod_int_int > product_prod_int_int ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
    produc8739625826339149834_nat_o: ( nat > nat > product_prod_nat_nat > $o ) > product_prod_nat_nat > product_prod_nat_nat > $o ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    produc27273713700761075at_nat: ( nat > nat > product_prod_nat_nat > product_prod_nat_nat ) > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001_Eo,type,
    produc6081775807080527818_nat_o: ( nat > nat > $o ) > product_prod_nat_nat > $o ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__List__Olist_It__Nat__Onat_J,type,
    produc2761476792215241774st_nat: ( nat > nat > list_nat ) > product_prod_nat_nat > list_nat ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
    produc6842872674320459806at_nat: ( nat > nat > nat ) > product_prod_nat_nat > nat ).

thf(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: ( nat > nat > product_prod_nat_nat ) > product_prod_nat_nat > product_prod_nat_nat ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Num__Onum_001t__Option__Ooption_It__Num__Onum_J,type,
    produc478579273971653890on_num: ( nat > num > option_num ) > product_prod_nat_num > option_num ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Real__Oreal_001t__Real__Oreal_001_Eo,type,
    produc5414030515140494994real_o: ( real > real > $o ) > produc2422161461964618553l_real > $o ).

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

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

thf(sy_c_Product__Type_Oprod_Ofst_001t__Nat__Onat_001t__Nat__Onat,type,
    product_fst_nat_nat: product_prod_nat_nat > nat ).

thf(sy_c_Product__Type_Oprod_Osnd_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
    produc6174133586879617921nteger: produc8923325533196201883nteger > code_integer ).

thf(sy_c_Product__Type_Oprod_Osnd_001t__Int__Oint_001t__Int__Oint,type,
    product_snd_int_int: product_prod_int_int > int ).

thf(sy_c_Product__Type_Oprod_Osnd_001t__Nat__Onat_001t__Nat__Onat,type,
    product_snd_nat_nat: product_prod_nat_nat > nat ).

thf(sy_c_Rat_OAbs__Rat,type,
    abs_Rat: product_prod_int_int > rat ).

thf(sy_c_Rat_OFract,type,
    fract: int > int > rat ).

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

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

thf(sy_c_Rat_Ofield__char__0__class_ORats_001t__Real__Oreal,type,
    field_5140801741446780682s_real: set_real ).

thf(sy_c_Rat_Ofield__char__0__class_Oof__rat_001t__Real__Oreal,type,
    field_7254667332652039916t_real: rat > real ).

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

thf(sy_c_Rat_Opcr__rat,type,
    pcr_rat: product_prod_int_int > rat > $o ).

thf(sy_c_Rat_Opositive,type,
    positive: rat > $o ).

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

thf(sy_c_Rat_Oratrel,type,
    ratrel: product_prod_int_int > product_prod_int_int > $o ).

thf(sy_c_Real_OReal,type,
    real2: ( nat > rat ) > real ).

thf(sy_c_Real_Ocauchy,type,
    cauchy: ( nat > rat ) > $o ).

thf(sy_c_Real_Ocr__real,type,
    cr_real: ( nat > rat ) > real > $o ).

thf(sy_c_Real_Opcr__real,type,
    pcr_real: ( nat > rat ) > real > $o ).

thf(sy_c_Real_Opositive,type,
    positive2: real > $o ).

thf(sy_c_Real_Orealrel,type,
    realrel: ( nat > rat ) > ( nat > rat ) > $o ).

thf(sy_c_Real_Orep__real,type,
    rep_real: real > nat > rat ).

thf(sy_c_Real_Ovanishes,type,
    vanishes: ( nat > rat ) > $o ).

thf(sy_c_Real__Vector__Spaces_OReals_001t__Complex__Ocomplex,type,
    real_V2521375963428798218omplex: set_complex ).

thf(sy_c_Real__Vector__Spaces_Obounded__linear_001t__Real__Oreal_001t__Real__Oreal,type,
    real_V5970128139526366754l_real: ( real > real ) > $o ).

thf(sy_c_Real__Vector__Spaces_Odist__class_Odist_001t__Complex__Ocomplex,type,
    real_V3694042436643373181omplex: complex > complex > real ).

thf(sy_c_Real__Vector__Spaces_Odist__class_Odist_001t__Real__Oreal,type,
    real_V975177566351809787t_real: real > real > real ).

thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Complex__Ocomplex,type,
    real_V1022390504157884413omplex: complex > real ).

thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Real__Oreal,type,
    real_V7735802525324610683m_real: real > real ).

thf(sy_c_Real__Vector__Spaces_Oof__real_001t__Complex__Ocomplex,type,
    real_V4546457046886955230omplex: real > complex ).

thf(sy_c_Real__Vector__Spaces_Oof__real_001t__Real__Oreal,type,
    real_V1803761363581548252l_real: real > real ).

thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001t__Complex__Ocomplex,type,
    real_V2046097035970521341omplex: real > complex > complex ).

thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001t__Real__Oreal,type,
    real_V1485227260804924795R_real: real > real > real ).

thf(sy_c_Relation_Otransp_001_062_It__Nat__Onat_Mt__Rat__Orat_J,type,
    transp_nat_rat: ( ( nat > rat ) > ( nat > rat ) > $o ) > $o ).

thf(sy_c_Rings_Oalgebraic__semidom__class_Ocoprime_001t__Int__Oint,type,
    algebr932160517623751201me_int: int > int > $o ).

thf(sy_c_Rings_Oalgebraic__semidom__class_Ocoprime_001t__Nat__Onat,type,
    algebr934650988132801477me_nat: nat > nat > $o ).

thf(sy_c_Rings_Odivide__class_Odivide_001t__Code____Numeral__Ointeger,type,
    divide6298287555418463151nteger: code_integer > code_integer > code_integer ).

thf(sy_c_Rings_Odivide__class_Odivide_001t__Complex__Ocomplex,type,
    divide1717551699836669952omplex: complex > complex > complex ).

thf(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
    divide_divide_int: int > int > int ).

thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
    divide_divide_nat: nat > nat > nat ).

thf(sy_c_Rings_Odivide__class_Odivide_001t__Rat__Orat,type,
    divide_divide_rat: rat > rat > rat ).

thf(sy_c_Rings_Odivide__class_Odivide_001t__Real__Oreal,type,
    divide_divide_real: real > real > real ).

thf(sy_c_Rings_Odvd__class_Odvd_001t__Code____Numeral__Ointeger,type,
    dvd_dvd_Code_integer: code_integer > code_integer > $o ).

thf(sy_c_Rings_Odvd__class_Odvd_001t__Complex__Ocomplex,type,
    dvd_dvd_complex: complex > complex > $o ).

thf(sy_c_Rings_Odvd__class_Odvd_001t__Int__Oint,type,
    dvd_dvd_int: int > int > $o ).

thf(sy_c_Rings_Odvd__class_Odvd_001t__Nat__Onat,type,
    dvd_dvd_nat: nat > nat > $o ).

thf(sy_c_Rings_Odvd__class_Odvd_001t__Rat__Orat,type,
    dvd_dvd_rat: rat > rat > $o ).

thf(sy_c_Rings_Odvd__class_Odvd_001t__Real__Oreal,type,
    dvd_dvd_real: real > real > $o ).

thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Code____Numeral__Ointeger,type,
    modulo364778990260209775nteger: code_integer > code_integer > code_integer ).

thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Int__Oint,type,
    modulo_modulo_int: int > int > int ).

thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Nat__Onat,type,
    modulo_modulo_nat: nat > nat > nat ).

thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Code____Numeral__Ointeger,type,
    zero_n356916108424825756nteger: $o > code_integer ).

thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Complex__Ocomplex,type,
    zero_n1201886186963655149omplex: $o > complex ).

thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Int__Oint,type,
    zero_n2684676970156552555ol_int: $o > int ).

thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Nat__Onat,type,
    zero_n2687167440665602831ol_nat: $o > nat ).

thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Rat__Orat,type,
    zero_n2052037380579107095ol_rat: $o > rat ).

thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Real__Oreal,type,
    zero_n3304061248610475627l_real: $o > real ).

thf(sy_c_Series_Osuminf_001t__Complex__Ocomplex,type,
    suminf_complex: ( nat > complex ) > complex ).

thf(sy_c_Series_Osuminf_001t__Real__Oreal,type,
    suminf_real: ( nat > real ) > real ).

thf(sy_c_Series_Osummable_001t__Complex__Ocomplex,type,
    summable_complex: ( nat > complex ) > $o ).

thf(sy_c_Series_Osummable_001t__Real__Oreal,type,
    summable_real: ( nat > real ) > $o ).

thf(sy_c_Series_Osums_001t__Real__Oreal,type,
    sums_real: ( nat > real ) > real > $o ).

thf(sy_c_Set_OCollect_001_Eo,type,
    collect_o: ( $o > $o ) > set_o ).

thf(sy_c_Set_OCollect_001t__Code____Numeral__Ointeger,type,
    collect_Code_integer: ( code_integer > $o ) > set_Code_integer ).

thf(sy_c_Set_OCollect_001t__Complex__Ocomplex,type,
    collect_complex: ( complex > $o ) > set_complex ).

thf(sy_c_Set_OCollect_001t__Extended____Nat__Oenat,type,
    collec4429806609662206161d_enat: ( extended_enat > $o ) > set_Extended_enat ).

thf(sy_c_Set_OCollect_001t__Int__Oint,type,
    collect_int: ( int > $o ) > set_int ).

thf(sy_c_Set_OCollect_001t__List__Olist_It__Nat__Onat_J,type,
    collect_list_nat: ( list_nat > $o ) > set_list_nat ).

thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
    collect_nat: ( nat > $o ) > set_nat ).

thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J,type,
    collec8663557070575231912omplex: ( produc4411394909380815293omplex > $o ) > set_Pr5085853215250843933omplex ).

thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    collec213857154873943460nt_int: ( product_prod_int_int > $o ) > set_Pr958786334691620121nt_int ).

thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    collec3392354462482085612at_nat: ( product_prod_nat_nat > $o ) > set_Pr1261947904930325089at_nat ).

thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
    collec3799799289383736868l_real: ( produc2422161461964618553l_real > $o ) > set_Pr6218003697084177305l_real ).

thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
    collect_real: ( real > $o ) > set_real ).

thf(sy_c_Set_OCollect_001t__VEBT____Definitions__OVEBT,type,
    collect_VEBT_VEBT: ( vEBT_VEBT > $o ) > set_VEBT_VEBT ).

thf(sy_c_Set_Oimage_001t__Extended____Nat__Oenat_001t__Extended____Nat__Oenat,type,
    image_80655429650038917d_enat: ( extended_enat > extended_enat ) > set_Extended_enat > set_Extended_enat ).

thf(sy_c_Set_Oimage_001t__Int__Oint_001t__Int__Oint,type,
    image_int_int: ( int > int ) > set_int > set_int ).

thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Int__Oint,type,
    image_nat_int: ( nat > int ) > set_nat > set_int ).

thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
    image_nat_nat: ( nat > nat ) > set_nat > set_nat ).

thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Real__Oreal,type,
    image_nat_real: ( nat > real ) > set_nat > set_real ).

thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
    image_nat_set_nat: ( nat > set_nat ) > set_nat > set_set_nat ).

thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__String__Ochar,type,
    image_nat_char: ( nat > char ) > set_nat > set_char ).

thf(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: ( real > filter6041513312241820739omplex ) > set_real > set_fi4554929511873752355omplex ).

thf(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: ( real > filter2146258269922977983l_real ) > set_real > set_fi7789364187291644575l_real ).

thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Real__Oreal,type,
    image_real_real: ( real > real ) > set_real > set_real ).

thf(sy_c_Set_Oimage_001t__String__Ochar_001t__Nat__Onat,type,
    image_char_nat: ( char > nat ) > set_char > set_nat ).

thf(sy_c_Set_Oinsert_001t__Int__Oint,type,
    insert_int: int > set_int > set_int ).

thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
    insert_nat: nat > set_nat > set_nat ).

thf(sy_c_Set_Oinsert_001t__Real__Oreal,type,
    insert_real: real > set_real > set_real ).

thf(sy_c_Set_Ovimage_001t__Nat__Onat_001t__Nat__Onat,type,
    vimage_nat_nat: ( nat > nat ) > set_nat > set_nat ).

thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Complex__Ocomplex,type,
    set_fo1517530859248394432omplex: ( nat > complex > complex ) > nat > nat > complex > complex ).

thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Int__Oint,type,
    set_fo2581907887559384638at_int: ( nat > int > int ) > nat > nat > int > int ).

thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Nat__Onat,type,
    set_fo2584398358068434914at_nat: ( nat > nat > nat ) > nat > nat > nat > nat ).

thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Rat__Orat,type,
    set_fo1949268297981939178at_rat: ( nat > rat > rat ) > nat > nat > rat > rat ).

thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Real__Oreal,type,
    set_fo3111899725591712190t_real: ( nat > real > real ) > nat > nat > real > real ).

thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Extended____Nat__Oenat,type,
    set_or5403411693681687835d_enat: extended_enat > extended_enat > set_Extended_enat ).

thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Int__Oint,type,
    set_or1266510415728281911st_int: int > int > set_int ).

thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Nat__Onat,type,
    set_or1269000886237332187st_nat: nat > nat > set_nat ).

thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Num__Onum,type,
    set_or7049704709247886629st_num: num > num > set_num ).

thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Rat__Orat,type,
    set_or633870826150836451st_rat: rat > rat > set_rat ).

thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Real__Oreal,type,
    set_or1222579329274155063t_real: real > real > set_real ).

thf(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 ).

thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Int__Oint,type,
    set_or4662586982721622107an_int: int > int > set_int ).

thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Nat__Onat,type,
    set_or4665077453230672383an_nat: nat > nat > set_nat ).

thf(sy_c_Set__Interval_Oord__class_OatLeast_001t__Nat__Onat,type,
    set_ord_atLeast_nat: nat > set_nat ).

thf(sy_c_Set__Interval_Oord__class_OatLeast_001t__Real__Oreal,type,
    set_ord_atLeast_real: real > set_real ).

thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Nat__Onat,type,
    set_ord_atMost_nat: nat > set_nat ).

thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Int__Oint,type,
    set_or6656581121297822940st_int: int > int > set_int ).

thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Nat__Onat,type,
    set_or6659071591806873216st_nat: nat > nat > set_nat ).

thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Int__Oint,type,
    set_or5832277885323065728an_int: int > int > set_int ).

thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Nat__Onat,type,
    set_or5834768355832116004an_nat: nat > nat > set_nat ).

thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Real__Oreal,type,
    set_or1633881224788618240n_real: real > real > set_real ).

thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Nat__Onat,type,
    set_or1210151606488870762an_nat: nat > set_nat ).

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

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

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

thf(sy_c_String_OCode_Oabort_001t__Real__Oreal,type,
    abort_real: literal > ( product_unit > real ) > real ).

thf(sy_c_String_OLiteral,type,
    literal2: $o > $o > $o > $o > $o > $o > $o > literal > literal ).

thf(sy_c_String_Oascii__of,type,
    ascii_of: char > char ).

thf(sy_c_String_Ochar_OChar,type,
    char2: $o > $o > $o > $o > $o > $o > $o > $o > char ).

thf(sy_c_String_Ocomm__semiring__1__class_Oof__char_001t__Nat__Onat,type,
    comm_s629917340098488124ar_nat: char > nat ).

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

thf(sy_c_String_Ounique__euclidean__semiring__with__bit__operations__class_Ochar__of_001t__Nat__Onat,type,
    unique3096191561947761185of_nat: nat > char ).

thf(sy_c_Topological__Spaces_Ocontinuous_001t__Real__Oreal_001t__Real__Oreal,type,
    topolo4422821103128117721l_real: filter_real > ( real > real ) > $o ).

thf(sy_c_Topological__Spaces_Ocontinuous__on_001t__Real__Oreal_001t__Real__Oreal,type,
    topolo5044208981011980120l_real: set_real > ( real > real ) > $o ).

thf(sy_c_Topological__Spaces_Omonoseq_001t__Real__Oreal,type,
    topolo6980174941875973593q_real: ( nat > real ) > $o ).

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

thf(sy_c_Topological__Spaces_Otopological__space__class_Oconvergent_001t__Real__Oreal,type,
    topolo7531315842566124627t_real: ( nat > real ) > $o ).

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

thf(sy_c_Topological__Spaces_Ouniform__space__class_OCauchy_001t__Complex__Ocomplex,type,
    topolo6517432010174082258omplex: ( nat > complex ) > $o ).

thf(sy_c_Topological__Spaces_Ouniform__space__class_OCauchy_001t__Real__Oreal,type,
    topolo4055970368930404560y_real: ( nat > real ) > $o ).

thf(sy_c_Topological__Spaces_Ouniformity__class_Ouniformity_001t__Complex__Ocomplex,type,
    topolo896644834953643431omplex: filter6041513312241820739omplex ).

thf(sy_c_Topological__Spaces_Ouniformity__class_Ouniformity_001t__Real__Oreal,type,
    topolo1511823702728130853y_real: filter2146258269922977983l_real ).

thf(sy_c_Transcendental_Oarccos,type,
    arccos: real > real ).

thf(sy_c_Transcendental_Oarcosh_001t__Real__Oreal,type,
    arcosh_real: real > real ).

thf(sy_c_Transcendental_Oarcsin,type,
    arcsin: real > real ).

thf(sy_c_Transcendental_Oarctan,type,
    arctan: real > real ).

thf(sy_c_Transcendental_Oarsinh_001t__Real__Oreal,type,
    arsinh_real: real > real ).

thf(sy_c_Transcendental_Oartanh_001t__Real__Oreal,type,
    artanh_real: real > real ).

thf(sy_c_Transcendental_Ocos_001t__Complex__Ocomplex,type,
    cos_complex: complex > complex ).

thf(sy_c_Transcendental_Ocos_001t__Real__Oreal,type,
    cos_real: real > real ).

thf(sy_c_Transcendental_Ocos__coeff,type,
    cos_coeff: nat > real ).

thf(sy_c_Transcendental_Ocosh_001t__Complex__Ocomplex,type,
    cosh_complex: complex > complex ).

thf(sy_c_Transcendental_Ocosh_001t__Real__Oreal,type,
    cosh_real: real > real ).

thf(sy_c_Transcendental_Ocot_001t__Complex__Ocomplex,type,
    cot_complex: complex > complex ).

thf(sy_c_Transcendental_Ocot_001t__Real__Oreal,type,
    cot_real: real > real ).

thf(sy_c_Transcendental_Oexp_001t__Complex__Ocomplex,type,
    exp_complex: complex > complex ).

thf(sy_c_Transcendental_Oexp_001t__Real__Oreal,type,
    exp_real: real > real ).

thf(sy_c_Transcendental_Oln__class_Oln_001t__Real__Oreal,type,
    ln_ln_real: real > real ).

thf(sy_c_Transcendental_Olog,type,
    log: real > real > real ).

thf(sy_c_Transcendental_Opi,type,
    pi: real ).

thf(sy_c_Transcendental_Opowr_001t__Real__Oreal,type,
    powr_real: real > real > real ).

thf(sy_c_Transcendental_Opowr__real,type,
    powr_real2: real > real > real ).

thf(sy_c_Transcendental_Osin_001t__Complex__Ocomplex,type,
    sin_complex: complex > complex ).

thf(sy_c_Transcendental_Osin_001t__Real__Oreal,type,
    sin_real: real > real ).

thf(sy_c_Transcendental_Osin__coeff,type,
    sin_coeff: nat > real ).

thf(sy_c_Transcendental_Osinh_001t__Complex__Ocomplex,type,
    sinh_complex: complex > complex ).

thf(sy_c_Transcendental_Osinh_001t__Real__Oreal,type,
    sinh_real: real > real ).

thf(sy_c_Transcendental_Otan_001t__Complex__Ocomplex,type,
    tan_complex: complex > complex ).

thf(sy_c_Transcendental_Otan_001t__Real__Oreal,type,
    tan_real: real > real ).

thf(sy_c_Transcendental_Otanh_001t__Complex__Ocomplex,type,
    tanh_complex: complex > complex ).

thf(sy_c_Transcendental_Otanh_001t__Real__Oreal,type,
    tanh_real: real > real ).

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

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

thf(sy_c_VEBT__Definitions_OVEBT_OLeaf,type,
    vEBT_Leaf: $o > $o > vEBT_VEBT ).

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

thf(sy_c_VEBT__Definitions_OVEBT_Osize__VEBT,type,
    vEBT_size_VEBT: vEBT_VEBT > nat ).

thf(sy_c_VEBT__Definitions_OVEBT__internal_Oboth__member__options,type,
    vEBT_V8194947554948674370ptions: vEBT_VEBT > nat > $o ).

thf(sy_c_VEBT__Definitions_OVEBT__internal_Oelim__dead,type,
    vEBT_VEBT_elim_dead: vEBT_VEBT > extended_enat > vEBT_VEBT ).

thf(sy_c_VEBT__Definitions_OVEBT__internal_Oelim__dead__rel,type,
    vEBT_V312737461966249ad_rel: produc7272778201969148633d_enat > produc7272778201969148633d_enat > $o ).

thf(sy_c_VEBT__Definitions_OVEBT__internal_Ohigh,type,
    vEBT_VEBT_high: nat > nat > nat ).

thf(sy_c_VEBT__Definitions_OVEBT__internal_Oin__children,type,
    vEBT_V5917875025757280293ildren: nat > list_VEBT_VEBT > nat > $o ).

thf(sy_c_VEBT__Definitions_OVEBT__internal_Olow,type,
    vEBT_VEBT_low: nat > nat > nat ).

thf(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima,type,
    vEBT_VEBT_membermima: vEBT_VEBT > nat > $o ).

thf(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima__rel,type,
    vEBT_V4351362008482014158ma_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).

thf(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member,type,
    vEBT_V5719532721284313246member: vEBT_VEBT > nat > $o ).

thf(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member__rel,type,
    vEBT_V5765760719290551771er_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).

thf(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H,type,
    vEBT_VEBT_valid: vEBT_VEBT > nat > $o ).

thf(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H__rel,type,
    vEBT_VEBT_valid_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).

thf(sy_c_VEBT__Definitions_Oinvar__vebt,type,
    vEBT_invar_vebt: vEBT_VEBT > nat > $o ).

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

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

thf(sy_c_VEBT__Definitions_Ovebt__buildup__rel,type,
    vEBT_v4011308405150292612up_rel: nat > nat > $o ).

thf(sy_c_VEBT__Insert_Ovebt__insert,type,
    vEBT_vebt_insert: vEBT_VEBT > nat > vEBT_VEBT ).

thf(sy_c_VEBT__Insert_Ovebt__insert__rel,type,
    vEBT_vebt_insert_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).

thf(sy_c_VEBT__Member_OVEBT__internal_Obit__concat,type,
    vEBT_VEBT_bit_concat: nat > nat > nat > nat ).

thf(sy_c_VEBT__Member_OVEBT__internal_OminNull,type,
    vEBT_VEBT_minNull: vEBT_VEBT > $o ).

thf(sy_c_VEBT__Member_OVEBT__internal_Oset__vebt_H,type,
    vEBT_VEBT_set_vebt: vEBT_VEBT > set_nat ).

thf(sy_c_VEBT__Member_Ovebt__member,type,
    vEBT_vebt_member: vEBT_VEBT > nat > $o ).

thf(sy_c_VEBT__Member_Ovebt__member__rel,type,
    vEBT_vebt_member_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).

thf(sy_c_VEBT__MinMax_OVEBT__internal_Oadd,type,
    vEBT_VEBT_add: option_nat > option_nat > option_nat ).

thf(sy_c_VEBT__MinMax_OVEBT__internal_Ogreater,type,
    vEBT_VEBT_greater: option_nat > option_nat > $o ).

thf(sy_c_VEBT__MinMax_OVEBT__internal_Oless,type,
    vEBT_VEBT_less: option_nat > option_nat > $o ).

thf(sy_c_VEBT__MinMax_OVEBT__internal_Olesseq,type,
    vEBT_VEBT_lesseq: option_nat > option_nat > $o ).

thf(sy_c_VEBT__MinMax_OVEBT__internal_Omax__in__set,type,
    vEBT_VEBT_max_in_set: set_nat > nat > $o ).

thf(sy_c_VEBT__MinMax_OVEBT__internal_Omin__in__set,type,
    vEBT_VEBT_min_in_set: set_nat > nat > $o ).

thf(sy_c_VEBT__MinMax_OVEBT__internal_Omul,type,
    vEBT_VEBT_mul: option_nat > option_nat > option_nat ).

thf(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift_001t__Nat__Onat,type,
    vEBT_V4262088993061758097ft_nat: ( nat > nat > nat ) > option_nat > option_nat > option_nat ).

thf(sy_c_VEBT__MinMax_OVEBT__internal_Opower,type,
    vEBT_VEBT_power: option_nat > option_nat > option_nat ).

thf(sy_c_VEBT__MinMax_Ovebt__maxt,type,
    vEBT_vebt_maxt: vEBT_VEBT > option_nat ).

thf(sy_c_VEBT__MinMax_Ovebt__maxt__rel,type,
    vEBT_vebt_maxt_rel: vEBT_VEBT > vEBT_VEBT > $o ).

thf(sy_c_VEBT__MinMax_Ovebt__mint,type,
    vEBT_vebt_mint: vEBT_VEBT > option_nat ).

thf(sy_c_VEBT__MinMax_Ovebt__mint__rel,type,
    vEBT_vebt_mint_rel: vEBT_VEBT > vEBT_VEBT > $o ).

thf(sy_c_VEBT__Succ_Ois__succ__in__set,type,
    vEBT_is_succ_in_set: set_nat > nat > nat > $o ).

thf(sy_c_VEBT__Succ_Ovebt__succ,type,
    vEBT_vebt_succ: vEBT_VEBT > nat > option_nat ).

thf(sy_c_VEBT__Succ_Ovebt__succ__rel,type,
    vEBT_vebt_succ_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).

thf(sy_c_Wellfounded_Oaccp_001t__List__Olist_It__Nat__Onat_J,type,
    accp_list_nat: ( list_nat > list_nat > $o ) > list_nat > $o ).

thf(sy_c_Wellfounded_Oaccp_001t__Nat__Onat,type,
    accp_nat: ( nat > nat > $o ) > nat > $o ).

thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    accp_P1096762738010456898nt_int: ( product_prod_int_int > product_prod_int_int > $o ) > product_prod_int_int > $o ).

thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    accp_P4275260045618599050at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > product_prod_nat_nat > $o ).

thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J,type,
    accp_P3113834385874906142um_num: ( product_prod_num_num > product_prod_num_num > $o ) > product_prod_num_num > $o ).

thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Extended____Nat__Oenat_J,type,
    accp_P6183159247885693666d_enat: ( produc7272778201969148633d_enat > produc7272778201969148633d_enat > $o ) > produc7272778201969148633d_enat > $o ).

thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
    accp_P2887432264394892906BT_nat: ( produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ) > produc9072475918466114483BT_nat > $o ).

thf(sy_c_Wellfounded_Oaccp_001t__VEBT____Definitions__OVEBT,type,
    accp_VEBT_VEBT: ( vEBT_VEBT > vEBT_VEBT > $o ) > vEBT_VEBT > $o ).

thf(sy_c_Wellfounded_Oless__than,type,
    less_than: set_Pr1261947904930325089at_nat ).

thf(sy_c_Wellfounded_Opred__nat,type,
    pred_nat: set_Pr1261947904930325089at_nat ).

thf(sy_c_fChoice_001t__Real__Oreal,type,
    fChoice_real: ( real > $o ) > real ).

thf(sy_c_member_001_Eo,type,
    member_o: $o > set_o > $o ).

thf(sy_c_member_001t__Code____Numeral__Ointeger,type,
    member_Code_integer: code_integer > set_Code_integer > $o ).

thf(sy_c_member_001t__Complex__Ocomplex,type,
    member_complex: complex > set_complex > $o ).

thf(sy_c_member_001t__Extended____Nat__Oenat,type,
    member_Extended_enat: extended_enat > set_Extended_enat > $o ).

thf(sy_c_member_001t__Int__Oint,type,
    member_int: int > set_int > $o ).

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

thf(sy_c_member_001t__Nat__Onat,type,
    member_nat: nat > set_nat > $o ).

thf(sy_c_member_001t__Num__Onum,type,
    member_num: num > set_num > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    member8440522571783428010at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    member8206827879206165904at_nat: produc859450856879609959at_nat > set_Pr8693737435421807431at_nat > $o ).

thf(sy_c_member_001t__Rat__Orat,type,
    member_rat: rat > set_rat > $o ).

thf(sy_c_member_001t__Real__Oreal,type,
    member_real: real > set_real > $o ).

thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
    member_set_nat: set_nat > set_set_nat > $o ).

thf(sy_c_member_001t__VEBT____Definitions__OVEBT,type,
    member_VEBT_VEBT: vEBT_VEBT > set_VEBT_VEBT > $o ).

thf(sy_v_deg____,type,
    deg: nat ).

thf(sy_v_m____,type,
    m: nat ).

thf(sy_v_ma____,type,
    ma: nat ).

thf(sy_v_mi____,type,
    mi: nat ).

thf(sy_v_miny____,type,
    miny: nat ).

thf(sy_v_na____,type,
    na: nat ).

thf(sy_v_res____,type,
    res: nat ).

thf(sy_v_sc____,type,
    sc: nat ).

thf(sy_v_summary____,type,
    summary: vEBT_VEBT ).

thf(sy_v_treeList____,type,
    treeList: list_VEBT_VEBT ).

thf(sy_v_xa____,type,
    xa: nat ).

thf(sy_v_za____,type,
    za: nat ).

% Relevant facts (10210)
thf(fact_0_bbbb,axiom,
    ord_less_eq_nat @ ( vEBT_VEBT_high @ za @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% bbbb
thf(fact_1__092_060open_0622_A_092_060le_062_Adeg_092_060close_062,axiom,
    ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ deg ).

% \<open>2 \<le> deg\<close>
thf(fact_2_aaaa,axiom,
    ord_less_eq_nat @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_VEBT_high @ za @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% aaaa
thf(fact_3_max__in__set__def,axiom,
    ( vEBT_VEBT_max_in_set
    = ( ^ [Xs: set_nat,X: nat] :
          ( ( member_nat @ X @ Xs )
          & ! [Y: nat] :
              ( ( member_nat @ Y @ Xs )
             => ( ord_less_eq_nat @ Y @ X ) ) ) ) ) ).

% max_in_set_def
thf(fact_4_min__in__set__def,axiom,
    ( vEBT_VEBT_min_in_set
    = ( ^ [Xs: set_nat,X: nat] :
          ( ( member_nat @ X @ Xs )
          & ! [Y: nat] :
              ( ( member_nat @ Y @ Xs )
             => ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ).

% min_in_set_def
thf(fact_5__092_060open_062deg_Adiv_A2_A_061_An_092_060close_062,axiom,
    ( ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = na ) ).

% \<open>deg div 2 = n\<close>
thf(fact_6_semiring__norm_I85_J,axiom,
    ! [M: num] :
      ( ( bit0 @ M )
     != one ) ).

% semiring_norm(85)
thf(fact_7_semiring__norm_I83_J,axiom,
    ! [N: num] :
      ( one
     != ( bit0 @ N ) ) ).

% semiring_norm(83)
thf(fact_8_numeral__Bit0__div__2,axiom,
    ! [N: num] :
      ( ( divide_divide_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( numeral_numeral_nat @ N ) ) ).

% numeral_Bit0_div_2
thf(fact_9_numeral__Bit0__div__2,axiom,
    ! [N: num] :
      ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
      = ( numeral_numeral_int @ N ) ) ).

% numeral_Bit0_div_2
thf(fact_10_high__def,axiom,
    ( vEBT_VEBT_high
    = ( ^ [X: nat,N2: nat] : ( divide_divide_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% high_def
thf(fact_11__092_060open_062high_Az_A_Ideg_Adiv_A2_J_A_060_Asc_092_060close_062,axiom,
    ord_less_nat @ ( vEBT_VEBT_high @ za @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ sc ).

% \<open>high z (deg div 2) < sc\<close>
thf(fact_12_divide__numeral__1,axiom,
    ! [A: complex] :
      ( ( divide1717551699836669952omplex @ A @ ( numera6690914467698888265omplex @ one ) )
      = A ) ).

% divide_numeral_1
thf(fact_13_divide__numeral__1,axiom,
    ! [A: real] :
      ( ( divide_divide_real @ A @ ( numeral_numeral_real @ one ) )
      = A ) ).

% divide_numeral_1
thf(fact_14_divide__numeral__1,axiom,
    ! [A: rat] :
      ( ( divide_divide_rat @ A @ ( numeral_numeral_rat @ one ) )
      = A ) ).

% divide_numeral_1
thf(fact_15_verit__eq__simplify_I8_J,axiom,
    ! [X2: num,Y2: num] :
      ( ( ( bit0 @ X2 )
        = ( bit0 @ Y2 ) )
      = ( X2 = Y2 ) ) ).

% verit_eq_simplify(8)
thf(fact_16_semiring__norm_I87_J,axiom,
    ! [M: num,N: num] :
      ( ( ( bit0 @ M )
        = ( bit0 @ N ) )
      = ( M = N ) ) ).

% semiring_norm(87)
thf(fact_17_numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( ( numera6690914467698888265omplex @ M )
        = ( numera6690914467698888265omplex @ N ) )
      = ( M = N ) ) ).

% numeral_eq_iff
thf(fact_18_numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( ( numeral_numeral_real @ M )
        = ( numeral_numeral_real @ N ) )
      = ( M = N ) ) ).

% numeral_eq_iff
thf(fact_19_numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( ( numeral_numeral_rat @ M )
        = ( numeral_numeral_rat @ N ) )
      = ( M = N ) ) ).

% numeral_eq_iff
thf(fact_20_numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( ( numeral_numeral_nat @ M )
        = ( numeral_numeral_nat @ N ) )
      = ( M = N ) ) ).

% numeral_eq_iff
thf(fact_21_numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( ( numeral_numeral_int @ M )
        = ( numeral_numeral_int @ N ) )
      = ( M = N ) ) ).

% numeral_eq_iff
thf(fact_22_numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
      = ( ord_less_eq_num @ M @ N ) ) ).

% numeral_le_iff
thf(fact_23_numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
      = ( ord_less_eq_num @ M @ N ) ) ).

% numeral_le_iff
thf(fact_24_numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
      = ( ord_less_eq_num @ M @ N ) ) ).

% numeral_le_iff
thf(fact_25_numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
      = ( ord_less_eq_num @ M @ N ) ) ).

% numeral_le_iff
thf(fact_26_numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
      = ( ord_less_num @ M @ N ) ) ).

% numeral_less_iff
thf(fact_27_numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
      = ( ord_less_num @ M @ N ) ) ).

% numeral_less_iff
thf(fact_28_numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
      = ( ord_less_num @ M @ N ) ) ).

% numeral_less_iff
thf(fact_29_numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
      = ( ord_less_num @ M @ N ) ) ).

% numeral_less_iff
thf(fact_30__092_060open_062z_A_060_A2_A_094_Adeg_092_060close_062,axiom,
    ord_less_nat @ za @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ deg ) ).

% \<open>z < 2 ^ deg\<close>
thf(fact_31__C5_Ohyps_C_I10_J,axiom,
    ord_less_nat @ ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ deg ) ).

% "5.hyps"(10)
thf(fact_32_verit__comp__simplify1_I3_J,axiom,
    ! [B: real,A2: real] :
      ( ( ~ ( ord_less_eq_real @ B @ A2 ) )
      = ( ord_less_real @ A2 @ B ) ) ).

% verit_comp_simplify1(3)
thf(fact_33_verit__comp__simplify1_I3_J,axiom,
    ! [B: rat,A2: rat] :
      ( ( ~ ( ord_less_eq_rat @ B @ A2 ) )
      = ( ord_less_rat @ A2 @ B ) ) ).

% verit_comp_simplify1(3)
thf(fact_34_verit__comp__simplify1_I3_J,axiom,
    ! [B: num,A2: num] :
      ( ( ~ ( ord_less_eq_num @ B @ A2 ) )
      = ( ord_less_num @ A2 @ B ) ) ).

% verit_comp_simplify1(3)
thf(fact_35_verit__comp__simplify1_I3_J,axiom,
    ! [B: nat,A2: nat] :
      ( ( ~ ( ord_less_eq_nat @ B @ A2 ) )
      = ( ord_less_nat @ A2 @ B ) ) ).

% verit_comp_simplify1(3)
thf(fact_36_verit__comp__simplify1_I3_J,axiom,
    ! [B: int,A2: int] :
      ( ( ~ ( ord_less_eq_int @ B @ A2 ) )
      = ( ord_less_int @ A2 @ B ) ) ).

% verit_comp_simplify1(3)
thf(fact_37_verit__comp__simplify1_I2_J,axiom,
    ! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_38_verit__comp__simplify1_I2_J,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_39_verit__comp__simplify1_I2_J,axiom,
    ! [A: num] : ( ord_less_eq_num @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_40_verit__comp__simplify1_I2_J,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_41_verit__comp__simplify1_I2_J,axiom,
    ! [A: int] : ( ord_less_eq_int @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_42_verit__comp__simplify1_I1_J,axiom,
    ! [A: real] :
      ~ ( ord_less_real @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_43_verit__comp__simplify1_I1_J,axiom,
    ! [A: rat] :
      ~ ( ord_less_rat @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_44_verit__comp__simplify1_I1_J,axiom,
    ! [A: num] :
      ~ ( ord_less_num @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_45_verit__comp__simplify1_I1_J,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_46_verit__comp__simplify1_I1_J,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_47_verit__la__disequality,axiom,
    ! [A: rat,B2: rat] :
      ( ( A = B2 )
      | ~ ( ord_less_eq_rat @ A @ B2 )
      | ~ ( ord_less_eq_rat @ B2 @ A ) ) ).

% verit_la_disequality
thf(fact_48_verit__la__disequality,axiom,
    ! [A: num,B2: num] :
      ( ( A = B2 )
      | ~ ( ord_less_eq_num @ A @ B2 )
      | ~ ( ord_less_eq_num @ B2 @ A ) ) ).

% verit_la_disequality
thf(fact_49_verit__la__disequality,axiom,
    ! [A: nat,B2: nat] :
      ( ( A = B2 )
      | ~ ( ord_less_eq_nat @ A @ B2 )
      | ~ ( ord_less_eq_nat @ B2 @ A ) ) ).

% verit_la_disequality
thf(fact_50_verit__la__disequality,axiom,
    ! [A: int,B2: int] :
      ( ( A = B2 )
      | ~ ( ord_less_eq_int @ A @ B2 )
      | ~ ( ord_less_eq_int @ B2 @ A ) ) ).

% verit_la_disequality
thf(fact_51_div__le__dividend,axiom,
    ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ M ) ).

% div_le_dividend
thf(fact_52_div__le__mono,axiom,
    ! [M: nat,N: nat,K: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_nat @ ( divide_divide_nat @ M @ K ) @ ( divide_divide_nat @ N @ K ) ) ) ).

% div_le_mono
thf(fact_53_verit__eq__simplify_I10_J,axiom,
    ! [X2: num] :
      ( one
     != ( bit0 @ X2 ) ) ).

% verit_eq_simplify(10)
thf(fact_54__092_060open_062high_Ares_A_Ideg_Adiv_A2_J_A_061_Asc_092_060close_062,axiom,
    ( ( vEBT_VEBT_high @ res @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
    = sc ) ).

% \<open>high res (deg div 2) = sc\<close>
thf(fact_55__092_060open_062x_A_060_Ares_092_060close_062,axiom,
    ord_less_nat @ xa @ res ).

% \<open>x < res\<close>
thf(fact_56__092_060open_062sc_A_060_A2_A_094_Am_092_060close_062,axiom,
    ord_less_nat @ sc @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) ).

% \<open>sc < 2 ^ m\<close>
thf(fact_57_power2__nat__le__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% power2_nat_le_imp_le
thf(fact_58_power2__nat__le__eq__le,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% power2_nat_le_eq_le
thf(fact_59_self__le__ge2__pow,axiom,
    ! [K: nat,M: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
     => ( ord_less_eq_nat @ M @ ( power_power_nat @ K @ M ) ) ) ).

% self_le_ge2_pow
thf(fact_60_less__exp,axiom,
    ! [N: nat] : ( ord_less_nat @ N @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).

% less_exp
thf(fact_61_high__bound__aux,axiom,
    ! [Ma: nat,N: nat,M: nat] :
      ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) )
     => ( ord_less_nat @ ( vEBT_VEBT_high @ Ma @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).

% high_bound_aux
thf(fact_62_power__minus__is__div,axiom,
    ! [B2: nat,A: nat] :
      ( ( ord_less_eq_nat @ B2 @ A )
     => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ A @ B2 ) )
        = ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) ) ) ).

% power_minus_is_div
thf(fact_63_pow__sum,axiom,
    ! [A: nat,B2: nat] :
      ( ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) ).

% pow_sum
thf(fact_64__092_060open_0621_A_092_060le_062_An_092_060close_062,axiom,
    ord_less_eq_nat @ one_one_nat @ na ).

% \<open>1 \<le> n\<close>
thf(fact_65__092_060open_062mi_A_092_060le_062_Ax_092_060close_062,axiom,
    ord_less_eq_nat @ mi @ xa ).

% \<open>mi \<le> x\<close>
thf(fact_66__C5_Ohyps_C_I9_J,axiom,
    ord_less_eq_nat @ mi @ ma ).

% "5.hyps"(9)
thf(fact_67__C5_Ohyps_C_I6_J,axiom,
    ( deg
    = ( plus_plus_nat @ na @ m ) ) ).

% "5.hyps"(6)
thf(fact_68__092_060open_062z_A_092_060noteq_062_Ami_092_060close_062,axiom,
    za != mi ).

% \<open>z \<noteq> mi\<close>
thf(fact_69__092_060open_062mi_A_092_060noteq_062_Ama_092_060close_062,axiom,
    mi != ma ).

% \<open>mi \<noteq> ma\<close>
thf(fact_70__092_060open_062res_A_092_060le_062_Ama_092_060close_062,axiom,
    ord_less_eq_nat @ res @ ma ).

% \<open>res \<le> ma\<close>
thf(fact_71_mem__Collect__eq,axiom,
    ! [A: extended_enat,P: extended_enat > $o] :
      ( ( member_Extended_enat @ A @ ( collec4429806609662206161d_enat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_72_mem__Collect__eq,axiom,
    ! [A: complex,P: complex > $o] :
      ( ( member_complex @ A @ ( collect_complex @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_73_mem__Collect__eq,axiom,
    ! [A: real,P: real > $o] :
      ( ( member_real @ A @ ( collect_real @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_74_mem__Collect__eq,axiom,
    ! [A: list_nat,P: list_nat > $o] :
      ( ( member_list_nat @ A @ ( collect_list_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_75_mem__Collect__eq,axiom,
    ! [A: nat,P: nat > $o] :
      ( ( member_nat @ A @ ( collect_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_76_mem__Collect__eq,axiom,
    ! [A: int,P: int > $o] :
      ( ( member_int @ A @ ( collect_int @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_77_Collect__mem__eq,axiom,
    ! [A3: set_Extended_enat] :
      ( ( collec4429806609662206161d_enat
        @ ^ [X: extended_enat] : ( member_Extended_enat @ X @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_78_Collect__mem__eq,axiom,
    ! [A3: set_complex] :
      ( ( collect_complex
        @ ^ [X: complex] : ( member_complex @ X @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_79_Collect__mem__eq,axiom,
    ! [A3: set_real] :
      ( ( collect_real
        @ ^ [X: real] : ( member_real @ X @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_80_Collect__mem__eq,axiom,
    ! [A3: set_list_nat] :
      ( ( collect_list_nat
        @ ^ [X: list_nat] : ( member_list_nat @ X @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_81_Collect__mem__eq,axiom,
    ! [A3: set_nat] :
      ( ( collect_nat
        @ ^ [X: nat] : ( member_nat @ X @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_82_Collect__mem__eq,axiom,
    ! [A3: set_int] :
      ( ( collect_int
        @ ^ [X: int] : ( member_int @ X @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_83_Collect__cong,axiom,
    ! [P: complex > $o,Q: complex > $o] :
      ( ! [X3: complex] :
          ( ( P @ X3 )
          = ( Q @ X3 ) )
     => ( ( collect_complex @ P )
        = ( collect_complex @ Q ) ) ) ).

% Collect_cong
thf(fact_84_Collect__cong,axiom,
    ! [P: real > $o,Q: real > $o] :
      ( ! [X3: real] :
          ( ( P @ X3 )
          = ( Q @ X3 ) )
     => ( ( collect_real @ P )
        = ( collect_real @ Q ) ) ) ).

% Collect_cong
thf(fact_85_Collect__cong,axiom,
    ! [P: list_nat > $o,Q: list_nat > $o] :
      ( ! [X3: list_nat] :
          ( ( P @ X3 )
          = ( Q @ X3 ) )
     => ( ( collect_list_nat @ P )
        = ( collect_list_nat @ Q ) ) ) ).

% Collect_cong
thf(fact_86_Collect__cong,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ! [X3: nat] :
          ( ( P @ X3 )
          = ( Q @ X3 ) )
     => ( ( collect_nat @ P )
        = ( collect_nat @ Q ) ) ) ).

% Collect_cong
thf(fact_87_Collect__cong,axiom,
    ! [P: int > $o,Q: int > $o] :
      ( ! [X3: int] :
          ( ( P @ X3 )
          = ( Q @ X3 ) )
     => ( ( collect_int @ P )
        = ( collect_int @ Q ) ) ) ).

% Collect_cong
thf(fact_88__092_060open_062mi_A_060_Ares_092_060close_062,axiom,
    ord_less_nat @ mi @ res ).

% \<open>mi < res\<close>
thf(fact_89_add__numeral__left,axiom,
    ! [V: num,W: num,Z: complex] :
      ( ( plus_plus_complex @ ( numera6690914467698888265omplex @ V ) @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ W ) @ Z ) )
      = ( plus_plus_complex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).

% add_numeral_left
thf(fact_90_add__numeral__left,axiom,
    ! [V: num,W: num,Z: real] :
      ( ( plus_plus_real @ ( numeral_numeral_real @ V ) @ ( plus_plus_real @ ( numeral_numeral_real @ W ) @ Z ) )
      = ( plus_plus_real @ ( numeral_numeral_real @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).

% add_numeral_left
thf(fact_91_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 @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).

% add_numeral_left
thf(fact_92_add__numeral__left,axiom,
    ! [V: num,W: num,Z: nat] :
      ( ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ ( numeral_numeral_nat @ W ) @ Z ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).

% add_numeral_left
thf(fact_93_add__numeral__left,axiom,
    ! [V: num,W: num,Z: int] :
      ( ( plus_plus_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ ( numeral_numeral_int @ W ) @ Z ) )
      = ( plus_plus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).

% add_numeral_left
thf(fact_94_numeral__plus__numeral,axiom,
    ! [M: num,N: num] :
      ( ( plus_plus_complex @ ( numera6690914467698888265omplex @ M ) @ ( numera6690914467698888265omplex @ N ) )
      = ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ N ) ) ) ).

% numeral_plus_numeral
thf(fact_95_numeral__plus__numeral,axiom,
    ! [M: num,N: num] :
      ( ( plus_plus_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
      = ( numeral_numeral_real @ ( plus_plus_num @ M @ N ) ) ) ).

% numeral_plus_numeral
thf(fact_96_numeral__plus__numeral,axiom,
    ! [M: num,N: num] :
      ( ( plus_plus_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
      = ( numeral_numeral_rat @ ( plus_plus_num @ M @ N ) ) ) ).

% numeral_plus_numeral
thf(fact_97_numeral__plus__numeral,axiom,
    ! [M: num,N: num] :
      ( ( plus_plus_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
      = ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ).

% numeral_plus_numeral
thf(fact_98_numeral__plus__numeral,axiom,
    ! [M: num,N: num] :
      ( ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
      = ( numeral_numeral_int @ ( plus_plus_num @ M @ N ) ) ) ).

% numeral_plus_numeral
thf(fact_99_power__one,axiom,
    ! [N: nat] :
      ( ( power_power_rat @ one_one_rat @ N )
      = one_one_rat ) ).

% power_one
thf(fact_100_power__one,axiom,
    ! [N: nat] :
      ( ( power_power_nat @ one_one_nat @ N )
      = one_one_nat ) ).

% power_one
thf(fact_101_power__one,axiom,
    ! [N: nat] :
      ( ( power_power_real @ one_one_real @ N )
      = one_one_real ) ).

% power_one
thf(fact_102_power__one,axiom,
    ! [N: nat] :
      ( ( power_power_int @ one_one_int @ N )
      = one_one_int ) ).

% power_one
thf(fact_103_power__one,axiom,
    ! [N: nat] :
      ( ( power_power_complex @ one_one_complex @ N )
      = one_one_complex ) ).

% power_one
thf(fact_104_power__one__right,axiom,
    ! [A: nat] :
      ( ( power_power_nat @ A @ one_one_nat )
      = A ) ).

% power_one_right
thf(fact_105_power__one__right,axiom,
    ! [A: real] :
      ( ( power_power_real @ A @ one_one_nat )
      = A ) ).

% power_one_right
thf(fact_106_power__one__right,axiom,
    ! [A: int] :
      ( ( power_power_int @ A @ one_one_nat )
      = A ) ).

% power_one_right
thf(fact_107_power__one__right,axiom,
    ! [A: complex] :
      ( ( power_power_complex @ A @ one_one_nat )
      = A ) ).

% power_one_right
thf(fact_108_semiring__norm_I78_J,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
      = ( ord_less_num @ M @ N ) ) ).

% semiring_norm(78)
thf(fact_109_semiring__norm_I71_J,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
      = ( ord_less_eq_num @ M @ N ) ) ).

% semiring_norm(71)
thf(fact_110_semiring__norm_I75_J,axiom,
    ! [M: num] :
      ~ ( ord_less_num @ M @ one ) ).

% semiring_norm(75)
thf(fact_111_semiring__norm_I68_J,axiom,
    ! [N: num] : ( ord_less_eq_num @ one @ N ) ).

% semiring_norm(68)
thf(fact_112_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( ( numera6690914467698888265omplex @ N )
        = one_one_complex )
      = ( N = one ) ) ).

% numeral_eq_one_iff
thf(fact_113_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( ( numeral_numeral_real @ N )
        = one_one_real )
      = ( N = one ) ) ).

% numeral_eq_one_iff
thf(fact_114_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( ( numeral_numeral_rat @ N )
        = one_one_rat )
      = ( N = one ) ) ).

% numeral_eq_one_iff
thf(fact_115_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( ( numeral_numeral_nat @ N )
        = one_one_nat )
      = ( N = one ) ) ).

% numeral_eq_one_iff
thf(fact_116_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( ( numeral_numeral_int @ N )
        = one_one_int )
      = ( N = one ) ) ).

% numeral_eq_one_iff
thf(fact_117_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_complex
        = ( numera6690914467698888265omplex @ N ) )
      = ( one = N ) ) ).

% one_eq_numeral_iff
thf(fact_118_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_real
        = ( numeral_numeral_real @ N ) )
      = ( one = N ) ) ).

% one_eq_numeral_iff
thf(fact_119_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_rat
        = ( numeral_numeral_rat @ N ) )
      = ( one = N ) ) ).

% one_eq_numeral_iff
thf(fact_120_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_nat
        = ( numeral_numeral_nat @ N ) )
      = ( one = N ) ) ).

% one_eq_numeral_iff
thf(fact_121_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_int
        = ( numeral_numeral_int @ N ) )
      = ( one = N ) ) ).

% one_eq_numeral_iff
thf(fact_122_power__inject__exp,axiom,
    ! [A: real,M: nat,N: nat] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ( power_power_real @ A @ M )
          = ( power_power_real @ A @ N ) )
        = ( M = N ) ) ) ).

% power_inject_exp
thf(fact_123_power__inject__exp,axiom,
    ! [A: rat,M: nat,N: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ( ( power_power_rat @ A @ M )
          = ( power_power_rat @ A @ N ) )
        = ( M = N ) ) ) ).

% power_inject_exp
thf(fact_124_power__inject__exp,axiom,
    ! [A: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ( ( power_power_nat @ A @ M )
          = ( power_power_nat @ A @ N ) )
        = ( M = N ) ) ) ).

% power_inject_exp
thf(fact_125_power__inject__exp,axiom,
    ! [A: int,M: nat,N: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ( ( power_power_int @ A @ M )
          = ( power_power_int @ A @ N ) )
        = ( M = N ) ) ) ).

% power_inject_exp
thf(fact_126_semiring__norm_I76_J,axiom,
    ! [N: num] : ( ord_less_num @ one @ ( bit0 @ N ) ) ).

% semiring_norm(76)
thf(fact_127_semiring__norm_I69_J,axiom,
    ! [M: num] :
      ~ ( ord_less_eq_num @ ( bit0 @ M ) @ one ) ).

% semiring_norm(69)
thf(fact_128__C5_Ohyps_C_I5_J,axiom,
    ( m
    = ( suc @ na ) ) ).

% "5.hyps"(5)
thf(fact_129_numeral__plus__one,axiom,
    ! [N: num] :
      ( ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ one_one_complex )
      = ( numera6690914467698888265omplex @ ( plus_plus_num @ N @ one ) ) ) ).

% numeral_plus_one
thf(fact_130_numeral__plus__one,axiom,
    ! [N: num] :
      ( ( plus_plus_real @ ( numeral_numeral_real @ N ) @ one_one_real )
      = ( numeral_numeral_real @ ( plus_plus_num @ N @ one ) ) ) ).

% numeral_plus_one
thf(fact_131_numeral__plus__one,axiom,
    ! [N: num] :
      ( ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ one_one_rat )
      = ( numeral_numeral_rat @ ( plus_plus_num @ N @ one ) ) ) ).

% numeral_plus_one
thf(fact_132_numeral__plus__one,axiom,
    ! [N: num] :
      ( ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat )
      = ( numeral_numeral_nat @ ( plus_plus_num @ N @ one ) ) ) ).

% numeral_plus_one
thf(fact_133_numeral__plus__one,axiom,
    ! [N: num] :
      ( ( plus_plus_int @ ( numeral_numeral_int @ N ) @ one_one_int )
      = ( numeral_numeral_int @ ( plus_plus_num @ N @ one ) ) ) ).

% numeral_plus_one
thf(fact_134_one__plus__numeral,axiom,
    ! [N: num] :
      ( ( plus_plus_complex @ one_one_complex @ ( numera6690914467698888265omplex @ N ) )
      = ( numera6690914467698888265omplex @ ( plus_plus_num @ one @ N ) ) ) ).

% one_plus_numeral
thf(fact_135_one__plus__numeral,axiom,
    ! [N: num] :
      ( ( plus_plus_real @ one_one_real @ ( numeral_numeral_real @ N ) )
      = ( numeral_numeral_real @ ( plus_plus_num @ one @ N ) ) ) ).

% one_plus_numeral
thf(fact_136_one__plus__numeral,axiom,
    ! [N: num] :
      ( ( plus_plus_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) )
      = ( numeral_numeral_rat @ ( plus_plus_num @ one @ N ) ) ) ).

% one_plus_numeral
thf(fact_137_one__plus__numeral,axiom,
    ! [N: num] :
      ( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) )
      = ( numeral_numeral_nat @ ( plus_plus_num @ one @ N ) ) ) ).

% one_plus_numeral
thf(fact_138_one__plus__numeral,axiom,
    ! [N: num] :
      ( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ N ) )
      = ( numeral_numeral_int @ ( plus_plus_num @ one @ N ) ) ) ).

% one_plus_numeral
thf(fact_139_power__strict__increasing__iff,axiom,
    ! [B2: real,X4: nat,Y3: nat] :
      ( ( ord_less_real @ one_one_real @ B2 )
     => ( ( ord_less_real @ ( power_power_real @ B2 @ X4 ) @ ( power_power_real @ B2 @ Y3 ) )
        = ( ord_less_nat @ X4 @ Y3 ) ) ) ).

% power_strict_increasing_iff
thf(fact_140_power__strict__increasing__iff,axiom,
    ! [B2: rat,X4: nat,Y3: nat] :
      ( ( ord_less_rat @ one_one_rat @ B2 )
     => ( ( ord_less_rat @ ( power_power_rat @ B2 @ X4 ) @ ( power_power_rat @ B2 @ Y3 ) )
        = ( ord_less_nat @ X4 @ Y3 ) ) ) ).

% power_strict_increasing_iff
thf(fact_141_power__strict__increasing__iff,axiom,
    ! [B2: nat,X4: nat,Y3: nat] :
      ( ( ord_less_nat @ one_one_nat @ B2 )
     => ( ( ord_less_nat @ ( power_power_nat @ B2 @ X4 ) @ ( power_power_nat @ B2 @ Y3 ) )
        = ( ord_less_nat @ X4 @ Y3 ) ) ) ).

% power_strict_increasing_iff
thf(fact_142_power__strict__increasing__iff,axiom,
    ! [B2: int,X4: nat,Y3: nat] :
      ( ( ord_less_int @ one_one_int @ B2 )
     => ( ( ord_less_int @ ( power_power_int @ B2 @ X4 ) @ ( power_power_int @ B2 @ Y3 ) )
        = ( ord_less_nat @ X4 @ Y3 ) ) ) ).

% power_strict_increasing_iff
thf(fact_143_one__add__one,axiom,
    ( ( plus_plus_complex @ one_one_complex @ one_one_complex )
    = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_144_one__add__one,axiom,
    ( ( plus_plus_real @ one_one_real @ one_one_real )
    = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_145_one__add__one,axiom,
    ( ( plus_plus_rat @ one_one_rat @ one_one_rat )
    = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_146_one__add__one,axiom,
    ( ( plus_plus_nat @ one_one_nat @ one_one_nat )
    = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_147_one__add__one,axiom,
    ( ( plus_plus_int @ one_one_int @ one_one_int )
    = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_148_power__increasing__iff,axiom,
    ! [B2: real,X4: nat,Y3: nat] :
      ( ( ord_less_real @ one_one_real @ B2 )
     => ( ( ord_less_eq_real @ ( power_power_real @ B2 @ X4 ) @ ( power_power_real @ B2 @ Y3 ) )
        = ( ord_less_eq_nat @ X4 @ Y3 ) ) ) ).

% power_increasing_iff
thf(fact_149_power__increasing__iff,axiom,
    ! [B2: rat,X4: nat,Y3: nat] :
      ( ( ord_less_rat @ one_one_rat @ B2 )
     => ( ( ord_less_eq_rat @ ( power_power_rat @ B2 @ X4 ) @ ( power_power_rat @ B2 @ Y3 ) )
        = ( ord_less_eq_nat @ X4 @ Y3 ) ) ) ).

% power_increasing_iff
thf(fact_150_power__increasing__iff,axiom,
    ! [B2: nat,X4: nat,Y3: nat] :
      ( ( ord_less_nat @ one_one_nat @ B2 )
     => ( ( ord_less_eq_nat @ ( power_power_nat @ B2 @ X4 ) @ ( power_power_nat @ B2 @ Y3 ) )
        = ( ord_less_eq_nat @ X4 @ Y3 ) ) ) ).

% power_increasing_iff
thf(fact_151_power__increasing__iff,axiom,
    ! [B2: int,X4: nat,Y3: nat] :
      ( ( ord_less_int @ one_one_int @ B2 )
     => ( ( ord_less_eq_int @ ( power_power_int @ B2 @ X4 ) @ ( power_power_int @ B2 @ Y3 ) )
        = ( ord_less_eq_nat @ X4 @ Y3 ) ) ) ).

% power_increasing_iff
thf(fact_152_add__self__div__2,axiom,
    ! [M: nat] :
      ( ( divide_divide_nat @ ( plus_plus_nat @ M @ M ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = M ) ).

% add_self_div_2
thf(fact_153_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ one_one_real )
      = ( ord_less_eq_num @ N @ one ) ) ).

% numeral_le_one_iff
thf(fact_154_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ N ) @ one_one_rat )
      = ( ord_less_eq_num @ N @ one ) ) ).

% numeral_le_one_iff
thf(fact_155_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat )
      = ( ord_less_eq_num @ N @ one ) ) ).

% numeral_le_one_iff
thf(fact_156_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ one_one_int )
      = ( ord_less_eq_num @ N @ one ) ) ).

% numeral_le_one_iff
thf(fact_157_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( ( ord_less_real @ one_one_real @ ( numeral_numeral_real @ N ) )
      = ( ord_less_num @ one @ N ) ) ).

% one_less_numeral_iff
thf(fact_158_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( ( ord_less_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) )
      = ( ord_less_num @ one @ N ) ) ).

% one_less_numeral_iff
thf(fact_159_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( ( ord_less_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) )
      = ( ord_less_num @ one @ N ) ) ).

% one_less_numeral_iff
thf(fact_160_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( ( ord_less_int @ one_one_int @ ( numeral_numeral_int @ N ) )
      = ( ord_less_num @ one @ N ) ) ).

% one_less_numeral_iff
thf(fact_161_is__num__normalize_I1_J,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( plus_plus_real @ ( plus_plus_real @ A @ B2 ) @ C )
      = ( plus_plus_real @ A @ ( plus_plus_real @ B2 @ C ) ) ) ).

% is_num_normalize(1)
thf(fact_162_is__num__normalize_I1_J,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( plus_plus_rat @ ( plus_plus_rat @ A @ B2 ) @ C )
      = ( plus_plus_rat @ A @ ( plus_plus_rat @ B2 @ C ) ) ) ).

% is_num_normalize(1)
thf(fact_163_is__num__normalize_I1_J,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ A @ B2 ) @ C )
      = ( plus_plus_int @ A @ ( plus_plus_int @ B2 @ C ) ) ) ).

% is_num_normalize(1)
thf(fact_164_one__plus__numeral__commute,axiom,
    ! [X4: num] :
      ( ( plus_plus_complex @ one_one_complex @ ( numera6690914467698888265omplex @ X4 ) )
      = ( plus_plus_complex @ ( numera6690914467698888265omplex @ X4 ) @ one_one_complex ) ) ).

% one_plus_numeral_commute
thf(fact_165_one__plus__numeral__commute,axiom,
    ! [X4: num] :
      ( ( plus_plus_real @ one_one_real @ ( numeral_numeral_real @ X4 ) )
      = ( plus_plus_real @ ( numeral_numeral_real @ X4 ) @ one_one_real ) ) ).

% one_plus_numeral_commute
thf(fact_166_one__plus__numeral__commute,axiom,
    ! [X4: num] :
      ( ( plus_plus_rat @ one_one_rat @ ( numeral_numeral_rat @ X4 ) )
      = ( plus_plus_rat @ ( numeral_numeral_rat @ X4 ) @ one_one_rat ) ) ).

% one_plus_numeral_commute
thf(fact_167_one__plus__numeral__commute,axiom,
    ! [X4: num] :
      ( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ X4 ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ X4 ) @ one_one_nat ) ) ).

% one_plus_numeral_commute
thf(fact_168_one__plus__numeral__commute,axiom,
    ! [X4: num] :
      ( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ X4 ) )
      = ( plus_plus_int @ ( numeral_numeral_int @ X4 ) @ one_one_int ) ) ).

% one_plus_numeral_commute
thf(fact_169_le__num__One__iff,axiom,
    ! [X4: num] :
      ( ( ord_less_eq_num @ X4 @ one )
      = ( X4 = one ) ) ).

% le_num_One_iff
thf(fact_170_le__numeral__extra_I4_J,axiom,
    ord_less_eq_real @ one_one_real @ one_one_real ).

% le_numeral_extra(4)
thf(fact_171_le__numeral__extra_I4_J,axiom,
    ord_less_eq_rat @ one_one_rat @ one_one_rat ).

% le_numeral_extra(4)
thf(fact_172_le__numeral__extra_I4_J,axiom,
    ord_less_eq_nat @ one_one_nat @ one_one_nat ).

% le_numeral_extra(4)
thf(fact_173_le__numeral__extra_I4_J,axiom,
    ord_less_eq_int @ one_one_int @ one_one_int ).

% le_numeral_extra(4)
thf(fact_174_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_real @ one_one_real @ one_one_real ) ).

% less_numeral_extra(4)
thf(fact_175_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_rat @ one_one_rat @ one_one_rat ) ).

% less_numeral_extra(4)
thf(fact_176_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).

% less_numeral_extra(4)
thf(fact_177_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_int @ one_one_int @ one_one_int ) ).

% less_numeral_extra(4)
thf(fact_178_one__le__power,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_eq_real @ one_one_real @ A )
     => ( ord_less_eq_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ).

% one_le_power
thf(fact_179_one__le__power,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_eq_rat @ one_one_rat @ A )
     => ( ord_less_eq_rat @ one_one_rat @ ( power_power_rat @ A @ N ) ) ) ).

% one_le_power
thf(fact_180_one__le__power,axiom,
    ! [A: nat,N: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ A )
     => ( ord_less_eq_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ).

% one_le_power
thf(fact_181_one__le__power,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_eq_int @ one_one_int @ A )
     => ( ord_less_eq_int @ one_one_int @ ( power_power_int @ A @ N ) ) ) ).

% one_le_power
thf(fact_182_power__one__over,axiom,
    ! [A: complex,N: nat] :
      ( ( power_power_complex @ ( divide1717551699836669952omplex @ one_one_complex @ A ) @ N )
      = ( divide1717551699836669952omplex @ one_one_complex @ ( power_power_complex @ A @ N ) ) ) ).

% power_one_over
thf(fact_183_power__one__over,axiom,
    ! [A: real,N: nat] :
      ( ( power_power_real @ ( divide_divide_real @ one_one_real @ A ) @ N )
      = ( divide_divide_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ).

% power_one_over
thf(fact_184_power__one__over,axiom,
    ! [A: rat,N: nat] :
      ( ( power_power_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ N )
      = ( divide_divide_rat @ one_one_rat @ ( power_power_rat @ A @ N ) ) ) ).

% power_one_over
thf(fact_185_nat__1__add__1,axiom,
    ( ( plus_plus_nat @ one_one_nat @ one_one_nat )
    = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% nat_1_add_1
thf(fact_186_numeral__Bit0,axiom,
    ! [N: num] :
      ( ( numera6690914467698888265omplex @ ( bit0 @ N ) )
      = ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ ( numera6690914467698888265omplex @ N ) ) ) ).

% numeral_Bit0
thf(fact_187_numeral__Bit0,axiom,
    ! [N: num] :
      ( ( numeral_numeral_real @ ( bit0 @ N ) )
      = ( plus_plus_real @ ( numeral_numeral_real @ N ) @ ( numeral_numeral_real @ N ) ) ) ).

% numeral_Bit0
thf(fact_188_numeral__Bit0,axiom,
    ! [N: num] :
      ( ( numeral_numeral_rat @ ( bit0 @ N ) )
      = ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ ( numeral_numeral_rat @ N ) ) ) ).

% numeral_Bit0
thf(fact_189_numeral__Bit0,axiom,
    ! [N: num] :
      ( ( numeral_numeral_nat @ ( bit0 @ N ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) ) ).

% numeral_Bit0
thf(fact_190_numeral__Bit0,axiom,
    ! [N: num] :
      ( ( numeral_numeral_int @ ( bit0 @ N ) )
      = ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) ) ).

% numeral_Bit0
thf(fact_191_power__strict__increasing,axiom,
    ! [N: nat,N3: nat,A: real] :
      ( ( ord_less_nat @ N @ N3 )
     => ( ( ord_less_real @ one_one_real @ A )
       => ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ A @ N3 ) ) ) ) ).

% power_strict_increasing
thf(fact_192_power__strict__increasing,axiom,
    ! [N: nat,N3: nat,A: rat] :
      ( ( ord_less_nat @ N @ N3 )
     => ( ( ord_less_rat @ one_one_rat @ A )
       => ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ A @ N3 ) ) ) ) ).

% power_strict_increasing
thf(fact_193_power__strict__increasing,axiom,
    ! [N: nat,N3: nat,A: nat] :
      ( ( ord_less_nat @ N @ N3 )
     => ( ( ord_less_nat @ one_one_nat @ A )
       => ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N3 ) ) ) ) ).

% power_strict_increasing
thf(fact_194_power__strict__increasing,axiom,
    ! [N: nat,N3: nat,A: int] :
      ( ( ord_less_nat @ N @ N3 )
     => ( ( ord_less_int @ one_one_int @ A )
       => ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N3 ) ) ) ) ).

% power_strict_increasing
thf(fact_195_power__less__imp__less__exp,axiom,
    ! [A: real,M: nat,N: nat] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) )
       => ( ord_less_nat @ M @ N ) ) ) ).

% power_less_imp_less_exp
thf(fact_196_power__less__imp__less__exp,axiom,
    ! [A: rat,M: nat,N: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ( ord_less_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N ) )
       => ( ord_less_nat @ M @ N ) ) ) ).

% power_less_imp_less_exp
thf(fact_197_power__less__imp__less__exp,axiom,
    ! [A: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ( ord_less_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
       => ( ord_less_nat @ M @ N ) ) ) ).

% power_less_imp_less_exp
thf(fact_198_power__less__imp__less__exp,axiom,
    ! [A: int,M: nat,N: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ( ord_less_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
       => ( ord_less_nat @ M @ N ) ) ) ).

% power_less_imp_less_exp
thf(fact_199_power__increasing,axiom,
    ! [N: nat,N3: nat,A: real] :
      ( ( ord_less_eq_nat @ N @ N3 )
     => ( ( ord_less_eq_real @ one_one_real @ A )
       => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ A @ N3 ) ) ) ) ).

% power_increasing
thf(fact_200_power__increasing,axiom,
    ! [N: nat,N3: nat,A: rat] :
      ( ( ord_less_eq_nat @ N @ N3 )
     => ( ( ord_less_eq_rat @ one_one_rat @ A )
       => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ A @ N3 ) ) ) ) ).

% power_increasing
thf(fact_201_power__increasing,axiom,
    ! [N: nat,N3: nat,A: nat] :
      ( ( ord_less_eq_nat @ N @ N3 )
     => ( ( ord_less_eq_nat @ one_one_nat @ A )
       => ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N3 ) ) ) ) ).

% power_increasing
thf(fact_202_power__increasing,axiom,
    ! [N: nat,N3: nat,A: int] :
      ( ( ord_less_eq_nat @ N @ N3 )
     => ( ( ord_less_eq_int @ one_one_int @ A )
       => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N3 ) ) ) ) ).

% power_increasing
thf(fact_203_one__le__numeral,axiom,
    ! [N: num] : ( ord_less_eq_real @ one_one_real @ ( numeral_numeral_real @ N ) ) ).

% one_le_numeral
thf(fact_204_one__le__numeral,axiom,
    ! [N: num] : ( ord_less_eq_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) ) ).

% one_le_numeral
thf(fact_205_one__le__numeral,axiom,
    ! [N: num] : ( ord_less_eq_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) ) ).

% one_le_numeral
thf(fact_206_one__le__numeral,axiom,
    ! [N: num] : ( ord_less_eq_int @ one_one_int @ ( numeral_numeral_int @ N ) ) ).

% one_le_numeral
thf(fact_207_not__numeral__less__one,axiom,
    ! [N: num] :
      ~ ( ord_less_real @ ( numeral_numeral_real @ N ) @ one_one_real ) ).

% not_numeral_less_one
thf(fact_208_not__numeral__less__one,axiom,
    ! [N: num] :
      ~ ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ one_one_rat ) ).

% not_numeral_less_one
thf(fact_209_not__numeral__less__one,axiom,
    ! [N: num] :
      ~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat ) ).

% not_numeral_less_one
thf(fact_210_not__numeral__less__one,axiom,
    ! [N: num] :
      ~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ one_one_int ) ).

% not_numeral_less_one
thf(fact_211_numeral__One,axiom,
    ( ( numera6690914467698888265omplex @ one )
    = one_one_complex ) ).

% numeral_One
thf(fact_212_numeral__One,axiom,
    ( ( numeral_numeral_real @ one )
    = one_one_real ) ).

% numeral_One
thf(fact_213_numeral__One,axiom,
    ( ( numeral_numeral_rat @ one )
    = one_one_rat ) ).

% numeral_One
thf(fact_214_numeral__One,axiom,
    ( ( numeral_numeral_nat @ one )
    = one_one_nat ) ).

% numeral_One
thf(fact_215_numeral__One,axiom,
    ( ( numeral_numeral_int @ one )
    = one_one_int ) ).

% numeral_One
thf(fact_216_numerals_I1_J,axiom,
    ( ( numeral_numeral_nat @ one )
    = one_one_nat ) ).

% numerals(1)
thf(fact_217_ex__power__ivl1,axiom,
    ! [B2: nat,K: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 )
     => ( ( ord_less_eq_nat @ one_one_nat @ K )
       => ? [N4: nat] :
            ( ( ord_less_eq_nat @ ( power_power_nat @ B2 @ N4 ) @ K )
            & ( ord_less_nat @ K @ ( power_power_nat @ B2 @ ( plus_plus_nat @ N4 @ one_one_nat ) ) ) ) ) ) ).

% ex_power_ivl1
thf(fact_218_ex__power__ivl2,axiom,
    ! [B2: nat,K: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 )
     => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
       => ? [N4: nat] :
            ( ( ord_less_nat @ ( power_power_nat @ B2 @ N4 ) @ K )
            & ( ord_less_eq_nat @ K @ ( power_power_nat @ B2 @ ( plus_plus_nat @ N4 @ one_one_nat ) ) ) ) ) ) ).

% ex_power_ivl2
thf(fact_219_power2__commute,axiom,
    ! [X4: complex,Y3: complex] :
      ( ( power_power_complex @ ( minus_minus_complex @ X4 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_power_complex @ ( minus_minus_complex @ Y3 @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_commute
thf(fact_220_power2__commute,axiom,
    ! [X4: real,Y3: real] :
      ( ( power_power_real @ ( minus_minus_real @ X4 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_power_real @ ( minus_minus_real @ Y3 @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_commute
thf(fact_221_power2__commute,axiom,
    ! [X4: rat,Y3: rat] :
      ( ( power_power_rat @ ( minus_minus_rat @ X4 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_power_rat @ ( minus_minus_rat @ Y3 @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_commute
thf(fact_222_power2__commute,axiom,
    ! [X4: int,Y3: int] :
      ( ( power_power_int @ ( minus_minus_int @ X4 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_power_int @ ( minus_minus_int @ Y3 @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_commute
thf(fact_223_power__le__imp__le__exp,axiom,
    ! [A: real,M: nat,N: nat] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_eq_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) )
       => ( ord_less_eq_nat @ M @ N ) ) ) ).

% power_le_imp_le_exp
thf(fact_224_power__le__imp__le__exp,axiom,
    ! [A: rat,M: nat,N: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ( ord_less_eq_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N ) )
       => ( ord_less_eq_nat @ M @ N ) ) ) ).

% power_le_imp_le_exp
thf(fact_225_power__le__imp__le__exp,axiom,
    ! [A: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ( ord_less_eq_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
       => ( ord_less_eq_nat @ M @ N ) ) ) ).

% power_le_imp_le_exp
thf(fact_226_power__le__imp__le__exp,axiom,
    ! [A: int,M: nat,N: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ( ord_less_eq_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
       => ( ord_less_eq_nat @ M @ N ) ) ) ).

% power_le_imp_le_exp
thf(fact_227_one__power2,axiom,
    ( ( power_power_rat @ one_one_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_rat ) ).

% one_power2
thf(fact_228_one__power2,axiom,
    ( ( power_power_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_nat ) ).

% one_power2
thf(fact_229_one__power2,axiom,
    ( ( power_power_real @ one_one_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_real ) ).

% one_power2
thf(fact_230_one__power2,axiom,
    ( ( power_power_int @ one_one_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_int ) ).

% one_power2
thf(fact_231_one__power2,axiom,
    ( ( power_power_complex @ one_one_complex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_complex ) ).

% one_power2
thf(fact_232_diff__le__diff__pow,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ ( minus_minus_nat @ ( power_power_nat @ K @ M ) @ ( power_power_nat @ K @ N ) ) ) ) ).

% diff_le_diff_pow
thf(fact_233_power__divide,axiom,
    ! [A: complex,B2: complex,N: nat] :
      ( ( power_power_complex @ ( divide1717551699836669952omplex @ A @ B2 ) @ N )
      = ( divide1717551699836669952omplex @ ( power_power_complex @ A @ N ) @ ( power_power_complex @ B2 @ N ) ) ) ).

% power_divide
thf(fact_234_power__divide,axiom,
    ! [A: real,B2: real,N: nat] :
      ( ( power_power_real @ ( divide_divide_real @ A @ B2 ) @ N )
      = ( divide_divide_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B2 @ N ) ) ) ).

% power_divide
thf(fact_235_power__divide,axiom,
    ! [A: rat,B2: rat,N: nat] :
      ( ( power_power_rat @ ( divide_divide_rat @ A @ B2 ) @ N )
      = ( divide_divide_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B2 @ N ) ) ) ).

% power_divide
thf(fact_236_Nat_Oadd__diff__assoc,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).

% Nat.add_diff_assoc
thf(fact_237_Nat_Oadd__diff__assoc2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
        = ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).

% Nat.add_diff_assoc2
thf(fact_238_Nat_Odiff__diff__right,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).

% Nat.diff_diff_right
thf(fact_239__092_060open_062high_Ax_An_A_060_A2_A_094_Am_A_092_060and_062_Alow_Ax_An_A_060_A2_A_094_An_092_060close_062,axiom,
    ( ( ord_less_nat @ ( vEBT_VEBT_high @ xa @ na ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
    & ( ord_less_nat @ ( vEBT_VEBT_low @ xa @ na ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) ) ).

% \<open>high x n < 2 ^ m \<and> low x n < 2 ^ n\<close>
thf(fact_240_le__add__diff__inverse,axiom,
    ! [B2: real,A: real] :
      ( ( ord_less_eq_real @ B2 @ A )
     => ( ( plus_plus_real @ B2 @ ( minus_minus_real @ A @ B2 ) )
        = A ) ) ).

% le_add_diff_inverse
thf(fact_241_le__add__diff__inverse,axiom,
    ! [B2: rat,A: rat] :
      ( ( ord_less_eq_rat @ B2 @ A )
     => ( ( plus_plus_rat @ B2 @ ( minus_minus_rat @ A @ B2 ) )
        = A ) ) ).

% le_add_diff_inverse
thf(fact_242_le__add__diff__inverse,axiom,
    ! [B2: nat,A: nat] :
      ( ( ord_less_eq_nat @ B2 @ A )
     => ( ( plus_plus_nat @ B2 @ ( minus_minus_nat @ A @ B2 ) )
        = A ) ) ).

% le_add_diff_inverse
thf(fact_243_le__add__diff__inverse,axiom,
    ! [B2: int,A: int] :
      ( ( ord_less_eq_int @ B2 @ A )
     => ( ( plus_plus_int @ B2 @ ( minus_minus_int @ A @ B2 ) )
        = A ) ) ).

% le_add_diff_inverse
thf(fact_244_le__add__diff__inverse2,axiom,
    ! [B2: real,A: real] :
      ( ( ord_less_eq_real @ B2 @ A )
     => ( ( plus_plus_real @ ( minus_minus_real @ A @ B2 ) @ B2 )
        = A ) ) ).

% le_add_diff_inverse2
thf(fact_245_le__add__diff__inverse2,axiom,
    ! [B2: rat,A: rat] :
      ( ( ord_less_eq_rat @ B2 @ A )
     => ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B2 ) @ B2 )
        = A ) ) ).

% le_add_diff_inverse2
thf(fact_246_le__add__diff__inverse2,axiom,
    ! [B2: nat,A: nat] :
      ( ( ord_less_eq_nat @ B2 @ A )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B2 ) @ B2 )
        = A ) ) ).

% le_add_diff_inverse2
thf(fact_247_le__add__diff__inverse2,axiom,
    ! [B2: int,A: int] :
      ( ( ord_less_eq_int @ B2 @ A )
     => ( ( plus_plus_int @ ( minus_minus_int @ A @ B2 ) @ B2 )
        = A ) ) ).

% le_add_diff_inverse2
thf(fact_248_div__exp__eq,axiom,
    ! [A: nat,M: nat,N: nat] :
      ( ( divide_divide_nat @ ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) ) ) ) ).

% div_exp_eq
thf(fact_249_div__exp__eq,axiom,
    ! [A: int,M: nat,N: nat] :
      ( ( divide_divide_int @ ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
      = ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) ) ) ) ).

% div_exp_eq
thf(fact_250_field__less__half__sum,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_real @ X4 @ Y3 )
     => ( ord_less_real @ X4 @ ( divide_divide_real @ ( plus_plus_real @ X4 @ Y3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% field_less_half_sum
thf(fact_251_field__less__half__sum,axiom,
    ! [X4: rat,Y3: rat] :
      ( ( ord_less_rat @ X4 @ Y3 )
     => ( ord_less_rat @ X4 @ ( divide_divide_rat @ ( plus_plus_rat @ X4 @ Y3 ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).

% field_less_half_sum
thf(fact_252_high__inv,axiom,
    ! [X4: nat,N: nat,Y3: nat] :
      ( ( ord_less_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
     => ( ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ Y3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ X4 ) @ N )
        = Y3 ) ) ).

% high_inv
thf(fact_253_diff__diff__left,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
      = ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).

% diff_diff_left
thf(fact_254_diff__diff__cancel,axiom,
    ! [I: nat,N: nat] :
      ( ( ord_less_eq_nat @ I @ N )
     => ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
        = I ) ) ).

% diff_diff_cancel
thf(fact_255_nat__add__left__cancel__le,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% nat_add_left_cancel_le
thf(fact_256_even__odd__cases,axiom,
    ! [X4: nat] :
      ( ! [N4: nat] :
          ( X4
         != ( plus_plus_nat @ N4 @ N4 ) )
     => ~ ! [N4: nat] :
            ( X4
           != ( plus_plus_nat @ N4 @ ( suc @ N4 ) ) ) ) ).

% even_odd_cases
thf(fact_257_bit__split__inv,axiom,
    ! [X4: nat,D: nat] :
      ( ( vEBT_VEBT_bit_concat @ ( vEBT_VEBT_high @ X4 @ D ) @ ( vEBT_VEBT_low @ X4 @ D ) @ D )
      = X4 ) ).

% bit_split_inv
thf(fact_258_nat_Oinject,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( ( suc @ X2 )
        = ( suc @ Y2 ) )
      = ( X2 = Y2 ) ) ).

% nat.inject
thf(fact_259_old_Onat_Oinject,axiom,
    ! [Nat: nat,Nat2: nat] :
      ( ( ( suc @ Nat )
        = ( suc @ Nat2 ) )
      = ( Nat = Nat2 ) ) ).

% old.nat.inject
thf(fact_260_numeral__times__numeral,axiom,
    ! [M: num,N: num] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ M ) @ ( numera6690914467698888265omplex @ N ) )
      = ( numera6690914467698888265omplex @ ( times_times_num @ M @ N ) ) ) ).

% numeral_times_numeral
thf(fact_261_numeral__times__numeral,axiom,
    ! [M: num,N: num] :
      ( ( times_times_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
      = ( numeral_numeral_real @ ( times_times_num @ M @ N ) ) ) ).

% numeral_times_numeral
thf(fact_262_numeral__times__numeral,axiom,
    ! [M: num,N: num] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
      = ( numeral_numeral_rat @ ( times_times_num @ M @ N ) ) ) ).

% numeral_times_numeral
thf(fact_263_numeral__times__numeral,axiom,
    ! [M: num,N: num] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
      = ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ).

% numeral_times_numeral
thf(fact_264_numeral__times__numeral,axiom,
    ! [M: num,N: num] :
      ( ( times_times_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
      = ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ).

% numeral_times_numeral
thf(fact_265_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: complex] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ Z ) )
      = ( times_times_complex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) @ Z ) ) ).

% mult_numeral_left_semiring_numeral
thf(fact_266_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( times_times_real @ ( numeral_numeral_real @ W ) @ Z ) )
      = ( times_times_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) @ Z ) ) ).

% mult_numeral_left_semiring_numeral
thf(fact_267_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ Z ) )
      = ( times_times_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) @ Z ) ) ).

% mult_numeral_left_semiring_numeral
thf(fact_268_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( times_times_nat @ ( numeral_numeral_nat @ W ) @ Z ) )
      = ( times_times_nat @ ( numeral_numeral_nat @ ( times_times_num @ V @ W ) ) @ Z ) ) ).

% mult_numeral_left_semiring_numeral
thf(fact_269_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( times_times_int @ ( numeral_numeral_int @ W ) @ Z ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) @ Z ) ) ).

% mult_numeral_left_semiring_numeral
thf(fact_270_low__inv,axiom,
    ! [X4: nat,N: nat,Y3: nat] :
      ( ( ord_less_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
     => ( ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ Y3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ X4 ) @ N )
        = X4 ) ) ).

% low_inv
thf(fact_271_bits__div__by__1,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ A @ one_one_nat )
      = A ) ).

% bits_div_by_1
thf(fact_272_bits__div__by__1,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ A @ one_one_int )
      = A ) ).

% bits_div_by_1
thf(fact_273_div__by__1,axiom,
    ! [A: complex] :
      ( ( divide1717551699836669952omplex @ A @ one_one_complex )
      = A ) ).

% div_by_1
thf(fact_274_div__by__1,axiom,
    ! [A: real] :
      ( ( divide_divide_real @ A @ one_one_real )
      = A ) ).

% div_by_1
thf(fact_275_div__by__1,axiom,
    ! [A: rat] :
      ( ( divide_divide_rat @ A @ one_one_rat )
      = A ) ).

% div_by_1
thf(fact_276_div__by__1,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ A @ one_one_nat )
      = A ) ).

% div_by_1
thf(fact_277_div__by__1,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ A @ one_one_int )
      = A ) ).

% div_by_1
thf(fact_278_lessI,axiom,
    ! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).

% lessI
thf(fact_279_Suc__mono,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).

% Suc_mono
thf(fact_280_Suc__less__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% Suc_less_eq
thf(fact_281_Suc__le__mono,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
      = ( ord_less_eq_nat @ N @ M ) ) ).

% Suc_le_mono
thf(fact_282_add__Suc__right,axiom,
    ! [M: nat,N: nat] :
      ( ( plus_plus_nat @ M @ ( suc @ N ) )
      = ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).

% add_Suc_right
thf(fact_283_nat__add__left__cancel__less,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% nat_add_left_cancel_less
thf(fact_284_diff__Suc__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
      = ( minus_minus_nat @ M @ N ) ) ).

% diff_Suc_Suc
thf(fact_285_Suc__diff__diff,axiom,
    ! [M: nat,N: nat,K: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
      = ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).

% Suc_diff_diff
thf(fact_286_nat__1__eq__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( one_one_nat
        = ( times_times_nat @ M @ N ) )
      = ( ( M = one_one_nat )
        & ( N = one_one_nat ) ) ) ).

% nat_1_eq_mult_iff
thf(fact_287_nat__mult__eq__1__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ( times_times_nat @ M @ N )
        = one_one_nat )
      = ( ( M = one_one_nat )
        & ( N = one_one_nat ) ) ) ).

% nat_mult_eq_1_iff
thf(fact_288_semiring__norm_I6_J,axiom,
    ! [M: num,N: num] :
      ( ( plus_plus_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
      = ( bit0 @ ( plus_plus_num @ M @ N ) ) ) ).

% semiring_norm(6)
thf(fact_289_bit__concat__def,axiom,
    ( vEBT_VEBT_bit_concat
    = ( ^ [H: nat,L: nat,D2: nat] : ( plus_plus_nat @ ( times_times_nat @ H @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ D2 ) ) @ L ) ) ) ).

% bit_concat_def
thf(fact_290_distrib__left__numeral,axiom,
    ! [V: num,B2: complex,C: complex] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( plus_plus_complex @ B2 @ C ) )
      = ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ B2 ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ C ) ) ) ).

% distrib_left_numeral
thf(fact_291_distrib__left__numeral,axiom,
    ! [V: num,B2: real,C: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( plus_plus_real @ B2 @ C ) )
      = ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ V ) @ B2 ) @ ( times_times_real @ ( numeral_numeral_real @ V ) @ C ) ) ) ).

% distrib_left_numeral
thf(fact_292_distrib__left__numeral,axiom,
    ! [V: num,B2: rat,C: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( plus_plus_rat @ B2 @ C ) )
      = ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ B2 ) @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ C ) ) ) ).

% distrib_left_numeral
thf(fact_293_distrib__left__numeral,axiom,
    ! [V: num,B2: nat,C: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ B2 @ C ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ B2 ) @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ C ) ) ) ).

% distrib_left_numeral
thf(fact_294_distrib__left__numeral,axiom,
    ! [V: num,B2: int,C: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ B2 @ C ) )
      = ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ V ) @ B2 ) @ ( times_times_int @ ( numeral_numeral_int @ V ) @ C ) ) ) ).

% distrib_left_numeral
thf(fact_295_distrib__right__numeral,axiom,
    ! [A: complex,B2: complex,V: num] :
      ( ( times_times_complex @ ( plus_plus_complex @ A @ B2 ) @ ( numera6690914467698888265omplex @ V ) )
      = ( plus_plus_complex @ ( times_times_complex @ A @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ B2 @ ( numera6690914467698888265omplex @ V ) ) ) ) ).

% distrib_right_numeral
thf(fact_296_distrib__right__numeral,axiom,
    ! [A: real,B2: real,V: num] :
      ( ( times_times_real @ ( plus_plus_real @ A @ B2 ) @ ( numeral_numeral_real @ V ) )
      = ( plus_plus_real @ ( times_times_real @ A @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ B2 @ ( numeral_numeral_real @ V ) ) ) ) ).

% distrib_right_numeral
thf(fact_297_distrib__right__numeral,axiom,
    ! [A: rat,B2: rat,V: num] :
      ( ( times_times_rat @ ( plus_plus_rat @ A @ B2 ) @ ( numeral_numeral_rat @ V ) )
      = ( plus_plus_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ B2 @ ( numeral_numeral_rat @ V ) ) ) ) ).

% distrib_right_numeral
thf(fact_298_distrib__right__numeral,axiom,
    ! [A: nat,B2: nat,V: num] :
      ( ( times_times_nat @ ( plus_plus_nat @ A @ B2 ) @ ( numeral_numeral_nat @ V ) )
      = ( plus_plus_nat @ ( times_times_nat @ A @ ( numeral_numeral_nat @ V ) ) @ ( times_times_nat @ B2 @ ( numeral_numeral_nat @ V ) ) ) ) ).

% distrib_right_numeral
thf(fact_299_distrib__right__numeral,axiom,
    ! [A: int,B2: int,V: num] :
      ( ( times_times_int @ ( plus_plus_int @ A @ B2 ) @ ( numeral_numeral_int @ V ) )
      = ( plus_plus_int @ ( times_times_int @ A @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ B2 @ ( numeral_numeral_int @ V ) ) ) ) ).

% distrib_right_numeral
thf(fact_300_left__diff__distrib__numeral,axiom,
    ! [A: complex,B2: complex,V: num] :
      ( ( times_times_complex @ ( minus_minus_complex @ A @ B2 ) @ ( numera6690914467698888265omplex @ V ) )
      = ( minus_minus_complex @ ( times_times_complex @ A @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ B2 @ ( numera6690914467698888265omplex @ V ) ) ) ) ).

% left_diff_distrib_numeral
thf(fact_301_left__diff__distrib__numeral,axiom,
    ! [A: real,B2: real,V: num] :
      ( ( times_times_real @ ( minus_minus_real @ A @ B2 ) @ ( numeral_numeral_real @ V ) )
      = ( minus_minus_real @ ( times_times_real @ A @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ B2 @ ( numeral_numeral_real @ V ) ) ) ) ).

% left_diff_distrib_numeral
thf(fact_302_left__diff__distrib__numeral,axiom,
    ! [A: rat,B2: rat,V: num] :
      ( ( times_times_rat @ ( minus_minus_rat @ A @ B2 ) @ ( numeral_numeral_rat @ V ) )
      = ( minus_minus_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ B2 @ ( numeral_numeral_rat @ V ) ) ) ) ).

% left_diff_distrib_numeral
thf(fact_303_left__diff__distrib__numeral,axiom,
    ! [A: int,B2: int,V: num] :
      ( ( times_times_int @ ( minus_minus_int @ A @ B2 ) @ ( numeral_numeral_int @ V ) )
      = ( minus_minus_int @ ( times_times_int @ A @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ B2 @ ( numeral_numeral_int @ V ) ) ) ) ).

% left_diff_distrib_numeral
thf(fact_304_right__diff__distrib__numeral,axiom,
    ! [V: num,B2: complex,C: complex] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( minus_minus_complex @ B2 @ C ) )
      = ( minus_minus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ B2 ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ C ) ) ) ).

% right_diff_distrib_numeral
thf(fact_305_right__diff__distrib__numeral,axiom,
    ! [V: num,B2: real,C: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( minus_minus_real @ B2 @ C ) )
      = ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ V ) @ B2 ) @ ( times_times_real @ ( numeral_numeral_real @ V ) @ C ) ) ) ).

% right_diff_distrib_numeral
thf(fact_306_right__diff__distrib__numeral,axiom,
    ! [V: num,B2: rat,C: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( minus_minus_rat @ B2 @ C ) )
      = ( minus_minus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ B2 ) @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ C ) ) ) ).

% right_diff_distrib_numeral
thf(fact_307_right__diff__distrib__numeral,axiom,
    ! [V: num,B2: int,C: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( minus_minus_int @ B2 @ C ) )
      = ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ V ) @ B2 ) @ ( times_times_int @ ( numeral_numeral_int @ V ) @ C ) ) ) ).

% right_diff_distrib_numeral
thf(fact_308_mult__Suc__right,axiom,
    ! [M: nat,N: nat] :
      ( ( times_times_nat @ M @ ( suc @ N ) )
      = ( plus_plus_nat @ M @ ( times_times_nat @ M @ N ) ) ) ).

% mult_Suc_right
thf(fact_309_diff__Suc__1,axiom,
    ! [N: nat] :
      ( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
      = N ) ).

% diff_Suc_1
thf(fact_310_semiring__norm_I2_J,axiom,
    ( ( plus_plus_num @ one @ one )
    = ( bit0 @ one ) ) ).

% semiring_norm(2)
thf(fact_311_le__divide__eq__numeral1_I1_J,axiom,
    ! [A: real,B2: real,W: num] :
      ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B2 @ ( numeral_numeral_real @ W ) ) )
      = ( ord_less_eq_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) @ B2 ) ) ).

% le_divide_eq_numeral1(1)
thf(fact_312_le__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B2: rat,W: num] :
      ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B2 @ ( numeral_numeral_rat @ W ) ) )
      = ( ord_less_eq_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) @ B2 ) ) ).

% le_divide_eq_numeral1(1)
thf(fact_313_divide__le__eq__numeral1_I1_J,axiom,
    ! [B2: real,W: num,A: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ B2 @ ( numeral_numeral_real @ W ) ) @ A )
      = ( ord_less_eq_real @ B2 @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) ) ) ).

% divide_le_eq_numeral1(1)
thf(fact_314_divide__le__eq__numeral1_I1_J,axiom,
    ! [B2: rat,W: num,A: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B2 @ ( numeral_numeral_rat @ W ) ) @ A )
      = ( ord_less_eq_rat @ B2 @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) ) ) ).

% divide_le_eq_numeral1(1)
thf(fact_315_less__divide__eq__numeral1_I1_J,axiom,
    ! [A: real,B2: real,W: num] :
      ( ( ord_less_real @ A @ ( divide_divide_real @ B2 @ ( numeral_numeral_real @ W ) ) )
      = ( ord_less_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) @ B2 ) ) ).

% less_divide_eq_numeral1(1)
thf(fact_316_less__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B2: rat,W: num] :
      ( ( ord_less_rat @ A @ ( divide_divide_rat @ B2 @ ( numeral_numeral_rat @ W ) ) )
      = ( ord_less_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) @ B2 ) ) ).

% less_divide_eq_numeral1(1)
thf(fact_317_divide__less__eq__numeral1_I1_J,axiom,
    ! [B2: real,W: num,A: real] :
      ( ( ord_less_real @ ( divide_divide_real @ B2 @ ( numeral_numeral_real @ W ) ) @ A )
      = ( ord_less_real @ B2 @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) ) ) ).

% divide_less_eq_numeral1(1)
thf(fact_318_divide__less__eq__numeral1_I1_J,axiom,
    ! [B2: rat,W: num,A: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ B2 @ ( numeral_numeral_rat @ W ) ) @ A )
      = ( ord_less_rat @ B2 @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) ) ) ).

% divide_less_eq_numeral1(1)
thf(fact_319_diff__Suc__diff__eq2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I )
        = ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I ) ) ) ) ).

% diff_Suc_diff_eq2
thf(fact_320_diff__Suc__diff__eq1,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ ( suc @ J ) ) ) ) ).

% diff_Suc_diff_eq1
thf(fact_321_power__add__numeral,axiom,
    ! [A: complex,M: num,N: num] :
      ( ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_complex @ A @ ( numeral_numeral_nat @ N ) ) )
      = ( power_power_complex @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).

% power_add_numeral
thf(fact_322_power__add__numeral,axiom,
    ! [A: real,M: num,N: num] :
      ( ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_real @ A @ ( numeral_numeral_nat @ N ) ) )
      = ( power_power_real @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).

% power_add_numeral
thf(fact_323_power__add__numeral,axiom,
    ! [A: rat,M: num,N: num] :
      ( ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_rat @ A @ ( numeral_numeral_nat @ N ) ) )
      = ( power_power_rat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).

% power_add_numeral
thf(fact_324_power__add__numeral,axiom,
    ! [A: nat,M: num,N: num] :
      ( ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_nat @ A @ ( numeral_numeral_nat @ N ) ) )
      = ( power_power_nat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).

% power_add_numeral
thf(fact_325_power__add__numeral,axiom,
    ! [A: int,M: num,N: num] :
      ( ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_int @ A @ ( numeral_numeral_nat @ N ) ) )
      = ( power_power_int @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).

% power_add_numeral
thf(fact_326_power__add__numeral2,axiom,
    ! [A: complex,M: num,N: num,B2: complex] :
      ( ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ N ) ) @ B2 ) )
      = ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B2 ) ) ).

% power_add_numeral2
thf(fact_327_power__add__numeral2,axiom,
    ! [A: real,M: num,N: num,B2: real] :
      ( ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ N ) ) @ B2 ) )
      = ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B2 ) ) ).

% power_add_numeral2
thf(fact_328_power__add__numeral2,axiom,
    ! [A: rat,M: num,N: num,B2: rat] :
      ( ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ N ) ) @ B2 ) )
      = ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B2 ) ) ).

% power_add_numeral2
thf(fact_329_power__add__numeral2,axiom,
    ! [A: nat,M: num,N: num,B2: nat] :
      ( ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ N ) ) @ B2 ) )
      = ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B2 ) ) ).

% power_add_numeral2
thf(fact_330_power__add__numeral2,axiom,
    ! [A: int,M: num,N: num,B2: int] :
      ( ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ N ) ) @ B2 ) )
      = ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B2 ) ) ).

% power_add_numeral2
thf(fact_331_Suc__numeral,axiom,
    ! [N: num] :
      ( ( suc @ ( numeral_numeral_nat @ N ) )
      = ( numeral_numeral_nat @ ( plus_plus_num @ N @ one ) ) ) ).

% Suc_numeral
thf(fact_332_add__2__eq__Suc_H,axiom,
    ! [N: nat] :
      ( ( plus_plus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( suc @ ( suc @ N ) ) ) ).

% add_2_eq_Suc'
thf(fact_333_add__2__eq__Suc,axiom,
    ! [N: nat] :
      ( ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
      = ( suc @ ( suc @ N ) ) ) ).

% add_2_eq_Suc
thf(fact_334_div2__Suc__Suc,axiom,
    ! [M: nat] :
      ( ( divide_divide_nat @ ( suc @ ( suc @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( suc @ ( divide_divide_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% div2_Suc_Suc
thf(fact_335_Suc__1,axiom,
    ( ( suc @ one_one_nat )
    = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% Suc_1
thf(fact_336_Suc__mult__le__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% Suc_mult_le_cancel1
thf(fact_337_Suc__mult__less__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% Suc_mult_less_cancel1
thf(fact_338_mult__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( times_times_nat @ ( suc @ M ) @ N )
      = ( plus_plus_nat @ N @ ( times_times_nat @ M @ N ) ) ) ).

% mult_Suc
thf(fact_339_Suc__inject,axiom,
    ! [X4: nat,Y3: nat] :
      ( ( ( suc @ X4 )
        = ( suc @ Y3 ) )
     => ( X4 = Y3 ) ) ).

% Suc_inject
thf(fact_340_n__not__Suc__n,axiom,
    ! [N: nat] :
      ( N
     != ( suc @ N ) ) ).

% n_not_Suc_n
thf(fact_341_Suc__mult__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ( times_times_nat @ ( suc @ K ) @ M )
        = ( times_times_nat @ ( suc @ K ) @ N ) )
      = ( M = N ) ) ).

% Suc_mult_cancel1
thf(fact_342_power__Suc2,axiom,
    ! [A: complex,N: nat] :
      ( ( power_power_complex @ A @ ( suc @ N ) )
      = ( times_times_complex @ ( power_power_complex @ A @ N ) @ A ) ) ).

% power_Suc2
thf(fact_343_power__Suc2,axiom,
    ! [A: real,N: nat] :
      ( ( power_power_real @ A @ ( suc @ N ) )
      = ( times_times_real @ ( power_power_real @ A @ N ) @ A ) ) ).

% power_Suc2
thf(fact_344_power__Suc2,axiom,
    ! [A: rat,N: nat] :
      ( ( power_power_rat @ A @ ( suc @ N ) )
      = ( times_times_rat @ ( power_power_rat @ A @ N ) @ A ) ) ).

% power_Suc2
thf(fact_345_power__Suc2,axiom,
    ! [A: nat,N: nat] :
      ( ( power_power_nat @ A @ ( suc @ N ) )
      = ( times_times_nat @ ( power_power_nat @ A @ N ) @ A ) ) ).

% power_Suc2
thf(fact_346_power__Suc2,axiom,
    ! [A: int,N: nat] :
      ( ( power_power_int @ A @ ( suc @ N ) )
      = ( times_times_int @ ( power_power_int @ A @ N ) @ A ) ) ).

% power_Suc2
thf(fact_347_power__Suc,axiom,
    ! [A: complex,N: nat] :
      ( ( power_power_complex @ A @ ( suc @ N ) )
      = ( times_times_complex @ A @ ( power_power_complex @ A @ N ) ) ) ).

% power_Suc
thf(fact_348_power__Suc,axiom,
    ! [A: real,N: nat] :
      ( ( power_power_real @ A @ ( suc @ N ) )
      = ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ).

% power_Suc
thf(fact_349_power__Suc,axiom,
    ! [A: rat,N: nat] :
      ( ( power_power_rat @ A @ ( suc @ N ) )
      = ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ).

% power_Suc
thf(fact_350_power__Suc,axiom,
    ! [A: nat,N: nat] :
      ( ( power_power_nat @ A @ ( suc @ N ) )
      = ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ).

% power_Suc
thf(fact_351_power__Suc,axiom,
    ! [A: int,N: nat] :
      ( ( power_power_int @ A @ ( suc @ N ) )
      = ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ).

% power_Suc
thf(fact_352_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( times_times_real @ ( plus_plus_real @ A @ B2 ) @ C )
      = ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) ) ) ).

% ring_class.ring_distribs(2)
thf(fact_353_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( times_times_rat @ ( plus_plus_rat @ A @ B2 ) @ C )
      = ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ C ) ) ) ).

% ring_class.ring_distribs(2)
thf(fact_354_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( times_times_int @ ( plus_plus_int @ A @ B2 ) @ C )
      = ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) ) ) ).

% ring_class.ring_distribs(2)
thf(fact_355_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( times_times_real @ A @ ( plus_plus_real @ B2 @ C ) )
      = ( plus_plus_real @ ( times_times_real @ A @ B2 ) @ ( times_times_real @ A @ C ) ) ) ).

% ring_class.ring_distribs(1)
thf(fact_356_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( times_times_rat @ A @ ( plus_plus_rat @ B2 @ C ) )
      = ( plus_plus_rat @ ( times_times_rat @ A @ B2 ) @ ( times_times_rat @ A @ C ) ) ) ).

% ring_class.ring_distribs(1)
thf(fact_357_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( times_times_int @ A @ ( plus_plus_int @ B2 @ C ) )
      = ( plus_plus_int @ ( times_times_int @ A @ B2 ) @ ( times_times_int @ A @ C ) ) ) ).

% ring_class.ring_distribs(1)
thf(fact_358_comm__semiring__class_Odistrib,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( times_times_real @ ( plus_plus_real @ A @ B2 ) @ C )
      = ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) ) ) ).

% comm_semiring_class.distrib
thf(fact_359_comm__semiring__class_Odistrib,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( times_times_rat @ ( plus_plus_rat @ A @ B2 ) @ C )
      = ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ C ) ) ) ).

% comm_semiring_class.distrib
thf(fact_360_comm__semiring__class_Odistrib,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( times_times_nat @ ( plus_plus_nat @ A @ B2 ) @ C )
      = ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ C ) ) ) ).

% comm_semiring_class.distrib
thf(fact_361_comm__semiring__class_Odistrib,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( times_times_int @ ( plus_plus_int @ A @ B2 ) @ C )
      = ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) ) ) ).

% comm_semiring_class.distrib
thf(fact_362_distrib__left,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( times_times_real @ A @ ( plus_plus_real @ B2 @ C ) )
      = ( plus_plus_real @ ( times_times_real @ A @ B2 ) @ ( times_times_real @ A @ C ) ) ) ).

% distrib_left
thf(fact_363_distrib__left,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( times_times_rat @ A @ ( plus_plus_rat @ B2 @ C ) )
      = ( plus_plus_rat @ ( times_times_rat @ A @ B2 ) @ ( times_times_rat @ A @ C ) ) ) ).

% distrib_left
thf(fact_364_distrib__left,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( times_times_nat @ A @ ( plus_plus_nat @ B2 @ C ) )
      = ( plus_plus_nat @ ( times_times_nat @ A @ B2 ) @ ( times_times_nat @ A @ C ) ) ) ).

% distrib_left
thf(fact_365_distrib__left,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( times_times_int @ A @ ( plus_plus_int @ B2 @ C ) )
      = ( plus_plus_int @ ( times_times_int @ A @ B2 ) @ ( times_times_int @ A @ C ) ) ) ).

% distrib_left
thf(fact_366_distrib__right,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( times_times_real @ ( plus_plus_real @ A @ B2 ) @ C )
      = ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) ) ) ).

% distrib_right
thf(fact_367_distrib__right,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( times_times_rat @ ( plus_plus_rat @ A @ B2 ) @ C )
      = ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ C ) ) ) ).

% distrib_right
thf(fact_368_distrib__right,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( times_times_nat @ ( plus_plus_nat @ A @ B2 ) @ C )
      = ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ C ) ) ) ).

% distrib_right
thf(fact_369_distrib__right,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( times_times_int @ ( plus_plus_int @ A @ B2 ) @ C )
      = ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) ) ) ).

% distrib_right
thf(fact_370_combine__common__factor,axiom,
    ! [A: real,E: real,B2: real,C: real] :
      ( ( plus_plus_real @ ( times_times_real @ A @ E ) @ ( plus_plus_real @ ( times_times_real @ B2 @ E ) @ C ) )
      = ( plus_plus_real @ ( times_times_real @ ( plus_plus_real @ A @ B2 ) @ E ) @ C ) ) ).

% combine_common_factor
thf(fact_371_combine__common__factor,axiom,
    ! [A: rat,E: rat,B2: rat,C: rat] :
      ( ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ ( plus_plus_rat @ ( times_times_rat @ B2 @ E ) @ C ) )
      = ( plus_plus_rat @ ( times_times_rat @ ( plus_plus_rat @ A @ B2 ) @ E ) @ C ) ) ).

% combine_common_factor
thf(fact_372_combine__common__factor,axiom,
    ! [A: nat,E: nat,B2: nat,C: nat] :
      ( ( plus_plus_nat @ ( times_times_nat @ A @ E ) @ ( plus_plus_nat @ ( times_times_nat @ B2 @ E ) @ C ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B2 ) @ E ) @ C ) ) ).

% combine_common_factor
thf(fact_373_combine__common__factor,axiom,
    ! [A: int,E: int,B2: int,C: int] :
      ( ( plus_plus_int @ ( times_times_int @ A @ E ) @ ( plus_plus_int @ ( times_times_int @ B2 @ E ) @ C ) )
      = ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A @ B2 ) @ E ) @ C ) ) ).

% combine_common_factor
thf(fact_374_left__diff__distrib,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( times_times_real @ ( minus_minus_real @ A @ B2 ) @ C )
      = ( minus_minus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) ) ) ).

% left_diff_distrib
thf(fact_375_left__diff__distrib,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( times_times_rat @ ( minus_minus_rat @ A @ B2 ) @ C )
      = ( minus_minus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ C ) ) ) ).

% left_diff_distrib
thf(fact_376_left__diff__distrib,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( times_times_int @ ( minus_minus_int @ A @ B2 ) @ C )
      = ( minus_minus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) ) ) ).

% left_diff_distrib
thf(fact_377_right__diff__distrib,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( times_times_real @ A @ ( minus_minus_real @ B2 @ C ) )
      = ( minus_minus_real @ ( times_times_real @ A @ B2 ) @ ( times_times_real @ A @ C ) ) ) ).

% right_diff_distrib
thf(fact_378_right__diff__distrib,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( times_times_rat @ A @ ( minus_minus_rat @ B2 @ C ) )
      = ( minus_minus_rat @ ( times_times_rat @ A @ B2 ) @ ( times_times_rat @ A @ C ) ) ) ).

% right_diff_distrib
thf(fact_379_right__diff__distrib,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( times_times_int @ A @ ( minus_minus_int @ B2 @ C ) )
      = ( minus_minus_int @ ( times_times_int @ A @ B2 ) @ ( times_times_int @ A @ C ) ) ) ).

% right_diff_distrib
thf(fact_380_left__diff__distrib_H,axiom,
    ! [B2: real,C: real,A: real] :
      ( ( times_times_real @ ( minus_minus_real @ B2 @ C ) @ A )
      = ( minus_minus_real @ ( times_times_real @ B2 @ A ) @ ( times_times_real @ C @ A ) ) ) ).

% left_diff_distrib'
thf(fact_381_left__diff__distrib_H,axiom,
    ! [B2: rat,C: rat,A: rat] :
      ( ( times_times_rat @ ( minus_minus_rat @ B2 @ C ) @ A )
      = ( minus_minus_rat @ ( times_times_rat @ B2 @ A ) @ ( times_times_rat @ C @ A ) ) ) ).

% left_diff_distrib'
thf(fact_382_left__diff__distrib_H,axiom,
    ! [B2: nat,C: nat,A: nat] :
      ( ( times_times_nat @ ( minus_minus_nat @ B2 @ C ) @ A )
      = ( minus_minus_nat @ ( times_times_nat @ B2 @ A ) @ ( times_times_nat @ C @ A ) ) ) ).

% left_diff_distrib'
thf(fact_383_left__diff__distrib_H,axiom,
    ! [B2: int,C: int,A: int] :
      ( ( times_times_int @ ( minus_minus_int @ B2 @ C ) @ A )
      = ( minus_minus_int @ ( times_times_int @ B2 @ A ) @ ( times_times_int @ C @ A ) ) ) ).

% left_diff_distrib'
thf(fact_384_right__diff__distrib_H,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( times_times_real @ A @ ( minus_minus_real @ B2 @ C ) )
      = ( minus_minus_real @ ( times_times_real @ A @ B2 ) @ ( times_times_real @ A @ C ) ) ) ).

% right_diff_distrib'
thf(fact_385_right__diff__distrib_H,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( times_times_rat @ A @ ( minus_minus_rat @ B2 @ C ) )
      = ( minus_minus_rat @ ( times_times_rat @ A @ B2 ) @ ( times_times_rat @ A @ C ) ) ) ).

% right_diff_distrib'
thf(fact_386_right__diff__distrib_H,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( times_times_nat @ A @ ( minus_minus_nat @ B2 @ C ) )
      = ( minus_minus_nat @ ( times_times_nat @ A @ B2 ) @ ( times_times_nat @ A @ C ) ) ) ).

% right_diff_distrib'
thf(fact_387_right__diff__distrib_H,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( times_times_int @ A @ ( minus_minus_int @ B2 @ C ) )
      = ( minus_minus_int @ ( times_times_int @ A @ B2 ) @ ( times_times_int @ A @ C ) ) ) ).

% right_diff_distrib'
thf(fact_388_Nat_OlessE,axiom,
    ! [I: nat,K: nat] :
      ( ( ord_less_nat @ I @ K )
     => ( ( K
         != ( suc @ I ) )
       => ~ ! [J2: nat] :
              ( ( ord_less_nat @ I @ J2 )
             => ( K
               != ( suc @ J2 ) ) ) ) ) ).

% Nat.lessE
thf(fact_389_Suc__lessD,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ M ) @ N )
     => ( ord_less_nat @ M @ N ) ) ).

% Suc_lessD
thf(fact_390_Suc__lessE,axiom,
    ! [I: nat,K: nat] :
      ( ( ord_less_nat @ ( suc @ I ) @ K )
     => ~ ! [J2: nat] :
            ( ( ord_less_nat @ I @ J2 )
           => ( K
             != ( suc @ J2 ) ) ) ) ).

% Suc_lessE
thf(fact_391_Suc__lessI,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ( ( suc @ M )
         != N )
       => ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).

% Suc_lessI
thf(fact_392_less__SucE,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N ) )
     => ( ~ ( ord_less_nat @ M @ N )
       => ( M = N ) ) ) ).

% less_SucE
thf(fact_393_less__SucI,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_nat @ M @ ( suc @ N ) ) ) ).

% less_SucI
thf(fact_394_Ex__less__Suc,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ? [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( suc @ N ) )
            & ( P @ I2 ) ) )
      = ( ( P @ N )
        | ? [I2: nat] :
            ( ( ord_less_nat @ I2 @ N )
            & ( P @ I2 ) ) ) ) ).

% Ex_less_Suc
thf(fact_395_less__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N ) )
      = ( ( ord_less_nat @ M @ N )
        | ( M = N ) ) ) ).

% less_Suc_eq
thf(fact_396_not__less__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ~ ( ord_less_nat @ M @ N ) )
      = ( ord_less_nat @ N @ ( suc @ M ) ) ) ).

% not_less_eq
thf(fact_397_All__less__Suc,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( suc @ N ) )
           => ( P @ I2 ) ) )
      = ( ( P @ N )
        & ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ N )
           => ( P @ I2 ) ) ) ) ).

% All_less_Suc
thf(fact_398_Suc__less__eq2,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ ( suc @ N ) @ M )
      = ( ? [M2: nat] :
            ( ( M
              = ( suc @ M2 ) )
            & ( ord_less_nat @ N @ M2 ) ) ) ) ).

% Suc_less_eq2
thf(fact_399_less__antisym,axiom,
    ! [N: nat,M: nat] :
      ( ~ ( ord_less_nat @ N @ M )
     => ( ( ord_less_nat @ N @ ( suc @ M ) )
       => ( M = N ) ) ) ).

% less_antisym
thf(fact_400_Suc__less__SucD,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
     => ( ord_less_nat @ M @ N ) ) ).

% Suc_less_SucD
thf(fact_401_less__trans__Suc,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( ord_less_nat @ J @ K )
       => ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).

% less_trans_Suc
thf(fact_402_less__Suc__induct,axiom,
    ! [I: nat,J: nat,P: nat > nat > $o] :
      ( ( ord_less_nat @ I @ J )
     => ( ! [I3: nat] : ( P @ I3 @ ( suc @ I3 ) )
       => ( ! [I3: nat,J2: nat,K2: nat] :
              ( ( ord_less_nat @ I3 @ J2 )
             => ( ( ord_less_nat @ J2 @ K2 )
               => ( ( P @ I3 @ J2 )
                 => ( ( P @ J2 @ K2 )
                   => ( P @ I3 @ K2 ) ) ) ) )
         => ( P @ I @ J ) ) ) ) ).

% less_Suc_induct
thf(fact_403_strict__inc__induct,axiom,
    ! [I: nat,J: nat,P: nat > $o] :
      ( ( ord_less_nat @ I @ J )
     => ( ! [I3: nat] :
            ( ( J
              = ( suc @ I3 ) )
           => ( P @ I3 ) )
       => ( ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ J )
             => ( ( P @ ( suc @ I3 ) )
               => ( P @ I3 ) ) )
         => ( P @ I ) ) ) ) ).

% strict_inc_induct
thf(fact_404_not__less__less__Suc__eq,axiom,
    ! [N: nat,M: nat] :
      ( ~ ( ord_less_nat @ N @ M )
     => ( ( ord_less_nat @ N @ ( suc @ M ) )
        = ( N = M ) ) ) ).

% not_less_less_Suc_eq
thf(fact_405_Suc__leD,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% Suc_leD
thf(fact_406_le__SucE,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
     => ( ~ ( ord_less_eq_nat @ M @ N )
       => ( M
          = ( suc @ N ) ) ) ) ).

% le_SucE
thf(fact_407_le__SucI,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).

% le_SucI
thf(fact_408_Suc__le__D,axiom,
    ! [N: nat,M3: nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ M3 )
     => ? [M4: nat] :
          ( M3
          = ( suc @ M4 ) ) ) ).

% Suc_le_D
thf(fact_409_le__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
      = ( ( ord_less_eq_nat @ M @ N )
        | ( M
          = ( suc @ N ) ) ) ) ).

% le_Suc_eq
thf(fact_410_Suc__n__not__le__n,axiom,
    ! [N: nat] :
      ~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).

% Suc_n_not_le_n
thf(fact_411_not__less__eq__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ~ ( ord_less_eq_nat @ M @ N ) )
      = ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).

% not_less_eq_eq
thf(fact_412_full__nat__induct,axiom,
    ! [P: nat > $o,N: nat] :
      ( ! [N4: nat] :
          ( ! [M5: nat] :
              ( ( ord_less_eq_nat @ ( suc @ M5 ) @ N4 )
             => ( P @ M5 ) )
         => ( P @ N4 ) )
     => ( P @ N ) ) ).

% full_nat_induct
thf(fact_413_nat__induct__at__least,axiom,
    ! [M: nat,N: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( P @ M )
       => ( ! [N4: nat] :
              ( ( ord_less_eq_nat @ M @ N4 )
             => ( ( P @ N4 )
               => ( P @ ( suc @ N4 ) ) ) )
         => ( P @ N ) ) ) ) ).

% nat_induct_at_least
thf(fact_414_transitive__stepwise__le,axiom,
    ! [M: nat,N: nat,R: nat > nat > $o] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ! [X3: nat] : ( R @ X3 @ X3 )
       => ( ! [X3: nat,Y4: nat,Z2: nat] :
              ( ( R @ X3 @ Y4 )
             => ( ( R @ Y4 @ Z2 )
               => ( R @ X3 @ Z2 ) ) )
         => ( ! [N4: nat] : ( R @ N4 @ ( suc @ N4 ) )
           => ( R @ M @ N ) ) ) ) ) ).

% transitive_stepwise_le
thf(fact_415_le__cube,axiom,
    ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).

% le_cube
thf(fact_416_le__square,axiom,
    ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).

% le_square
thf(fact_417_mult__le__mono,axiom,
    ! [I: nat,J: nat,K: nat,L2: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ord_less_eq_nat @ K @ L2 )
       => ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ L2 ) ) ) ) ).

% mult_le_mono
thf(fact_418_mult__le__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ).

% mult_le_mono1
thf(fact_419_mult__le__mono2,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ).

% mult_le_mono2
thf(fact_420_nat__arith_Osuc1,axiom,
    ! [A3: nat,K: nat,A: nat] :
      ( ( A3
        = ( plus_plus_nat @ K @ A ) )
     => ( ( suc @ A3 )
        = ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).

% nat_arith.suc1
thf(fact_421_add__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( plus_plus_nat @ ( suc @ M ) @ N )
      = ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).

% add_Suc
thf(fact_422_add__Suc__shift,axiom,
    ! [M: nat,N: nat] :
      ( ( plus_plus_nat @ ( suc @ M ) @ N )
      = ( plus_plus_nat @ M @ ( suc @ N ) ) ) ).

% add_Suc_shift
thf(fact_423_add__mult__distrib,axiom,
    ! [M: nat,N: nat,K: nat] :
      ( ( times_times_nat @ ( plus_plus_nat @ M @ N ) @ K )
      = ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).

% add_mult_distrib
thf(fact_424_add__mult__distrib2,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N ) )
      = ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).

% add_mult_distrib2
thf(fact_425_zero__induct__lemma,axiom,
    ! [P: nat > $o,K: nat,I: nat] :
      ( ( P @ K )
     => ( ! [N4: nat] :
            ( ( P @ ( suc @ N4 ) )
           => ( P @ N4 ) )
       => ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).

% zero_induct_lemma
thf(fact_426_diff__mult__distrib,axiom,
    ! [M: nat,N: nat,K: nat] :
      ( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K )
      = ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).

% diff_mult_distrib
thf(fact_427_diff__mult__distrib2,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N ) )
      = ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).

% diff_mult_distrib2
thf(fact_428_nat__mult__1,axiom,
    ! [N: nat] :
      ( ( times_times_nat @ one_one_nat @ N )
      = N ) ).

% nat_mult_1
thf(fact_429_nat__mult__1__right,axiom,
    ! [N: nat] :
      ( ( times_times_nat @ N @ one_one_nat )
      = N ) ).

% nat_mult_1_right
thf(fact_430_two__realpow__ge__one,axiom,
    ! [N: nat] : ( ord_less_eq_real @ one_one_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ).

% two_realpow_ge_one
thf(fact_431_add__One__commute,axiom,
    ! [N: num] :
      ( ( plus_plus_num @ one @ N )
      = ( plus_plus_num @ N @ one ) ) ).

% add_One_commute
thf(fact_432_power__commutes,axiom,
    ! [A: complex,N: nat] :
      ( ( times_times_complex @ ( power_power_complex @ A @ N ) @ A )
      = ( times_times_complex @ A @ ( power_power_complex @ A @ N ) ) ) ).

% power_commutes
thf(fact_433_power__commutes,axiom,
    ! [A: real,N: nat] :
      ( ( times_times_real @ ( power_power_real @ A @ N ) @ A )
      = ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ).

% power_commutes
thf(fact_434_power__commutes,axiom,
    ! [A: rat,N: nat] :
      ( ( times_times_rat @ ( power_power_rat @ A @ N ) @ A )
      = ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ).

% power_commutes
thf(fact_435_power__commutes,axiom,
    ! [A: nat,N: nat] :
      ( ( times_times_nat @ ( power_power_nat @ A @ N ) @ A )
      = ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ).

% power_commutes
thf(fact_436_power__commutes,axiom,
    ! [A: int,N: nat] :
      ( ( times_times_int @ ( power_power_int @ A @ N ) @ A )
      = ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ).

% power_commutes
thf(fact_437_power__mult__distrib,axiom,
    ! [A: complex,B2: complex,N: nat] :
      ( ( power_power_complex @ ( times_times_complex @ A @ B2 ) @ N )
      = ( times_times_complex @ ( power_power_complex @ A @ N ) @ ( power_power_complex @ B2 @ N ) ) ) ).

% power_mult_distrib
thf(fact_438_power__mult__distrib,axiom,
    ! [A: real,B2: real,N: nat] :
      ( ( power_power_real @ ( times_times_real @ A @ B2 ) @ N )
      = ( times_times_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B2 @ N ) ) ) ).

% power_mult_distrib
thf(fact_439_power__mult__distrib,axiom,
    ! [A: rat,B2: rat,N: nat] :
      ( ( power_power_rat @ ( times_times_rat @ A @ B2 ) @ N )
      = ( times_times_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B2 @ N ) ) ) ).

% power_mult_distrib
thf(fact_440_power__mult__distrib,axiom,
    ! [A: nat,B2: nat,N: nat] :
      ( ( power_power_nat @ ( times_times_nat @ A @ B2 ) @ N )
      = ( times_times_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B2 @ N ) ) ) ).

% power_mult_distrib
thf(fact_441_power__mult__distrib,axiom,
    ! [A: int,B2: int,N: nat] :
      ( ( power_power_int @ ( times_times_int @ A @ B2 ) @ N )
      = ( times_times_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B2 @ N ) ) ) ).

% power_mult_distrib
thf(fact_442_power__commuting__commutes,axiom,
    ! [X4: complex,Y3: complex,N: nat] :
      ( ( ( times_times_complex @ X4 @ Y3 )
        = ( times_times_complex @ Y3 @ X4 ) )
     => ( ( times_times_complex @ ( power_power_complex @ X4 @ N ) @ Y3 )
        = ( times_times_complex @ Y3 @ ( power_power_complex @ X4 @ N ) ) ) ) ).

% power_commuting_commutes
thf(fact_443_power__commuting__commutes,axiom,
    ! [X4: real,Y3: real,N: nat] :
      ( ( ( times_times_real @ X4 @ Y3 )
        = ( times_times_real @ Y3 @ X4 ) )
     => ( ( times_times_real @ ( power_power_real @ X4 @ N ) @ Y3 )
        = ( times_times_real @ Y3 @ ( power_power_real @ X4 @ N ) ) ) ) ).

% power_commuting_commutes
thf(fact_444_power__commuting__commutes,axiom,
    ! [X4: rat,Y3: rat,N: nat] :
      ( ( ( times_times_rat @ X4 @ Y3 )
        = ( times_times_rat @ Y3 @ X4 ) )
     => ( ( times_times_rat @ ( power_power_rat @ X4 @ N ) @ Y3 )
        = ( times_times_rat @ Y3 @ ( power_power_rat @ X4 @ N ) ) ) ) ).

% power_commuting_commutes
thf(fact_445_power__commuting__commutes,axiom,
    ! [X4: nat,Y3: nat,N: nat] :
      ( ( ( times_times_nat @ X4 @ Y3 )
        = ( times_times_nat @ Y3 @ X4 ) )
     => ( ( times_times_nat @ ( power_power_nat @ X4 @ N ) @ Y3 )
        = ( times_times_nat @ Y3 @ ( power_power_nat @ X4 @ N ) ) ) ) ).

% power_commuting_commutes
thf(fact_446_power__commuting__commutes,axiom,
    ! [X4: int,Y3: int,N: nat] :
      ( ( ( times_times_int @ X4 @ Y3 )
        = ( times_times_int @ Y3 @ X4 ) )
     => ( ( times_times_int @ ( power_power_int @ X4 @ N ) @ Y3 )
        = ( times_times_int @ Y3 @ ( power_power_int @ X4 @ N ) ) ) ) ).

% power_commuting_commutes
thf(fact_447_power__mult,axiom,
    ! [A: nat,M: nat,N: nat] :
      ( ( power_power_nat @ A @ ( times_times_nat @ M @ N ) )
      = ( power_power_nat @ ( power_power_nat @ A @ M ) @ N ) ) ).

% power_mult
thf(fact_448_power__mult,axiom,
    ! [A: real,M: nat,N: nat] :
      ( ( power_power_real @ A @ ( times_times_nat @ M @ N ) )
      = ( power_power_real @ ( power_power_real @ A @ M ) @ N ) ) ).

% power_mult
thf(fact_449_power__mult,axiom,
    ! [A: int,M: nat,N: nat] :
      ( ( power_power_int @ A @ ( times_times_nat @ M @ N ) )
      = ( power_power_int @ ( power_power_int @ A @ M ) @ N ) ) ).

% power_mult
thf(fact_450_power__mult,axiom,
    ! [A: complex,M: nat,N: nat] :
      ( ( power_power_complex @ A @ ( times_times_nat @ M @ N ) )
      = ( power_power_complex @ ( power_power_complex @ A @ M ) @ N ) ) ).

% power_mult
thf(fact_451_left__add__mult__distrib,axiom,
    ! [I: nat,U: nat,J: nat,K: nat] :
      ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ K ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I @ J ) @ U ) @ K ) ) ).

% left_add_mult_distrib
thf(fact_452_div__mult2__eq,axiom,
    ! [M: nat,N: nat,Q2: nat] :
      ( ( divide_divide_nat @ M @ ( times_times_nat @ N @ Q2 ) )
      = ( divide_divide_nat @ ( divide_divide_nat @ M @ N ) @ Q2 ) ) ).

% div_mult2_eq
thf(fact_453_power__odd__eq,axiom,
    ! [A: complex,N: nat] :
      ( ( power_power_complex @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( times_times_complex @ A @ ( power_power_complex @ ( power_power_complex @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% power_odd_eq
thf(fact_454_power__odd__eq,axiom,
    ! [A: real,N: nat] :
      ( ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( times_times_real @ A @ ( power_power_real @ ( power_power_real @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% power_odd_eq
thf(fact_455_power__odd__eq,axiom,
    ! [A: rat,N: nat] :
      ( ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( times_times_rat @ A @ ( power_power_rat @ ( power_power_rat @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% power_odd_eq
thf(fact_456_power__odd__eq,axiom,
    ! [A: nat,N: nat] :
      ( ( power_power_nat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( times_times_nat @ A @ ( power_power_nat @ ( power_power_nat @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% power_odd_eq
thf(fact_457_power__odd__eq,axiom,
    ! [A: int,N: nat] :
      ( ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( times_times_int @ A @ ( power_power_int @ ( power_power_int @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% power_odd_eq
thf(fact_458_less__1__mult,axiom,
    ! [M: real,N: real] :
      ( ( ord_less_real @ one_one_real @ M )
     => ( ( ord_less_real @ one_one_real @ N )
       => ( ord_less_real @ one_one_real @ ( times_times_real @ M @ N ) ) ) ) ).

% less_1_mult
thf(fact_459_less__1__mult,axiom,
    ! [M: rat,N: rat] :
      ( ( ord_less_rat @ one_one_rat @ M )
     => ( ( ord_less_rat @ one_one_rat @ N )
       => ( ord_less_rat @ one_one_rat @ ( times_times_rat @ M @ N ) ) ) ) ).

% less_1_mult
thf(fact_460_less__1__mult,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ M )
     => ( ( ord_less_nat @ one_one_nat @ N )
       => ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M @ N ) ) ) ) ).

% less_1_mult
thf(fact_461_less__1__mult,axiom,
    ! [M: int,N: int] :
      ( ( ord_less_int @ one_one_int @ M )
     => ( ( ord_less_int @ one_one_int @ N )
       => ( ord_less_int @ one_one_int @ ( times_times_int @ M @ N ) ) ) ) ).

% less_1_mult
thf(fact_462_eq__add__iff1,axiom,
    ! [A: real,E: real,C: real,B2: real,D: real] :
      ( ( ( plus_plus_real @ ( times_times_real @ A @ E ) @ C )
        = ( plus_plus_real @ ( times_times_real @ B2 @ E ) @ D ) )
      = ( ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B2 ) @ E ) @ C )
        = D ) ) ).

% eq_add_iff1
thf(fact_463_eq__add__iff1,axiom,
    ! [A: rat,E: rat,C: rat,B2: rat,D: rat] :
      ( ( ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C )
        = ( plus_plus_rat @ ( times_times_rat @ B2 @ E ) @ D ) )
      = ( ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B2 ) @ E ) @ C )
        = D ) ) ).

% eq_add_iff1
thf(fact_464_eq__add__iff1,axiom,
    ! [A: int,E: int,C: int,B2: int,D: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ A @ E ) @ C )
        = ( plus_plus_int @ ( times_times_int @ B2 @ E ) @ D ) )
      = ( ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B2 ) @ E ) @ C )
        = D ) ) ).

% eq_add_iff1
thf(fact_465_eq__add__iff2,axiom,
    ! [A: real,E: real,C: real,B2: real,D: real] :
      ( ( ( plus_plus_real @ ( times_times_real @ A @ E ) @ C )
        = ( plus_plus_real @ ( times_times_real @ B2 @ E ) @ D ) )
      = ( C
        = ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B2 @ A ) @ E ) @ D ) ) ) ).

% eq_add_iff2
thf(fact_466_eq__add__iff2,axiom,
    ! [A: rat,E: rat,C: rat,B2: rat,D: rat] :
      ( ( ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C )
        = ( plus_plus_rat @ ( times_times_rat @ B2 @ E ) @ D ) )
      = ( C
        = ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B2 @ A ) @ E ) @ D ) ) ) ).

% eq_add_iff2
thf(fact_467_eq__add__iff2,axiom,
    ! [A: int,E: int,C: int,B2: int,D: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ A @ E ) @ C )
        = ( plus_plus_int @ ( times_times_int @ B2 @ E ) @ D ) )
      = ( C
        = ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B2 @ A ) @ E ) @ D ) ) ) ).

% eq_add_iff2
thf(fact_468_square__diff__square__factored,axiom,
    ! [X4: real,Y3: real] :
      ( ( minus_minus_real @ ( times_times_real @ X4 @ X4 ) @ ( times_times_real @ Y3 @ Y3 ) )
      = ( times_times_real @ ( plus_plus_real @ X4 @ Y3 ) @ ( minus_minus_real @ X4 @ Y3 ) ) ) ).

% square_diff_square_factored
thf(fact_469_square__diff__square__factored,axiom,
    ! [X4: rat,Y3: rat] :
      ( ( minus_minus_rat @ ( times_times_rat @ X4 @ X4 ) @ ( times_times_rat @ Y3 @ Y3 ) )
      = ( times_times_rat @ ( plus_plus_rat @ X4 @ Y3 ) @ ( minus_minus_rat @ X4 @ Y3 ) ) ) ).

% square_diff_square_factored
thf(fact_470_square__diff__square__factored,axiom,
    ! [X4: int,Y3: int] :
      ( ( minus_minus_int @ ( times_times_int @ X4 @ X4 ) @ ( times_times_int @ Y3 @ Y3 ) )
      = ( times_times_int @ ( plus_plus_int @ X4 @ Y3 ) @ ( minus_minus_int @ X4 @ Y3 ) ) ) ).

% square_diff_square_factored
thf(fact_471_mult__diff__mult,axiom,
    ! [X4: real,Y3: real,A: real,B2: real] :
      ( ( minus_minus_real @ ( times_times_real @ X4 @ Y3 ) @ ( times_times_real @ A @ B2 ) )
      = ( plus_plus_real @ ( times_times_real @ X4 @ ( minus_minus_real @ Y3 @ B2 ) ) @ ( times_times_real @ ( minus_minus_real @ X4 @ A ) @ B2 ) ) ) ).

% mult_diff_mult
thf(fact_472_mult__diff__mult,axiom,
    ! [X4: rat,Y3: rat,A: rat,B2: rat] :
      ( ( minus_minus_rat @ ( times_times_rat @ X4 @ Y3 ) @ ( times_times_rat @ A @ B2 ) )
      = ( plus_plus_rat @ ( times_times_rat @ X4 @ ( minus_minus_rat @ Y3 @ B2 ) ) @ ( times_times_rat @ ( minus_minus_rat @ X4 @ A ) @ B2 ) ) ) ).

% mult_diff_mult
thf(fact_473_mult__diff__mult,axiom,
    ! [X4: int,Y3: int,A: int,B2: int] :
      ( ( minus_minus_int @ ( times_times_int @ X4 @ Y3 ) @ ( times_times_int @ A @ B2 ) )
      = ( plus_plus_int @ ( times_times_int @ X4 @ ( minus_minus_int @ Y3 @ B2 ) ) @ ( times_times_int @ ( minus_minus_int @ X4 @ A ) @ B2 ) ) ) ).

% mult_diff_mult
thf(fact_474_lift__Suc__mono__less,axiom,
    ! [F: nat > real,N: nat,N5: nat] :
      ( ! [N4: nat] : ( ord_less_real @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
     => ( ( ord_less_nat @ N @ N5 )
       => ( ord_less_real @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_475_lift__Suc__mono__less,axiom,
    ! [F: nat > rat,N: nat,N5: nat] :
      ( ! [N4: nat] : ( ord_less_rat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
     => ( ( ord_less_nat @ N @ N5 )
       => ( ord_less_rat @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_476_lift__Suc__mono__less,axiom,
    ! [F: nat > num,N: nat,N5: nat] :
      ( ! [N4: nat] : ( ord_less_num @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
     => ( ( ord_less_nat @ N @ N5 )
       => ( ord_less_num @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_477_lift__Suc__mono__less,axiom,
    ! [F: nat > nat,N: nat,N5: nat] :
      ( ! [N4: nat] : ( ord_less_nat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
     => ( ( ord_less_nat @ N @ N5 )
       => ( ord_less_nat @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_478_lift__Suc__mono__less,axiom,
    ! [F: nat > int,N: nat,N5: nat] :
      ( ! [N4: nat] : ( ord_less_int @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
     => ( ( ord_less_nat @ N @ N5 )
       => ( ord_less_int @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_479_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > real,N: nat,M: nat] :
      ( ! [N4: nat] : ( ord_less_real @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
     => ( ( ord_less_real @ ( F @ N ) @ ( F @ M ) )
        = ( ord_less_nat @ N @ M ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_480_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > rat,N: nat,M: nat] :
      ( ! [N4: nat] : ( ord_less_rat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
     => ( ( ord_less_rat @ ( F @ N ) @ ( F @ M ) )
        = ( ord_less_nat @ N @ M ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_481_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > num,N: nat,M: nat] :
      ( ! [N4: nat] : ( ord_less_num @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
     => ( ( ord_less_num @ ( F @ N ) @ ( F @ M ) )
        = ( ord_less_nat @ N @ M ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_482_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > nat,N: nat,M: nat] :
      ( ! [N4: nat] : ( ord_less_nat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
     => ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
        = ( ord_less_nat @ N @ M ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_483_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > int,N: nat,M: nat] :
      ( ! [N4: nat] : ( ord_less_int @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
     => ( ( ord_less_int @ ( F @ N ) @ ( F @ M ) )
        = ( ord_less_nat @ N @ M ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_484_lift__Suc__mono__le,axiom,
    ! [F: nat > set_nat,N: nat,N5: nat] :
      ( ! [N4: nat] : ( ord_less_eq_set_nat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
     => ( ( ord_less_eq_nat @ N @ N5 )
       => ( ord_less_eq_set_nat @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_485_lift__Suc__mono__le,axiom,
    ! [F: nat > rat,N: nat,N5: nat] :
      ( ! [N4: nat] : ( ord_less_eq_rat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
     => ( ( ord_less_eq_nat @ N @ N5 )
       => ( ord_less_eq_rat @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_486_lift__Suc__mono__le,axiom,
    ! [F: nat > num,N: nat,N5: nat] :
      ( ! [N4: nat] : ( ord_less_eq_num @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
     => ( ( ord_less_eq_nat @ N @ N5 )
       => ( ord_less_eq_num @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_487_lift__Suc__mono__le,axiom,
    ! [F: nat > nat,N: nat,N5: nat] :
      ( ! [N4: nat] : ( ord_less_eq_nat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
     => ( ( ord_less_eq_nat @ N @ N5 )
       => ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_488_lift__Suc__mono__le,axiom,
    ! [F: nat > int,N: nat,N5: nat] :
      ( ! [N4: nat] : ( ord_less_eq_int @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
     => ( ( ord_less_eq_nat @ N @ N5 )
       => ( ord_less_eq_int @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_489_lift__Suc__antimono__le,axiom,
    ! [F: nat > set_nat,N: nat,N5: nat] :
      ( ! [N4: nat] : ( ord_less_eq_set_nat @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
     => ( ( ord_less_eq_nat @ N @ N5 )
       => ( ord_less_eq_set_nat @ ( F @ N5 ) @ ( F @ N ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_490_lift__Suc__antimono__le,axiom,
    ! [F: nat > rat,N: nat,N5: nat] :
      ( ! [N4: nat] : ( ord_less_eq_rat @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
     => ( ( ord_less_eq_nat @ N @ N5 )
       => ( ord_less_eq_rat @ ( F @ N5 ) @ ( F @ N ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_491_lift__Suc__antimono__le,axiom,
    ! [F: nat > num,N: nat,N5: nat] :
      ( ! [N4: nat] : ( ord_less_eq_num @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
     => ( ( ord_less_eq_nat @ N @ N5 )
       => ( ord_less_eq_num @ ( F @ N5 ) @ ( F @ N ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_492_lift__Suc__antimono__le,axiom,
    ! [F: nat > nat,N: nat,N5: nat] :
      ( ! [N4: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
     => ( ( ord_less_eq_nat @ N @ N5 )
       => ( ord_less_eq_nat @ ( F @ N5 ) @ ( F @ N ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_493_lift__Suc__antimono__le,axiom,
    ! [F: nat > int,N: nat,N5: nat] :
      ( ! [N4: nat] : ( ord_less_eq_int @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
     => ( ( ord_less_eq_nat @ N @ N5 )
       => ( ord_less_eq_int @ ( F @ N5 ) @ ( F @ N ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_494_Suc__leI,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).

% Suc_leI
thf(fact_495_Suc__le__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
      = ( ord_less_nat @ M @ N ) ) ).

% Suc_le_eq
thf(fact_496_dec__induct,axiom,
    ! [I: nat,J: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( P @ I )
       => ( ! [N4: nat] :
              ( ( ord_less_eq_nat @ I @ N4 )
             => ( ( ord_less_nat @ N4 @ J )
               => ( ( P @ N4 )
                 => ( P @ ( suc @ N4 ) ) ) ) )
         => ( P @ J ) ) ) ) ).

% dec_induct
thf(fact_497_inc__induct,axiom,
    ! [I: nat,J: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( P @ J )
       => ( ! [N4: nat] :
              ( ( ord_less_eq_nat @ I @ N4 )
             => ( ( ord_less_nat @ N4 @ J )
               => ( ( P @ ( suc @ N4 ) )
                 => ( P @ N4 ) ) ) )
         => ( P @ I ) ) ) ) ).

% inc_induct
thf(fact_498_Suc__le__lessD,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
     => ( ord_less_nat @ M @ N ) ) ).

% Suc_le_lessD
thf(fact_499_le__less__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( ord_less_nat @ N @ ( suc @ M ) )
        = ( N = M ) ) ) ).

% le_less_Suc_eq
thf(fact_500_less__Suc__eq__le,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% less_Suc_eq_le
thf(fact_501_less__eq__Suc__le,axiom,
    ( ord_less_nat
    = ( ^ [N2: nat] : ( ord_less_eq_nat @ ( suc @ N2 ) ) ) ) ).

% less_eq_Suc_le
thf(fact_502_le__imp__less__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_nat @ M @ ( suc @ N ) ) ) ).

% le_imp_less_Suc
thf(fact_503_less__imp__Suc__add,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ? [K2: nat] :
          ( N
          = ( suc @ ( plus_plus_nat @ M @ K2 ) ) ) ) ).

% less_imp_Suc_add
thf(fact_504_less__iff__Suc__add,axiom,
    ( ord_less_nat
    = ( ^ [M6: nat,N2: nat] :
        ? [K3: nat] :
          ( N2
          = ( suc @ ( plus_plus_nat @ M6 @ K3 ) ) ) ) ) ).

% less_iff_Suc_add
thf(fact_505_less__add__Suc2,axiom,
    ! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).

% less_add_Suc2
thf(fact_506_less__add__Suc1,axiom,
    ! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).

% less_add_Suc1
thf(fact_507_less__natE,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ~ ! [Q3: nat] :
            ( N
           != ( suc @ ( plus_plus_nat @ M @ Q3 ) ) ) ) ).

% less_natE
thf(fact_508_diff__less__Suc,axiom,
    ! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).

% diff_less_Suc
thf(fact_509_Suc__diff__Suc,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ N @ M )
     => ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N ) ) )
        = ( minus_minus_nat @ M @ N ) ) ) ).

% Suc_diff_Suc
thf(fact_510_Suc__diff__le,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( minus_minus_nat @ ( suc @ M ) @ N )
        = ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).

% Suc_diff_le
thf(fact_511_Suc__eq__plus1,axiom,
    ( suc
    = ( ^ [N2: nat] : ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ).

% Suc_eq_plus1
thf(fact_512_plus__1__eq__Suc,axiom,
    ( ( plus_plus_nat @ one_one_nat )
    = suc ) ).

% plus_1_eq_Suc
thf(fact_513_Suc__eq__plus1__left,axiom,
    ( suc
    = ( plus_plus_nat @ one_one_nat ) ) ).

% Suc_eq_plus1_left
thf(fact_514_diff__Suc__eq__diff__pred,axiom,
    ! [M: nat,N: nat] :
      ( ( minus_minus_nat @ M @ ( suc @ N ) )
      = ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ).

% diff_Suc_eq_diff_pred
thf(fact_515_Suc__nat__number__of__add,axiom,
    ! [V: num,N: nat] :
      ( ( suc @ ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ N ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ one ) ) @ N ) ) ).

% Suc_nat_number_of_add
thf(fact_516_div__nat__eqI,axiom,
    ! [N: nat,Q2: nat,M: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q2 ) @ M )
     => ( ( ord_less_nat @ M @ ( times_times_nat @ N @ ( suc @ Q2 ) ) )
       => ( ( divide_divide_nat @ M @ N )
          = Q2 ) ) ) ).

% div_nat_eqI
thf(fact_517_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: real,E: real,C: real,B2: real,D: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B2 @ E ) @ D ) )
      = ( ord_less_eq_real @ C @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B2 @ A ) @ E ) @ D ) ) ) ).

% ordered_ring_class.le_add_iff2
thf(fact_518_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: rat,E: rat,C: rat,B2: rat,D: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B2 @ E ) @ D ) )
      = ( ord_less_eq_rat @ C @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B2 @ A ) @ E ) @ D ) ) ) ).

% ordered_ring_class.le_add_iff2
thf(fact_519_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: int,E: int,C: int,B2: int,D: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B2 @ E ) @ D ) )
      = ( ord_less_eq_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B2 @ A ) @ E ) @ D ) ) ) ).

% ordered_ring_class.le_add_iff2
thf(fact_520_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: real,E: real,C: real,B2: real,D: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B2 @ E ) @ D ) )
      = ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B2 ) @ E ) @ C ) @ D ) ) ).

% ordered_ring_class.le_add_iff1
thf(fact_521_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: rat,E: rat,C: rat,B2: rat,D: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B2 @ E ) @ D ) )
      = ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B2 ) @ E ) @ C ) @ D ) ) ).

% ordered_ring_class.le_add_iff1
thf(fact_522_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: int,E: int,C: int,B2: int,D: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B2 @ E ) @ D ) )
      = ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B2 ) @ E ) @ C ) @ D ) ) ).

% ordered_ring_class.le_add_iff1
thf(fact_523_less__add__iff2,axiom,
    ! [A: real,E: real,C: real,B2: real,D: real] :
      ( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B2 @ E ) @ D ) )
      = ( ord_less_real @ C @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B2 @ A ) @ E ) @ D ) ) ) ).

% less_add_iff2
thf(fact_524_less__add__iff2,axiom,
    ! [A: rat,E: rat,C: rat,B2: rat,D: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B2 @ E ) @ D ) )
      = ( ord_less_rat @ C @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B2 @ A ) @ E ) @ D ) ) ) ).

% less_add_iff2
thf(fact_525_less__add__iff2,axiom,
    ! [A: int,E: int,C: int,B2: int,D: int] :
      ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B2 @ E ) @ D ) )
      = ( ord_less_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B2 @ A ) @ E ) @ D ) ) ) ).

% less_add_iff2
thf(fact_526_less__add__iff1,axiom,
    ! [A: real,E: real,C: real,B2: real,D: real] :
      ( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B2 @ E ) @ D ) )
      = ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B2 ) @ E ) @ C ) @ D ) ) ).

% less_add_iff1
thf(fact_527_less__add__iff1,axiom,
    ! [A: rat,E: rat,C: rat,B2: rat,D: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B2 @ E ) @ D ) )
      = ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B2 ) @ E ) @ C ) @ D ) ) ).

% less_add_iff1
thf(fact_528_less__add__iff1,axiom,
    ! [A: int,E: int,C: int,B2: int,D: int] :
      ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B2 @ E ) @ D ) )
      = ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B2 ) @ E ) @ C ) @ D ) ) ).

% less_add_iff1
thf(fact_529_square__diff__one__factored,axiom,
    ! [X4: complex] :
      ( ( minus_minus_complex @ ( times_times_complex @ X4 @ X4 ) @ one_one_complex )
      = ( times_times_complex @ ( plus_plus_complex @ X4 @ one_one_complex ) @ ( minus_minus_complex @ X4 @ one_one_complex ) ) ) ).

% square_diff_one_factored
thf(fact_530_square__diff__one__factored,axiom,
    ! [X4: real] :
      ( ( minus_minus_real @ ( times_times_real @ X4 @ X4 ) @ one_one_real )
      = ( times_times_real @ ( plus_plus_real @ X4 @ one_one_real ) @ ( minus_minus_real @ X4 @ one_one_real ) ) ) ).

% square_diff_one_factored
thf(fact_531_square__diff__one__factored,axiom,
    ! [X4: rat] :
      ( ( minus_minus_rat @ ( times_times_rat @ X4 @ X4 ) @ one_one_rat )
      = ( times_times_rat @ ( plus_plus_rat @ X4 @ one_one_rat ) @ ( minus_minus_rat @ X4 @ one_one_rat ) ) ) ).

% square_diff_one_factored
thf(fact_532_square__diff__one__factored,axiom,
    ! [X4: int] :
      ( ( minus_minus_int @ ( times_times_int @ X4 @ X4 ) @ one_one_int )
      = ( times_times_int @ ( plus_plus_int @ X4 @ one_one_int ) @ ( minus_minus_int @ X4 @ one_one_int ) ) ) ).

% square_diff_one_factored
thf(fact_533_mult__numeral__1__right,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ A @ ( numera6690914467698888265omplex @ one ) )
      = A ) ).

% mult_numeral_1_right
thf(fact_534_mult__numeral__1__right,axiom,
    ! [A: real] :
      ( ( times_times_real @ A @ ( numeral_numeral_real @ one ) )
      = A ) ).

% mult_numeral_1_right
thf(fact_535_mult__numeral__1__right,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ A @ ( numeral_numeral_rat @ one ) )
      = A ) ).

% mult_numeral_1_right
thf(fact_536_mult__numeral__1__right,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ ( numeral_numeral_nat @ one ) )
      = A ) ).

% mult_numeral_1_right
thf(fact_537_mult__numeral__1__right,axiom,
    ! [A: int] :
      ( ( times_times_int @ A @ ( numeral_numeral_int @ one ) )
      = A ) ).

% mult_numeral_1_right
thf(fact_538_mult__numeral__1,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ one ) @ A )
      = A ) ).

% mult_numeral_1
thf(fact_539_mult__numeral__1,axiom,
    ! [A: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ one ) @ A )
      = A ) ).

% mult_numeral_1
thf(fact_540_mult__numeral__1,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ one ) @ A )
      = A ) ).

% mult_numeral_1
thf(fact_541_mult__numeral__1,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ one ) @ A )
      = A ) ).

% mult_numeral_1
thf(fact_542_mult__numeral__1,axiom,
    ! [A: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ one ) @ A )
      = A ) ).

% mult_numeral_1
thf(fact_543_left__right__inverse__power,axiom,
    ! [X4: complex,Y3: complex,N: nat] :
      ( ( ( times_times_complex @ X4 @ Y3 )
        = one_one_complex )
     => ( ( times_times_complex @ ( power_power_complex @ X4 @ N ) @ ( power_power_complex @ Y3 @ N ) )
        = one_one_complex ) ) ).

% left_right_inverse_power
thf(fact_544_left__right__inverse__power,axiom,
    ! [X4: real,Y3: real,N: nat] :
      ( ( ( times_times_real @ X4 @ Y3 )
        = one_one_real )
     => ( ( times_times_real @ ( power_power_real @ X4 @ N ) @ ( power_power_real @ Y3 @ N ) )
        = one_one_real ) ) ).

% left_right_inverse_power
thf(fact_545_left__right__inverse__power,axiom,
    ! [X4: rat,Y3: rat,N: nat] :
      ( ( ( times_times_rat @ X4 @ Y3 )
        = one_one_rat )
     => ( ( times_times_rat @ ( power_power_rat @ X4 @ N ) @ ( power_power_rat @ Y3 @ N ) )
        = one_one_rat ) ) ).

% left_right_inverse_power
thf(fact_546_left__right__inverse__power,axiom,
    ! [X4: nat,Y3: nat,N: nat] :
      ( ( ( times_times_nat @ X4 @ Y3 )
        = one_one_nat )
     => ( ( times_times_nat @ ( power_power_nat @ X4 @ N ) @ ( power_power_nat @ Y3 @ N ) )
        = one_one_nat ) ) ).

% left_right_inverse_power
thf(fact_547_left__right__inverse__power,axiom,
    ! [X4: int,Y3: int,N: nat] :
      ( ( ( times_times_int @ X4 @ Y3 )
        = one_one_int )
     => ( ( times_times_int @ ( power_power_int @ X4 @ N ) @ ( power_power_int @ Y3 @ N ) )
        = one_one_int ) ) ).

% left_right_inverse_power
thf(fact_548_power__add,axiom,
    ! [A: complex,M: nat,N: nat] :
      ( ( power_power_complex @ A @ ( plus_plus_nat @ M @ N ) )
      = ( times_times_complex @ ( power_power_complex @ A @ M ) @ ( power_power_complex @ A @ N ) ) ) ).

% power_add
thf(fact_549_power__add,axiom,
    ! [A: real,M: nat,N: nat] :
      ( ( power_power_real @ A @ ( plus_plus_nat @ M @ N ) )
      = ( times_times_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) ) ) ).

% power_add
thf(fact_550_power__add,axiom,
    ! [A: rat,M: nat,N: nat] :
      ( ( power_power_rat @ A @ ( plus_plus_nat @ M @ N ) )
      = ( times_times_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N ) ) ) ).

% power_add
thf(fact_551_power__add,axiom,
    ! [A: nat,M: nat,N: nat] :
      ( ( power_power_nat @ A @ ( plus_plus_nat @ M @ N ) )
      = ( times_times_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) ) ) ).

% power_add
thf(fact_552_power__add,axiom,
    ! [A: int,M: nat,N: nat] :
      ( ( power_power_int @ A @ ( plus_plus_nat @ M @ N ) )
      = ( times_times_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) ) ) ).

% power_add
thf(fact_553_Suc__div__le__mono,axiom,
    ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ ( divide_divide_nat @ ( suc @ M ) @ N ) ) ).

% Suc_div_le_mono
thf(fact_554_less__mult__imp__div__less,axiom,
    ! [M: nat,I: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( times_times_nat @ I @ N ) )
     => ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ I ) ) ).

% less_mult_imp_div_less
thf(fact_555_times__div__less__eq__dividend,axiom,
    ! [N: nat,M: nat] : ( ord_less_eq_nat @ ( times_times_nat @ N @ ( divide_divide_nat @ M @ N ) ) @ M ) ).

% times_div_less_eq_dividend
thf(fact_556_div__times__less__eq__dividend,axiom,
    ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( times_times_nat @ ( divide_divide_nat @ M @ N ) @ N ) @ M ) ).

% div_times_less_eq_dividend
thf(fact_557_power__gt1__lemma,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ord_less_real @ one_one_real @ ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ) ).

% power_gt1_lemma
thf(fact_558_power__gt1__lemma,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ord_less_rat @ one_one_rat @ ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ) ).

% power_gt1_lemma
thf(fact_559_power__gt1__lemma,axiom,
    ! [A: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ord_less_nat @ one_one_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).

% power_gt1_lemma
thf(fact_560_power__gt1__lemma,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ord_less_int @ one_one_int @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).

% power_gt1_lemma
thf(fact_561_power__less__power__Suc,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ord_less_real @ ( power_power_real @ A @ N ) @ ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ) ).

% power_less_power_Suc
thf(fact_562_power__less__power__Suc,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ) ).

% power_less_power_Suc
thf(fact_563_power__less__power__Suc,axiom,
    ! [A: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).

% power_less_power_Suc
thf(fact_564_power__less__power__Suc,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ord_less_int @ ( power_power_int @ A @ N ) @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).

% power_less_power_Suc
thf(fact_565_power__gt1,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ord_less_real @ one_one_real @ ( power_power_real @ A @ ( suc @ N ) ) ) ) ).

% power_gt1
thf(fact_566_power__gt1,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A @ ( suc @ N ) ) ) ) ).

% power_gt1
thf(fact_567_power__gt1,axiom,
    ! [A: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ ( suc @ N ) ) ) ) ).

% power_gt1
thf(fact_568_power__gt1,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ord_less_int @ one_one_int @ ( power_power_int @ A @ ( suc @ N ) ) ) ) ).

% power_gt1
thf(fact_569_linorder__neqE__linordered__idom,axiom,
    ! [X4: real,Y3: real] :
      ( ( X4 != Y3 )
     => ( ~ ( ord_less_real @ X4 @ Y3 )
       => ( ord_less_real @ Y3 @ X4 ) ) ) ).

% linorder_neqE_linordered_idom
thf(fact_570_linorder__neqE__linordered__idom,axiom,
    ! [X4: rat,Y3: rat] :
      ( ( X4 != Y3 )
     => ( ~ ( ord_less_rat @ X4 @ Y3 )
       => ( ord_less_rat @ Y3 @ X4 ) ) ) ).

% linorder_neqE_linordered_idom
thf(fact_571_linorder__neqE__linordered__idom,axiom,
    ! [X4: int,Y3: int] :
      ( ( X4 != Y3 )
     => ( ~ ( ord_less_int @ X4 @ Y3 )
       => ( ord_less_int @ Y3 @ X4 ) ) ) ).

% linorder_neqE_linordered_idom
thf(fact_572_nat__diff__add__eq2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( minus_minus_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).

% nat_diff_add_eq2
thf(fact_573_nat__diff__add__eq1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ J @ I )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).

% nat_diff_add_eq1
thf(fact_574_nat__le__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( ord_less_eq_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).

% nat_le_add_iff2
thf(fact_575_nat__le__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ J @ I )
     => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).

% nat_le_add_iff1
thf(fact_576_nat__eq__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
          = ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( M
          = ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).

% nat_eq_add_iff2
thf(fact_577_nat__eq__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ J @ I )
     => ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
          = ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M )
          = N ) ) ) ).

% nat_eq_add_iff1
thf(fact_578_nat__neq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( M != N )
      = ( ( ord_less_nat @ M @ N )
        | ( ord_less_nat @ N @ M ) ) ) ).

% nat_neq_iff
thf(fact_579_less__not__refl,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ N ) ).

% less_not_refl
thf(fact_580_less__not__refl2,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ N @ M )
     => ( M != N ) ) ).

% less_not_refl2
thf(fact_581_less__not__refl3,axiom,
    ! [S: nat,T: nat] :
      ( ( ord_less_nat @ S @ T )
     => ( S != T ) ) ).

% less_not_refl3
thf(fact_582_less__irrefl__nat,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ N ) ).

% less_irrefl_nat
thf(fact_583_nat__less__induct,axiom,
    ! [P: nat > $o,N: nat] :
      ( ! [N4: nat] :
          ( ! [M5: nat] :
              ( ( ord_less_nat @ M5 @ N4 )
             => ( P @ M5 ) )
         => ( P @ N4 ) )
     => ( P @ N ) ) ).

% nat_less_induct
thf(fact_584_infinite__descent,axiom,
    ! [P: nat > $o,N: nat] :
      ( ! [N4: nat] :
          ( ~ ( P @ N4 )
         => ? [M5: nat] :
              ( ( ord_less_nat @ M5 @ N4 )
              & ~ ( P @ M5 ) ) )
     => ( P @ N ) ) ).

% infinite_descent
thf(fact_585_linorder__neqE__nat,axiom,
    ! [X4: nat,Y3: nat] :
      ( ( X4 != Y3 )
     => ( ~ ( ord_less_nat @ X4 @ Y3 )
       => ( ord_less_nat @ Y3 @ X4 ) ) ) ).

% linorder_neqE_nat
thf(fact_586_le__refl,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).

% le_refl
thf(fact_587_le__trans,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ord_less_eq_nat @ J @ K )
       => ( ord_less_eq_nat @ I @ K ) ) ) ).

% le_trans
thf(fact_588_eq__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( M = N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% eq_imp_le
thf(fact_589_le__antisym,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( M = N ) ) ) ).

% le_antisym
thf(fact_590_nat__le__linear,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
      | ( ord_less_eq_nat @ N @ M ) ) ).

% nat_le_linear
thf(fact_591_Nat_Oex__has__greatest__nat,axiom,
    ! [P: nat > $o,K: nat,B2: nat] :
      ( ( P @ K )
     => ( ! [Y4: nat] :
            ( ( P @ Y4 )
           => ( ord_less_eq_nat @ Y4 @ B2 ) )
       => ? [X3: nat] :
            ( ( P @ X3 )
            & ! [Y5: nat] :
                ( ( P @ Y5 )
               => ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ) ).

% Nat.ex_has_greatest_nat
thf(fact_592_diff__commute,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
      = ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).

% diff_commute
thf(fact_593_left__add__twice,axiom,
    ! [A: complex,B2: complex] :
      ( ( plus_plus_complex @ A @ ( plus_plus_complex @ A @ B2 ) )
      = ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ A ) @ B2 ) ) ).

% left_add_twice
thf(fact_594_left__add__twice,axiom,
    ! [A: real,B2: real] :
      ( ( plus_plus_real @ A @ ( plus_plus_real @ A @ B2 ) )
      = ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ A ) @ B2 ) ) ).

% left_add_twice
thf(fact_595_left__add__twice,axiom,
    ! [A: rat,B2: rat] :
      ( ( plus_plus_rat @ A @ ( plus_plus_rat @ A @ B2 ) )
      = ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ A ) @ B2 ) ) ).

% left_add_twice
thf(fact_596_left__add__twice,axiom,
    ! [A: nat,B2: nat] :
      ( ( plus_plus_nat @ A @ ( plus_plus_nat @ A @ B2 ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ B2 ) ) ).

% left_add_twice
thf(fact_597_left__add__twice,axiom,
    ! [A: int,B2: int] :
      ( ( plus_plus_int @ A @ ( plus_plus_int @ A @ B2 ) )
      = ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) @ B2 ) ) ).

% left_add_twice
thf(fact_598_mult__2__right,axiom,
    ! [Z: complex] :
      ( ( times_times_complex @ Z @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) )
      = ( plus_plus_complex @ Z @ Z ) ) ).

% mult_2_right
thf(fact_599_mult__2__right,axiom,
    ! [Z: real] :
      ( ( times_times_real @ Z @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
      = ( plus_plus_real @ Z @ Z ) ) ).

% mult_2_right
thf(fact_600_mult__2__right,axiom,
    ! [Z: rat] :
      ( ( times_times_rat @ Z @ ( numeral_numeral_rat @ ( bit0 @ one ) ) )
      = ( plus_plus_rat @ Z @ Z ) ) ).

% mult_2_right
thf(fact_601_mult__2__right,axiom,
    ! [Z: nat] :
      ( ( times_times_nat @ Z @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_nat @ Z @ Z ) ) ).

% mult_2_right
thf(fact_602_mult__2__right,axiom,
    ! [Z: int] :
      ( ( times_times_int @ Z @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
      = ( plus_plus_int @ Z @ Z ) ) ).

% mult_2_right
thf(fact_603_mult__2,axiom,
    ! [Z: complex] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ Z )
      = ( plus_plus_complex @ Z @ Z ) ) ).

% mult_2
thf(fact_604_mult__2,axiom,
    ! [Z: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z )
      = ( plus_plus_real @ Z @ Z ) ) ).

% mult_2
thf(fact_605_mult__2,axiom,
    ! [Z: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ Z )
      = ( plus_plus_rat @ Z @ Z ) ) ).

% mult_2
thf(fact_606_mult__2,axiom,
    ! [Z: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Z )
      = ( plus_plus_nat @ Z @ Z ) ) ).

% mult_2
thf(fact_607_mult__2,axiom,
    ! [Z: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Z )
      = ( plus_plus_int @ Z @ Z ) ) ).

% mult_2
thf(fact_608_power2__eq__square,axiom,
    ! [A: complex] :
      ( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( times_times_complex @ A @ A ) ) ).

% power2_eq_square
thf(fact_609_power2__eq__square,axiom,
    ! [A: real] :
      ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( times_times_real @ A @ A ) ) ).

% power2_eq_square
thf(fact_610_power2__eq__square,axiom,
    ! [A: rat] :
      ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( times_times_rat @ A @ A ) ) ).

% power2_eq_square
thf(fact_611_power2__eq__square,axiom,
    ! [A: nat] :
      ( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( times_times_nat @ A @ A ) ) ).

% power2_eq_square
thf(fact_612_power2__eq__square,axiom,
    ! [A: int] :
      ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( times_times_int @ A @ A ) ) ).

% power2_eq_square
thf(fact_613_power4__eq__xxxx,axiom,
    ! [X4: complex] :
      ( ( power_power_complex @ X4 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
      = ( times_times_complex @ ( times_times_complex @ ( times_times_complex @ X4 @ X4 ) @ X4 ) @ X4 ) ) ).

% power4_eq_xxxx
thf(fact_614_power4__eq__xxxx,axiom,
    ! [X4: real] :
      ( ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
      = ( times_times_real @ ( times_times_real @ ( times_times_real @ X4 @ X4 ) @ X4 ) @ X4 ) ) ).

% power4_eq_xxxx
thf(fact_615_power4__eq__xxxx,axiom,
    ! [X4: rat] :
      ( ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
      = ( times_times_rat @ ( times_times_rat @ ( times_times_rat @ X4 @ X4 ) @ X4 ) @ X4 ) ) ).

% power4_eq_xxxx
thf(fact_616_power4__eq__xxxx,axiom,
    ! [X4: nat] :
      ( ( power_power_nat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
      = ( times_times_nat @ ( times_times_nat @ ( times_times_nat @ X4 @ X4 ) @ X4 ) @ X4 ) ) ).

% power4_eq_xxxx
thf(fact_617_power4__eq__xxxx,axiom,
    ! [X4: int] :
      ( ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
      = ( times_times_int @ ( times_times_int @ ( times_times_int @ X4 @ X4 ) @ X4 ) @ X4 ) ) ).

% power4_eq_xxxx
thf(fact_618_power__even__eq,axiom,
    ! [A: nat,N: nat] :
      ( ( power_power_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( power_power_nat @ ( power_power_nat @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power_even_eq
thf(fact_619_power__even__eq,axiom,
    ! [A: real,N: nat] :
      ( ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( power_power_real @ ( power_power_real @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power_even_eq
thf(fact_620_power__even__eq,axiom,
    ! [A: int,N: nat] :
      ( ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( power_power_int @ ( power_power_int @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power_even_eq
thf(fact_621_power__even__eq,axiom,
    ! [A: complex,N: nat] :
      ( ( power_power_complex @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( power_power_complex @ ( power_power_complex @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power_even_eq
thf(fact_622_nat__less__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ J @ I )
     => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).

% nat_less_add_iff1
thf(fact_623_nat__less__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( ord_less_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).

% nat_less_add_iff2
thf(fact_624_power2__sum,axiom,
    ! [X4: complex,Y3: complex] :
      ( ( power_power_complex @ ( plus_plus_complex @ X4 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_complex @ ( plus_plus_complex @ ( power_power_complex @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X4 ) @ Y3 ) ) ) ).

% power2_sum
thf(fact_625_power2__sum,axiom,
    ! [X4: real,Y3: real] :
      ( ( power_power_real @ ( plus_plus_real @ X4 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_real @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X4 ) @ Y3 ) ) ) ).

% power2_sum
thf(fact_626_power2__sum,axiom,
    ! [X4: rat,Y3: rat] :
      ( ( power_power_rat @ ( plus_plus_rat @ X4 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_rat @ ( plus_plus_rat @ ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X4 ) @ Y3 ) ) ) ).

% power2_sum
thf(fact_627_power2__sum,axiom,
    ! [X4: nat,Y3: nat] :
      ( ( power_power_nat @ ( plus_plus_nat @ X4 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_nat @ ( plus_plus_nat @ ( power_power_nat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X4 ) @ Y3 ) ) ) ).

% power2_sum
thf(fact_628_power2__sum,axiom,
    ! [X4: int,Y3: int] :
      ( ( power_power_int @ ( plus_plus_int @ X4 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_int @ ( plus_plus_int @ ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X4 ) @ Y3 ) ) ) ).

% power2_sum
thf(fact_629_add__diff__add,axiom,
    ! [A: real,C: real,B2: real,D: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ D ) )
      = ( plus_plus_real @ ( minus_minus_real @ A @ B2 ) @ ( minus_minus_real @ C @ D ) ) ) ).

% add_diff_add
thf(fact_630_add__diff__add,axiom,
    ! [A: rat,C: rat,B2: rat,D: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B2 @ D ) )
      = ( plus_plus_rat @ ( minus_minus_rat @ A @ B2 ) @ ( minus_minus_rat @ C @ D ) ) ) ).

% add_diff_add
thf(fact_631_add__diff__add,axiom,
    ! [A: int,C: int,B2: int,D: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ D ) )
      = ( plus_plus_int @ ( minus_minus_int @ A @ B2 ) @ ( minus_minus_int @ C @ D ) ) ) ).

% add_diff_add
thf(fact_632_nat__less__le,axiom,
    ( ord_less_nat
    = ( ^ [M6: nat,N2: nat] :
          ( ( ord_less_eq_nat @ M6 @ N2 )
          & ( M6 != N2 ) ) ) ) ).

% nat_less_le
thf(fact_633_less__imp__le__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% less_imp_le_nat
thf(fact_634_le__eq__less__or__eq,axiom,
    ( ord_less_eq_nat
    = ( ^ [M6: nat,N2: nat] :
          ( ( ord_less_nat @ M6 @ N2 )
          | ( M6 = N2 ) ) ) ) ).

% le_eq_less_or_eq
thf(fact_635_less__or__eq__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( ( ord_less_nat @ M @ N )
        | ( M = N ) )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% less_or_eq_imp_le
thf(fact_636_le__neq__implies__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( M != N )
       => ( ord_less_nat @ M @ N ) ) ) ).

% le_neq_implies_less
thf(fact_637_less__mono__imp__le__mono,axiom,
    ! [F: nat > nat,I: nat,J: nat] :
      ( ! [I3: nat,J2: nat] :
          ( ( ord_less_nat @ I3 @ J2 )
         => ( ord_less_nat @ ( F @ I3 ) @ ( F @ J2 ) ) )
     => ( ( ord_less_eq_nat @ I @ J )
       => ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).

% less_mono_imp_le_mono
thf(fact_638_add__lessD1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
     => ( ord_less_nat @ I @ K ) ) ).

% add_lessD1
thf(fact_639_add__less__mono,axiom,
    ! [I: nat,J: nat,K: nat,L2: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( ord_less_nat @ K @ L2 )
       => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ) ).

% add_less_mono
thf(fact_640_not__add__less1,axiom,
    ! [I: nat,J: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).

% not_add_less1
thf(fact_641_not__add__less2,axiom,
    ! [J: nat,I: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).

% not_add_less2
thf(fact_642_add__less__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).

% add_less_mono1
thf(fact_643_trans__less__add1,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).

% trans_less_add1
thf(fact_644_trans__less__add2,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).

% trans_less_add2
thf(fact_645_less__add__eq__less,axiom,
    ! [K: nat,L2: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ K @ L2 )
     => ( ( ( plus_plus_nat @ M @ L2 )
          = ( plus_plus_nat @ K @ N ) )
       => ( ord_less_nat @ M @ N ) ) ) ).

% less_add_eq_less
thf(fact_646_add__leE,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
     => ~ ( ( ord_less_eq_nat @ M @ N )
         => ~ ( ord_less_eq_nat @ K @ N ) ) ) ).

% add_leE
thf(fact_647_le__add1,axiom,
    ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).

% le_add1
thf(fact_648_le__add2,axiom,
    ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).

% le_add2
thf(fact_649_add__leD1,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% add_leD1
thf(fact_650_add__leD2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
     => ( ord_less_eq_nat @ K @ N ) ) ).

% add_leD2
thf(fact_651_le__Suc__ex,axiom,
    ! [K: nat,L2: nat] :
      ( ( ord_less_eq_nat @ K @ L2 )
     => ? [N4: nat] :
          ( L2
          = ( plus_plus_nat @ K @ N4 ) ) ) ).

% le_Suc_ex
thf(fact_652_add__le__mono,axiom,
    ! [I: nat,J: nat,K: nat,L2: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ord_less_eq_nat @ K @ L2 )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ) ).

% add_le_mono
thf(fact_653_add__le__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).

% add_le_mono1
thf(fact_654_trans__le__add1,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).

% trans_le_add1
thf(fact_655_trans__le__add2,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).

% trans_le_add2
thf(fact_656_nat__le__iff__add,axiom,
    ( ord_less_eq_nat
    = ( ^ [M6: nat,N2: nat] :
        ? [K3: nat] :
          ( N2
          = ( plus_plus_nat @ M6 @ K3 ) ) ) ) ).

% nat_le_iff_add
thf(fact_657_diff__less__mono2,axiom,
    ! [M: nat,N: nat,L2: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ( ord_less_nat @ M @ L2 )
       => ( ord_less_nat @ ( minus_minus_nat @ L2 @ N ) @ ( minus_minus_nat @ L2 @ M ) ) ) ) ).

% diff_less_mono2
thf(fact_658_less__imp__diff__less,axiom,
    ! [J: nat,K: nat,N: nat] :
      ( ( ord_less_nat @ J @ K )
     => ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).

% less_imp_diff_less
thf(fact_659_eq__diff__iff,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( ( ( minus_minus_nat @ M @ K )
            = ( minus_minus_nat @ N @ K ) )
          = ( M = N ) ) ) ) ).

% eq_diff_iff
thf(fact_660_le__diff__iff,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
          = ( ord_less_eq_nat @ M @ N ) ) ) ) ).

% le_diff_iff
thf(fact_661_Nat_Odiff__diff__eq,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
          = ( minus_minus_nat @ M @ N ) ) ) ) ).

% Nat.diff_diff_eq
thf(fact_662_diff__le__mono,axiom,
    ! [M: nat,N: nat,L2: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L2 ) @ ( minus_minus_nat @ N @ L2 ) ) ) ).

% diff_le_mono
thf(fact_663_diff__le__self,axiom,
    ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).

% diff_le_self
thf(fact_664_le__diff__iff_H,axiom,
    ! [A: nat,C: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A @ C )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B2 ) )
          = ( ord_less_eq_nat @ B2 @ A ) ) ) ) ).

% le_diff_iff'
thf(fact_665_diff__le__mono2,axiom,
    ! [M: nat,N: nat,L2: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ L2 @ N ) @ ( minus_minus_nat @ L2 @ M ) ) ) ).

% diff_le_mono2
thf(fact_666_power2__diff,axiom,
    ! [X4: complex,Y3: complex] :
      ( ( power_power_complex @ ( minus_minus_complex @ X4 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_complex @ ( plus_plus_complex @ ( power_power_complex @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X4 ) @ Y3 ) ) ) ).

% power2_diff
thf(fact_667_power2__diff,axiom,
    ! [X4: real,Y3: real] :
      ( ( power_power_real @ ( minus_minus_real @ X4 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_real @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X4 ) @ Y3 ) ) ) ).

% power2_diff
thf(fact_668_power2__diff,axiom,
    ! [X4: rat,Y3: rat] :
      ( ( power_power_rat @ ( minus_minus_rat @ X4 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_rat @ ( plus_plus_rat @ ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X4 ) @ Y3 ) ) ) ).

% power2_diff
thf(fact_669_power2__diff,axiom,
    ! [X4: int,Y3: int] :
      ( ( power_power_int @ ( minus_minus_int @ X4 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_int @ ( plus_plus_int @ ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X4 ) @ Y3 ) ) ) ).

% power2_diff
thf(fact_670_Nat_Odiff__cancel,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
      = ( minus_minus_nat @ M @ N ) ) ).

% Nat.diff_cancel
thf(fact_671_diff__cancel2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
      = ( minus_minus_nat @ M @ N ) ) ).

% diff_cancel2
thf(fact_672_diff__add__inverse,axiom,
    ! [N: nat,M: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
      = M ) ).

% diff_add_inverse
thf(fact_673_diff__add__inverse2,axiom,
    ! [M: nat,N: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
      = M ) ).

% diff_add_inverse2
thf(fact_674_add__le__add__imp__diff__le,axiom,
    ! [I: real,K: real,N: real,J: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N )
     => ( ( ord_less_eq_real @ N @ ( plus_plus_real @ J @ K ) )
       => ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N )
         => ( ( ord_less_eq_real @ N @ ( plus_plus_real @ J @ K ) )
           => ( ord_less_eq_real @ ( minus_minus_real @ N @ K ) @ J ) ) ) ) ) ).

% add_le_add_imp_diff_le
thf(fact_675_add__le__add__imp__diff__le,axiom,
    ! [I: rat,K: rat,N: rat,J: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N )
     => ( ( ord_less_eq_rat @ N @ ( plus_plus_rat @ J @ K ) )
       => ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N )
         => ( ( ord_less_eq_rat @ N @ ( plus_plus_rat @ J @ K ) )
           => ( ord_less_eq_rat @ ( minus_minus_rat @ N @ K ) @ J ) ) ) ) ) ).

% add_le_add_imp_diff_le
thf(fact_676_add__le__add__imp__diff__le,axiom,
    ! [I: nat,K: nat,N: nat,J: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
     => ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
       => ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
         => ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
           => ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K ) @ J ) ) ) ) ) ).

% add_le_add_imp_diff_le
thf(fact_677_add__le__add__imp__diff__le,axiom,
    ! [I: int,K: int,N: int,J: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
     => ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
       => ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
         => ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
           => ( ord_less_eq_int @ ( minus_minus_int @ N @ K ) @ J ) ) ) ) ) ).

% add_le_add_imp_diff_le
thf(fact_678_add__le__imp__le__diff,axiom,
    ! [I: real,K: real,N: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N )
     => ( ord_less_eq_real @ I @ ( minus_minus_real @ N @ K ) ) ) ).

% add_le_imp_le_diff
thf(fact_679_add__le__imp__le__diff,axiom,
    ! [I: rat,K: rat,N: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N )
     => ( ord_less_eq_rat @ I @ ( minus_minus_rat @ N @ K ) ) ) ).

% add_le_imp_le_diff
thf(fact_680_add__le__imp__le__diff,axiom,
    ! [I: nat,K: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
     => ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N @ K ) ) ) ).

% add_le_imp_le_diff
thf(fact_681_add__le__imp__le__diff,axiom,
    ! [I: int,K: int,N: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
     => ( ord_less_eq_int @ I @ ( minus_minus_int @ N @ K ) ) ) ).

% add_le_imp_le_diff
thf(fact_682_add__mono1,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ord_less_real @ ( plus_plus_real @ A @ one_one_real ) @ ( plus_plus_real @ B2 @ one_one_real ) ) ) ).

% add_mono1
thf(fact_683_add__mono1,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_rat @ A @ B2 )
     => ( ord_less_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( plus_plus_rat @ B2 @ one_one_rat ) ) ) ).

% add_mono1
thf(fact_684_add__mono1,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( ord_less_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( plus_plus_nat @ B2 @ one_one_nat ) ) ) ).

% add_mono1
thf(fact_685_add__mono1,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_int @ A @ B2 )
     => ( ord_less_int @ ( plus_plus_int @ A @ one_one_int ) @ ( plus_plus_int @ B2 @ one_one_int ) ) ) ).

% add_mono1
thf(fact_686_less__add__one,axiom,
    ! [A: real] : ( ord_less_real @ A @ ( plus_plus_real @ A @ one_one_real ) ) ).

% less_add_one
thf(fact_687_less__add__one,axiom,
    ! [A: rat] : ( ord_less_rat @ A @ ( plus_plus_rat @ A @ one_one_rat ) ) ).

% less_add_one
thf(fact_688_less__add__one,axiom,
    ! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).

% less_add_one
thf(fact_689_less__add__one,axiom,
    ! [A: int] : ( ord_less_int @ A @ ( plus_plus_int @ A @ one_one_int ) ) ).

% less_add_one
thf(fact_690_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: real,B2: real] :
      ( ~ ( ord_less_real @ A @ B2 )
     => ( ( plus_plus_real @ B2 @ ( minus_minus_real @ A @ B2 ) )
        = A ) ) ).

% linordered_semidom_class.add_diff_inverse
thf(fact_691_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: rat,B2: rat] :
      ( ~ ( ord_less_rat @ A @ B2 )
     => ( ( plus_plus_rat @ B2 @ ( minus_minus_rat @ A @ B2 ) )
        = A ) ) ).

% linordered_semidom_class.add_diff_inverse
thf(fact_692_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: nat,B2: nat] :
      ( ~ ( ord_less_nat @ A @ B2 )
     => ( ( plus_plus_nat @ B2 @ ( minus_minus_nat @ A @ B2 ) )
        = A ) ) ).

% linordered_semidom_class.add_diff_inverse
thf(fact_693_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: int,B2: int] :
      ( ~ ( ord_less_int @ A @ B2 )
     => ( ( plus_plus_int @ B2 @ ( minus_minus_int @ A @ B2 ) )
        = A ) ) ).

% linordered_semidom_class.add_diff_inverse
thf(fact_694_mono__nat__linear__lb,axiom,
    ! [F: nat > nat,M: nat,K: nat] :
      ( ! [M4: nat,N4: nat] :
          ( ( ord_less_nat @ M4 @ N4 )
         => ( ord_less_nat @ ( F @ M4 ) @ ( F @ N4 ) ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).

% mono_nat_linear_lb
thf(fact_695_less__diff__iff,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
          = ( ord_less_nat @ M @ N ) ) ) ) ).

% less_diff_iff
thf(fact_696_diff__less__mono,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( ( ord_less_eq_nat @ C @ A )
       => ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B2 @ C ) ) ) ) ).

% diff_less_mono
thf(fact_697_add__diff__inverse__nat,axiom,
    ! [M: nat,N: nat] :
      ( ~ ( ord_less_nat @ M @ N )
     => ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
        = M ) ) ).

% add_diff_inverse_nat
thf(fact_698_less__diff__conv,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K ) )
      = ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ).

% less_diff_conv
thf(fact_699_Nat_Ole__imp__diff__is__add,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ( minus_minus_nat @ J @ I )
          = K )
        = ( J
          = ( plus_plus_nat @ K @ I ) ) ) ) ).

% Nat.le_imp_diff_is_add
thf(fact_700_Nat_Odiff__add__assoc2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
        = ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).

% Nat.diff_add_assoc2
thf(fact_701_Nat_Odiff__add__assoc,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
        = ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).

% Nat.diff_add_assoc
thf(fact_702_Nat_Ole__diff__conv2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
        = ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).

% Nat.le_diff_conv2
thf(fact_703_le__diff__conv,axiom,
    ! [J: nat,K: nat,I: nat] :
      ( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
      = ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).

% le_diff_conv
thf(fact_704_less__diff__conv2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I )
        = ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ) ).

% less_diff_conv2
thf(fact_705_field__sum__of__halves,axiom,
    ! [X4: real] :
      ( ( plus_plus_real @ ( divide_divide_real @ X4 @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( divide_divide_real @ X4 @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
      = X4 ) ).

% field_sum_of_halves
thf(fact_706_field__sum__of__halves,axiom,
    ! [X4: rat] :
      ( ( plus_plus_rat @ ( divide_divide_rat @ X4 @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( divide_divide_rat @ X4 @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
      = X4 ) ).

% field_sum_of_halves
thf(fact_707__092_060open_062res_A_061_A2_A_094_A_Ideg_Adiv_A2_J_A_K_Asc_A_L_Aminy_092_060close_062,axiom,
    ( res
    = ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ sc ) @ miny ) ) ).

% \<open>res = 2 ^ (deg div 2) * sc + miny\<close>
thf(fact_708_sum__squares__bound,axiom,
    ! [X4: real,Y3: real] : ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X4 ) @ Y3 ) @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sum_squares_bound
thf(fact_709_sum__squares__bound,axiom,
    ! [X4: rat,Y3: rat] : ( ord_less_eq_rat @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X4 ) @ Y3 ) @ ( plus_plus_rat @ ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sum_squares_bound
thf(fact_710_enat__ord__number_I1_J,axiom,
    ! [M: num,N: num] :
      ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
      = ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) ) ) ).

% enat_ord_number(1)
thf(fact_711_enat__ord__number_I2_J,axiom,
    ! [M: num,N: num] :
      ( ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
      = ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) ) ) ).

% enat_ord_number(2)
thf(fact_712_double__not__eq__Suc__double,axiom,
    ! [M: nat,N: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
     != ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% double_not_eq_Suc_double
thf(fact_713_Suc__double__not__eq__double,axiom,
    ! [M: nat,N: nat] :
      ( ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
     != ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).

% Suc_double_not_eq_double
thf(fact_714_add__diff__cancel,axiom,
    ! [A: real,B2: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ A @ B2 ) @ B2 )
      = A ) ).

% add_diff_cancel
thf(fact_715_add__diff__cancel,axiom,
    ! [A: rat,B2: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B2 ) @ B2 )
      = A ) ).

% add_diff_cancel
thf(fact_716_add__diff__cancel,axiom,
    ! [A: int,B2: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A @ B2 ) @ B2 )
      = A ) ).

% add_diff_cancel
thf(fact_717_diff__add__cancel,axiom,
    ! [A: real,B2: real] :
      ( ( plus_plus_real @ ( minus_minus_real @ A @ B2 ) @ B2 )
      = A ) ).

% diff_add_cancel
thf(fact_718_diff__add__cancel,axiom,
    ! [A: rat,B2: rat] :
      ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B2 ) @ B2 )
      = A ) ).

% diff_add_cancel
thf(fact_719_diff__add__cancel,axiom,
    ! [A: int,B2: int] :
      ( ( plus_plus_int @ ( minus_minus_int @ A @ B2 ) @ B2 )
      = A ) ).

% diff_add_cancel
thf(fact_720_add__diff__cancel__left,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B2 ) )
      = ( minus_minus_real @ A @ B2 ) ) ).

% add_diff_cancel_left
thf(fact_721_add__diff__cancel__left,axiom,
    ! [C: rat,A: rat,B2: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B2 ) )
      = ( minus_minus_rat @ A @ B2 ) ) ).

% add_diff_cancel_left
thf(fact_722_add__diff__cancel__left,axiom,
    ! [C: nat,A: nat,B2: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) )
      = ( minus_minus_nat @ A @ B2 ) ) ).

% add_diff_cancel_left
thf(fact_723_add__diff__cancel__left,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B2 ) )
      = ( minus_minus_int @ A @ B2 ) ) ).

% add_diff_cancel_left
thf(fact_724_add__diff__cancel__left_H,axiom,
    ! [A: real,B2: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ A @ B2 ) @ A )
      = B2 ) ).

% add_diff_cancel_left'
thf(fact_725_add__diff__cancel__left_H,axiom,
    ! [A: rat,B2: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B2 ) @ A )
      = B2 ) ).

% add_diff_cancel_left'
thf(fact_726_add__diff__cancel__left_H,axiom,
    ! [A: nat,B2: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B2 ) @ A )
      = B2 ) ).

% add_diff_cancel_left'
thf(fact_727_add__diff__cancel__left_H,axiom,
    ! [A: int,B2: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A @ B2 ) @ A )
      = B2 ) ).

% add_diff_cancel_left'
thf(fact_728_add__diff__cancel__right,axiom,
    ! [A: real,C: real,B2: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ C ) )
      = ( minus_minus_real @ A @ B2 ) ) ).

% add_diff_cancel_right
thf(fact_729_add__diff__cancel__right,axiom,
    ! [A: rat,C: rat,B2: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B2 @ C ) )
      = ( minus_minus_rat @ A @ B2 ) ) ).

% add_diff_cancel_right
thf(fact_730_add__diff__cancel__right,axiom,
    ! [A: nat,C: nat,B2: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) )
      = ( minus_minus_nat @ A @ B2 ) ) ).

% add_diff_cancel_right
thf(fact_731_add__diff__cancel__right,axiom,
    ! [A: int,C: int,B2: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ C ) )
      = ( minus_minus_int @ A @ B2 ) ) ).

% add_diff_cancel_right
thf(fact_732_add__diff__cancel__right_H,axiom,
    ! [A: real,B2: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ A @ B2 ) @ B2 )
      = A ) ).

% add_diff_cancel_right'
thf(fact_733_add__diff__cancel__right_H,axiom,
    ! [A: rat,B2: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B2 ) @ B2 )
      = A ) ).

% add_diff_cancel_right'
thf(fact_734_add__diff__cancel__right_H,axiom,
    ! [A: nat,B2: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B2 ) @ B2 )
      = A ) ).

% add_diff_cancel_right'
thf(fact_735_add__diff__cancel__right_H,axiom,
    ! [A: int,B2: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A @ B2 ) @ B2 )
      = A ) ).

% add_diff_cancel_right'
thf(fact_736_add__right__cancel,axiom,
    ! [B2: real,A: real,C: real] :
      ( ( ( plus_plus_real @ B2 @ A )
        = ( plus_plus_real @ C @ A ) )
      = ( B2 = C ) ) ).

% add_right_cancel
thf(fact_737_add__right__cancel,axiom,
    ! [B2: rat,A: rat,C: rat] :
      ( ( ( plus_plus_rat @ B2 @ A )
        = ( plus_plus_rat @ C @ A ) )
      = ( B2 = C ) ) ).

% add_right_cancel
thf(fact_738_add__right__cancel,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( ( ( plus_plus_nat @ B2 @ A )
        = ( plus_plus_nat @ C @ A ) )
      = ( B2 = C ) ) ).

% add_right_cancel
thf(fact_739_add__right__cancel,axiom,
    ! [B2: int,A: int,C: int] :
      ( ( ( plus_plus_int @ B2 @ A )
        = ( plus_plus_int @ C @ A ) )
      = ( B2 = C ) ) ).

% add_right_cancel
thf(fact_740_add__left__cancel,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ( plus_plus_real @ A @ B2 )
        = ( plus_plus_real @ A @ C ) )
      = ( B2 = C ) ) ).

% add_left_cancel
thf(fact_741_add__left__cancel,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( ( plus_plus_rat @ A @ B2 )
        = ( plus_plus_rat @ A @ C ) )
      = ( B2 = C ) ) ).

% add_left_cancel
thf(fact_742_add__left__cancel,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ( plus_plus_nat @ A @ B2 )
        = ( plus_plus_nat @ A @ C ) )
      = ( B2 = C ) ) ).

% add_left_cancel
thf(fact_743_add__left__cancel,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( ( plus_plus_int @ A @ B2 )
        = ( plus_plus_int @ A @ C ) )
      = ( B2 = C ) ) ).

% add_left_cancel
thf(fact_744_add__le__cancel__right,axiom,
    ! [A: real,C: real,B2: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ C ) )
      = ( ord_less_eq_real @ A @ B2 ) ) ).

% add_le_cancel_right
thf(fact_745_add__le__cancel__right,axiom,
    ! [A: rat,C: rat,B2: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B2 @ C ) )
      = ( ord_less_eq_rat @ A @ B2 ) ) ).

% add_le_cancel_right
thf(fact_746_add__le__cancel__right,axiom,
    ! [A: nat,C: nat,B2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) )
      = ( ord_less_eq_nat @ A @ B2 ) ) ).

% add_le_cancel_right
thf(fact_747_add__le__cancel__right,axiom,
    ! [A: int,C: int,B2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ C ) )
      = ( ord_less_eq_int @ A @ B2 ) ) ).

% add_le_cancel_right
thf(fact_748_add__le__cancel__left,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B2 ) )
      = ( ord_less_eq_real @ A @ B2 ) ) ).

% add_le_cancel_left
thf(fact_749_add__le__cancel__left,axiom,
    ! [C: rat,A: rat,B2: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B2 ) )
      = ( ord_less_eq_rat @ A @ B2 ) ) ).

% add_le_cancel_left
thf(fact_750_add__le__cancel__left,axiom,
    ! [C: nat,A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) )
      = ( ord_less_eq_nat @ A @ B2 ) ) ).

% add_le_cancel_left
thf(fact_751_add__le__cancel__left,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B2 ) )
      = ( ord_less_eq_int @ A @ B2 ) ) ).

% add_le_cancel_left
thf(fact_752_add__less__cancel__right,axiom,
    ! [A: real,C: real,B2: real] :
      ( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ C ) )
      = ( ord_less_real @ A @ B2 ) ) ).

% add_less_cancel_right
thf(fact_753_add__less__cancel__right,axiom,
    ! [A: rat,C: rat,B2: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B2 @ C ) )
      = ( ord_less_rat @ A @ B2 ) ) ).

% add_less_cancel_right
thf(fact_754_add__less__cancel__right,axiom,
    ! [A: nat,C: nat,B2: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) )
      = ( ord_less_nat @ A @ B2 ) ) ).

% add_less_cancel_right
thf(fact_755_add__less__cancel__right,axiom,
    ! [A: int,C: int,B2: int] :
      ( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ C ) )
      = ( ord_less_int @ A @ B2 ) ) ).

% add_less_cancel_right
thf(fact_756_add__less__cancel__left,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B2 ) )
      = ( ord_less_real @ A @ B2 ) ) ).

% add_less_cancel_left
thf(fact_757_add__less__cancel__left,axiom,
    ! [C: rat,A: rat,B2: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B2 ) )
      = ( ord_less_rat @ A @ B2 ) ) ).

% add_less_cancel_left
thf(fact_758_add__less__cancel__left,axiom,
    ! [C: nat,A: nat,B2: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) )
      = ( ord_less_nat @ A @ B2 ) ) ).

% add_less_cancel_left
thf(fact_759_add__less__cancel__left,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B2 ) )
      = ( ord_less_int @ A @ B2 ) ) ).

% add_less_cancel_left
thf(fact_760_mult__1,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ one_one_complex @ A )
      = A ) ).

% mult_1
thf(fact_761_mult__1,axiom,
    ! [A: real] :
      ( ( times_times_real @ one_one_real @ A )
      = A ) ).

% mult_1
thf(fact_762_mult__1,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ one_one_rat @ A )
      = A ) ).

% mult_1
thf(fact_763_mult__1,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ one_one_nat @ A )
      = A ) ).

% mult_1
thf(fact_764_mult__1,axiom,
    ! [A: int] :
      ( ( times_times_int @ one_one_int @ A )
      = A ) ).

% mult_1
thf(fact_765_mult_Oright__neutral,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ A @ one_one_complex )
      = A ) ).

% mult.right_neutral
thf(fact_766_mult_Oright__neutral,axiom,
    ! [A: real] :
      ( ( times_times_real @ A @ one_one_real )
      = A ) ).

% mult.right_neutral
thf(fact_767_mult_Oright__neutral,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ A @ one_one_rat )
      = A ) ).

% mult.right_neutral
thf(fact_768_mult_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ one_one_nat )
      = A ) ).

% mult.right_neutral
thf(fact_769_mult_Oright__neutral,axiom,
    ! [A: int] :
      ( ( times_times_int @ A @ one_one_int )
      = A ) ).

% mult.right_neutral
thf(fact_770_semiring__norm_I13_J,axiom,
    ! [M: num,N: num] :
      ( ( times_times_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
      = ( bit0 @ ( bit0 @ ( times_times_num @ M @ N ) ) ) ) ).

% semiring_norm(13)
thf(fact_771_semiring__norm_I12_J,axiom,
    ! [N: num] :
      ( ( times_times_num @ one @ N )
      = N ) ).

% semiring_norm(12)
thf(fact_772_semiring__norm_I11_J,axiom,
    ! [M: num] :
      ( ( times_times_num @ M @ one )
      = M ) ).

% semiring_norm(11)
thf(fact_773_num__double,axiom,
    ! [N: num] :
      ( ( times_times_num @ ( bit0 @ one ) @ N )
      = ( bit0 @ N ) ) ).

% num_double
thf(fact_774_power__mult__numeral,axiom,
    ! [A: nat,M: num,N: num] :
      ( ( power_power_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
      = ( power_power_nat @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).

% power_mult_numeral
thf(fact_775_power__mult__numeral,axiom,
    ! [A: real,M: num,N: num] :
      ( ( power_power_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
      = ( power_power_real @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).

% power_mult_numeral
thf(fact_776_power__mult__numeral,axiom,
    ! [A: int,M: num,N: num] :
      ( ( power_power_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
      = ( power_power_int @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).

% power_mult_numeral
thf(fact_777_power__mult__numeral,axiom,
    ! [A: complex,M: num,N: num] :
      ( ( power_power_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
      = ( power_power_complex @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).

% power_mult_numeral
thf(fact_778_complete__real,axiom,
    ! [S2: set_real] :
      ( ? [X5: real] : ( member_real @ X5 @ S2 )
     => ( ? [Z3: real] :
          ! [X3: real] :
            ( ( member_real @ X3 @ S2 )
           => ( ord_less_eq_real @ X3 @ Z3 ) )
       => ? [Y4: real] :
            ( ! [X5: real] :
                ( ( member_real @ X5 @ S2 )
               => ( ord_less_eq_real @ X5 @ Y4 ) )
            & ! [Z3: real] :
                ( ! [X3: real] :
                    ( ( member_real @ X3 @ S2 )
                   => ( ord_less_eq_real @ X3 @ Z3 ) )
               => ( ord_less_eq_real @ Y4 @ Z3 ) ) ) ) ) ).

% complete_real
thf(fact_779_real__arch__pow,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_real @ one_one_real @ X4 )
     => ? [N4: nat] : ( ord_less_real @ Y3 @ ( power_power_real @ X4 @ N4 ) ) ) ).

% real_arch_pow
thf(fact_780_less__eq__real__def,axiom,
    ( ord_less_eq_real
    = ( ^ [X: real,Y: real] :
          ( ( ord_less_real @ X @ Y )
          | ( X = Y ) ) ) ) ).

% less_eq_real_def
thf(fact_781_add__diff__assoc__enat,axiom,
    ! [Z: extended_enat,Y3: extended_enat,X4: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ Z @ Y3 )
     => ( ( plus_p3455044024723400733d_enat @ X4 @ ( minus_3235023915231533773d_enat @ Y3 @ Z ) )
        = ( minus_3235023915231533773d_enat @ ( plus_p3455044024723400733d_enat @ X4 @ Y3 ) @ Z ) ) ) ).

% add_diff_assoc_enat
thf(fact_782_enat__less__induct,axiom,
    ! [P: extended_enat > $o,N: extended_enat] :
      ( ! [N4: extended_enat] :
          ( ! [M5: extended_enat] :
              ( ( ord_le72135733267957522d_enat @ M5 @ N4 )
             => ( P @ M5 ) )
         => ( P @ N4 ) )
     => ( P @ N ) ) ).

% enat_less_induct
thf(fact_783_four__x__squared,axiom,
    ! [X4: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( power_power_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% four_x_squared
thf(fact_784_L2__set__mult__ineq__lemma,axiom,
    ! [A: real,C: real,B2: real,D: real] : ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( times_times_real @ A @ C ) ) @ ( times_times_real @ B2 @ D ) ) @ ( plus_plus_real @ ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ D @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( power_power_real @ B2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ C @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% L2_set_mult_ineq_lemma
thf(fact_785_div__mult2__numeral__eq,axiom,
    ! [A: nat,K: num,L2: num] :
      ( ( divide_divide_nat @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ L2 ) )
      = ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( times_times_num @ K @ L2 ) ) ) ) ).

% div_mult2_numeral_eq
thf(fact_786_div__mult2__numeral__eq,axiom,
    ! [A: int,K: num,L2: num] :
      ( ( divide_divide_int @ ( divide_divide_int @ A @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ L2 ) )
      = ( divide_divide_int @ A @ ( numeral_numeral_int @ ( times_times_num @ K @ L2 ) ) ) ) ).

% div_mult2_numeral_eq
thf(fact_787_mult_Oleft__commute,axiom,
    ! [B2: real,A: real,C: real] :
      ( ( times_times_real @ B2 @ ( times_times_real @ A @ C ) )
      = ( times_times_real @ A @ ( times_times_real @ B2 @ C ) ) ) ).

% mult.left_commute
thf(fact_788_mult_Oleft__commute,axiom,
    ! [B2: rat,A: rat,C: rat] :
      ( ( times_times_rat @ B2 @ ( times_times_rat @ A @ C ) )
      = ( times_times_rat @ A @ ( times_times_rat @ B2 @ C ) ) ) ).

% mult.left_commute
thf(fact_789_mult_Oleft__commute,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( ( times_times_nat @ B2 @ ( times_times_nat @ A @ C ) )
      = ( times_times_nat @ A @ ( times_times_nat @ B2 @ C ) ) ) ).

% mult.left_commute
thf(fact_790_mult_Oleft__commute,axiom,
    ! [B2: int,A: int,C: int] :
      ( ( times_times_int @ B2 @ ( times_times_int @ A @ C ) )
      = ( times_times_int @ A @ ( times_times_int @ B2 @ C ) ) ) ).

% mult.left_commute
thf(fact_791_mult_Ocommute,axiom,
    ( times_times_real
    = ( ^ [A4: real,B3: real] : ( times_times_real @ B3 @ A4 ) ) ) ).

% mult.commute
thf(fact_792_mult_Ocommute,axiom,
    ( times_times_rat
    = ( ^ [A4: rat,B3: rat] : ( times_times_rat @ B3 @ A4 ) ) ) ).

% mult.commute
thf(fact_793_mult_Ocommute,axiom,
    ( times_times_nat
    = ( ^ [A4: nat,B3: nat] : ( times_times_nat @ B3 @ A4 ) ) ) ).

% mult.commute
thf(fact_794_mult_Ocommute,axiom,
    ( times_times_int
    = ( ^ [A4: int,B3: int] : ( times_times_int @ B3 @ A4 ) ) ) ).

% mult.commute
thf(fact_795_mult_Oassoc,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( times_times_real @ ( times_times_real @ A @ B2 ) @ C )
      = ( times_times_real @ A @ ( times_times_real @ B2 @ C ) ) ) ).

% mult.assoc
thf(fact_796_mult_Oassoc,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( times_times_rat @ ( times_times_rat @ A @ B2 ) @ C )
      = ( times_times_rat @ A @ ( times_times_rat @ B2 @ C ) ) ) ).

% mult.assoc
thf(fact_797_mult_Oassoc,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( times_times_nat @ ( times_times_nat @ A @ B2 ) @ C )
      = ( times_times_nat @ A @ ( times_times_nat @ B2 @ C ) ) ) ).

% mult.assoc
thf(fact_798_mult_Oassoc,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( times_times_int @ ( times_times_int @ A @ B2 ) @ C )
      = ( times_times_int @ A @ ( times_times_int @ B2 @ C ) ) ) ).

% mult.assoc
thf(fact_799_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( times_times_real @ ( times_times_real @ A @ B2 ) @ C )
      = ( times_times_real @ A @ ( times_times_real @ B2 @ C ) ) ) ).

% ab_semigroup_mult_class.mult_ac(1)
thf(fact_800_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( times_times_rat @ ( times_times_rat @ A @ B2 ) @ C )
      = ( times_times_rat @ A @ ( times_times_rat @ B2 @ C ) ) ) ).

% ab_semigroup_mult_class.mult_ac(1)
thf(fact_801_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( times_times_nat @ ( times_times_nat @ A @ B2 ) @ C )
      = ( times_times_nat @ A @ ( times_times_nat @ B2 @ C ) ) ) ).

% ab_semigroup_mult_class.mult_ac(1)
thf(fact_802_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( times_times_int @ ( times_times_int @ A @ B2 ) @ C )
      = ( times_times_int @ A @ ( times_times_int @ B2 @ C ) ) ) ).

% ab_semigroup_mult_class.mult_ac(1)
thf(fact_803_one__reorient,axiom,
    ! [X4: complex] :
      ( ( one_one_complex = X4 )
      = ( X4 = one_one_complex ) ) ).

% one_reorient
thf(fact_804_one__reorient,axiom,
    ! [X4: real] :
      ( ( one_one_real = X4 )
      = ( X4 = one_one_real ) ) ).

% one_reorient
thf(fact_805_one__reorient,axiom,
    ! [X4: rat] :
      ( ( one_one_rat = X4 )
      = ( X4 = one_one_rat ) ) ).

% one_reorient
thf(fact_806_one__reorient,axiom,
    ! [X4: nat] :
      ( ( one_one_nat = X4 )
      = ( X4 = one_one_nat ) ) ).

% one_reorient
thf(fact_807_one__reorient,axiom,
    ! [X4: int] :
      ( ( one_one_int = X4 )
      = ( X4 = one_one_int ) ) ).

% one_reorient
thf(fact_808_add__right__imp__eq,axiom,
    ! [B2: real,A: real,C: real] :
      ( ( ( plus_plus_real @ B2 @ A )
        = ( plus_plus_real @ C @ A ) )
     => ( B2 = C ) ) ).

% add_right_imp_eq
thf(fact_809_add__right__imp__eq,axiom,
    ! [B2: rat,A: rat,C: rat] :
      ( ( ( plus_plus_rat @ B2 @ A )
        = ( plus_plus_rat @ C @ A ) )
     => ( B2 = C ) ) ).

% add_right_imp_eq
thf(fact_810_add__right__imp__eq,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( ( ( plus_plus_nat @ B2 @ A )
        = ( plus_plus_nat @ C @ A ) )
     => ( B2 = C ) ) ).

% add_right_imp_eq
thf(fact_811_add__right__imp__eq,axiom,
    ! [B2: int,A: int,C: int] :
      ( ( ( plus_plus_int @ B2 @ A )
        = ( plus_plus_int @ C @ A ) )
     => ( B2 = C ) ) ).

% add_right_imp_eq
thf(fact_812_add__left__imp__eq,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ( plus_plus_real @ A @ B2 )
        = ( plus_plus_real @ A @ C ) )
     => ( B2 = C ) ) ).

% add_left_imp_eq
thf(fact_813_add__left__imp__eq,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( ( plus_plus_rat @ A @ B2 )
        = ( plus_plus_rat @ A @ C ) )
     => ( B2 = C ) ) ).

% add_left_imp_eq
thf(fact_814_add__left__imp__eq,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ( plus_plus_nat @ A @ B2 )
        = ( plus_plus_nat @ A @ C ) )
     => ( B2 = C ) ) ).

% add_left_imp_eq
thf(fact_815_add__left__imp__eq,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( ( plus_plus_int @ A @ B2 )
        = ( plus_plus_int @ A @ C ) )
     => ( B2 = C ) ) ).

% add_left_imp_eq
thf(fact_816_add_Oleft__commute,axiom,
    ! [B2: real,A: real,C: real] :
      ( ( plus_plus_real @ B2 @ ( plus_plus_real @ A @ C ) )
      = ( plus_plus_real @ A @ ( plus_plus_real @ B2 @ C ) ) ) ).

% add.left_commute
thf(fact_817_add_Oleft__commute,axiom,
    ! [B2: rat,A: rat,C: rat] :
      ( ( plus_plus_rat @ B2 @ ( plus_plus_rat @ A @ C ) )
      = ( plus_plus_rat @ A @ ( plus_plus_rat @ B2 @ C ) ) ) ).

% add.left_commute
thf(fact_818_add_Oleft__commute,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( ( plus_plus_nat @ B2 @ ( plus_plus_nat @ A @ C ) )
      = ( plus_plus_nat @ A @ ( plus_plus_nat @ B2 @ C ) ) ) ).

% add.left_commute
thf(fact_819_add_Oleft__commute,axiom,
    ! [B2: int,A: int,C: int] :
      ( ( plus_plus_int @ B2 @ ( plus_plus_int @ A @ C ) )
      = ( plus_plus_int @ A @ ( plus_plus_int @ B2 @ C ) ) ) ).

% add.left_commute
thf(fact_820_add_Ocommute,axiom,
    ( plus_plus_real
    = ( ^ [A4: real,B3: real] : ( plus_plus_real @ B3 @ A4 ) ) ) ).

% add.commute
thf(fact_821_add_Ocommute,axiom,
    ( plus_plus_rat
    = ( ^ [A4: rat,B3: rat] : ( plus_plus_rat @ B3 @ A4 ) ) ) ).

% add.commute
thf(fact_822_add_Ocommute,axiom,
    ( plus_plus_nat
    = ( ^ [A4: nat,B3: nat] : ( plus_plus_nat @ B3 @ A4 ) ) ) ).

% add.commute
thf(fact_823_add_Ocommute,axiom,
    ( plus_plus_int
    = ( ^ [A4: int,B3: int] : ( plus_plus_int @ B3 @ A4 ) ) ) ).

% add.commute
thf(fact_824_add_Oright__cancel,axiom,
    ! [B2: real,A: real,C: real] :
      ( ( ( plus_plus_real @ B2 @ A )
        = ( plus_plus_real @ C @ A ) )
      = ( B2 = C ) ) ).

% add.right_cancel
thf(fact_825_add_Oright__cancel,axiom,
    ! [B2: rat,A: rat,C: rat] :
      ( ( ( plus_plus_rat @ B2 @ A )
        = ( plus_plus_rat @ C @ A ) )
      = ( B2 = C ) ) ).

% add.right_cancel
thf(fact_826_add_Oright__cancel,axiom,
    ! [B2: int,A: int,C: int] :
      ( ( ( plus_plus_int @ B2 @ A )
        = ( plus_plus_int @ C @ A ) )
      = ( B2 = C ) ) ).

% add.right_cancel
thf(fact_827_add_Oleft__cancel,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ( plus_plus_real @ A @ B2 )
        = ( plus_plus_real @ A @ C ) )
      = ( B2 = C ) ) ).

% add.left_cancel
thf(fact_828_add_Oleft__cancel,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( ( plus_plus_rat @ A @ B2 )
        = ( plus_plus_rat @ A @ C ) )
      = ( B2 = C ) ) ).

% add.left_cancel
thf(fact_829_add_Oleft__cancel,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( ( plus_plus_int @ A @ B2 )
        = ( plus_plus_int @ A @ C ) )
      = ( B2 = C ) ) ).

% add.left_cancel
thf(fact_830_add_Oassoc,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( plus_plus_real @ ( plus_plus_real @ A @ B2 ) @ C )
      = ( plus_plus_real @ A @ ( plus_plus_real @ B2 @ C ) ) ) ).

% add.assoc
thf(fact_831_add_Oassoc,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( plus_plus_rat @ ( plus_plus_rat @ A @ B2 ) @ C )
      = ( plus_plus_rat @ A @ ( plus_plus_rat @ B2 @ C ) ) ) ).

% add.assoc
thf(fact_832_add_Oassoc,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B2 ) @ C )
      = ( plus_plus_nat @ A @ ( plus_plus_nat @ B2 @ C ) ) ) ).

% add.assoc
thf(fact_833_add_Oassoc,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ A @ B2 ) @ C )
      = ( plus_plus_int @ A @ ( plus_plus_int @ B2 @ C ) ) ) ).

% add.assoc
thf(fact_834_group__cancel_Oadd2,axiom,
    ! [B4: real,K: real,B2: real,A: real] :
      ( ( B4
        = ( plus_plus_real @ K @ B2 ) )
     => ( ( plus_plus_real @ A @ B4 )
        = ( plus_plus_real @ K @ ( plus_plus_real @ A @ B2 ) ) ) ) ).

% group_cancel.add2
thf(fact_835_group__cancel_Oadd2,axiom,
    ! [B4: rat,K: rat,B2: rat,A: rat] :
      ( ( B4
        = ( plus_plus_rat @ K @ B2 ) )
     => ( ( plus_plus_rat @ A @ B4 )
        = ( plus_plus_rat @ K @ ( plus_plus_rat @ A @ B2 ) ) ) ) ).

% group_cancel.add2
thf(fact_836_group__cancel_Oadd2,axiom,
    ! [B4: nat,K: nat,B2: nat,A: nat] :
      ( ( B4
        = ( plus_plus_nat @ K @ B2 ) )
     => ( ( plus_plus_nat @ A @ B4 )
        = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B2 ) ) ) ) ).

% group_cancel.add2
thf(fact_837_group__cancel_Oadd2,axiom,
    ! [B4: int,K: int,B2: int,A: int] :
      ( ( B4
        = ( plus_plus_int @ K @ B2 ) )
     => ( ( plus_plus_int @ A @ B4 )
        = ( plus_plus_int @ K @ ( plus_plus_int @ A @ B2 ) ) ) ) ).

% group_cancel.add2
thf(fact_838_group__cancel_Oadd1,axiom,
    ! [A3: real,K: real,A: real,B2: real] :
      ( ( A3
        = ( plus_plus_real @ K @ A ) )
     => ( ( plus_plus_real @ A3 @ B2 )
        = ( plus_plus_real @ K @ ( plus_plus_real @ A @ B2 ) ) ) ) ).

% group_cancel.add1
thf(fact_839_group__cancel_Oadd1,axiom,
    ! [A3: rat,K: rat,A: rat,B2: rat] :
      ( ( A3
        = ( plus_plus_rat @ K @ A ) )
     => ( ( plus_plus_rat @ A3 @ B2 )
        = ( plus_plus_rat @ K @ ( plus_plus_rat @ A @ B2 ) ) ) ) ).

% group_cancel.add1
thf(fact_840_group__cancel_Oadd1,axiom,
    ! [A3: nat,K: nat,A: nat,B2: nat] :
      ( ( A3
        = ( plus_plus_nat @ K @ A ) )
     => ( ( plus_plus_nat @ A3 @ B2 )
        = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B2 ) ) ) ) ).

% group_cancel.add1
thf(fact_841_group__cancel_Oadd1,axiom,
    ! [A3: int,K: int,A: int,B2: int] :
      ( ( A3
        = ( plus_plus_int @ K @ A ) )
     => ( ( plus_plus_int @ A3 @ B2 )
        = ( plus_plus_int @ K @ ( plus_plus_int @ A @ B2 ) ) ) ) ).

% group_cancel.add1
thf(fact_842_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: real,J: real,K: real,L2: real] :
      ( ( ( I = J )
        & ( K = L2 ) )
     => ( ( plus_plus_real @ I @ K )
        = ( plus_plus_real @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_843_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: rat,J: rat,K: rat,L2: rat] :
      ( ( ( I = J )
        & ( K = L2 ) )
     => ( ( plus_plus_rat @ I @ K )
        = ( plus_plus_rat @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_844_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: nat,J: nat,K: nat,L2: nat] :
      ( ( ( I = J )
        & ( K = L2 ) )
     => ( ( plus_plus_nat @ I @ K )
        = ( plus_plus_nat @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_845_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: int,J: int,K: int,L2: int] :
      ( ( ( I = J )
        & ( K = L2 ) )
     => ( ( plus_plus_int @ I @ K )
        = ( plus_plus_int @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_846_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( plus_plus_real @ ( plus_plus_real @ A @ B2 ) @ C )
      = ( plus_plus_real @ A @ ( plus_plus_real @ B2 @ C ) ) ) ).

% ab_semigroup_add_class.add_ac(1)
thf(fact_847_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( plus_plus_rat @ ( plus_plus_rat @ A @ B2 ) @ C )
      = ( plus_plus_rat @ A @ ( plus_plus_rat @ B2 @ C ) ) ) ).

% ab_semigroup_add_class.add_ac(1)
thf(fact_848_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B2 ) @ C )
      = ( plus_plus_nat @ A @ ( plus_plus_nat @ B2 @ C ) ) ) ).

% ab_semigroup_add_class.add_ac(1)
thf(fact_849_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ A @ B2 ) @ C )
      = ( plus_plus_int @ A @ ( plus_plus_int @ B2 @ C ) ) ) ).

% ab_semigroup_add_class.add_ac(1)
thf(fact_850_diff__right__commute,axiom,
    ! [A: real,C: real,B2: real] :
      ( ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B2 )
      = ( minus_minus_real @ ( minus_minus_real @ A @ B2 ) @ C ) ) ).

% diff_right_commute
thf(fact_851_diff__right__commute,axiom,
    ! [A: rat,C: rat,B2: rat] :
      ( ( minus_minus_rat @ ( minus_minus_rat @ A @ C ) @ B2 )
      = ( minus_minus_rat @ ( minus_minus_rat @ A @ B2 ) @ C ) ) ).

% diff_right_commute
thf(fact_852_diff__right__commute,axiom,
    ! [A: nat,C: nat,B2: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B2 )
      = ( minus_minus_nat @ ( minus_minus_nat @ A @ B2 ) @ C ) ) ).

% diff_right_commute
thf(fact_853_diff__right__commute,axiom,
    ! [A: int,C: int,B2: int] :
      ( ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B2 )
      = ( minus_minus_int @ ( minus_minus_int @ A @ B2 ) @ C ) ) ).

% diff_right_commute
thf(fact_854_diff__eq__diff__eq,axiom,
    ! [A: real,B2: real,C: real,D: real] :
      ( ( ( minus_minus_real @ A @ B2 )
        = ( minus_minus_real @ C @ D ) )
     => ( ( A = B2 )
        = ( C = D ) ) ) ).

% diff_eq_diff_eq
thf(fact_855_diff__eq__diff__eq,axiom,
    ! [A: rat,B2: rat,C: rat,D: rat] :
      ( ( ( minus_minus_rat @ A @ B2 )
        = ( minus_minus_rat @ C @ D ) )
     => ( ( A = B2 )
        = ( C = D ) ) ) ).

% diff_eq_diff_eq
thf(fact_856_diff__eq__diff__eq,axiom,
    ! [A: int,B2: int,C: int,D: int] :
      ( ( ( minus_minus_int @ A @ B2 )
        = ( minus_minus_int @ C @ D ) )
     => ( ( A = B2 )
        = ( C = D ) ) ) ).

% diff_eq_diff_eq
thf(fact_857_add__le__imp__le__right,axiom,
    ! [A: real,C: real,B2: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ C ) )
     => ( ord_less_eq_real @ A @ B2 ) ) ).

% add_le_imp_le_right
thf(fact_858_add__le__imp__le__right,axiom,
    ! [A: rat,C: rat,B2: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B2 @ C ) )
     => ( ord_less_eq_rat @ A @ B2 ) ) ).

% add_le_imp_le_right
thf(fact_859_add__le__imp__le__right,axiom,
    ! [A: nat,C: nat,B2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) )
     => ( ord_less_eq_nat @ A @ B2 ) ) ).

% add_le_imp_le_right
thf(fact_860_add__le__imp__le__right,axiom,
    ! [A: int,C: int,B2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ C ) )
     => ( ord_less_eq_int @ A @ B2 ) ) ).

% add_le_imp_le_right
thf(fact_861_add__le__imp__le__left,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B2 ) )
     => ( ord_less_eq_real @ A @ B2 ) ) ).

% add_le_imp_le_left
thf(fact_862_add__le__imp__le__left,axiom,
    ! [C: rat,A: rat,B2: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B2 ) )
     => ( ord_less_eq_rat @ A @ B2 ) ) ).

% add_le_imp_le_left
thf(fact_863_add__le__imp__le__left,axiom,
    ! [C: nat,A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) )
     => ( ord_less_eq_nat @ A @ B2 ) ) ).

% add_le_imp_le_left
thf(fact_864_add__le__imp__le__left,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B2 ) )
     => ( ord_less_eq_int @ A @ B2 ) ) ).

% add_le_imp_le_left
thf(fact_865_le__iff__add,axiom,
    ( ord_less_eq_nat
    = ( ^ [A4: nat,B3: nat] :
        ? [C2: nat] :
          ( B3
          = ( plus_plus_nat @ A4 @ C2 ) ) ) ) ).

% le_iff_add
thf(fact_866_add__right__mono,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ord_less_eq_real @ A @ B2 )
     => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ C ) ) ) ).

% add_right_mono
thf(fact_867_add__right__mono,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B2 )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B2 @ C ) ) ) ).

% add_right_mono
thf(fact_868_add__right__mono,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) ) ) ).

% add_right_mono
thf(fact_869_add__right__mono,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( ord_less_eq_int @ A @ B2 )
     => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ C ) ) ) ).

% add_right_mono
thf(fact_870_less__eqE,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ~ ! [C3: nat] :
            ( B2
           != ( plus_plus_nat @ A @ C3 ) ) ) ).

% less_eqE
thf(fact_871_add__left__mono,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ord_less_eq_real @ A @ B2 )
     => ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B2 ) ) ) ).

% add_left_mono
thf(fact_872_add__left__mono,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B2 )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B2 ) ) ) ).

% add_left_mono
thf(fact_873_add__left__mono,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) ) ) ).

% add_left_mono
thf(fact_874_add__left__mono,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( ord_less_eq_int @ A @ B2 )
     => ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B2 ) ) ) ).

% add_left_mono
thf(fact_875_add__mono,axiom,
    ! [A: real,B2: real,C: real,D: real] :
      ( ( ord_less_eq_real @ A @ B2 )
     => ( ( ord_less_eq_real @ C @ D )
       => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ D ) ) ) ) ).

% add_mono
thf(fact_876_add__mono,axiom,
    ! [A: rat,B2: rat,C: rat,D: rat] :
      ( ( ord_less_eq_rat @ A @ B2 )
     => ( ( ord_less_eq_rat @ C @ D )
       => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B2 @ D ) ) ) ) ).

% add_mono
thf(fact_877_add__mono,axiom,
    ! [A: nat,B2: nat,C: nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( ord_less_eq_nat @ C @ D )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ D ) ) ) ) ).

% add_mono
thf(fact_878_add__mono,axiom,
    ! [A: int,B2: int,C: int,D: int] :
      ( ( ord_less_eq_int @ A @ B2 )
     => ( ( ord_less_eq_int @ C @ D )
       => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ D ) ) ) ) ).

% add_mono
thf(fact_879_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: real,J: real,K: real,L2: real] :
      ( ( ( ord_less_eq_real @ I @ J )
        & ( ord_less_eq_real @ K @ L2 ) )
     => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_880_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: rat,J: rat,K: rat,L2: rat] :
      ( ( ( ord_less_eq_rat @ I @ J )
        & ( ord_less_eq_rat @ K @ L2 ) )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_881_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: nat,J: nat,K: nat,L2: nat] :
      ( ( ( ord_less_eq_nat @ I @ J )
        & ( ord_less_eq_nat @ K @ L2 ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_882_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: int,J: int,K: int,L2: int] :
      ( ( ( ord_less_eq_int @ I @ J )
        & ( ord_less_eq_int @ K @ L2 ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_883_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: real,J: real,K: real,L2: real] :
      ( ( ( I = J )
        & ( ord_less_eq_real @ K @ L2 ) )
     => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_884_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: rat,J: rat,K: rat,L2: rat] :
      ( ( ( I = J )
        & ( ord_less_eq_rat @ K @ L2 ) )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_885_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: nat,J: nat,K: nat,L2: nat] :
      ( ( ( I = J )
        & ( ord_less_eq_nat @ K @ L2 ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_886_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: int,J: int,K: int,L2: int] :
      ( ( ( I = J )
        & ( ord_less_eq_int @ K @ L2 ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_887_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: real,J: real,K: real,L2: real] :
      ( ( ( ord_less_eq_real @ I @ J )
        & ( K = L2 ) )
     => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_888_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: rat,J: rat,K: rat,L2: rat] :
      ( ( ( ord_less_eq_rat @ I @ J )
        & ( K = L2 ) )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_889_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: nat,J: nat,K: nat,L2: nat] :
      ( ( ( ord_less_eq_nat @ I @ J )
        & ( K = L2 ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_890_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: int,J: int,K: int,L2: int] :
      ( ( ( ord_less_eq_int @ I @ J )
        & ( K = L2 ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_891_diff__eq__diff__less__eq,axiom,
    ! [A: real,B2: real,C: real,D: real] :
      ( ( ( minus_minus_real @ A @ B2 )
        = ( minus_minus_real @ C @ D ) )
     => ( ( ord_less_eq_real @ A @ B2 )
        = ( ord_less_eq_real @ C @ D ) ) ) ).

% diff_eq_diff_less_eq
thf(fact_892_diff__eq__diff__less__eq,axiom,
    ! [A: rat,B2: rat,C: rat,D: rat] :
      ( ( ( minus_minus_rat @ A @ B2 )
        = ( minus_minus_rat @ C @ D ) )
     => ( ( ord_less_eq_rat @ A @ B2 )
        = ( ord_less_eq_rat @ C @ D ) ) ) ).

% diff_eq_diff_less_eq
thf(fact_893_diff__eq__diff__less__eq,axiom,
    ! [A: int,B2: int,C: int,D: int] :
      ( ( ( minus_minus_int @ A @ B2 )
        = ( minus_minus_int @ C @ D ) )
     => ( ( ord_less_eq_int @ A @ B2 )
        = ( ord_less_eq_int @ C @ D ) ) ) ).

% diff_eq_diff_less_eq
thf(fact_894_diff__right__mono,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ord_less_eq_real @ A @ B2 )
     => ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B2 @ C ) ) ) ).

% diff_right_mono
thf(fact_895_diff__right__mono,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B2 )
     => ( ord_less_eq_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B2 @ C ) ) ) ).

% diff_right_mono
thf(fact_896_diff__right__mono,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( ord_less_eq_int @ A @ B2 )
     => ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B2 @ C ) ) ) ).

% diff_right_mono
thf(fact_897_diff__left__mono,axiom,
    ! [B2: real,A: real,C: real] :
      ( ( ord_less_eq_real @ B2 @ A )
     => ( ord_less_eq_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B2 ) ) ) ).

% diff_left_mono
thf(fact_898_diff__left__mono,axiom,
    ! [B2: rat,A: rat,C: rat] :
      ( ( ord_less_eq_rat @ B2 @ A )
     => ( ord_less_eq_rat @ ( minus_minus_rat @ C @ A ) @ ( minus_minus_rat @ C @ B2 ) ) ) ).

% diff_left_mono
thf(fact_899_diff__left__mono,axiom,
    ! [B2: int,A: int,C: int] :
      ( ( ord_less_eq_int @ B2 @ A )
     => ( ord_less_eq_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B2 ) ) ) ).

% diff_left_mono
thf(fact_900_diff__mono,axiom,
    ! [A: real,B2: real,D: real,C: real] :
      ( ( ord_less_eq_real @ A @ B2 )
     => ( ( ord_less_eq_real @ D @ C )
       => ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B2 @ D ) ) ) ) ).

% diff_mono
thf(fact_901_diff__mono,axiom,
    ! [A: rat,B2: rat,D: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B2 )
     => ( ( ord_less_eq_rat @ D @ C )
       => ( ord_less_eq_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B2 @ D ) ) ) ) ).

% diff_mono
thf(fact_902_diff__mono,axiom,
    ! [A: int,B2: int,D: int,C: int] :
      ( ( ord_less_eq_int @ A @ B2 )
     => ( ( ord_less_eq_int @ D @ C )
       => ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B2 @ D ) ) ) ) ).

% diff_mono
thf(fact_903_add__less__imp__less__right,axiom,
    ! [A: real,C: real,B2: real] :
      ( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ C ) )
     => ( ord_less_real @ A @ B2 ) ) ).

% add_less_imp_less_right
thf(fact_904_add__less__imp__less__right,axiom,
    ! [A: rat,C: rat,B2: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B2 @ C ) )
     => ( ord_less_rat @ A @ B2 ) ) ).

% add_less_imp_less_right
thf(fact_905_add__less__imp__less__right,axiom,
    ! [A: nat,C: nat,B2: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) )
     => ( ord_less_nat @ A @ B2 ) ) ).

% add_less_imp_less_right
thf(fact_906_add__less__imp__less__right,axiom,
    ! [A: int,C: int,B2: int] :
      ( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ C ) )
     => ( ord_less_int @ A @ B2 ) ) ).

% add_less_imp_less_right
thf(fact_907_add__less__imp__less__left,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B2 ) )
     => ( ord_less_real @ A @ B2 ) ) ).

% add_less_imp_less_left
thf(fact_908_add__less__imp__less__left,axiom,
    ! [C: rat,A: rat,B2: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B2 ) )
     => ( ord_less_rat @ A @ B2 ) ) ).

% add_less_imp_less_left
thf(fact_909_add__less__imp__less__left,axiom,
    ! [C: nat,A: nat,B2: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) )
     => ( ord_less_nat @ A @ B2 ) ) ).

% add_less_imp_less_left
thf(fact_910_add__less__imp__less__left,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B2 ) )
     => ( ord_less_int @ A @ B2 ) ) ).

% add_less_imp_less_left
thf(fact_911_add__strict__right__mono,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ C ) ) ) ).

% add_strict_right_mono
thf(fact_912_add__strict__right__mono,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( ord_less_rat @ A @ B2 )
     => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B2 @ C ) ) ) ).

% add_strict_right_mono
thf(fact_913_add__strict__right__mono,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) ) ) ).

% add_strict_right_mono
thf(fact_914_add__strict__right__mono,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( ord_less_int @ A @ B2 )
     => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ C ) ) ) ).

% add_strict_right_mono
thf(fact_915_add__strict__left__mono,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B2 ) ) ) ).

% add_strict_left_mono
thf(fact_916_add__strict__left__mono,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( ord_less_rat @ A @ B2 )
     => ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B2 ) ) ) ).

% add_strict_left_mono
thf(fact_917_add__strict__left__mono,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) ) ) ).

% add_strict_left_mono
thf(fact_918_add__strict__left__mono,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( ord_less_int @ A @ B2 )
     => ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B2 ) ) ) ).

% add_strict_left_mono
thf(fact_919_add__strict__mono,axiom,
    ! [A: real,B2: real,C: real,D: real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ( ord_less_real @ C @ D )
       => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ D ) ) ) ) ).

% add_strict_mono
thf(fact_920_add__strict__mono,axiom,
    ! [A: rat,B2: rat,C: rat,D: rat] :
      ( ( ord_less_rat @ A @ B2 )
     => ( ( ord_less_rat @ C @ D )
       => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B2 @ D ) ) ) ) ).

% add_strict_mono
thf(fact_921_add__strict__mono,axiom,
    ! [A: nat,B2: nat,C: nat,D: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( ( ord_less_nat @ C @ D )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ D ) ) ) ) ).

% add_strict_mono
thf(fact_922_add__strict__mono,axiom,
    ! [A: int,B2: int,C: int,D: int] :
      ( ( ord_less_int @ A @ B2 )
     => ( ( ord_less_int @ C @ D )
       => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ D ) ) ) ) ).

% add_strict_mono
thf(fact_923_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: real,J: real,K: real,L2: real] :
      ( ( ( ord_less_real @ I @ J )
        & ( K = L2 ) )
     => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_924_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: rat,J: rat,K: rat,L2: rat] :
      ( ( ( ord_less_rat @ I @ J )
        & ( K = L2 ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_925_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: nat,J: nat,K: nat,L2: nat] :
      ( ( ( ord_less_nat @ I @ J )
        & ( K = L2 ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_926_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: int,J: int,K: int,L2: int] :
      ( ( ( ord_less_int @ I @ J )
        & ( K = L2 ) )
     => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_927_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: real,J: real,K: real,L2: real] :
      ( ( ( I = J )
        & ( ord_less_real @ K @ L2 ) )
     => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_928_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: rat,J: rat,K: rat,L2: rat] :
      ( ( ( I = J )
        & ( ord_less_rat @ K @ L2 ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_929_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: nat,J: nat,K: nat,L2: nat] :
      ( ( ( I = J )
        & ( ord_less_nat @ K @ L2 ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_930_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: int,J: int,K: int,L2: int] :
      ( ( ( I = J )
        & ( ord_less_int @ K @ L2 ) )
     => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_931_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: real,J: real,K: real,L2: real] :
      ( ( ( ord_less_real @ I @ J )
        & ( ord_less_real @ K @ L2 ) )
     => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_932_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: rat,J: rat,K: rat,L2: rat] :
      ( ( ( ord_less_rat @ I @ J )
        & ( ord_less_rat @ K @ L2 ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_933_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: nat,J: nat,K: nat,L2: nat] :
      ( ( ( ord_less_nat @ I @ J )
        & ( ord_less_nat @ K @ L2 ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_934_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: int,J: int,K: int,L2: int] :
      ( ( ( ord_less_int @ I @ J )
        & ( ord_less_int @ K @ L2 ) )
     => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_935_diff__strict__right__mono,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B2 @ C ) ) ) ).

% diff_strict_right_mono
thf(fact_936_diff__strict__right__mono,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( ord_less_rat @ A @ B2 )
     => ( ord_less_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B2 @ C ) ) ) ).

% diff_strict_right_mono
thf(fact_937_diff__strict__right__mono,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( ord_less_int @ A @ B2 )
     => ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B2 @ C ) ) ) ).

% diff_strict_right_mono
thf(fact_938_diff__strict__left__mono,axiom,
    ! [B2: real,A: real,C: real] :
      ( ( ord_less_real @ B2 @ A )
     => ( ord_less_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B2 ) ) ) ).

% diff_strict_left_mono
thf(fact_939_diff__strict__left__mono,axiom,
    ! [B2: rat,A: rat,C: rat] :
      ( ( ord_less_rat @ B2 @ A )
     => ( ord_less_rat @ ( minus_minus_rat @ C @ A ) @ ( minus_minus_rat @ C @ B2 ) ) ) ).

% diff_strict_left_mono
thf(fact_940_diff__strict__left__mono,axiom,
    ! [B2: int,A: int,C: int] :
      ( ( ord_less_int @ B2 @ A )
     => ( ord_less_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B2 ) ) ) ).

% diff_strict_left_mono
thf(fact_941_diff__eq__diff__less,axiom,
    ! [A: real,B2: real,C: real,D: real] :
      ( ( ( minus_minus_real @ A @ B2 )
        = ( minus_minus_real @ C @ D ) )
     => ( ( ord_less_real @ A @ B2 )
        = ( ord_less_real @ C @ D ) ) ) ).

% diff_eq_diff_less
thf(fact_942_diff__eq__diff__less,axiom,
    ! [A: rat,B2: rat,C: rat,D: rat] :
      ( ( ( minus_minus_rat @ A @ B2 )
        = ( minus_minus_rat @ C @ D ) )
     => ( ( ord_less_rat @ A @ B2 )
        = ( ord_less_rat @ C @ D ) ) ) ).

% diff_eq_diff_less
thf(fact_943_diff__eq__diff__less,axiom,
    ! [A: int,B2: int,C: int,D: int] :
      ( ( ( minus_minus_int @ A @ B2 )
        = ( minus_minus_int @ C @ D ) )
     => ( ( ord_less_int @ A @ B2 )
        = ( ord_less_int @ C @ D ) ) ) ).

% diff_eq_diff_less
thf(fact_944_diff__strict__mono,axiom,
    ! [A: real,B2: real,D: real,C: real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ( ord_less_real @ D @ C )
       => ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B2 @ D ) ) ) ) ).

% diff_strict_mono
thf(fact_945_diff__strict__mono,axiom,
    ! [A: rat,B2: rat,D: rat,C: rat] :
      ( ( ord_less_rat @ A @ B2 )
     => ( ( ord_less_rat @ D @ C )
       => ( ord_less_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B2 @ D ) ) ) ) ).

% diff_strict_mono
thf(fact_946_diff__strict__mono,axiom,
    ! [A: int,B2: int,D: int,C: int] :
      ( ( ord_less_int @ A @ B2 )
     => ( ( ord_less_int @ D @ C )
       => ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B2 @ D ) ) ) ) ).

% diff_strict_mono
thf(fact_947_mult_Ocomm__neutral,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ A @ one_one_complex )
      = A ) ).

% mult.comm_neutral
thf(fact_948_mult_Ocomm__neutral,axiom,
    ! [A: real] :
      ( ( times_times_real @ A @ one_one_real )
      = A ) ).

% mult.comm_neutral
thf(fact_949_mult_Ocomm__neutral,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ A @ one_one_rat )
      = A ) ).

% mult.comm_neutral
thf(fact_950_mult_Ocomm__neutral,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ one_one_nat )
      = A ) ).

% mult.comm_neutral
thf(fact_951_mult_Ocomm__neutral,axiom,
    ! [A: int] :
      ( ( times_times_int @ A @ one_one_int )
      = A ) ).

% mult.comm_neutral
thf(fact_952_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ one_one_complex @ A )
      = A ) ).

% comm_monoid_mult_class.mult_1
thf(fact_953_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: real] :
      ( ( times_times_real @ one_one_real @ A )
      = A ) ).

% comm_monoid_mult_class.mult_1
thf(fact_954_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ one_one_rat @ A )
      = A ) ).

% comm_monoid_mult_class.mult_1
thf(fact_955_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ one_one_nat @ A )
      = A ) ).

% comm_monoid_mult_class.mult_1
thf(fact_956_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: int] :
      ( ( times_times_int @ one_one_int @ A )
      = A ) ).

% comm_monoid_mult_class.mult_1
thf(fact_957_diff__diff__eq,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( minus_minus_real @ ( minus_minus_real @ A @ B2 ) @ C )
      = ( minus_minus_real @ A @ ( plus_plus_real @ B2 @ C ) ) ) ).

% diff_diff_eq
thf(fact_958_diff__diff__eq,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( minus_minus_rat @ ( minus_minus_rat @ A @ B2 ) @ C )
      = ( minus_minus_rat @ A @ ( plus_plus_rat @ B2 @ C ) ) ) ).

% diff_diff_eq
thf(fact_959_diff__diff__eq,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ A @ B2 ) @ C )
      = ( minus_minus_nat @ A @ ( plus_plus_nat @ B2 @ C ) ) ) ).

% diff_diff_eq
thf(fact_960_diff__diff__eq,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( minus_minus_int @ ( minus_minus_int @ A @ B2 ) @ C )
      = ( minus_minus_int @ A @ ( plus_plus_int @ B2 @ C ) ) ) ).

% diff_diff_eq
thf(fact_961_add__implies__diff,axiom,
    ! [C: real,B2: real,A: real] :
      ( ( ( plus_plus_real @ C @ B2 )
        = A )
     => ( C
        = ( minus_minus_real @ A @ B2 ) ) ) ).

% add_implies_diff
thf(fact_962_add__implies__diff,axiom,
    ! [C: rat,B2: rat,A: rat] :
      ( ( ( plus_plus_rat @ C @ B2 )
        = A )
     => ( C
        = ( minus_minus_rat @ A @ B2 ) ) ) ).

% add_implies_diff
thf(fact_963_add__implies__diff,axiom,
    ! [C: nat,B2: nat,A: nat] :
      ( ( ( plus_plus_nat @ C @ B2 )
        = A )
     => ( C
        = ( minus_minus_nat @ A @ B2 ) ) ) ).

% add_implies_diff
thf(fact_964_add__implies__diff,axiom,
    ! [C: int,B2: int,A: int] :
      ( ( ( plus_plus_int @ C @ B2 )
        = A )
     => ( C
        = ( minus_minus_int @ A @ B2 ) ) ) ).

% add_implies_diff
thf(fact_965_diff__add__eq__diff__diff__swap,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( minus_minus_real @ A @ ( plus_plus_real @ B2 @ C ) )
      = ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B2 ) ) ).

% diff_add_eq_diff_diff_swap
thf(fact_966_diff__add__eq__diff__diff__swap,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( minus_minus_rat @ A @ ( plus_plus_rat @ B2 @ C ) )
      = ( minus_minus_rat @ ( minus_minus_rat @ A @ C ) @ B2 ) ) ).

% diff_add_eq_diff_diff_swap
thf(fact_967_diff__add__eq__diff__diff__swap,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( minus_minus_int @ A @ ( plus_plus_int @ B2 @ C ) )
      = ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B2 ) ) ).

% diff_add_eq_diff_diff_swap
thf(fact_968_diff__add__eq,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( plus_plus_real @ ( minus_minus_real @ A @ B2 ) @ C )
      = ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B2 ) ) ).

% diff_add_eq
thf(fact_969_diff__add__eq,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B2 ) @ C )
      = ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ B2 ) ) ).

% diff_add_eq
thf(fact_970_diff__add__eq,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( plus_plus_int @ ( minus_minus_int @ A @ B2 ) @ C )
      = ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B2 ) ) ).

% diff_add_eq
thf(fact_971_diff__diff__eq2,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( minus_minus_real @ A @ ( minus_minus_real @ B2 @ C ) )
      = ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B2 ) ) ).

% diff_diff_eq2
thf(fact_972_diff__diff__eq2,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( minus_minus_rat @ A @ ( minus_minus_rat @ B2 @ C ) )
      = ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ B2 ) ) ).

% diff_diff_eq2
thf(fact_973_diff__diff__eq2,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( minus_minus_int @ A @ ( minus_minus_int @ B2 @ C ) )
      = ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B2 ) ) ).

% diff_diff_eq2
thf(fact_974_add__diff__eq,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( plus_plus_real @ A @ ( minus_minus_real @ B2 @ C ) )
      = ( minus_minus_real @ ( plus_plus_real @ A @ B2 ) @ C ) ) ).

% add_diff_eq
thf(fact_975_add__diff__eq,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( plus_plus_rat @ A @ ( minus_minus_rat @ B2 @ C ) )
      = ( minus_minus_rat @ ( plus_plus_rat @ A @ B2 ) @ C ) ) ).

% add_diff_eq
thf(fact_976_add__diff__eq,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( plus_plus_int @ A @ ( minus_minus_int @ B2 @ C ) )
      = ( minus_minus_int @ ( plus_plus_int @ A @ B2 ) @ C ) ) ).

% add_diff_eq
thf(fact_977_eq__diff__eq,axiom,
    ! [A: real,C: real,B2: real] :
      ( ( A
        = ( minus_minus_real @ C @ B2 ) )
      = ( ( plus_plus_real @ A @ B2 )
        = C ) ) ).

% eq_diff_eq
thf(fact_978_eq__diff__eq,axiom,
    ! [A: rat,C: rat,B2: rat] :
      ( ( A
        = ( minus_minus_rat @ C @ B2 ) )
      = ( ( plus_plus_rat @ A @ B2 )
        = C ) ) ).

% eq_diff_eq
thf(fact_979_eq__diff__eq,axiom,
    ! [A: int,C: int,B2: int] :
      ( ( A
        = ( minus_minus_int @ C @ B2 ) )
      = ( ( plus_plus_int @ A @ B2 )
        = C ) ) ).

% eq_diff_eq
thf(fact_980_diff__eq__eq,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ( minus_minus_real @ A @ B2 )
        = C )
      = ( A
        = ( plus_plus_real @ C @ B2 ) ) ) ).

% diff_eq_eq
thf(fact_981_diff__eq__eq,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( ( minus_minus_rat @ A @ B2 )
        = C )
      = ( A
        = ( plus_plus_rat @ C @ B2 ) ) ) ).

% diff_eq_eq
thf(fact_982_diff__eq__eq,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( ( minus_minus_int @ A @ B2 )
        = C )
      = ( A
        = ( plus_plus_int @ C @ B2 ) ) ) ).

% diff_eq_eq
thf(fact_983_group__cancel_Osub1,axiom,
    ! [A3: real,K: real,A: real,B2: real] :
      ( ( A3
        = ( plus_plus_real @ K @ A ) )
     => ( ( minus_minus_real @ A3 @ B2 )
        = ( plus_plus_real @ K @ ( minus_minus_real @ A @ B2 ) ) ) ) ).

% group_cancel.sub1
thf(fact_984_group__cancel_Osub1,axiom,
    ! [A3: rat,K: rat,A: rat,B2: rat] :
      ( ( A3
        = ( plus_plus_rat @ K @ A ) )
     => ( ( minus_minus_rat @ A3 @ B2 )
        = ( plus_plus_rat @ K @ ( minus_minus_rat @ A @ B2 ) ) ) ) ).

% group_cancel.sub1
thf(fact_985_group__cancel_Osub1,axiom,
    ! [A3: int,K: int,A: int,B2: int] :
      ( ( A3
        = ( plus_plus_int @ K @ A ) )
     => ( ( minus_minus_int @ A3 @ B2 )
        = ( plus_plus_int @ K @ ( minus_minus_int @ A @ B2 ) ) ) ) ).

% group_cancel.sub1
thf(fact_986_add__less__le__mono,axiom,
    ! [A: real,B2: real,C: real,D: real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ( ord_less_eq_real @ C @ D )
       => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ D ) ) ) ) ).

% add_less_le_mono
thf(fact_987_add__less__le__mono,axiom,
    ! [A: rat,B2: rat,C: rat,D: rat] :
      ( ( ord_less_rat @ A @ B2 )
     => ( ( ord_less_eq_rat @ C @ D )
       => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B2 @ D ) ) ) ) ).

% add_less_le_mono
thf(fact_988_add__less__le__mono,axiom,
    ! [A: nat,B2: nat,C: nat,D: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( ( ord_less_eq_nat @ C @ D )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ D ) ) ) ) ).

% add_less_le_mono
thf(fact_989_add__less__le__mono,axiom,
    ! [A: int,B2: int,C: int,D: int] :
      ( ( ord_less_int @ A @ B2 )
     => ( ( ord_less_eq_int @ C @ D )
       => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ D ) ) ) ) ).

% add_less_le_mono
thf(fact_990_add__le__less__mono,axiom,
    ! [A: real,B2: real,C: real,D: real] :
      ( ( ord_less_eq_real @ A @ B2 )
     => ( ( ord_less_real @ C @ D )
       => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ D ) ) ) ) ).

% add_le_less_mono
thf(fact_991_add__le__less__mono,axiom,
    ! [A: rat,B2: rat,C: rat,D: rat] :
      ( ( ord_less_eq_rat @ A @ B2 )
     => ( ( ord_less_rat @ C @ D )
       => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B2 @ D ) ) ) ) ).

% add_le_less_mono
thf(fact_992_add__le__less__mono,axiom,
    ! [A: nat,B2: nat,C: nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( ord_less_nat @ C @ D )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ D ) ) ) ) ).

% add_le_less_mono
thf(fact_993_add__le__less__mono,axiom,
    ! [A: int,B2: int,C: int,D: int] :
      ( ( ord_less_eq_int @ A @ B2 )
     => ( ( ord_less_int @ C @ D )
       => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ D ) ) ) ) ).

% add_le_less_mono
thf(fact_994_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: real,J: real,K: real,L2: real] :
      ( ( ( ord_less_real @ I @ J )
        & ( ord_less_eq_real @ K @ L2 ) )
     => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_995_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: rat,J: rat,K: rat,L2: rat] :
      ( ( ( ord_less_rat @ I @ J )
        & ( ord_less_eq_rat @ K @ L2 ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_996_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: nat,J: nat,K: nat,L2: nat] :
      ( ( ( ord_less_nat @ I @ J )
        & ( ord_less_eq_nat @ K @ L2 ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_997_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: int,J: int,K: int,L2: int] :
      ( ( ( ord_less_int @ I @ J )
        & ( ord_less_eq_int @ K @ L2 ) )
     => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_998_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: real,J: real,K: real,L2: real] :
      ( ( ( ord_less_eq_real @ I @ J )
        & ( ord_less_real @ K @ L2 ) )
     => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_999_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: rat,J: rat,K: rat,L2: rat] :
      ( ( ( ord_less_eq_rat @ I @ J )
        & ( ord_less_rat @ K @ L2 ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_1000_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: nat,J: nat,K: nat,L2: nat] :
      ( ( ( ord_less_eq_nat @ I @ J )
        & ( ord_less_nat @ K @ L2 ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_1001_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: int,J: int,K: int,L2: int] :
      ( ( ( ord_less_eq_int @ I @ J )
        & ( ord_less_int @ K @ L2 ) )
     => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_1002_diff__le__eq,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ord_less_eq_real @ ( minus_minus_real @ A @ B2 ) @ C )
      = ( ord_less_eq_real @ A @ ( plus_plus_real @ C @ B2 ) ) ) ).

% diff_le_eq
thf(fact_1003_diff__le__eq,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( ord_less_eq_rat @ ( minus_minus_rat @ A @ B2 ) @ C )
      = ( ord_less_eq_rat @ A @ ( plus_plus_rat @ C @ B2 ) ) ) ).

% diff_le_eq
thf(fact_1004_diff__le__eq,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( ord_less_eq_int @ ( minus_minus_int @ A @ B2 ) @ C )
      = ( ord_less_eq_int @ A @ ( plus_plus_int @ C @ B2 ) ) ) ).

% diff_le_eq
thf(fact_1005_le__diff__eq,axiom,
    ! [A: real,C: real,B2: real] :
      ( ( ord_less_eq_real @ A @ ( minus_minus_real @ C @ B2 ) )
      = ( ord_less_eq_real @ ( plus_plus_real @ A @ B2 ) @ C ) ) ).

% le_diff_eq
thf(fact_1006_le__diff__eq,axiom,
    ! [A: rat,C: rat,B2: rat] :
      ( ( ord_less_eq_rat @ A @ ( minus_minus_rat @ C @ B2 ) )
      = ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B2 ) @ C ) ) ).

% le_diff_eq
thf(fact_1007_le__diff__eq,axiom,
    ! [A: int,C: int,B2: int] :
      ( ( ord_less_eq_int @ A @ ( minus_minus_int @ C @ B2 ) )
      = ( ord_less_eq_int @ ( plus_plus_int @ A @ B2 ) @ C ) ) ).

% le_diff_eq
thf(fact_1008_diff__add,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ B2 @ A ) @ A )
        = B2 ) ) ).

% diff_add
thf(fact_1009_le__add__diff,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B2 @ C ) @ A ) ) ) ).

% le_add_diff
thf(fact_1010_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B2 @ A ) )
        = ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B2 ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_1011_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B2 @ A ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ C @ B2 ) @ A ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_1012_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B2 ) @ A )
        = ( plus_plus_nat @ C @ ( minus_minus_nat @ B2 @ A ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_1013_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ B2 @ A ) @ C )
        = ( minus_minus_nat @ ( plus_plus_nat @ B2 @ C ) @ A ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_1014_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ B2 @ C ) @ A )
        = ( plus_plus_nat @ ( minus_minus_nat @ B2 @ A ) @ C ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_1015_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B2 @ A ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B2 ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_1016_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B2 @ A ) )
        = B2 ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_1017_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( ord_less_eq_nat @ A @ B2 )
       => ( ( ( minus_minus_nat @ B2 @ A )
            = C )
          = ( B2
            = ( plus_plus_nat @ C @ A ) ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_1018_diff__less__eq,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ord_less_real @ ( minus_minus_real @ A @ B2 ) @ C )
      = ( ord_less_real @ A @ ( plus_plus_real @ C @ B2 ) ) ) ).

% diff_less_eq
thf(fact_1019_diff__less__eq,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( ord_less_rat @ ( minus_minus_rat @ A @ B2 ) @ C )
      = ( ord_less_rat @ A @ ( plus_plus_rat @ C @ B2 ) ) ) ).

% diff_less_eq
thf(fact_1020_diff__less__eq,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( ord_less_int @ ( minus_minus_int @ A @ B2 ) @ C )
      = ( ord_less_int @ A @ ( plus_plus_int @ C @ B2 ) ) ) ).

% diff_less_eq
thf(fact_1021_less__diff__eq,axiom,
    ! [A: real,C: real,B2: real] :
      ( ( ord_less_real @ A @ ( minus_minus_real @ C @ B2 ) )
      = ( ord_less_real @ ( plus_plus_real @ A @ B2 ) @ C ) ) ).

% less_diff_eq
thf(fact_1022_less__diff__eq,axiom,
    ! [A: rat,C: rat,B2: rat] :
      ( ( ord_less_rat @ A @ ( minus_minus_rat @ C @ B2 ) )
      = ( ord_less_rat @ ( plus_plus_rat @ A @ B2 ) @ C ) ) ).

% less_diff_eq
thf(fact_1023_less__diff__eq,axiom,
    ! [A: int,C: int,B2: int] :
      ( ( ord_less_int @ A @ ( minus_minus_int @ C @ B2 ) )
      = ( ord_less_int @ ( plus_plus_int @ A @ B2 ) @ C ) ) ).

% less_diff_eq
thf(fact_1024_real__average__minus__second,axiom,
    ! [B2: real,A: real] :
      ( ( minus_minus_real @ ( divide_divide_real @ ( plus_plus_real @ B2 @ A ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ A )
      = ( divide_divide_real @ ( minus_minus_real @ B2 @ A ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% real_average_minus_second
thf(fact_1025_real__average__minus__first,axiom,
    ! [A: real,B2: real] :
      ( ( minus_minus_real @ ( divide_divide_real @ ( plus_plus_real @ A @ B2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ A )
      = ( divide_divide_real @ ( minus_minus_real @ B2 @ A ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% real_average_minus_first
thf(fact_1026_times__divide__eq__left,axiom,
    ! [B2: complex,C: complex,A: complex] :
      ( ( times_times_complex @ ( divide1717551699836669952omplex @ B2 @ C ) @ A )
      = ( divide1717551699836669952omplex @ ( times_times_complex @ B2 @ A ) @ C ) ) ).

% times_divide_eq_left
thf(fact_1027_times__divide__eq__left,axiom,
    ! [B2: real,C: real,A: real] :
      ( ( times_times_real @ ( divide_divide_real @ B2 @ C ) @ A )
      = ( divide_divide_real @ ( times_times_real @ B2 @ A ) @ C ) ) ).

% times_divide_eq_left
thf(fact_1028_times__divide__eq__left,axiom,
    ! [B2: rat,C: rat,A: rat] :
      ( ( times_times_rat @ ( divide_divide_rat @ B2 @ C ) @ A )
      = ( divide_divide_rat @ ( times_times_rat @ B2 @ A ) @ C ) ) ).

% times_divide_eq_left
thf(fact_1029_divide__divide__eq__left,axiom,
    ! [A: complex,B2: complex,C: complex] :
      ( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ A @ B2 ) @ C )
      = ( divide1717551699836669952omplex @ A @ ( times_times_complex @ B2 @ C ) ) ) ).

% divide_divide_eq_left
thf(fact_1030_divide__divide__eq__left,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( divide_divide_real @ ( divide_divide_real @ A @ B2 ) @ C )
      = ( divide_divide_real @ A @ ( times_times_real @ B2 @ C ) ) ) ).

% divide_divide_eq_left
thf(fact_1031_divide__divide__eq__left,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( divide_divide_rat @ ( divide_divide_rat @ A @ B2 ) @ C )
      = ( divide_divide_rat @ A @ ( times_times_rat @ B2 @ C ) ) ) ).

% divide_divide_eq_left
thf(fact_1032_divide__divide__eq__right,axiom,
    ! [A: complex,B2: complex,C: complex] :
      ( ( divide1717551699836669952omplex @ A @ ( divide1717551699836669952omplex @ B2 @ C ) )
      = ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C ) @ B2 ) ) ).

% divide_divide_eq_right
thf(fact_1033_divide__divide__eq__right,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( divide_divide_real @ A @ ( divide_divide_real @ B2 @ C ) )
      = ( divide_divide_real @ ( times_times_real @ A @ C ) @ B2 ) ) ).

% divide_divide_eq_right
thf(fact_1034_divide__divide__eq__right,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( divide_divide_rat @ A @ ( divide_divide_rat @ B2 @ C ) )
      = ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ B2 ) ) ).

% divide_divide_eq_right
thf(fact_1035_times__divide__eq__right,axiom,
    ! [A: complex,B2: complex,C: complex] :
      ( ( times_times_complex @ A @ ( divide1717551699836669952omplex @ B2 @ C ) )
      = ( divide1717551699836669952omplex @ ( times_times_complex @ A @ B2 ) @ C ) ) ).

% times_divide_eq_right
thf(fact_1036_times__divide__eq__right,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( times_times_real @ A @ ( divide_divide_real @ B2 @ C ) )
      = ( divide_divide_real @ ( times_times_real @ A @ B2 ) @ C ) ) ).

% times_divide_eq_right
thf(fact_1037_times__divide__eq__right,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( times_times_rat @ A @ ( divide_divide_rat @ B2 @ C ) )
      = ( divide_divide_rat @ ( times_times_rat @ A @ B2 ) @ C ) ) ).

% times_divide_eq_right
thf(fact_1038_less__half__sum,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ord_less_real @ A @ ( divide_divide_real @ ( plus_plus_real @ A @ B2 ) @ ( plus_plus_real @ one_one_real @ one_one_real ) ) ) ) ).

% less_half_sum
thf(fact_1039_less__half__sum,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_rat @ A @ B2 )
     => ( ord_less_rat @ A @ ( divide_divide_rat @ ( plus_plus_rat @ A @ B2 ) @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ) ) ) ).

% less_half_sum
thf(fact_1040_gt__half__sum,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ord_less_real @ ( divide_divide_real @ ( plus_plus_real @ A @ B2 ) @ ( plus_plus_real @ one_one_real @ one_one_real ) ) @ B2 ) ) ).

% gt_half_sum
thf(fact_1041_gt__half__sum,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_rat @ A @ B2 )
     => ( ord_less_rat @ ( divide_divide_rat @ ( plus_plus_rat @ A @ B2 ) @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ) @ B2 ) ) ).

% gt_half_sum
thf(fact_1042_discrete,axiom,
    ( ord_less_nat
    = ( ^ [A4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ A4 @ one_one_nat ) ) ) ) ).

% discrete
thf(fact_1043_discrete,axiom,
    ( ord_less_int
    = ( ^ [A4: int] : ( ord_less_eq_int @ ( plus_plus_int @ A4 @ one_one_int ) ) ) ) ).

% discrete
thf(fact_1044_low__def,axiom,
    ( vEBT_VEBT_low
    = ( ^ [X: nat,N2: nat] : ( modulo_modulo_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% low_def
thf(fact_1045_zdiv__numeral__Bit0,axiom,
    ! [V: num,W: num] :
      ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit0 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
      = ( divide_divide_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) ).

% zdiv_numeral_Bit0
thf(fact_1046_mod__mod__trivial,axiom,
    ! [A: nat,B2: nat] :
      ( ( modulo_modulo_nat @ ( modulo_modulo_nat @ A @ B2 ) @ B2 )
      = ( modulo_modulo_nat @ A @ B2 ) ) ).

% mod_mod_trivial
thf(fact_1047_mod__mod__trivial,axiom,
    ! [A: int,B2: int] :
      ( ( modulo_modulo_int @ ( modulo_modulo_int @ A @ B2 ) @ B2 )
      = ( modulo_modulo_int @ A @ B2 ) ) ).

% mod_mod_trivial
thf(fact_1048_mod__mod__trivial,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( modulo364778990260209775nteger @ ( modulo364778990260209775nteger @ A @ B2 ) @ B2 )
      = ( modulo364778990260209775nteger @ A @ B2 ) ) ).

% mod_mod_trivial
thf(fact_1049_mod__add__self2,axiom,
    ! [A: nat,B2: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B2 ) @ B2 )
      = ( modulo_modulo_nat @ A @ B2 ) ) ).

% mod_add_self2
thf(fact_1050_mod__add__self2,axiom,
    ! [A: int,B2: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ A @ B2 ) @ B2 )
      = ( modulo_modulo_int @ A @ B2 ) ) ).

% mod_add_self2
thf(fact_1051_mod__add__self2,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B2 ) @ B2 )
      = ( modulo364778990260209775nteger @ A @ B2 ) ) ).

% mod_add_self2
thf(fact_1052_mod__add__self1,axiom,
    ! [B2: nat,A: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ B2 @ A ) @ B2 )
      = ( modulo_modulo_nat @ A @ B2 ) ) ).

% mod_add_self1
thf(fact_1053_mod__add__self1,axiom,
    ! [B2: int,A: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ B2 @ A ) @ B2 )
      = ( modulo_modulo_int @ A @ B2 ) ) ).

% mod_add_self1
thf(fact_1054_mod__add__self1,axiom,
    ! [B2: code_integer,A: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ B2 @ A ) @ B2 )
      = ( modulo364778990260209775nteger @ A @ B2 ) ) ).

% mod_add_self1
thf(fact_1055_minus__mod__self2,axiom,
    ! [A: int,B2: int] :
      ( ( modulo_modulo_int @ ( minus_minus_int @ A @ B2 ) @ B2 )
      = ( modulo_modulo_int @ A @ B2 ) ) ).

% minus_mod_self2
thf(fact_1056_minus__mod__self2,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B2 ) @ B2 )
      = ( modulo364778990260209775nteger @ A @ B2 ) ) ).

% minus_mod_self2
thf(fact_1057_real__divide__square__eq,axiom,
    ! [R2: real,A: real] :
      ( ( divide_divide_real @ ( times_times_real @ R2 @ A ) @ ( times_times_real @ R2 @ R2 ) )
      = ( divide_divide_real @ A @ R2 ) ) ).

% real_divide_square_eq
thf(fact_1058_mod__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ( modulo_modulo_nat @ M @ N )
        = M ) ) ).

% mod_less
thf(fact_1059_mod__mult__self4,axiom,
    ! [B2: nat,C: nat,A: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ B2 @ C ) @ A ) @ B2 )
      = ( modulo_modulo_nat @ A @ B2 ) ) ).

% mod_mult_self4
thf(fact_1060_mod__mult__self4,axiom,
    ! [B2: int,C: int,A: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ B2 @ C ) @ A ) @ B2 )
      = ( modulo_modulo_int @ A @ B2 ) ) ).

% mod_mult_self4
thf(fact_1061_mod__mult__self4,axiom,
    ! [B2: code_integer,C: code_integer,A: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B2 @ C ) @ A ) @ B2 )
      = ( modulo364778990260209775nteger @ A @ B2 ) ) ).

% mod_mult_self4
thf(fact_1062_mod__mult__self3,axiom,
    ! [C: nat,B2: nat,A: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B2 ) @ A ) @ B2 )
      = ( modulo_modulo_nat @ A @ B2 ) ) ).

% mod_mult_self3
thf(fact_1063_mod__mult__self3,axiom,
    ! [C: int,B2: int,A: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ C @ B2 ) @ A ) @ B2 )
      = ( modulo_modulo_int @ A @ B2 ) ) ).

% mod_mult_self3
thf(fact_1064_mod__mult__self3,axiom,
    ! [C: code_integer,B2: code_integer,A: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ C @ B2 ) @ A ) @ B2 )
      = ( modulo364778990260209775nteger @ A @ B2 ) ) ).

% mod_mult_self3
thf(fact_1065_mod__mult__self2,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B2 @ C ) ) @ B2 )
      = ( modulo_modulo_nat @ A @ B2 ) ) ).

% mod_mult_self2
thf(fact_1066_mod__mult__self2,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( times_times_int @ B2 @ C ) ) @ B2 )
      = ( modulo_modulo_int @ A @ B2 ) ) ).

% mod_mult_self2
thf(fact_1067_mod__mult__self2,axiom,
    ! [A: code_integer,B2: code_integer,C: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ B2 @ C ) ) @ B2 )
      = ( modulo364778990260209775nteger @ A @ B2 ) ) ).

% mod_mult_self2
thf(fact_1068_mod__mult__self1,axiom,
    ! [A: nat,C: nat,B2: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C @ B2 ) ) @ B2 )
      = ( modulo_modulo_nat @ A @ B2 ) ) ).

% mod_mult_self1
thf(fact_1069_mod__mult__self1,axiom,
    ! [A: int,C: int,B2: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( times_times_int @ C @ B2 ) ) @ B2 )
      = ( modulo_modulo_int @ A @ B2 ) ) ).

% mod_mult_self1
thf(fact_1070_mod__mult__self1,axiom,
    ! [A: code_integer,C: code_integer,B2: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ C @ B2 ) ) @ B2 )
      = ( modulo364778990260209775nteger @ A @ B2 ) ) ).

% mod_mult_self1
thf(fact_1071_Suc__mod__mult__self1,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M @ ( times_times_nat @ K @ N ) ) ) @ N )
      = ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).

% Suc_mod_mult_self1
thf(fact_1072_Suc__mod__mult__self2,axiom,
    ! [M: nat,N: nat,K: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M @ ( times_times_nat @ N @ K ) ) ) @ N )
      = ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).

% Suc_mod_mult_self2
thf(fact_1073_Suc__mod__mult__self3,axiom,
    ! [K: nat,N: nat,M: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ K @ N ) @ M ) ) @ N )
      = ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).

% Suc_mod_mult_self3
thf(fact_1074_Suc__mod__mult__self4,axiom,
    ! [N: nat,K: nat,M: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ N @ K ) @ M ) ) @ N )
      = ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).

% Suc_mod_mult_self4
thf(fact_1075_bits__one__mod__two__eq__one,axiom,
    ( ( modulo_modulo_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_nat ) ).

% bits_one_mod_two_eq_one
thf(fact_1076_bits__one__mod__two__eq__one,axiom,
    ( ( modulo_modulo_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
    = one_one_int ) ).

% bits_one_mod_two_eq_one
thf(fact_1077_bits__one__mod__two__eq__one,axiom,
    ( ( modulo364778990260209775nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
    = one_one_Code_integer ) ).

% bits_one_mod_two_eq_one
thf(fact_1078_one__mod__two__eq__one,axiom,
    ( ( modulo_modulo_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_nat ) ).

% one_mod_two_eq_one
thf(fact_1079_one__mod__two__eq__one,axiom,
    ( ( modulo_modulo_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
    = one_one_int ) ).

% one_mod_two_eq_one
thf(fact_1080_one__mod__two__eq__one,axiom,
    ( ( modulo364778990260209775nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
    = one_one_Code_integer ) ).

% one_mod_two_eq_one
thf(fact_1081_mod2__Suc__Suc,axiom,
    ! [M: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( suc @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% mod2_Suc_Suc
thf(fact_1082_Suc__times__numeral__mod__eq,axiom,
    ! [K: num,N: nat] :
      ( ( ( numeral_numeral_nat @ K )
       != one_one_nat )
     => ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ K ) @ N ) ) @ ( numeral_numeral_nat @ K ) )
        = one_one_nat ) ) ).

% Suc_times_numeral_mod_eq
thf(fact_1083_mod__mult__right__eq,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( modulo_modulo_nat @ ( times_times_nat @ A @ ( modulo_modulo_nat @ B2 @ C ) ) @ C )
      = ( modulo_modulo_nat @ ( times_times_nat @ A @ B2 ) @ C ) ) ).

% mod_mult_right_eq
thf(fact_1084_mod__mult__right__eq,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( modulo_modulo_int @ ( times_times_int @ A @ ( modulo_modulo_int @ B2 @ C ) ) @ C )
      = ( modulo_modulo_int @ ( times_times_int @ A @ B2 ) @ C ) ) ).

% mod_mult_right_eq
thf(fact_1085_mod__mult__right__eq,axiom,
    ! [A: code_integer,B2: code_integer,C: code_integer] :
      ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ ( modulo364778990260209775nteger @ B2 @ C ) ) @ C )
      = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B2 ) @ C ) ) ).

% mod_mult_right_eq
thf(fact_1086_mod__mult__left__eq,axiom,
    ! [A: nat,C: nat,B2: nat] :
      ( ( modulo_modulo_nat @ ( times_times_nat @ ( modulo_modulo_nat @ A @ C ) @ B2 ) @ C )
      = ( modulo_modulo_nat @ ( times_times_nat @ A @ B2 ) @ C ) ) ).

% mod_mult_left_eq
thf(fact_1087_mod__mult__left__eq,axiom,
    ! [A: int,C: int,B2: int] :
      ( ( modulo_modulo_int @ ( times_times_int @ ( modulo_modulo_int @ A @ C ) @ B2 ) @ C )
      = ( modulo_modulo_int @ ( times_times_int @ A @ B2 ) @ C ) ) ).

% mod_mult_left_eq
thf(fact_1088_mod__mult__left__eq,axiom,
    ! [A: code_integer,C: code_integer,B2: code_integer] :
      ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ C ) @ B2 ) @ C )
      = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B2 ) @ C ) ) ).

% mod_mult_left_eq
thf(fact_1089_mult__mod__right,axiom,
    ! [C: nat,A: nat,B2: nat] :
      ( ( times_times_nat @ C @ ( modulo_modulo_nat @ A @ B2 ) )
      = ( modulo_modulo_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B2 ) ) ) ).

% mult_mod_right
thf(fact_1090_mult__mod__right,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( times_times_int @ C @ ( modulo_modulo_int @ A @ B2 ) )
      = ( modulo_modulo_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) ) ) ).

% mult_mod_right
thf(fact_1091_mult__mod__right,axiom,
    ! [C: code_integer,A: code_integer,B2: code_integer] :
      ( ( times_3573771949741848930nteger @ C @ ( modulo364778990260209775nteger @ A @ B2 ) )
      = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B2 ) ) ) ).

% mult_mod_right
thf(fact_1092_mod__mult__mult2,axiom,
    ! [A: nat,C: nat,B2: nat] :
      ( ( modulo_modulo_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ C ) )
      = ( times_times_nat @ ( modulo_modulo_nat @ A @ B2 ) @ C ) ) ).

% mod_mult_mult2
thf(fact_1093_mod__mult__mult2,axiom,
    ! [A: int,C: int,B2: int] :
      ( ( modulo_modulo_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) )
      = ( times_times_int @ ( modulo_modulo_int @ A @ B2 ) @ C ) ) ).

% mod_mult_mult2
thf(fact_1094_mod__mult__mult2,axiom,
    ! [A: code_integer,C: code_integer,B2: code_integer] :
      ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B2 @ C ) )
      = ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ B2 ) @ C ) ) ).

% mod_mult_mult2
thf(fact_1095_mod__mult__cong,axiom,
    ! [A: nat,C: nat,A2: nat,B2: nat,B: nat] :
      ( ( ( modulo_modulo_nat @ A @ C )
        = ( modulo_modulo_nat @ A2 @ C ) )
     => ( ( ( modulo_modulo_nat @ B2 @ C )
          = ( modulo_modulo_nat @ B @ C ) )
       => ( ( modulo_modulo_nat @ ( times_times_nat @ A @ B2 ) @ C )
          = ( modulo_modulo_nat @ ( times_times_nat @ A2 @ B ) @ C ) ) ) ) ).

% mod_mult_cong
thf(fact_1096_mod__mult__cong,axiom,
    ! [A: int,C: int,A2: int,B2: int,B: int] :
      ( ( ( modulo_modulo_int @ A @ C )
        = ( modulo_modulo_int @ A2 @ C ) )
     => ( ( ( modulo_modulo_int @ B2 @ C )
          = ( modulo_modulo_int @ B @ C ) )
       => ( ( modulo_modulo_int @ ( times_times_int @ A @ B2 ) @ C )
          = ( modulo_modulo_int @ ( times_times_int @ A2 @ B ) @ C ) ) ) ) ).

% mod_mult_cong
thf(fact_1097_mod__mult__cong,axiom,
    ! [A: code_integer,C: code_integer,A2: code_integer,B2: code_integer,B: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ C )
        = ( modulo364778990260209775nteger @ A2 @ C ) )
     => ( ( ( modulo364778990260209775nteger @ B2 @ C )
          = ( modulo364778990260209775nteger @ B @ C ) )
       => ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B2 ) @ C )
          = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A2 @ B ) @ C ) ) ) ) ).

% mod_mult_cong
thf(fact_1098_mod__mult__eq,axiom,
    ! [A: nat,C: nat,B2: nat] :
      ( ( modulo_modulo_nat @ ( times_times_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B2 @ C ) ) @ C )
      = ( modulo_modulo_nat @ ( times_times_nat @ A @ B2 ) @ C ) ) ).

% mod_mult_eq
thf(fact_1099_mod__mult__eq,axiom,
    ! [A: int,C: int,B2: int] :
      ( ( modulo_modulo_int @ ( times_times_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B2 @ C ) ) @ C )
      = ( modulo_modulo_int @ ( times_times_int @ A @ B2 ) @ C ) ) ).

% mod_mult_eq
thf(fact_1100_mod__mult__eq,axiom,
    ! [A: code_integer,C: code_integer,B2: code_integer] :
      ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B2 @ C ) ) @ C )
      = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B2 ) @ C ) ) ).

% mod_mult_eq
thf(fact_1101_mod__add__right__eq,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( modulo_modulo_nat @ B2 @ C ) ) @ C )
      = ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B2 ) @ C ) ) ).

% mod_add_right_eq
thf(fact_1102_mod__add__right__eq,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( modulo_modulo_int @ B2 @ C ) ) @ C )
      = ( modulo_modulo_int @ ( plus_plus_int @ A @ B2 ) @ C ) ) ).

% mod_add_right_eq
thf(fact_1103_mod__add__right__eq,axiom,
    ! [A: code_integer,B2: code_integer,C: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ ( modulo364778990260209775nteger @ B2 @ C ) ) @ C )
      = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B2 ) @ C ) ) ).

% mod_add_right_eq
thf(fact_1104_mod__add__left__eq,axiom,
    ! [A: nat,C: nat,B2: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ B2 ) @ C )
      = ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B2 ) @ C ) ) ).

% mod_add_left_eq
thf(fact_1105_mod__add__left__eq,axiom,
    ! [A: int,C: int,B2: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ B2 ) @ C )
      = ( modulo_modulo_int @ ( plus_plus_int @ A @ B2 ) @ C ) ) ).

% mod_add_left_eq
thf(fact_1106_mod__add__left__eq,axiom,
    ! [A: code_integer,C: code_integer,B2: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ C ) @ B2 ) @ C )
      = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B2 ) @ C ) ) ).

% mod_add_left_eq
thf(fact_1107_mod__add__cong,axiom,
    ! [A: nat,C: nat,A2: nat,B2: nat,B: nat] :
      ( ( ( modulo_modulo_nat @ A @ C )
        = ( modulo_modulo_nat @ A2 @ C ) )
     => ( ( ( modulo_modulo_nat @ B2 @ C )
          = ( modulo_modulo_nat @ B @ C ) )
       => ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B2 ) @ C )
          = ( modulo_modulo_nat @ ( plus_plus_nat @ A2 @ B ) @ C ) ) ) ) ).

% mod_add_cong
thf(fact_1108_mod__add__cong,axiom,
    ! [A: int,C: int,A2: int,B2: int,B: int] :
      ( ( ( modulo_modulo_int @ A @ C )
        = ( modulo_modulo_int @ A2 @ C ) )
     => ( ( ( modulo_modulo_int @ B2 @ C )
          = ( modulo_modulo_int @ B @ C ) )
       => ( ( modulo_modulo_int @ ( plus_plus_int @ A @ B2 ) @ C )
          = ( modulo_modulo_int @ ( plus_plus_int @ A2 @ B ) @ C ) ) ) ) ).

% mod_add_cong
thf(fact_1109_mod__add__cong,axiom,
    ! [A: code_integer,C: code_integer,A2: code_integer,B2: code_integer,B: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ C )
        = ( modulo364778990260209775nteger @ A2 @ C ) )
     => ( ( ( modulo364778990260209775nteger @ B2 @ C )
          = ( modulo364778990260209775nteger @ B @ C ) )
       => ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B2 ) @ C )
          = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A2 @ B ) @ C ) ) ) ) ).

% mod_add_cong
thf(fact_1110_mod__add__eq,axiom,
    ! [A: nat,C: nat,B2: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B2 @ C ) ) @ C )
      = ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B2 ) @ C ) ) ).

% mod_add_eq
thf(fact_1111_mod__add__eq,axiom,
    ! [A: int,C: int,B2: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B2 @ C ) ) @ C )
      = ( modulo_modulo_int @ ( plus_plus_int @ A @ B2 ) @ C ) ) ).

% mod_add_eq
thf(fact_1112_mod__add__eq,axiom,
    ! [A: code_integer,C: code_integer,B2: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B2 @ C ) ) @ C )
      = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B2 ) @ C ) ) ).

% mod_add_eq
thf(fact_1113_mod__diff__right__eq,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( modulo_modulo_int @ ( minus_minus_int @ A @ ( modulo_modulo_int @ B2 @ C ) ) @ C )
      = ( modulo_modulo_int @ ( minus_minus_int @ A @ B2 ) @ C ) ) ).

% mod_diff_right_eq
thf(fact_1114_mod__diff__right__eq,axiom,
    ! [A: code_integer,B2: code_integer,C: code_integer] :
      ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ B2 @ C ) ) @ C )
      = ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B2 ) @ C ) ) ).

% mod_diff_right_eq
thf(fact_1115_mod__diff__left__eq,axiom,
    ! [A: int,C: int,B2: int] :
      ( ( modulo_modulo_int @ ( minus_minus_int @ ( modulo_modulo_int @ A @ C ) @ B2 ) @ C )
      = ( modulo_modulo_int @ ( minus_minus_int @ A @ B2 ) @ C ) ) ).

% mod_diff_left_eq
thf(fact_1116_mod__diff__left__eq,axiom,
    ! [A: code_integer,C: code_integer,B2: code_integer] :
      ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ ( modulo364778990260209775nteger @ A @ C ) @ B2 ) @ C )
      = ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B2 ) @ C ) ) ).

% mod_diff_left_eq
thf(fact_1117_mod__diff__cong,axiom,
    ! [A: int,C: int,A2: int,B2: int,B: int] :
      ( ( ( modulo_modulo_int @ A @ C )
        = ( modulo_modulo_int @ A2 @ C ) )
     => ( ( ( modulo_modulo_int @ B2 @ C )
          = ( modulo_modulo_int @ B @ C ) )
       => ( ( modulo_modulo_int @ ( minus_minus_int @ A @ B2 ) @ C )
          = ( modulo_modulo_int @ ( minus_minus_int @ A2 @ B ) @ C ) ) ) ) ).

% mod_diff_cong
thf(fact_1118_mod__diff__cong,axiom,
    ! [A: code_integer,C: code_integer,A2: code_integer,B2: code_integer,B: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ C )
        = ( modulo364778990260209775nteger @ A2 @ C ) )
     => ( ( ( modulo364778990260209775nteger @ B2 @ C )
          = ( modulo364778990260209775nteger @ B @ C ) )
       => ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B2 ) @ C )
          = ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A2 @ B ) @ C ) ) ) ) ).

% mod_diff_cong
thf(fact_1119_mod__diff__eq,axiom,
    ! [A: int,C: int,B2: int] :
      ( ( modulo_modulo_int @ ( minus_minus_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B2 @ C ) ) @ C )
      = ( modulo_modulo_int @ ( minus_minus_int @ A @ B2 ) @ C ) ) ).

% mod_diff_eq
thf(fact_1120_mod__diff__eq,axiom,
    ! [A: code_integer,C: code_integer,B2: code_integer] :
      ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B2 @ C ) ) @ C )
      = ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B2 ) @ C ) ) ).

% mod_diff_eq
thf(fact_1121_power__mod,axiom,
    ! [A: nat,B2: nat,N: nat] :
      ( ( modulo_modulo_nat @ ( power_power_nat @ ( modulo_modulo_nat @ A @ B2 ) @ N ) @ B2 )
      = ( modulo_modulo_nat @ ( power_power_nat @ A @ N ) @ B2 ) ) ).

% power_mod
thf(fact_1122_power__mod,axiom,
    ! [A: int,B2: int,N: nat] :
      ( ( modulo_modulo_int @ ( power_power_int @ ( modulo_modulo_int @ A @ B2 ) @ N ) @ B2 )
      = ( modulo_modulo_int @ ( power_power_int @ A @ N ) @ B2 ) ) ).

% power_mod
thf(fact_1123_power__mod,axiom,
    ! [A: code_integer,B2: code_integer,N: nat] :
      ( ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ ( modulo364778990260209775nteger @ A @ B2 ) @ N ) @ B2 )
      = ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ A @ N ) @ B2 ) ) ).

% power_mod
thf(fact_1124_mod__Suc__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( suc @ ( modulo_modulo_nat @ M @ N ) ) ) @ N )
      = ( modulo_modulo_nat @ ( suc @ ( suc @ M ) ) @ N ) ) ).

% mod_Suc_Suc_eq
thf(fact_1125_mod__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( modulo_modulo_nat @ M @ N ) ) @ N )
      = ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).

% mod_Suc_eq
thf(fact_1126_mod__less__eq__dividend,axiom,
    ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ N ) @ M ) ).

% mod_less_eq_dividend
thf(fact_1127_cong__exp__iff__simps_I9_J,axiom,
    ! [M: num,Q2: num,N: num] :
      ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) )
      = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q2 ) )
        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(9)
thf(fact_1128_cong__exp__iff__simps_I9_J,axiom,
    ! [M: num,Q2: num,N: num] :
      ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
        = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) )
      = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q2 ) )
        = ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(9)
thf(fact_1129_cong__exp__iff__simps_I9_J,axiom,
    ! [M: num,Q2: num,N: num] :
      ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
        = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) )
      = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ Q2 ) )
        = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(9)
thf(fact_1130_cong__exp__iff__simps_I4_J,axiom,
    ! [M: num,N: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ one ) )
      = ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ one ) ) ) ).

% cong_exp_iff_simps(4)
thf(fact_1131_cong__exp__iff__simps_I4_J,axiom,
    ! [M: num,N: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ one ) )
      = ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ one ) ) ) ).

% cong_exp_iff_simps(4)
thf(fact_1132_cong__exp__iff__simps_I4_J,axiom,
    ! [M: num,N: num] :
      ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ one ) )
      = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ one ) ) ) ).

% cong_exp_iff_simps(4)
thf(fact_1133_mod__geq,axiom,
    ! [M: nat,N: nat] :
      ( ~ ( ord_less_nat @ M @ N )
     => ( ( modulo_modulo_nat @ M @ N )
        = ( modulo_modulo_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ).

% mod_geq
thf(fact_1134_nat__mod__eq__iff,axiom,
    ! [X4: nat,N: nat,Y3: nat] :
      ( ( ( modulo_modulo_nat @ X4 @ N )
        = ( modulo_modulo_nat @ Y3 @ N ) )
      = ( ? [Q1: nat,Q22: nat] :
            ( ( plus_plus_nat @ X4 @ ( times_times_nat @ N @ Q1 ) )
            = ( plus_plus_nat @ Y3 @ ( times_times_nat @ N @ Q22 ) ) ) ) ) ).

% nat_mod_eq_iff
thf(fact_1135_cong__exp__iff__simps_I6_J,axiom,
    ! [Q2: num,N: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
     != ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(6)
thf(fact_1136_cong__exp__iff__simps_I6_J,axiom,
    ! [Q2: num,N: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
     != ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(6)
thf(fact_1137_cong__exp__iff__simps_I6_J,axiom,
    ! [Q2: num,N: num] :
      ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
     != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(6)
thf(fact_1138_cong__exp__iff__simps_I8_J,axiom,
    ! [M: num,Q2: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
     != ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(8)
thf(fact_1139_cong__exp__iff__simps_I8_J,axiom,
    ! [M: num,Q2: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
     != ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(8)
thf(fact_1140_cong__exp__iff__simps_I8_J,axiom,
    ! [M: num,Q2: num] :
      ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
     != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(8)
thf(fact_1141_mod__eqE,axiom,
    ! [A: int,C: int,B2: int] :
      ( ( ( modulo_modulo_int @ A @ C )
        = ( modulo_modulo_int @ B2 @ C ) )
     => ~ ! [D3: int] :
            ( B2
           != ( plus_plus_int @ A @ ( times_times_int @ C @ D3 ) ) ) ) ).

% mod_eqE
thf(fact_1142_mod__eqE,axiom,
    ! [A: code_integer,C: code_integer,B2: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ C )
        = ( modulo364778990260209775nteger @ B2 @ C ) )
     => ~ ! [D3: code_integer] :
            ( B2
           != ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ C @ D3 ) ) ) ) ).

% mod_eqE
thf(fact_1143_div__add1__eq,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B2 ) @ C )
      = ( plus_plus_nat @ ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B2 @ C ) ) @ ( divide_divide_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B2 @ C ) ) @ C ) ) ) ).

% div_add1_eq
thf(fact_1144_div__add1__eq,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( divide_divide_int @ ( plus_plus_int @ A @ B2 ) @ C )
      = ( plus_plus_int @ ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B2 @ C ) ) @ ( divide_divide_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B2 @ C ) ) @ C ) ) ) ).

% div_add1_eq
thf(fact_1145_div__add1__eq,axiom,
    ! [A: code_integer,B2: code_integer,C: code_integer] :
      ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B2 ) @ C )
      = ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B2 @ C ) ) @ ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B2 @ C ) ) @ C ) ) ) ).

% div_add1_eq
thf(fact_1146_mod__induct,axiom,
    ! [P: nat > $o,N: nat,P2: nat,M: nat] :
      ( ( P @ N )
     => ( ( ord_less_nat @ N @ P2 )
       => ( ( ord_less_nat @ M @ P2 )
         => ( ! [N4: nat] :
                ( ( ord_less_nat @ N4 @ P2 )
               => ( ( P @ N4 )
                 => ( P @ ( modulo_modulo_nat @ ( suc @ N4 ) @ P2 ) ) ) )
           => ( P @ M ) ) ) ) ) ).

% mod_induct
thf(fact_1147_mod__Suc__le__divisor,axiom,
    ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ ( suc @ N ) ) @ N ) ).

% mod_Suc_le_divisor
thf(fact_1148_nat__mod__eq__lemma,axiom,
    ! [X4: nat,N: nat,Y3: nat] :
      ( ( ( modulo_modulo_nat @ X4 @ N )
        = ( modulo_modulo_nat @ Y3 @ N ) )
     => ( ( ord_less_eq_nat @ Y3 @ X4 )
       => ? [Q3: nat] :
            ( X4
            = ( plus_plus_nat @ Y3 @ ( times_times_nat @ N @ Q3 ) ) ) ) ) ).

% nat_mod_eq_lemma
thf(fact_1149_mod__eq__nat2E,axiom,
    ! [M: nat,Q2: nat,N: nat] :
      ( ( ( modulo_modulo_nat @ M @ Q2 )
        = ( modulo_modulo_nat @ N @ Q2 ) )
     => ( ( ord_less_eq_nat @ M @ N )
       => ~ ! [S3: nat] :
              ( N
             != ( plus_plus_nat @ M @ ( times_times_nat @ Q2 @ S3 ) ) ) ) ) ).

% mod_eq_nat2E
thf(fact_1150_mod__eq__nat1E,axiom,
    ! [M: nat,Q2: nat,N: nat] :
      ( ( ( modulo_modulo_nat @ M @ Q2 )
        = ( modulo_modulo_nat @ N @ Q2 ) )
     => ( ( ord_less_eq_nat @ N @ M )
       => ~ ! [S3: nat] :
              ( M
             != ( plus_plus_nat @ N @ ( times_times_nat @ Q2 @ S3 ) ) ) ) ) ).

% mod_eq_nat1E
thf(fact_1151_mod__if,axiom,
    ( modulo_modulo_nat
    = ( ^ [M6: nat,N2: nat] : ( if_nat @ ( ord_less_nat @ M6 @ N2 ) @ M6 @ ( modulo_modulo_nat @ ( minus_minus_nat @ M6 @ N2 ) @ N2 ) ) ) ) ).

% mod_if
thf(fact_1152_le__mod__geq,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( modulo_modulo_nat @ M @ N )
        = ( modulo_modulo_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ).

% le_mod_geq
thf(fact_1153_mult__div__mod__eq,axiom,
    ! [B2: nat,A: nat] :
      ( ( plus_plus_nat @ ( times_times_nat @ B2 @ ( divide_divide_nat @ A @ B2 ) ) @ ( modulo_modulo_nat @ A @ B2 ) )
      = A ) ).

% mult_div_mod_eq
thf(fact_1154_mult__div__mod__eq,axiom,
    ! [B2: int,A: int] :
      ( ( plus_plus_int @ ( times_times_int @ B2 @ ( divide_divide_int @ A @ B2 ) ) @ ( modulo_modulo_int @ A @ B2 ) )
      = A ) ).

% mult_div_mod_eq
thf(fact_1155_mult__div__mod__eq,axiom,
    ! [B2: code_integer,A: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B2 @ ( divide6298287555418463151nteger @ A @ B2 ) ) @ ( modulo364778990260209775nteger @ A @ B2 ) )
      = A ) ).

% mult_div_mod_eq
thf(fact_1156_mod__mult__div__eq,axiom,
    ! [A: nat,B2: nat] :
      ( ( plus_plus_nat @ ( modulo_modulo_nat @ A @ B2 ) @ ( times_times_nat @ B2 @ ( divide_divide_nat @ A @ B2 ) ) )
      = A ) ).

% mod_mult_div_eq
thf(fact_1157_mod__mult__div__eq,axiom,
    ! [A: int,B2: int] :
      ( ( plus_plus_int @ ( modulo_modulo_int @ A @ B2 ) @ ( times_times_int @ B2 @ ( divide_divide_int @ A @ B2 ) ) )
      = A ) ).

% mod_mult_div_eq
thf(fact_1158_mod__mult__div__eq,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ B2 ) @ ( times_3573771949741848930nteger @ B2 @ ( divide6298287555418463151nteger @ A @ B2 ) ) )
      = A ) ).

% mod_mult_div_eq
thf(fact_1159_mod__div__mult__eq,axiom,
    ! [A: nat,B2: nat] :
      ( ( plus_plus_nat @ ( modulo_modulo_nat @ A @ B2 ) @ ( times_times_nat @ ( divide_divide_nat @ A @ B2 ) @ B2 ) )
      = A ) ).

% mod_div_mult_eq
thf(fact_1160_mod__div__mult__eq,axiom,
    ! [A: int,B2: int] :
      ( ( plus_plus_int @ ( modulo_modulo_int @ A @ B2 ) @ ( times_times_int @ ( divide_divide_int @ A @ B2 ) @ B2 ) )
      = A ) ).

% mod_div_mult_eq
thf(fact_1161_mod__div__mult__eq,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ B2 ) @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B2 ) @ B2 ) )
      = A ) ).

% mod_div_mult_eq
thf(fact_1162_div__mult__mod__eq,axiom,
    ! [A: nat,B2: nat] :
      ( ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B2 ) @ B2 ) @ ( modulo_modulo_nat @ A @ B2 ) )
      = A ) ).

% div_mult_mod_eq
thf(fact_1163_div__mult__mod__eq,axiom,
    ! [A: int,B2: int] :
      ( ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B2 ) @ B2 ) @ ( modulo_modulo_int @ A @ B2 ) )
      = A ) ).

% div_mult_mod_eq
thf(fact_1164_div__mult__mod__eq,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B2 ) @ B2 ) @ ( modulo364778990260209775nteger @ A @ B2 ) )
      = A ) ).

% div_mult_mod_eq
thf(fact_1165_mod__div__decomp,axiom,
    ! [A: nat,B2: nat] :
      ( A
      = ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B2 ) @ B2 ) @ ( modulo_modulo_nat @ A @ B2 ) ) ) ).

% mod_div_decomp
thf(fact_1166_mod__div__decomp,axiom,
    ! [A: int,B2: int] :
      ( A
      = ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B2 ) @ B2 ) @ ( modulo_modulo_int @ A @ B2 ) ) ) ).

% mod_div_decomp
thf(fact_1167_mod__div__decomp,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( A
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B2 ) @ B2 ) @ ( modulo364778990260209775nteger @ A @ B2 ) ) ) ).

% mod_div_decomp
thf(fact_1168_cancel__div__mod__rules_I1_J,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B2 ) @ B2 ) @ ( modulo_modulo_nat @ A @ B2 ) ) @ C )
      = ( plus_plus_nat @ A @ C ) ) ).

% cancel_div_mod_rules(1)
thf(fact_1169_cancel__div__mod__rules_I1_J,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B2 ) @ B2 ) @ ( modulo_modulo_int @ A @ B2 ) ) @ C )
      = ( plus_plus_int @ A @ C ) ) ).

% cancel_div_mod_rules(1)
thf(fact_1170_cancel__div__mod__rules_I1_J,axiom,
    ! [A: code_integer,B2: code_integer,C: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B2 ) @ B2 ) @ ( modulo364778990260209775nteger @ A @ B2 ) ) @ C )
      = ( plus_p5714425477246183910nteger @ A @ C ) ) ).

% cancel_div_mod_rules(1)
thf(fact_1171_cancel__div__mod__rules_I2_J,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ ( times_times_nat @ B2 @ ( divide_divide_nat @ A @ B2 ) ) @ ( modulo_modulo_nat @ A @ B2 ) ) @ C )
      = ( plus_plus_nat @ A @ C ) ) ).

% cancel_div_mod_rules(2)
thf(fact_1172_cancel__div__mod__rules_I2_J,axiom,
    ! [B2: int,A: int,C: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ B2 @ ( divide_divide_int @ A @ B2 ) ) @ ( modulo_modulo_int @ A @ B2 ) ) @ C )
      = ( plus_plus_int @ A @ C ) ) ).

% cancel_div_mod_rules(2)
thf(fact_1173_cancel__div__mod__rules_I2_J,axiom,
    ! [B2: code_integer,A: code_integer,C: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B2 @ ( divide6298287555418463151nteger @ A @ B2 ) ) @ ( modulo364778990260209775nteger @ A @ B2 ) ) @ C )
      = ( plus_p5714425477246183910nteger @ A @ C ) ) ).

% cancel_div_mod_rules(2)
thf(fact_1174_div__mult1__eq,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( divide_divide_nat @ ( times_times_nat @ A @ B2 ) @ C )
      = ( plus_plus_nat @ ( times_times_nat @ A @ ( divide_divide_nat @ B2 @ C ) ) @ ( divide_divide_nat @ ( times_times_nat @ A @ ( modulo_modulo_nat @ B2 @ C ) ) @ C ) ) ) ).

% div_mult1_eq
thf(fact_1175_div__mult1__eq,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( divide_divide_int @ ( times_times_int @ A @ B2 ) @ C )
      = ( plus_plus_int @ ( times_times_int @ A @ ( divide_divide_int @ B2 @ C ) ) @ ( divide_divide_int @ ( times_times_int @ A @ ( modulo_modulo_int @ B2 @ C ) ) @ C ) ) ) ).

% div_mult1_eq
thf(fact_1176_div__mult1__eq,axiom,
    ! [A: code_integer,B2: code_integer,C: code_integer] :
      ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B2 ) @ C )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B2 @ C ) ) @ ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ ( modulo364778990260209775nteger @ B2 @ C ) ) @ C ) ) ) ).

% div_mult1_eq
thf(fact_1177_minus__mult__div__eq__mod,axiom,
    ! [A: nat,B2: nat] :
      ( ( minus_minus_nat @ A @ ( times_times_nat @ B2 @ ( divide_divide_nat @ A @ B2 ) ) )
      = ( modulo_modulo_nat @ A @ B2 ) ) ).

% minus_mult_div_eq_mod
thf(fact_1178_minus__mult__div__eq__mod,axiom,
    ! [A: int,B2: int] :
      ( ( minus_minus_int @ A @ ( times_times_int @ B2 @ ( divide_divide_int @ A @ B2 ) ) )
      = ( modulo_modulo_int @ A @ B2 ) ) ).

% minus_mult_div_eq_mod
thf(fact_1179_minus__mult__div__eq__mod,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( minus_8373710615458151222nteger @ A @ ( times_3573771949741848930nteger @ B2 @ ( divide6298287555418463151nteger @ A @ B2 ) ) )
      = ( modulo364778990260209775nteger @ A @ B2 ) ) ).

% minus_mult_div_eq_mod
thf(fact_1180_minus__mod__eq__mult__div,axiom,
    ! [A: nat,B2: nat] :
      ( ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B2 ) )
      = ( times_times_nat @ B2 @ ( divide_divide_nat @ A @ B2 ) ) ) ).

% minus_mod_eq_mult_div
thf(fact_1181_minus__mod__eq__mult__div,axiom,
    ! [A: int,B2: int] :
      ( ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B2 ) )
      = ( times_times_int @ B2 @ ( divide_divide_int @ A @ B2 ) ) ) ).

% minus_mod_eq_mult_div
thf(fact_1182_minus__mod__eq__mult__div,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ A @ B2 ) )
      = ( times_3573771949741848930nteger @ B2 @ ( divide6298287555418463151nteger @ A @ B2 ) ) ) ).

% minus_mod_eq_mult_div
thf(fact_1183_minus__mod__eq__div__mult,axiom,
    ! [A: nat,B2: nat] :
      ( ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B2 ) )
      = ( times_times_nat @ ( divide_divide_nat @ A @ B2 ) @ B2 ) ) ).

% minus_mod_eq_div_mult
thf(fact_1184_minus__mod__eq__div__mult,axiom,
    ! [A: int,B2: int] :
      ( ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B2 ) )
      = ( times_times_int @ ( divide_divide_int @ A @ B2 ) @ B2 ) ) ).

% minus_mod_eq_div_mult
thf(fact_1185_minus__mod__eq__div__mult,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ A @ B2 ) )
      = ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B2 ) @ B2 ) ) ).

% minus_mod_eq_div_mult
thf(fact_1186_minus__div__mult__eq__mod,axiom,
    ! [A: nat,B2: nat] :
      ( ( minus_minus_nat @ A @ ( times_times_nat @ ( divide_divide_nat @ A @ B2 ) @ B2 ) )
      = ( modulo_modulo_nat @ A @ B2 ) ) ).

% minus_div_mult_eq_mod
thf(fact_1187_minus__div__mult__eq__mod,axiom,
    ! [A: int,B2: int] :
      ( ( minus_minus_int @ A @ ( times_times_int @ ( divide_divide_int @ A @ B2 ) @ B2 ) )
      = ( modulo_modulo_int @ A @ B2 ) ) ).

% minus_div_mult_eq_mod
thf(fact_1188_minus__div__mult__eq__mod,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( minus_8373710615458151222nteger @ A @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B2 ) @ B2 ) )
      = ( modulo364778990260209775nteger @ A @ B2 ) ) ).

% minus_div_mult_eq_mod
thf(fact_1189_mod__mult2__eq,axiom,
    ! [M: nat,N: nat,Q2: nat] :
      ( ( modulo_modulo_nat @ M @ ( times_times_nat @ N @ Q2 ) )
      = ( plus_plus_nat @ ( times_times_nat @ N @ ( modulo_modulo_nat @ ( divide_divide_nat @ M @ N ) @ Q2 ) ) @ ( modulo_modulo_nat @ M @ N ) ) ) ).

% mod_mult2_eq
thf(fact_1190_div__mod__decomp,axiom,
    ! [A3: nat,N: nat] :
      ( A3
      = ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A3 @ N ) @ N ) @ ( modulo_modulo_nat @ A3 @ N ) ) ) ).

% div_mod_decomp
thf(fact_1191_modulo__nat__def,axiom,
    ( modulo_modulo_nat
    = ( ^ [M6: nat,N2: nat] : ( minus_minus_nat @ M6 @ ( times_times_nat @ ( divide_divide_nat @ M6 @ N2 ) @ N2 ) ) ) ) ).

% modulo_nat_def
thf(fact_1192_linordered__field__no__lb,axiom,
    ! [X5: real] :
    ? [Y4: real] : ( ord_less_real @ Y4 @ X5 ) ).

% linordered_field_no_lb
thf(fact_1193_linordered__field__no__lb,axiom,
    ! [X5: rat] :
    ? [Y4: rat] : ( ord_less_rat @ Y4 @ X5 ) ).

% linordered_field_no_lb
thf(fact_1194_linordered__field__no__ub,axiom,
    ! [X5: real] :
    ? [X_1: real] : ( ord_less_real @ X5 @ X_1 ) ).

% linordered_field_no_ub
thf(fact_1195_linordered__field__no__ub,axiom,
    ! [X5: rat] :
    ? [X_1: rat] : ( ord_less_rat @ X5 @ X_1 ) ).

% linordered_field_no_ub
thf(fact_1196_bounded__Max__nat,axiom,
    ! [P: nat > $o,X4: nat,M7: nat] :
      ( ( P @ X4 )
     => ( ! [X3: nat] :
            ( ( P @ X3 )
           => ( ord_less_eq_nat @ X3 @ M7 ) )
       => ~ ! [M4: nat] :
              ( ( P @ M4 )
             => ~ ! [X5: nat] :
                    ( ( P @ X5 )
                   => ( ord_less_eq_nat @ X5 @ M4 ) ) ) ) ) ).

% bounded_Max_nat
thf(fact_1197_div__exp__mod__exp__eq,axiom,
    ! [A: nat,N: nat,M: nat] :
      ( ( modulo_modulo_nat @ ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
      = ( divide_divide_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% div_exp_mod_exp_eq
thf(fact_1198_div__exp__mod__exp__eq,axiom,
    ! [A: int,N: nat,M: nat] :
      ( ( modulo_modulo_int @ ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
      = ( divide_divide_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).

% div_exp_mod_exp_eq
thf(fact_1199_div__exp__mod__exp__eq,axiom,
    ! [A: code_integer,N: nat,M: nat] :
      ( ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
      = ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ).

% div_exp_mod_exp_eq
thf(fact_1200_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,N: nat,A: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( modulo_modulo_nat @ ( times_times_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
        = ( times_times_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) ).

% mult_exp_mod_exp_eq
thf(fact_1201_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,N: nat,A: int] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( modulo_modulo_int @ ( times_times_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
        = ( times_times_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) ) ) ).

% mult_exp_mod_exp_eq
thf(fact_1202_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,N: nat,A: code_integer] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
        = ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) ) ) ).

% mult_exp_mod_exp_eq
thf(fact_1203_divide__divide__eq__left_H,axiom,
    ! [A: complex,B2: complex,C: complex] :
      ( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ A @ B2 ) @ C )
      = ( divide1717551699836669952omplex @ A @ ( times_times_complex @ C @ B2 ) ) ) ).

% divide_divide_eq_left'
thf(fact_1204_divide__divide__eq__left_H,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( divide_divide_real @ ( divide_divide_real @ A @ B2 ) @ C )
      = ( divide_divide_real @ A @ ( times_times_real @ C @ B2 ) ) ) ).

% divide_divide_eq_left'
thf(fact_1205_divide__divide__eq__left_H,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( divide_divide_rat @ ( divide_divide_rat @ A @ B2 ) @ C )
      = ( divide_divide_rat @ A @ ( times_times_rat @ C @ B2 ) ) ) ).

% divide_divide_eq_left'
thf(fact_1206_divide__divide__times__eq,axiom,
    ! [X4: complex,Y3: complex,Z: complex,W: complex] :
      ( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ X4 @ Y3 ) @ ( divide1717551699836669952omplex @ Z @ W ) )
      = ( divide1717551699836669952omplex @ ( times_times_complex @ X4 @ W ) @ ( times_times_complex @ Y3 @ Z ) ) ) ).

% divide_divide_times_eq
thf(fact_1207_divide__divide__times__eq,axiom,
    ! [X4: real,Y3: real,Z: real,W: real] :
      ( ( divide_divide_real @ ( divide_divide_real @ X4 @ Y3 ) @ ( divide_divide_real @ Z @ W ) )
      = ( divide_divide_real @ ( times_times_real @ X4 @ W ) @ ( times_times_real @ Y3 @ Z ) ) ) ).

% divide_divide_times_eq
thf(fact_1208_divide__divide__times__eq,axiom,
    ! [X4: rat,Y3: rat,Z: rat,W: rat] :
      ( ( divide_divide_rat @ ( divide_divide_rat @ X4 @ Y3 ) @ ( divide_divide_rat @ Z @ W ) )
      = ( divide_divide_rat @ ( times_times_rat @ X4 @ W ) @ ( times_times_rat @ Y3 @ Z ) ) ) ).

% divide_divide_times_eq
thf(fact_1209_times__divide__times__eq,axiom,
    ! [X4: complex,Y3: complex,Z: complex,W: complex] :
      ( ( times_times_complex @ ( divide1717551699836669952omplex @ X4 @ Y3 ) @ ( divide1717551699836669952omplex @ Z @ W ) )
      = ( divide1717551699836669952omplex @ ( times_times_complex @ X4 @ Z ) @ ( times_times_complex @ Y3 @ W ) ) ) ).

% times_divide_times_eq
thf(fact_1210_times__divide__times__eq,axiom,
    ! [X4: real,Y3: real,Z: real,W: real] :
      ( ( times_times_real @ ( divide_divide_real @ X4 @ Y3 ) @ ( divide_divide_real @ Z @ W ) )
      = ( divide_divide_real @ ( times_times_real @ X4 @ Z ) @ ( times_times_real @ Y3 @ W ) ) ) ).

% times_divide_times_eq
thf(fact_1211_times__divide__times__eq,axiom,
    ! [X4: rat,Y3: rat,Z: rat,W: rat] :
      ( ( times_times_rat @ ( divide_divide_rat @ X4 @ Y3 ) @ ( divide_divide_rat @ Z @ W ) )
      = ( divide_divide_rat @ ( times_times_rat @ X4 @ Z ) @ ( times_times_rat @ Y3 @ W ) ) ) ).

% times_divide_times_eq
thf(fact_1212_add__divide__distrib,axiom,
    ! [A: complex,B2: complex,C: complex] :
      ( ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ B2 ) @ C )
      = ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ C ) @ ( divide1717551699836669952omplex @ B2 @ C ) ) ) ).

% add_divide_distrib
thf(fact_1213_add__divide__distrib,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( divide_divide_real @ ( plus_plus_real @ A @ B2 ) @ C )
      = ( plus_plus_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B2 @ C ) ) ) ).

% add_divide_distrib
thf(fact_1214_add__divide__distrib,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( divide_divide_rat @ ( plus_plus_rat @ A @ B2 ) @ C )
      = ( plus_plus_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B2 @ C ) ) ) ).

% add_divide_distrib
thf(fact_1215_diff__divide__distrib,axiom,
    ! [A: complex,B2: complex,C: complex] :
      ( ( divide1717551699836669952omplex @ ( minus_minus_complex @ A @ B2 ) @ C )
      = ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ C ) @ ( divide1717551699836669952omplex @ B2 @ C ) ) ) ).

% diff_divide_distrib
thf(fact_1216_diff__divide__distrib,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( divide_divide_real @ ( minus_minus_real @ A @ B2 ) @ C )
      = ( minus_minus_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B2 @ C ) ) ) ).

% diff_divide_distrib
thf(fact_1217_diff__divide__distrib,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( divide_divide_rat @ ( minus_minus_rat @ A @ B2 ) @ C )
      = ( minus_minus_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B2 @ C ) ) ) ).

% diff_divide_distrib
thf(fact_1218_unset__bit__Suc,axiom,
    ! [N: nat,A: code_integer] :
      ( ( bit_se8260200283734997820nteger @ ( suc @ N ) @ A )
      = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se8260200283734997820nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% unset_bit_Suc
thf(fact_1219_unset__bit__Suc,axiom,
    ! [N: nat,A: int] :
      ( ( bit_se4203085406695923979it_int @ ( suc @ N ) @ A )
      = ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se4203085406695923979it_int @ N @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).

% unset_bit_Suc
thf(fact_1220_unset__bit__Suc,axiom,
    ! [N: nat,A: nat] :
      ( ( bit_se4205575877204974255it_nat @ ( suc @ N ) @ A )
      = ( plus_plus_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se4205575877204974255it_nat @ N @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% unset_bit_Suc
thf(fact_1221_flip__bit__Suc,axiom,
    ! [N: nat,A: code_integer] :
      ( ( bit_se1345352211410354436nteger @ ( suc @ N ) @ A )
      = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1345352211410354436nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% flip_bit_Suc
thf(fact_1222_flip__bit__Suc,axiom,
    ! [N: nat,A: int] :
      ( ( bit_se2159334234014336723it_int @ ( suc @ N ) @ A )
      = ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2159334234014336723it_int @ N @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).

% flip_bit_Suc
thf(fact_1223_flip__bit__Suc,axiom,
    ! [N: nat,A: nat] :
      ( ( bit_se2161824704523386999it_nat @ ( suc @ N ) @ A )
      = ( plus_plus_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2161824704523386999it_nat @ N @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% flip_bit_Suc
thf(fact_1224_set__bit__Suc,axiom,
    ! [N: nat,A: code_integer] :
      ( ( bit_se2793503036327961859nteger @ ( suc @ N ) @ A )
      = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se2793503036327961859nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% set_bit_Suc
thf(fact_1225_set__bit__Suc,axiom,
    ! [N: nat,A: int] :
      ( ( bit_se7879613467334960850it_int @ ( suc @ N ) @ A )
      = ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se7879613467334960850it_int @ N @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).

% set_bit_Suc
thf(fact_1226_set__bit__Suc,axiom,
    ! [N: nat,A: nat] :
      ( ( bit_se7882103937844011126it_nat @ ( suc @ N ) @ A )
      = ( plus_plus_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se7882103937844011126it_nat @ N @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% set_bit_Suc
thf(fact_1227_dbl__simps_I3_J,axiom,
    ( ( neg_nu7009210354673126013omplex @ one_one_complex )
    = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).

% dbl_simps(3)
thf(fact_1228_dbl__simps_I3_J,axiom,
    ( ( neg_numeral_dbl_real @ one_one_real )
    = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).

% dbl_simps(3)
thf(fact_1229_dbl__simps_I3_J,axiom,
    ( ( neg_numeral_dbl_rat @ one_one_rat )
    = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).

% dbl_simps(3)
thf(fact_1230_dbl__simps_I3_J,axiom,
    ( ( neg_numeral_dbl_int @ one_one_int )
    = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).

% dbl_simps(3)
thf(fact_1231_divmod__digit__1_I1_J,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B2 )
       => ( ( ord_le3102999989581377725nteger @ B2 @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) ) )
         => ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) ) ) @ one_one_Code_integer )
            = ( divide6298287555418463151nteger @ A @ B2 ) ) ) ) ) ).

% divmod_digit_1(1)
thf(fact_1232_divmod__digit__1_I1_J,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ B2 )
       => ( ( ord_less_eq_nat @ B2 @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) )
         => ( ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) ) @ one_one_nat )
            = ( divide_divide_nat @ A @ B2 ) ) ) ) ) ).

% divmod_digit_1(1)
thf(fact_1233_divmod__digit__1_I1_J,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ zero_zero_int @ B2 )
       => ( ( ord_less_eq_int @ B2 @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) )
         => ( ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) ) @ one_one_int )
            = ( divide_divide_int @ A @ B2 ) ) ) ) ) ).

% divmod_digit_1(1)
thf(fact_1234_signed__take__bit__Suc,axiom,
    ! [N: nat,A: code_integer] :
      ( ( bit_ri6519982836138164636nteger @ ( suc @ N ) @ A )
      = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% signed_take_bit_Suc
thf(fact_1235_signed__take__bit__Suc,axiom,
    ! [N: nat,A: int] :
      ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ A )
      = ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ N @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).

% signed_take_bit_Suc
thf(fact_1236_power__numeral,axiom,
    ! [K: num,L2: num] :
      ( ( power_power_complex @ ( numera6690914467698888265omplex @ K ) @ ( numeral_numeral_nat @ L2 ) )
      = ( numera6690914467698888265omplex @ ( pow @ K @ L2 ) ) ) ).

% power_numeral
thf(fact_1237_power__numeral,axiom,
    ! [K: num,L2: num] :
      ( ( power_power_real @ ( numeral_numeral_real @ K ) @ ( numeral_numeral_nat @ L2 ) )
      = ( numeral_numeral_real @ ( pow @ K @ L2 ) ) ) ).

% power_numeral
thf(fact_1238_power__numeral,axiom,
    ! [K: num,L2: num] :
      ( ( power_power_rat @ ( numeral_numeral_rat @ K ) @ ( numeral_numeral_nat @ L2 ) )
      = ( numeral_numeral_rat @ ( pow @ K @ L2 ) ) ) ).

% power_numeral
thf(fact_1239_power__numeral,axiom,
    ! [K: num,L2: num] :
      ( ( power_power_nat @ ( numeral_numeral_nat @ K ) @ ( numeral_numeral_nat @ L2 ) )
      = ( numeral_numeral_nat @ ( pow @ K @ L2 ) ) ) ).

% power_numeral
thf(fact_1240_power__numeral,axiom,
    ! [K: num,L2: num] :
      ( ( power_power_int @ ( numeral_numeral_int @ K ) @ ( numeral_numeral_nat @ L2 ) )
      = ( numeral_numeral_int @ ( pow @ K @ L2 ) ) ) ).

% power_numeral
thf(fact_1241_arith__geo__mean,axiom,
    ! [U: real,X4: real,Y3: real] :
      ( ( ( power_power_real @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( times_times_real @ X4 @ Y3 ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ X4 )
       => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
         => ( ord_less_eq_real @ U @ ( divide_divide_real @ ( plus_plus_real @ X4 @ Y3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).

% arith_geo_mean
thf(fact_1242_arith__geo__mean,axiom,
    ! [U: rat,X4: rat,Y3: rat] :
      ( ( ( power_power_rat @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( times_times_rat @ X4 @ Y3 ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ X4 )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
         => ( ord_less_eq_rat @ U @ ( divide_divide_rat @ ( plus_plus_rat @ X4 @ Y3 ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% arith_geo_mean
thf(fact_1243_odd__two__times__div__two__nat,axiom,
    ! [N: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = ( minus_minus_nat @ N @ one_one_nat ) ) ) ).

% odd_two_times_div_two_nat
thf(fact_1244_less__nat__zero__code,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% less_nat_zero_code
thf(fact_1245_neq0__conv,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
      = ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% neq0_conv
thf(fact_1246_bot__nat__0_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
      = ( ord_less_nat @ zero_zero_nat @ A ) ) ).

% bot_nat_0.not_eq_extremum
thf(fact_1247_le0,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).

% le0
thf(fact_1248_bot__nat__0_Oextremum,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).

% bot_nat_0.extremum
thf(fact_1249_Nat_Oadd__0__right,axiom,
    ! [M: nat] :
      ( ( plus_plus_nat @ M @ zero_zero_nat )
      = M ) ).

% Nat.add_0_right
thf(fact_1250_add__is__0,axiom,
    ! [M: nat,N: nat] :
      ( ( ( plus_plus_nat @ M @ N )
        = zero_zero_nat )
      = ( ( M = zero_zero_nat )
        & ( N = zero_zero_nat ) ) ) ).

% add_is_0
thf(fact_1251_diff__self__eq__0,axiom,
    ! [M: nat] :
      ( ( minus_minus_nat @ M @ M )
      = zero_zero_nat ) ).

% diff_self_eq_0
thf(fact_1252_diff__0__eq__0,axiom,
    ! [N: nat] :
      ( ( minus_minus_nat @ zero_zero_nat @ N )
      = zero_zero_nat ) ).

% diff_0_eq_0
thf(fact_1253_mult__cancel2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( ( times_times_nat @ M @ K )
        = ( times_times_nat @ N @ K ) )
      = ( ( M = N )
        | ( K = zero_zero_nat ) ) ) ).

% mult_cancel2
thf(fact_1254_mult__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ( times_times_nat @ K @ M )
        = ( times_times_nat @ K @ N ) )
      = ( ( M = N )
        | ( K = zero_zero_nat ) ) ) ).

% mult_cancel1
thf(fact_1255_mult__0__right,axiom,
    ! [M: nat] :
      ( ( times_times_nat @ M @ zero_zero_nat )
      = zero_zero_nat ) ).

% mult_0_right
thf(fact_1256_mult__is__0,axiom,
    ! [M: nat,N: nat] :
      ( ( ( times_times_nat @ M @ N )
        = zero_zero_nat )
      = ( ( M = zero_zero_nat )
        | ( N = zero_zero_nat ) ) ) ).

% mult_is_0
thf(fact_1257_div__pos__pos__trivial,axiom,
    ! [K: int,L2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ( ( ord_less_int @ K @ L2 )
       => ( ( divide_divide_int @ K @ L2 )
          = zero_zero_int ) ) ) ).

% div_pos_pos_trivial
thf(fact_1258_div__neg__neg__trivial,axiom,
    ! [K: int,L2: int] :
      ( ( ord_less_eq_int @ K @ zero_zero_int )
     => ( ( ord_less_int @ L2 @ K )
       => ( ( divide_divide_int @ K @ L2 )
          = zero_zero_int ) ) ) ).

% div_neg_neg_trivial
thf(fact_1259_i0__less,axiom,
    ! [N: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N )
      = ( N != zero_z5237406670263579293d_enat ) ) ).

% i0_less
thf(fact_1260_idiff__0,axiom,
    ! [N: extended_enat] :
      ( ( minus_3235023915231533773d_enat @ zero_z5237406670263579293d_enat @ N )
      = zero_z5237406670263579293d_enat ) ).

% idiff_0
thf(fact_1261_idiff__0__right,axiom,
    ! [N: extended_enat] :
      ( ( minus_3235023915231533773d_enat @ N @ zero_z5237406670263579293d_enat )
      = N ) ).

% idiff_0_right
thf(fact_1262_le__zero__eq,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ N @ zero_zero_nat )
      = ( N = zero_zero_nat ) ) ).

% le_zero_eq
thf(fact_1263_not__gr__zero,axiom,
    ! [N: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
      = ( N = zero_zero_nat ) ) ).

% not_gr_zero
thf(fact_1264_mult__cancel__right,axiom,
    ! [A: real,C: real,B2: real] :
      ( ( ( times_times_real @ A @ C )
        = ( times_times_real @ B2 @ C ) )
      = ( ( C = zero_zero_real )
        | ( A = B2 ) ) ) ).

% mult_cancel_right
thf(fact_1265_mult__cancel__right,axiom,
    ! [A: rat,C: rat,B2: rat] :
      ( ( ( times_times_rat @ A @ C )
        = ( times_times_rat @ B2 @ C ) )
      = ( ( C = zero_zero_rat )
        | ( A = B2 ) ) ) ).

% mult_cancel_right
thf(fact_1266_mult__cancel__right,axiom,
    ! [A: nat,C: nat,B2: nat] :
      ( ( ( times_times_nat @ A @ C )
        = ( times_times_nat @ B2 @ C ) )
      = ( ( C = zero_zero_nat )
        | ( A = B2 ) ) ) ).

% mult_cancel_right
thf(fact_1267_mult__cancel__right,axiom,
    ! [A: int,C: int,B2: int] :
      ( ( ( times_times_int @ A @ C )
        = ( times_times_int @ B2 @ C ) )
      = ( ( C = zero_zero_int )
        | ( A = B2 ) ) ) ).

% mult_cancel_right
thf(fact_1268_mult__cancel__left,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( ( times_times_real @ C @ A )
        = ( times_times_real @ C @ B2 ) )
      = ( ( C = zero_zero_real )
        | ( A = B2 ) ) ) ).

% mult_cancel_left
thf(fact_1269_mult__cancel__left,axiom,
    ! [C: rat,A: rat,B2: rat] :
      ( ( ( times_times_rat @ C @ A )
        = ( times_times_rat @ C @ B2 ) )
      = ( ( C = zero_zero_rat )
        | ( A = B2 ) ) ) ).

% mult_cancel_left
thf(fact_1270_mult__cancel__left,axiom,
    ! [C: nat,A: nat,B2: nat] :
      ( ( ( times_times_nat @ C @ A )
        = ( times_times_nat @ C @ B2 ) )
      = ( ( C = zero_zero_nat )
        | ( A = B2 ) ) ) ).

% mult_cancel_left
thf(fact_1271_mult__cancel__left,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( ( times_times_int @ C @ A )
        = ( times_times_int @ C @ B2 ) )
      = ( ( C = zero_zero_int )
        | ( A = B2 ) ) ) ).

% mult_cancel_left
thf(fact_1272_mult__eq__0__iff,axiom,
    ! [A: real,B2: real] :
      ( ( ( times_times_real @ A @ B2 )
        = zero_zero_real )
      = ( ( A = zero_zero_real )
        | ( B2 = zero_zero_real ) ) ) ).

% mult_eq_0_iff
thf(fact_1273_mult__eq__0__iff,axiom,
    ! [A: rat,B2: rat] :
      ( ( ( times_times_rat @ A @ B2 )
        = zero_zero_rat )
      = ( ( A = zero_zero_rat )
        | ( B2 = zero_zero_rat ) ) ) ).

% mult_eq_0_iff
thf(fact_1274_mult__eq__0__iff,axiom,
    ! [A: nat,B2: nat] :
      ( ( ( times_times_nat @ A @ B2 )
        = zero_zero_nat )
      = ( ( A = zero_zero_nat )
        | ( B2 = zero_zero_nat ) ) ) ).

% mult_eq_0_iff
thf(fact_1275_mult__eq__0__iff,axiom,
    ! [A: int,B2: int] :
      ( ( ( times_times_int @ A @ B2 )
        = zero_zero_int )
      = ( ( A = zero_zero_int )
        | ( B2 = zero_zero_int ) ) ) ).

% mult_eq_0_iff
thf(fact_1276_mult__zero__right,axiom,
    ! [A: real] :
      ( ( times_times_real @ A @ zero_zero_real )
      = zero_zero_real ) ).

% mult_zero_right
thf(fact_1277_mult__zero__right,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ A @ zero_zero_rat )
      = zero_zero_rat ) ).

% mult_zero_right
thf(fact_1278_mult__zero__right,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ zero_zero_nat )
      = zero_zero_nat ) ).

% mult_zero_right
thf(fact_1279_mult__zero__right,axiom,
    ! [A: int] :
      ( ( times_times_int @ A @ zero_zero_int )
      = zero_zero_int ) ).

% mult_zero_right
thf(fact_1280_mult__zero__left,axiom,
    ! [A: real] :
      ( ( times_times_real @ zero_zero_real @ A )
      = zero_zero_real ) ).

% mult_zero_left
thf(fact_1281_mult__zero__left,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ zero_zero_rat @ A )
      = zero_zero_rat ) ).

% mult_zero_left
thf(fact_1282_mult__zero__left,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% mult_zero_left
thf(fact_1283_mult__zero__left,axiom,
    ! [A: int] :
      ( ( times_times_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% mult_zero_left
thf(fact_1284_add__0,axiom,
    ! [A: literal] :
      ( ( plus_plus_literal @ zero_zero_literal @ A )
      = A ) ).

% add_0
thf(fact_1285_add__0,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ zero_zero_real @ A )
      = A ) ).

% add_0
thf(fact_1286_add__0,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ zero_zero_rat @ A )
      = A ) ).

% add_0
thf(fact_1287_add__0,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ A )
      = A ) ).

% add_0
thf(fact_1288_add__0,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ zero_zero_int @ A )
      = A ) ).

% add_0
thf(fact_1289_zero__eq__add__iff__both__eq__0,axiom,
    ! [X4: nat,Y3: nat] :
      ( ( zero_zero_nat
        = ( plus_plus_nat @ X4 @ Y3 ) )
      = ( ( X4 = zero_zero_nat )
        & ( Y3 = zero_zero_nat ) ) ) ).

% zero_eq_add_iff_both_eq_0
thf(fact_1290_add__eq__0__iff__both__eq__0,axiom,
    ! [X4: nat,Y3: nat] :
      ( ( ( plus_plus_nat @ X4 @ Y3 )
        = zero_zero_nat )
      = ( ( X4 = zero_zero_nat )
        & ( Y3 = zero_zero_nat ) ) ) ).

% add_eq_0_iff_both_eq_0
thf(fact_1291_add__cancel__right__right,axiom,
    ! [A: real,B2: real] :
      ( ( A
        = ( plus_plus_real @ A @ B2 ) )
      = ( B2 = zero_zero_real ) ) ).

% add_cancel_right_right
thf(fact_1292_add__cancel__right__right,axiom,
    ! [A: rat,B2: rat] :
      ( ( A
        = ( plus_plus_rat @ A @ B2 ) )
      = ( B2 = zero_zero_rat ) ) ).

% add_cancel_right_right
thf(fact_1293_add__cancel__right__right,axiom,
    ! [A: nat,B2: nat] :
      ( ( A
        = ( plus_plus_nat @ A @ B2 ) )
      = ( B2 = zero_zero_nat ) ) ).

% add_cancel_right_right
thf(fact_1294_add__cancel__right__right,axiom,
    ! [A: int,B2: int] :
      ( ( A
        = ( plus_plus_int @ A @ B2 ) )
      = ( B2 = zero_zero_int ) ) ).

% add_cancel_right_right
thf(fact_1295_add__cancel__right__left,axiom,
    ! [A: real,B2: real] :
      ( ( A
        = ( plus_plus_real @ B2 @ A ) )
      = ( B2 = zero_zero_real ) ) ).

% add_cancel_right_left
thf(fact_1296_add__cancel__right__left,axiom,
    ! [A: rat,B2: rat] :
      ( ( A
        = ( plus_plus_rat @ B2 @ A ) )
      = ( B2 = zero_zero_rat ) ) ).

% add_cancel_right_left
thf(fact_1297_add__cancel__right__left,axiom,
    ! [A: nat,B2: nat] :
      ( ( A
        = ( plus_plus_nat @ B2 @ A ) )
      = ( B2 = zero_zero_nat ) ) ).

% add_cancel_right_left
thf(fact_1298_add__cancel__right__left,axiom,
    ! [A: int,B2: int] :
      ( ( A
        = ( plus_plus_int @ B2 @ A ) )
      = ( B2 = zero_zero_int ) ) ).

% add_cancel_right_left
thf(fact_1299_add__cancel__left__right,axiom,
    ! [A: real,B2: real] :
      ( ( ( plus_plus_real @ A @ B2 )
        = A )
      = ( B2 = zero_zero_real ) ) ).

% add_cancel_left_right
thf(fact_1300_add__cancel__left__right,axiom,
    ! [A: rat,B2: rat] :
      ( ( ( plus_plus_rat @ A @ B2 )
        = A )
      = ( B2 = zero_zero_rat ) ) ).

% add_cancel_left_right
thf(fact_1301_add__cancel__left__right,axiom,
    ! [A: nat,B2: nat] :
      ( ( ( plus_plus_nat @ A @ B2 )
        = A )
      = ( B2 = zero_zero_nat ) ) ).

% add_cancel_left_right
thf(fact_1302_add__cancel__left__right,axiom,
    ! [A: int,B2: int] :
      ( ( ( plus_plus_int @ A @ B2 )
        = A )
      = ( B2 = zero_zero_int ) ) ).

% add_cancel_left_right
thf(fact_1303_add__cancel__left__left,axiom,
    ! [B2: real,A: real] :
      ( ( ( plus_plus_real @ B2 @ A )
        = A )
      = ( B2 = zero_zero_real ) ) ).

% add_cancel_left_left
thf(fact_1304_add__cancel__left__left,axiom,
    ! [B2: rat,A: rat] :
      ( ( ( plus_plus_rat @ B2 @ A )
        = A )
      = ( B2 = zero_zero_rat ) ) ).

% add_cancel_left_left
thf(fact_1305_add__cancel__left__left,axiom,
    ! [B2: nat,A: nat] :
      ( ( ( plus_plus_nat @ B2 @ A )
        = A )
      = ( B2 = zero_zero_nat ) ) ).

% add_cancel_left_left
thf(fact_1306_add__cancel__left__left,axiom,
    ! [B2: int,A: int] :
      ( ( ( plus_plus_int @ B2 @ A )
        = A )
      = ( B2 = zero_zero_int ) ) ).

% add_cancel_left_left
thf(fact_1307_double__zero__sym,axiom,
    ! [A: real] :
      ( ( zero_zero_real
        = ( plus_plus_real @ A @ A ) )
      = ( A = zero_zero_real ) ) ).

% double_zero_sym
thf(fact_1308_double__zero__sym,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat
        = ( plus_plus_rat @ A @ A ) )
      = ( A = zero_zero_rat ) ) ).

% double_zero_sym
thf(fact_1309_double__zero__sym,axiom,
    ! [A: int] :
      ( ( zero_zero_int
        = ( plus_plus_int @ A @ A ) )
      = ( A = zero_zero_int ) ) ).

% double_zero_sym
thf(fact_1310_add_Oright__neutral,axiom,
    ! [A: literal] :
      ( ( plus_plus_literal @ A @ zero_zero_literal )
      = A ) ).

% add.right_neutral
thf(fact_1311_add_Oright__neutral,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ A @ zero_zero_real )
      = A ) ).

% add.right_neutral
thf(fact_1312_add_Oright__neutral,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ A @ zero_zero_rat )
      = A ) ).

% add.right_neutral
thf(fact_1313_add_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% add.right_neutral
thf(fact_1314_add_Oright__neutral,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ zero_zero_int )
      = A ) ).

% add.right_neutral
thf(fact_1315_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: real] :
      ( ( minus_minus_real @ A @ A )
      = zero_zero_real ) ).

% cancel_comm_monoid_add_class.diff_cancel
thf(fact_1316_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
thf(fact_1317_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: nat] :
      ( ( minus_minus_nat @ A @ A )
      = zero_zero_nat ) ).

% cancel_comm_monoid_add_class.diff_cancel
thf(fact_1318_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ A @ A )
      = zero_zero_int ) ).

% cancel_comm_monoid_add_class.diff_cancel
thf(fact_1319_diff__zero,axiom,
    ! [A: real] :
      ( ( minus_minus_real @ A @ zero_zero_real )
      = A ) ).

% diff_zero
thf(fact_1320_diff__zero,axiom,
    ! [A: rat] :
      ( ( minus_minus_rat @ A @ zero_zero_rat )
      = A ) ).

% diff_zero
thf(fact_1321_diff__zero,axiom,
    ! [A: nat] :
      ( ( minus_minus_nat @ A @ zero_zero_nat )
      = A ) ).

% diff_zero
thf(fact_1322_diff__zero,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ A @ zero_zero_int )
      = A ) ).

% diff_zero
thf(fact_1323_zero__diff,axiom,
    ! [A: nat] :
      ( ( minus_minus_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% zero_diff
thf(fact_1324_diff__0__right,axiom,
    ! [A: real] :
      ( ( minus_minus_real @ A @ zero_zero_real )
      = A ) ).

% diff_0_right
thf(fact_1325_diff__0__right,axiom,
    ! [A: rat] :
      ( ( minus_minus_rat @ A @ zero_zero_rat )
      = A ) ).

% diff_0_right
thf(fact_1326_diff__0__right,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ A @ zero_zero_int )
      = A ) ).

% diff_0_right
thf(fact_1327_diff__self,axiom,
    ! [A: real] :
      ( ( minus_minus_real @ A @ A )
      = zero_zero_real ) ).

% diff_self
thf(fact_1328_diff__self,axiom,
    ! [A: rat] :
      ( ( minus_minus_rat @ A @ A )
      = zero_zero_rat ) ).

% diff_self
thf(fact_1329_diff__self,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ A @ A )
      = zero_zero_int ) ).

% diff_self
thf(fact_1330_div__0,axiom,
    ! [A: complex] :
      ( ( divide1717551699836669952omplex @ zero_zero_complex @ A )
      = zero_zero_complex ) ).

% div_0
thf(fact_1331_div__0,axiom,
    ! [A: real] :
      ( ( divide_divide_real @ zero_zero_real @ A )
      = zero_zero_real ) ).

% div_0
thf(fact_1332_div__0,axiom,
    ! [A: rat] :
      ( ( divide_divide_rat @ zero_zero_rat @ A )
      = zero_zero_rat ) ).

% div_0
thf(fact_1333_div__0,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% div_0
thf(fact_1334_div__0,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% div_0
thf(fact_1335_div__by__0,axiom,
    ! [A: complex] :
      ( ( divide1717551699836669952omplex @ A @ zero_zero_complex )
      = zero_zero_complex ) ).

% div_by_0
thf(fact_1336_div__by__0,axiom,
    ! [A: real] :
      ( ( divide_divide_real @ A @ zero_zero_real )
      = zero_zero_real ) ).

% div_by_0
thf(fact_1337_div__by__0,axiom,
    ! [A: rat] :
      ( ( divide_divide_rat @ A @ zero_zero_rat )
      = zero_zero_rat ) ).

% div_by_0
thf(fact_1338_div__by__0,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ A @ zero_zero_nat )
      = zero_zero_nat ) ).

% div_by_0
thf(fact_1339_div__by__0,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ A @ zero_zero_int )
      = zero_zero_int ) ).

% div_by_0
thf(fact_1340_bits__div__0,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% bits_div_0
thf(fact_1341_bits__div__0,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% bits_div_0
thf(fact_1342_bits__div__by__0,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ A @ zero_zero_nat )
      = zero_zero_nat ) ).

% bits_div_by_0
thf(fact_1343_bits__div__by__0,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ A @ zero_zero_int )
      = zero_zero_int ) ).

% bits_div_by_0
thf(fact_1344_divide__eq__0__iff,axiom,
    ! [A: complex,B2: complex] :
      ( ( ( divide1717551699836669952omplex @ A @ B2 )
        = zero_zero_complex )
      = ( ( A = zero_zero_complex )
        | ( B2 = zero_zero_complex ) ) ) ).

% divide_eq_0_iff
thf(fact_1345_divide__eq__0__iff,axiom,
    ! [A: real,B2: real] :
      ( ( ( divide_divide_real @ A @ B2 )
        = zero_zero_real )
      = ( ( A = zero_zero_real )
        | ( B2 = zero_zero_real ) ) ) ).

% divide_eq_0_iff
thf(fact_1346_divide__eq__0__iff,axiom,
    ! [A: rat,B2: rat] :
      ( ( ( divide_divide_rat @ A @ B2 )
        = zero_zero_rat )
      = ( ( A = zero_zero_rat )
        | ( B2 = zero_zero_rat ) ) ) ).

% divide_eq_0_iff
thf(fact_1347_divide__cancel__left,axiom,
    ! [C: complex,A: complex,B2: complex] :
      ( ( ( divide1717551699836669952omplex @ C @ A )
        = ( divide1717551699836669952omplex @ C @ B2 ) )
      = ( ( C = zero_zero_complex )
        | ( A = B2 ) ) ) ).

% divide_cancel_left
thf(fact_1348_divide__cancel__left,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( ( divide_divide_real @ C @ A )
        = ( divide_divide_real @ C @ B2 ) )
      = ( ( C = zero_zero_real )
        | ( A = B2 ) ) ) ).

% divide_cancel_left
thf(fact_1349_divide__cancel__left,axiom,
    ! [C: rat,A: rat,B2: rat] :
      ( ( ( divide_divide_rat @ C @ A )
        = ( divide_divide_rat @ C @ B2 ) )
      = ( ( C = zero_zero_rat )
        | ( A = B2 ) ) ) ).

% divide_cancel_left
thf(fact_1350_divide__cancel__right,axiom,
    ! [A: complex,C: complex,B2: complex] :
      ( ( ( divide1717551699836669952omplex @ A @ C )
        = ( divide1717551699836669952omplex @ B2 @ C ) )
      = ( ( C = zero_zero_complex )
        | ( A = B2 ) ) ) ).

% divide_cancel_right
thf(fact_1351_divide__cancel__right,axiom,
    ! [A: real,C: real,B2: real] :
      ( ( ( divide_divide_real @ A @ C )
        = ( divide_divide_real @ B2 @ C ) )
      = ( ( C = zero_zero_real )
        | ( A = B2 ) ) ) ).

% divide_cancel_right
thf(fact_1352_divide__cancel__right,axiom,
    ! [A: rat,C: rat,B2: rat] :
      ( ( ( divide_divide_rat @ A @ C )
        = ( divide_divide_rat @ B2 @ C ) )
      = ( ( C = zero_zero_rat )
        | ( A = B2 ) ) ) ).

% divide_cancel_right
thf(fact_1353_division__ring__divide__zero,axiom,
    ! [A: complex] :
      ( ( divide1717551699836669952omplex @ A @ zero_zero_complex )
      = zero_zero_complex ) ).

% division_ring_divide_zero
thf(fact_1354_division__ring__divide__zero,axiom,
    ! [A: real] :
      ( ( divide_divide_real @ A @ zero_zero_real )
      = zero_zero_real ) ).

% division_ring_divide_zero
thf(fact_1355_division__ring__divide__zero,axiom,
    ! [A: rat] :
      ( ( divide_divide_rat @ A @ zero_zero_rat )
      = zero_zero_rat ) ).

% division_ring_divide_zero
thf(fact_1356_dvd__0__right,axiom,
    ! [A: code_integer] : ( dvd_dvd_Code_integer @ A @ zero_z3403309356797280102nteger ) ).

% dvd_0_right
thf(fact_1357_dvd__0__right,axiom,
    ! [A: real] : ( dvd_dvd_real @ A @ zero_zero_real ) ).

% dvd_0_right
thf(fact_1358_dvd__0__right,axiom,
    ! [A: rat] : ( dvd_dvd_rat @ A @ zero_zero_rat ) ).

% dvd_0_right
thf(fact_1359_dvd__0__right,axiom,
    ! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).

% dvd_0_right
thf(fact_1360_dvd__0__right,axiom,
    ! [A: int] : ( dvd_dvd_int @ A @ zero_zero_int ) ).

% dvd_0_right
thf(fact_1361_dvd__0__left__iff,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ A )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% dvd_0_left_iff
thf(fact_1362_dvd__0__left__iff,axiom,
    ! [A: real] :
      ( ( dvd_dvd_real @ zero_zero_real @ A )
      = ( A = zero_zero_real ) ) ).

% dvd_0_left_iff
thf(fact_1363_dvd__0__left__iff,axiom,
    ! [A: rat] :
      ( ( dvd_dvd_rat @ zero_zero_rat @ A )
      = ( A = zero_zero_rat ) ) ).

% dvd_0_left_iff
thf(fact_1364_dvd__0__left__iff,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ zero_zero_nat @ A )
      = ( A = zero_zero_nat ) ) ).

% dvd_0_left_iff
thf(fact_1365_dvd__0__left__iff,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ zero_zero_int @ A )
      = ( A = zero_zero_int ) ) ).

% dvd_0_left_iff
thf(fact_1366_dvd__add__triv__right__iff,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B2 @ A ) )
      = ( dvd_dvd_Code_integer @ A @ B2 ) ) ).

% dvd_add_triv_right_iff
thf(fact_1367_dvd__add__triv__right__iff,axiom,
    ! [A: real,B2: real] :
      ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B2 @ A ) )
      = ( dvd_dvd_real @ A @ B2 ) ) ).

% dvd_add_triv_right_iff
thf(fact_1368_dvd__add__triv__right__iff,axiom,
    ! [A: rat,B2: rat] :
      ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B2 @ A ) )
      = ( dvd_dvd_rat @ A @ B2 ) ) ).

% dvd_add_triv_right_iff
thf(fact_1369_dvd__add__triv__right__iff,axiom,
    ! [A: nat,B2: nat] :
      ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B2 @ A ) )
      = ( dvd_dvd_nat @ A @ B2 ) ) ).

% dvd_add_triv_right_iff
thf(fact_1370_dvd__add__triv__right__iff,axiom,
    ! [A: int,B2: int] :
      ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B2 @ A ) )
      = ( dvd_dvd_int @ A @ B2 ) ) ).

% dvd_add_triv_right_iff
thf(fact_1371_dvd__add__triv__left__iff,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ A @ B2 ) )
      = ( dvd_dvd_Code_integer @ A @ B2 ) ) ).

% dvd_add_triv_left_iff
thf(fact_1372_dvd__add__triv__left__iff,axiom,
    ! [A: real,B2: real] :
      ( ( dvd_dvd_real @ A @ ( plus_plus_real @ A @ B2 ) )
      = ( dvd_dvd_real @ A @ B2 ) ) ).

% dvd_add_triv_left_iff
thf(fact_1373_dvd__add__triv__left__iff,axiom,
    ! [A: rat,B2: rat] :
      ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ A @ B2 ) )
      = ( dvd_dvd_rat @ A @ B2 ) ) ).

% dvd_add_triv_left_iff
thf(fact_1374_dvd__add__triv__left__iff,axiom,
    ! [A: nat,B2: nat] :
      ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ A @ B2 ) )
      = ( dvd_dvd_nat @ A @ B2 ) ) ).

% dvd_add_triv_left_iff
thf(fact_1375_dvd__add__triv__left__iff,axiom,
    ! [A: int,B2: int] :
      ( ( dvd_dvd_int @ A @ ( plus_plus_int @ A @ B2 ) )
      = ( dvd_dvd_int @ A @ B2 ) ) ).

% dvd_add_triv_left_iff
thf(fact_1376_power__Suc0__right,axiom,
    ! [A: nat] :
      ( ( power_power_nat @ A @ ( suc @ zero_zero_nat ) )
      = A ) ).

% power_Suc0_right
thf(fact_1377_power__Suc0__right,axiom,
    ! [A: real] :
      ( ( power_power_real @ A @ ( suc @ zero_zero_nat ) )
      = A ) ).

% power_Suc0_right
thf(fact_1378_power__Suc0__right,axiom,
    ! [A: int] :
      ( ( power_power_int @ A @ ( suc @ zero_zero_nat ) )
      = A ) ).

% power_Suc0_right
thf(fact_1379_power__Suc0__right,axiom,
    ! [A: complex] :
      ( ( power_power_complex @ A @ ( suc @ zero_zero_nat ) )
      = A ) ).

% power_Suc0_right
thf(fact_1380_mod__0,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% mod_0
thf(fact_1381_mod__0,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% mod_0
thf(fact_1382_mod__0,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger @ zero_z3403309356797280102nteger @ A )
      = zero_z3403309356797280102nteger ) ).

% mod_0
thf(fact_1383_mod__by__0,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ A @ zero_zero_nat )
      = A ) ).

% mod_by_0
thf(fact_1384_mod__by__0,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ A @ zero_zero_int )
      = A ) ).

% mod_by_0
thf(fact_1385_mod__by__0,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger @ A @ zero_z3403309356797280102nteger )
      = A ) ).

% mod_by_0
thf(fact_1386_mod__self,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ A @ A )
      = zero_zero_nat ) ).

% mod_self
thf(fact_1387_mod__self,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ A @ A )
      = zero_zero_int ) ).

% mod_self
thf(fact_1388_mod__self,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger @ A @ A )
      = zero_z3403309356797280102nteger ) ).

% mod_self
thf(fact_1389_bits__mod__0,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% bits_mod_0
thf(fact_1390_bits__mod__0,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% bits_mod_0
thf(fact_1391_bits__mod__0,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger @ zero_z3403309356797280102nteger @ A )
      = zero_z3403309356797280102nteger ) ).

% bits_mod_0
thf(fact_1392_div__dvd__div,axiom,
    ! [A: code_integer,B2: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ B2 )
     => ( ( dvd_dvd_Code_integer @ A @ C )
       => ( ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ B2 @ A ) @ ( divide6298287555418463151nteger @ C @ A ) )
          = ( dvd_dvd_Code_integer @ B2 @ C ) ) ) ) ).

% div_dvd_div
thf(fact_1393_div__dvd__div,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ B2 )
     => ( ( dvd_dvd_nat @ A @ C )
       => ( ( dvd_dvd_nat @ ( divide_divide_nat @ B2 @ A ) @ ( divide_divide_nat @ C @ A ) )
          = ( dvd_dvd_nat @ B2 @ C ) ) ) ) ).

% div_dvd_div
thf(fact_1394_div__dvd__div,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( dvd_dvd_int @ A @ B2 )
     => ( ( dvd_dvd_int @ A @ C )
       => ( ( dvd_dvd_int @ ( divide_divide_int @ B2 @ A ) @ ( divide_divide_int @ C @ A ) )
          = ( dvd_dvd_int @ B2 @ C ) ) ) ) ).

% div_dvd_div
thf(fact_1395_zero__less__Suc,axiom,
    ! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).

% zero_less_Suc
thf(fact_1396_less__Suc0,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
      = ( N = zero_zero_nat ) ) ).

% less_Suc0
thf(fact_1397_add__gr__0,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ M )
        | ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% add_gr_0
thf(fact_1398_one__eq__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ( suc @ zero_zero_nat )
        = ( times_times_nat @ M @ N ) )
      = ( ( M
          = ( suc @ zero_zero_nat ) )
        & ( N
          = ( suc @ zero_zero_nat ) ) ) ) ).

% one_eq_mult_iff
thf(fact_1399_mult__eq__1__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ( times_times_nat @ M @ N )
        = ( suc @ zero_zero_nat ) )
      = ( ( M
          = ( suc @ zero_zero_nat ) )
        & ( N
          = ( suc @ zero_zero_nat ) ) ) ) ).

% mult_eq_1_iff
thf(fact_1400_div__by__Suc__0,axiom,
    ! [M: nat] :
      ( ( divide_divide_nat @ M @ ( suc @ zero_zero_nat ) )
      = M ) ).

% div_by_Suc_0
thf(fact_1401_zero__less__diff,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
      = ( ord_less_nat @ M @ N ) ) ).

% zero_less_diff
thf(fact_1402_nat__mult__less__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K )
        & ( ord_less_nat @ M @ N ) ) ) ).

% nat_mult_less_cancel_disj
thf(fact_1403_nat__0__less__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ M )
        & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% nat_0_less_mult_iff
thf(fact_1404_mult__less__cancel2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K )
        & ( ord_less_nat @ M @ N ) ) ) ).

% mult_less_cancel2
thf(fact_1405_div__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ( divide_divide_nat @ M @ N )
        = zero_zero_nat ) ) ).

% div_less
thf(fact_1406_not__real__square__gt__zero,axiom,
    ! [X4: real] :
      ( ( ~ ( ord_less_real @ zero_zero_real @ ( times_times_real @ X4 @ X4 ) ) )
      = ( X4 = zero_zero_real ) ) ).

% not_real_square_gt_zero
thf(fact_1407_diff__is__0__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ( minus_minus_nat @ M @ N )
        = zero_zero_nat )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% diff_is_0_eq
thf(fact_1408_diff__is__0__eq_H,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( minus_minus_nat @ M @ N )
        = zero_zero_nat ) ) ).

% diff_is_0_eq'
thf(fact_1409_dvd__1__iff__1,axiom,
    ! [M: nat] :
      ( ( dvd_dvd_nat @ M @ ( suc @ zero_zero_nat ) )
      = ( M
        = ( suc @ zero_zero_nat ) ) ) ).

% dvd_1_iff_1
thf(fact_1410_dvd__1__left,axiom,
    ! [K: nat] : ( dvd_dvd_nat @ ( suc @ zero_zero_nat ) @ K ) ).

% dvd_1_left
thf(fact_1411_less__one,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ N @ one_one_nat )
      = ( N = zero_zero_nat ) ) ).

% less_one
thf(fact_1412_power__Suc__0,axiom,
    ! [N: nat] :
      ( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N )
      = ( suc @ zero_zero_nat ) ) ).

% power_Suc_0
thf(fact_1413_nat__power__eq__Suc__0__iff,axiom,
    ! [X4: nat,M: nat] :
      ( ( ( power_power_nat @ X4 @ M )
        = ( suc @ zero_zero_nat ) )
      = ( ( M = zero_zero_nat )
        | ( X4
          = ( suc @ zero_zero_nat ) ) ) ) ).

% nat_power_eq_Suc_0_iff
thf(fact_1414_nat__zero__less__power__iff,axiom,
    ! [X4: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X4 @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X4 )
        | ( N = zero_zero_nat ) ) ) ).

% nat_zero_less_power_iff
thf(fact_1415_nat__mult__div__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ( K = zero_zero_nat )
       => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
          = zero_zero_nat ) )
      & ( ( K != zero_zero_nat )
       => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
          = ( divide_divide_nat @ M @ N ) ) ) ) ).

% nat_mult_div_cancel_disj
thf(fact_1416_mod__by__Suc__0,axiom,
    ! [M: nat] :
      ( ( modulo_modulo_nat @ M @ ( suc @ zero_zero_nat ) )
      = zero_zero_nat ) ).

% mod_by_Suc_0
thf(fact_1417_nat__mult__dvd__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
      = ( ( K = zero_zero_nat )
        | ( dvd_dvd_nat @ M @ N ) ) ) ).

% nat_mult_dvd_cancel_disj
thf(fact_1418_nat__dvd__1__iff__1,axiom,
    ! [M: nat] :
      ( ( dvd_dvd_nat @ M @ one_one_nat )
      = ( M = one_one_nat ) ) ).

% nat_dvd_1_iff_1
thf(fact_1419_signed__take__bit__of__0,axiom,
    ! [N: nat] :
      ( ( bit_ri631733984087533419it_int @ N @ zero_zero_int )
      = zero_zero_int ) ).

% signed_take_bit_of_0
thf(fact_1420_dbl__simps_I2_J,axiom,
    ( ( neg_numeral_dbl_real @ zero_zero_real )
    = zero_zero_real ) ).

% dbl_simps(2)
thf(fact_1421_dbl__simps_I2_J,axiom,
    ( ( neg_numeral_dbl_rat @ zero_zero_rat )
    = zero_zero_rat ) ).

% dbl_simps(2)
thf(fact_1422_dbl__simps_I2_J,axiom,
    ( ( neg_numeral_dbl_int @ zero_zero_int )
    = zero_zero_int ) ).

% dbl_simps(2)
thf(fact_1423_add__le__same__cancel1,axiom,
    ! [B2: real,A: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ B2 @ A ) @ B2 )
      = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).

% add_le_same_cancel1
thf(fact_1424_add__le__same__cancel1,axiom,
    ! [B2: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ B2 @ A ) @ B2 )
      = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).

% add_le_same_cancel1
thf(fact_1425_add__le__same__cancel1,axiom,
    ! [B2: nat,A: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ B2 @ A ) @ B2 )
      = ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).

% add_le_same_cancel1
thf(fact_1426_add__le__same__cancel1,axiom,
    ! [B2: int,A: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ B2 @ A ) @ B2 )
      = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).

% add_le_same_cancel1
thf(fact_1427_add__le__same__cancel2,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ A @ B2 ) @ B2 )
      = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).

% add_le_same_cancel2
thf(fact_1428_add__le__same__cancel2,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B2 ) @ B2 )
      = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).

% add_le_same_cancel2
thf(fact_1429_add__le__same__cancel2,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B2 ) @ B2 )
      = ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).

% add_le_same_cancel2
thf(fact_1430_add__le__same__cancel2,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ A @ B2 ) @ B2 )
      = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).

% add_le_same_cancel2
thf(fact_1431_le__add__same__cancel1,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ A @ ( plus_plus_real @ A @ B2 ) )
      = ( ord_less_eq_real @ zero_zero_real @ B2 ) ) ).

% le_add_same_cancel1
thf(fact_1432_le__add__same__cancel1,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_eq_rat @ A @ ( plus_plus_rat @ A @ B2 ) )
      = ( ord_less_eq_rat @ zero_zero_rat @ B2 ) ) ).

% le_add_same_cancel1
thf(fact_1433_le__add__same__cancel1,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B2 ) )
      = ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) ).

% le_add_same_cancel1
thf(fact_1434_le__add__same__cancel1,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ A @ ( plus_plus_int @ A @ B2 ) )
      = ( ord_less_eq_int @ zero_zero_int @ B2 ) ) ).

% le_add_same_cancel1
thf(fact_1435_le__add__same__cancel2,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ A @ ( plus_plus_real @ B2 @ A ) )
      = ( ord_less_eq_real @ zero_zero_real @ B2 ) ) ).

% le_add_same_cancel2
thf(fact_1436_le__add__same__cancel2,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_eq_rat @ A @ ( plus_plus_rat @ B2 @ A ) )
      = ( ord_less_eq_rat @ zero_zero_rat @ B2 ) ) ).

% le_add_same_cancel2
thf(fact_1437_le__add__same__cancel2,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B2 @ A ) )
      = ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) ).

% le_add_same_cancel2
thf(fact_1438_le__add__same__cancel2,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ A @ ( plus_plus_int @ B2 @ A ) )
      = ( ord_less_eq_int @ zero_zero_int @ B2 ) ) ).

% le_add_same_cancel2
thf(fact_1439_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
      = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).

% double_add_le_zero_iff_single_add_le_zero
thf(fact_1440_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ A ) @ zero_zero_rat )
      = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).

% double_add_le_zero_iff_single_add_le_zero
thf(fact_1441_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
      = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).

% double_add_le_zero_iff_single_add_le_zero
thf(fact_1442_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
      = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).

% zero_le_double_add_iff_zero_le_single_add
thf(fact_1443_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ A ) )
      = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).

% zero_le_double_add_iff_zero_le_single_add
thf(fact_1444_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
      = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).

% zero_le_double_add_iff_zero_le_single_add
thf(fact_1445_diff__ge__0__iff__ge,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A @ B2 ) )
      = ( ord_less_eq_real @ B2 @ A ) ) ).

% diff_ge_0_iff_ge
thf(fact_1446_diff__ge__0__iff__ge,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( minus_minus_rat @ A @ B2 ) )
      = ( ord_less_eq_rat @ B2 @ A ) ) ).

% diff_ge_0_iff_ge
thf(fact_1447_diff__ge__0__iff__ge,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A @ B2 ) )
      = ( ord_less_eq_int @ B2 @ A ) ) ).

% diff_ge_0_iff_ge
thf(fact_1448_add__less__same__cancel1,axiom,
    ! [B2: real,A: real] :
      ( ( ord_less_real @ ( plus_plus_real @ B2 @ A ) @ B2 )
      = ( ord_less_real @ A @ zero_zero_real ) ) ).

% add_less_same_cancel1
thf(fact_1449_add__less__same__cancel1,axiom,
    ! [B2: rat,A: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ B2 @ A ) @ B2 )
      = ( ord_less_rat @ A @ zero_zero_rat ) ) ).

% add_less_same_cancel1
thf(fact_1450_add__less__same__cancel1,axiom,
    ! [B2: nat,A: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ B2 @ A ) @ B2 )
      = ( ord_less_nat @ A @ zero_zero_nat ) ) ).

% add_less_same_cancel1
thf(fact_1451_add__less__same__cancel1,axiom,
    ! [B2: int,A: int] :
      ( ( ord_less_int @ ( plus_plus_int @ B2 @ A ) @ B2 )
      = ( ord_less_int @ A @ zero_zero_int ) ) ).

% add_less_same_cancel1
thf(fact_1452_add__less__same__cancel2,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ ( plus_plus_real @ A @ B2 ) @ B2 )
      = ( ord_less_real @ A @ zero_zero_real ) ) ).

% add_less_same_cancel2
thf(fact_1453_add__less__same__cancel2,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ A @ B2 ) @ B2 )
      = ( ord_less_rat @ A @ zero_zero_rat ) ) ).

% add_less_same_cancel2
thf(fact_1454_add__less__same__cancel2,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A @ B2 ) @ B2 )
      = ( ord_less_nat @ A @ zero_zero_nat ) ) ).

% add_less_same_cancel2
thf(fact_1455_add__less__same__cancel2,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_int @ ( plus_plus_int @ A @ B2 ) @ B2 )
      = ( ord_less_int @ A @ zero_zero_int ) ) ).

% add_less_same_cancel2
thf(fact_1456_less__add__same__cancel1,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ A @ ( plus_plus_real @ A @ B2 ) )
      = ( ord_less_real @ zero_zero_real @ B2 ) ) ).

% less_add_same_cancel1
thf(fact_1457_less__add__same__cancel1,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_rat @ A @ ( plus_plus_rat @ A @ B2 ) )
      = ( ord_less_rat @ zero_zero_rat @ B2 ) ) ).

% less_add_same_cancel1
thf(fact_1458_less__add__same__cancel1,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B2 ) )
      = ( ord_less_nat @ zero_zero_nat @ B2 ) ) ).

% less_add_same_cancel1
thf(fact_1459_less__add__same__cancel1,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_int @ A @ ( plus_plus_int @ A @ B2 ) )
      = ( ord_less_int @ zero_zero_int @ B2 ) ) ).

% less_add_same_cancel1
thf(fact_1460_less__add__same__cancel2,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ A @ ( plus_plus_real @ B2 @ A ) )
      = ( ord_less_real @ zero_zero_real @ B2 ) ) ).

% less_add_same_cancel2
thf(fact_1461_less__add__same__cancel2,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_rat @ A @ ( plus_plus_rat @ B2 @ A ) )
      = ( ord_less_rat @ zero_zero_rat @ B2 ) ) ).

% less_add_same_cancel2
thf(fact_1462_less__add__same__cancel2,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_nat @ A @ ( plus_plus_nat @ B2 @ A ) )
      = ( ord_less_nat @ zero_zero_nat @ B2 ) ) ).

% less_add_same_cancel2
thf(fact_1463_less__add__same__cancel2,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_int @ A @ ( plus_plus_int @ B2 @ A ) )
      = ( ord_less_int @ zero_zero_int @ B2 ) ) ).

% less_add_same_cancel2
thf(fact_1464_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: real] :
      ( ( ord_less_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
      = ( ord_less_real @ A @ zero_zero_real ) ) ).

% double_add_less_zero_iff_single_add_less_zero
thf(fact_1465_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ A @ A ) @ zero_zero_rat )
      = ( ord_less_rat @ A @ zero_zero_rat ) ) ).

% double_add_less_zero_iff_single_add_less_zero
thf(fact_1466_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: int] :
      ( ( ord_less_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
      = ( ord_less_int @ A @ zero_zero_int ) ) ).

% double_add_less_zero_iff_single_add_less_zero
thf(fact_1467_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
      = ( ord_less_real @ zero_zero_real @ A ) ) ).

% zero_less_double_add_iff_zero_less_single_add
thf(fact_1468_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ A ) )
      = ( ord_less_rat @ zero_zero_rat @ A ) ) ).

% zero_less_double_add_iff_zero_less_single_add
thf(fact_1469_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: int] :
      ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
      = ( ord_less_int @ zero_zero_int @ A ) ) ).

% zero_less_double_add_iff_zero_less_single_add
thf(fact_1470_diff__gt__0__iff__gt,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ zero_zero_real @ ( minus_minus_real @ A @ B2 ) )
      = ( ord_less_real @ B2 @ A ) ) ).

% diff_gt_0_iff_gt
thf(fact_1471_diff__gt__0__iff__gt,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( minus_minus_rat @ A @ B2 ) )
      = ( ord_less_rat @ B2 @ A ) ) ).

% diff_gt_0_iff_gt
thf(fact_1472_diff__gt__0__iff__gt,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A @ B2 ) )
      = ( ord_less_int @ B2 @ A ) ) ).

% diff_gt_0_iff_gt
thf(fact_1473_mult__cancel__left1,axiom,
    ! [C: complex,B2: complex] :
      ( ( C
        = ( times_times_complex @ C @ B2 ) )
      = ( ( C = zero_zero_complex )
        | ( B2 = one_one_complex ) ) ) ).

% mult_cancel_left1
thf(fact_1474_mult__cancel__left1,axiom,
    ! [C: real,B2: real] :
      ( ( C
        = ( times_times_real @ C @ B2 ) )
      = ( ( C = zero_zero_real )
        | ( B2 = one_one_real ) ) ) ).

% mult_cancel_left1
thf(fact_1475_mult__cancel__left1,axiom,
    ! [C: rat,B2: rat] :
      ( ( C
        = ( times_times_rat @ C @ B2 ) )
      = ( ( C = zero_zero_rat )
        | ( B2 = one_one_rat ) ) ) ).

% mult_cancel_left1
thf(fact_1476_mult__cancel__left1,axiom,
    ! [C: int,B2: int] :
      ( ( C
        = ( times_times_int @ C @ B2 ) )
      = ( ( C = zero_zero_int )
        | ( B2 = one_one_int ) ) ) ).

% mult_cancel_left1
thf(fact_1477_mult__cancel__left2,axiom,
    ! [C: complex,A: complex] :
      ( ( ( times_times_complex @ C @ A )
        = C )
      = ( ( C = zero_zero_complex )
        | ( A = one_one_complex ) ) ) ).

% mult_cancel_left2
thf(fact_1478_mult__cancel__left2,axiom,
    ! [C: real,A: real] :
      ( ( ( times_times_real @ C @ A )
        = C )
      = ( ( C = zero_zero_real )
        | ( A = one_one_real ) ) ) ).

% mult_cancel_left2
thf(fact_1479_mult__cancel__left2,axiom,
    ! [C: rat,A: rat] :
      ( ( ( times_times_rat @ C @ A )
        = C )
      = ( ( C = zero_zero_rat )
        | ( A = one_one_rat ) ) ) ).

% mult_cancel_left2
thf(fact_1480_mult__cancel__left2,axiom,
    ! [C: int,A: int] :
      ( ( ( times_times_int @ C @ A )
        = C )
      = ( ( C = zero_zero_int )
        | ( A = one_one_int ) ) ) ).

% mult_cancel_left2
thf(fact_1481_mult__cancel__right1,axiom,
    ! [C: complex,B2: complex] :
      ( ( C
        = ( times_times_complex @ B2 @ C ) )
      = ( ( C = zero_zero_complex )
        | ( B2 = one_one_complex ) ) ) ).

% mult_cancel_right1
thf(fact_1482_mult__cancel__right1,axiom,
    ! [C: real,B2: real] :
      ( ( C
        = ( times_times_real @ B2 @ C ) )
      = ( ( C = zero_zero_real )
        | ( B2 = one_one_real ) ) ) ).

% mult_cancel_right1
thf(fact_1483_mult__cancel__right1,axiom,
    ! [C: rat,B2: rat] :
      ( ( C
        = ( times_times_rat @ B2 @ C ) )
      = ( ( C = zero_zero_rat )
        | ( B2 = one_one_rat ) ) ) ).

% mult_cancel_right1
thf(fact_1484_mult__cancel__right1,axiom,
    ! [C: int,B2: int] :
      ( ( C
        = ( times_times_int @ B2 @ C ) )
      = ( ( C = zero_zero_int )
        | ( B2 = one_one_int ) ) ) ).

% mult_cancel_right1
thf(fact_1485_mult__cancel__right2,axiom,
    ! [A: complex,C: complex] :
      ( ( ( times_times_complex @ A @ C )
        = C )
      = ( ( C = zero_zero_complex )
        | ( A = one_one_complex ) ) ) ).

% mult_cancel_right2
thf(fact_1486_mult__cancel__right2,axiom,
    ! [A: real,C: real] :
      ( ( ( times_times_real @ A @ C )
        = C )
      = ( ( C = zero_zero_real )
        | ( A = one_one_real ) ) ) ).

% mult_cancel_right2
thf(fact_1487_mult__cancel__right2,axiom,
    ! [A: rat,C: rat] :
      ( ( ( times_times_rat @ A @ C )
        = C )
      = ( ( C = zero_zero_rat )
        | ( A = one_one_rat ) ) ) ).

% mult_cancel_right2
thf(fact_1488_mult__cancel__right2,axiom,
    ! [A: int,C: int] :
      ( ( ( times_times_int @ A @ C )
        = C )
      = ( ( C = zero_zero_int )
        | ( A = one_one_int ) ) ) ).

% mult_cancel_right2
thf(fact_1489_sum__squares__eq__zero__iff,axiom,
    ! [X4: real,Y3: real] :
      ( ( ( plus_plus_real @ ( times_times_real @ X4 @ X4 ) @ ( times_times_real @ Y3 @ Y3 ) )
        = zero_zero_real )
      = ( ( X4 = zero_zero_real )
        & ( Y3 = zero_zero_real ) ) ) ).

% sum_squares_eq_zero_iff
thf(fact_1490_sum__squares__eq__zero__iff,axiom,
    ! [X4: rat,Y3: rat] :
      ( ( ( plus_plus_rat @ ( times_times_rat @ X4 @ X4 ) @ ( times_times_rat @ Y3 @ Y3 ) )
        = zero_zero_rat )
      = ( ( X4 = zero_zero_rat )
        & ( Y3 = zero_zero_rat ) ) ) ).

% sum_squares_eq_zero_iff
thf(fact_1491_sum__squares__eq__zero__iff,axiom,
    ! [X4: int,Y3: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ X4 @ X4 ) @ ( times_times_int @ Y3 @ Y3 ) )
        = zero_zero_int )
      = ( ( X4 = zero_zero_int )
        & ( Y3 = zero_zero_int ) ) ) ).

% sum_squares_eq_zero_iff
thf(fact_1492_div__mult__mult1__if,axiom,
    ! [C: nat,A: nat,B2: nat] :
      ( ( ( C = zero_zero_nat )
       => ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B2 ) )
          = zero_zero_nat ) )
      & ( ( C != zero_zero_nat )
       => ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B2 ) )
          = ( divide_divide_nat @ A @ B2 ) ) ) ) ).

% div_mult_mult1_if
thf(fact_1493_div__mult__mult1__if,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( ( C = zero_zero_int )
       => ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
          = zero_zero_int ) )
      & ( ( C != zero_zero_int )
       => ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
          = ( divide_divide_int @ A @ B2 ) ) ) ) ).

% div_mult_mult1_if
thf(fact_1494_div__mult__mult2,axiom,
    ! [C: nat,A: nat,B2: nat] :
      ( ( C != zero_zero_nat )
     => ( ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ C ) )
        = ( divide_divide_nat @ A @ B2 ) ) ) ).

% div_mult_mult2
thf(fact_1495_div__mult__mult2,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( C != zero_zero_int )
     => ( ( divide_divide_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) )
        = ( divide_divide_int @ A @ B2 ) ) ) ).

% div_mult_mult2
thf(fact_1496_div__mult__mult1,axiom,
    ! [C: nat,A: nat,B2: nat] :
      ( ( C != zero_zero_nat )
     => ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B2 ) )
        = ( divide_divide_nat @ A @ B2 ) ) ) ).

% div_mult_mult1
thf(fact_1497_div__mult__mult1,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( C != zero_zero_int )
     => ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
        = ( divide_divide_int @ A @ B2 ) ) ) ).

% div_mult_mult1
thf(fact_1498_nonzero__mult__div__cancel__left,axiom,
    ! [A: complex,B2: complex] :
      ( ( A != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ B2 ) @ A )
        = B2 ) ) ).

% nonzero_mult_div_cancel_left
thf(fact_1499_nonzero__mult__div__cancel__left,axiom,
    ! [A: real,B2: real] :
      ( ( A != zero_zero_real )
     => ( ( divide_divide_real @ ( times_times_real @ A @ B2 ) @ A )
        = B2 ) ) ).

% nonzero_mult_div_cancel_left
thf(fact_1500_nonzero__mult__div__cancel__left,axiom,
    ! [A: rat,B2: rat] :
      ( ( A != zero_zero_rat )
     => ( ( divide_divide_rat @ ( times_times_rat @ A @ B2 ) @ A )
        = B2 ) ) ).

% nonzero_mult_div_cancel_left
thf(fact_1501_nonzero__mult__div__cancel__left,axiom,
    ! [A: nat,B2: nat] :
      ( ( A != zero_zero_nat )
     => ( ( divide_divide_nat @ ( times_times_nat @ A @ B2 ) @ A )
        = B2 ) ) ).

% nonzero_mult_div_cancel_left
thf(fact_1502_nonzero__mult__div__cancel__left,axiom,
    ! [A: int,B2: int] :
      ( ( A != zero_zero_int )
     => ( ( divide_divide_int @ ( times_times_int @ A @ B2 ) @ A )
        = B2 ) ) ).

% nonzero_mult_div_cancel_left
thf(fact_1503_nonzero__mult__div__cancel__right,axiom,
    ! [B2: complex,A: complex] :
      ( ( B2 != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ B2 ) @ B2 )
        = A ) ) ).

% nonzero_mult_div_cancel_right
thf(fact_1504_nonzero__mult__div__cancel__right,axiom,
    ! [B2: real,A: real] :
      ( ( B2 != zero_zero_real )
     => ( ( divide_divide_real @ ( times_times_real @ A @ B2 ) @ B2 )
        = A ) ) ).

% nonzero_mult_div_cancel_right
thf(fact_1505_nonzero__mult__div__cancel__right,axiom,
    ! [B2: rat,A: rat] :
      ( ( B2 != zero_zero_rat )
     => ( ( divide_divide_rat @ ( times_times_rat @ A @ B2 ) @ B2 )
        = A ) ) ).

% nonzero_mult_div_cancel_right
thf(fact_1506_nonzero__mult__div__cancel__right,axiom,
    ! [B2: nat,A: nat] :
      ( ( B2 != zero_zero_nat )
     => ( ( divide_divide_nat @ ( times_times_nat @ A @ B2 ) @ B2 )
        = A ) ) ).

% nonzero_mult_div_cancel_right
thf(fact_1507_nonzero__mult__div__cancel__right,axiom,
    ! [B2: int,A: int] :
      ( ( B2 != zero_zero_int )
     => ( ( divide_divide_int @ ( times_times_int @ A @ B2 ) @ B2 )
        = A ) ) ).

% nonzero_mult_div_cancel_right
thf(fact_1508_mult__divide__mult__cancel__left__if,axiom,
    ! [C: complex,A: complex,B2: complex] :
      ( ( ( C = zero_zero_complex )
       => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B2 ) )
          = zero_zero_complex ) )
      & ( ( C != zero_zero_complex )
       => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B2 ) )
          = ( divide1717551699836669952omplex @ A @ B2 ) ) ) ) ).

% mult_divide_mult_cancel_left_if
thf(fact_1509_mult__divide__mult__cancel__left__if,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( ( C = zero_zero_real )
       => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
          = zero_zero_real ) )
      & ( ( C != zero_zero_real )
       => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
          = ( divide_divide_real @ A @ B2 ) ) ) ) ).

% mult_divide_mult_cancel_left_if
thf(fact_1510_mult__divide__mult__cancel__left__if,axiom,
    ! [C: rat,A: rat,B2: rat] :
      ( ( ( C = zero_zero_rat )
       => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) )
          = zero_zero_rat ) )
      & ( ( C != zero_zero_rat )
       => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) )
          = ( divide_divide_rat @ A @ B2 ) ) ) ) ).

% mult_divide_mult_cancel_left_if
thf(fact_1511_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C: complex,A: complex,B2: complex] :
      ( ( C != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B2 ) )
        = ( divide1717551699836669952omplex @ A @ B2 ) ) ) ).

% nonzero_mult_divide_mult_cancel_left
thf(fact_1512_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( C != zero_zero_real )
     => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
        = ( divide_divide_real @ A @ B2 ) ) ) ).

% nonzero_mult_divide_mult_cancel_left
thf(fact_1513_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C: rat,A: rat,B2: rat] :
      ( ( C != zero_zero_rat )
     => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) )
        = ( divide_divide_rat @ A @ B2 ) ) ) ).

% nonzero_mult_divide_mult_cancel_left
thf(fact_1514_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C: complex,A: complex,B2: complex] :
      ( ( C != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ B2 @ C ) )
        = ( divide1717551699836669952omplex @ A @ B2 ) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
thf(fact_1515_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( C != zero_zero_real )
     => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ B2 @ C ) )
        = ( divide_divide_real @ A @ B2 ) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
thf(fact_1516_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C: rat,A: rat,B2: rat] :
      ( ( C != zero_zero_rat )
     => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ B2 @ C ) )
        = ( divide_divide_rat @ A @ B2 ) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
thf(fact_1517_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C: complex,A: complex,B2: complex] :
      ( ( C != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B2 @ C ) )
        = ( divide1717551699836669952omplex @ A @ B2 ) ) ) ).

% nonzero_mult_divide_mult_cancel_right
thf(fact_1518_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( C != zero_zero_real )
     => ( ( divide_divide_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) )
        = ( divide_divide_real @ A @ B2 ) ) ) ).

% nonzero_mult_divide_mult_cancel_right
thf(fact_1519_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C: rat,A: rat,B2: rat] :
      ( ( C != zero_zero_rat )
     => ( ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ C ) )
        = ( divide_divide_rat @ A @ B2 ) ) ) ).

% nonzero_mult_divide_mult_cancel_right
thf(fact_1520_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C: complex,A: complex,B2: complex] :
      ( ( C != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ C @ B2 ) )
        = ( divide1717551699836669952omplex @ A @ B2 ) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
thf(fact_1521_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( C != zero_zero_real )
     => ( ( divide_divide_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ C @ B2 ) )
        = ( divide_divide_real @ A @ B2 ) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
thf(fact_1522_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C: rat,A: rat,B2: rat] :
      ( ( C != zero_zero_rat )
     => ( ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ C @ B2 ) )
        = ( divide_divide_rat @ A @ B2 ) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
thf(fact_1523_diff__numeral__special_I9_J,axiom,
    ( ( minus_minus_complex @ one_one_complex @ one_one_complex )
    = zero_zero_complex ) ).

% diff_numeral_special(9)
thf(fact_1524_diff__numeral__special_I9_J,axiom,
    ( ( minus_minus_real @ one_one_real @ one_one_real )
    = zero_zero_real ) ).

% diff_numeral_special(9)
thf(fact_1525_diff__numeral__special_I9_J,axiom,
    ( ( minus_minus_rat @ one_one_rat @ one_one_rat )
    = zero_zero_rat ) ).

% diff_numeral_special(9)
thf(fact_1526_diff__numeral__special_I9_J,axiom,
    ( ( minus_minus_int @ one_one_int @ one_one_int )
    = zero_zero_int ) ).

% diff_numeral_special(9)
thf(fact_1527_diff__add__zero,axiom,
    ! [A: nat,B2: nat] :
      ( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B2 ) )
      = zero_zero_nat ) ).

% diff_add_zero
thf(fact_1528_div__self,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ A @ A )
        = one_one_complex ) ) ).

% div_self
thf(fact_1529_div__self,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( ( divide_divide_real @ A @ A )
        = one_one_real ) ) ).

% div_self
thf(fact_1530_div__self,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( ( divide_divide_rat @ A @ A )
        = one_one_rat ) ) ).

% div_self
thf(fact_1531_div__self,axiom,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
     => ( ( divide_divide_nat @ A @ A )
        = one_one_nat ) ) ).

% div_self
thf(fact_1532_div__self,axiom,
    ! [A: int] :
      ( ( A != zero_zero_int )
     => ( ( divide_divide_int @ A @ A )
        = one_one_int ) ) ).

% div_self
thf(fact_1533_divide__eq__1__iff,axiom,
    ! [A: complex,B2: complex] :
      ( ( ( divide1717551699836669952omplex @ A @ B2 )
        = one_one_complex )
      = ( ( B2 != zero_zero_complex )
        & ( A = B2 ) ) ) ).

% divide_eq_1_iff
thf(fact_1534_divide__eq__1__iff,axiom,
    ! [A: real,B2: real] :
      ( ( ( divide_divide_real @ A @ B2 )
        = one_one_real )
      = ( ( B2 != zero_zero_real )
        & ( A = B2 ) ) ) ).

% divide_eq_1_iff
thf(fact_1535_divide__eq__1__iff,axiom,
    ! [A: rat,B2: rat] :
      ( ( ( divide_divide_rat @ A @ B2 )
        = one_one_rat )
      = ( ( B2 != zero_zero_rat )
        & ( A = B2 ) ) ) ).

% divide_eq_1_iff
thf(fact_1536_one__eq__divide__iff,axiom,
    ! [A: complex,B2: complex] :
      ( ( one_one_complex
        = ( divide1717551699836669952omplex @ A @ B2 ) )
      = ( ( B2 != zero_zero_complex )
        & ( A = B2 ) ) ) ).

% one_eq_divide_iff
thf(fact_1537_one__eq__divide__iff,axiom,
    ! [A: real,B2: real] :
      ( ( one_one_real
        = ( divide_divide_real @ A @ B2 ) )
      = ( ( B2 != zero_zero_real )
        & ( A = B2 ) ) ) ).

% one_eq_divide_iff
thf(fact_1538_one__eq__divide__iff,axiom,
    ! [A: rat,B2: rat] :
      ( ( one_one_rat
        = ( divide_divide_rat @ A @ B2 ) )
      = ( ( B2 != zero_zero_rat )
        & ( A = B2 ) ) ) ).

% one_eq_divide_iff
thf(fact_1539_divide__self,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ A @ A )
        = one_one_complex ) ) ).

% divide_self
thf(fact_1540_divide__self,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( ( divide_divide_real @ A @ A )
        = one_one_real ) ) ).

% divide_self
thf(fact_1541_divide__self,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( ( divide_divide_rat @ A @ A )
        = one_one_rat ) ) ).

% divide_self
thf(fact_1542_divide__self__if,axiom,
    ! [A: complex] :
      ( ( ( A = zero_zero_complex )
       => ( ( divide1717551699836669952omplex @ A @ A )
          = zero_zero_complex ) )
      & ( ( A != zero_zero_complex )
       => ( ( divide1717551699836669952omplex @ A @ A )
          = one_one_complex ) ) ) ).

% divide_self_if
thf(fact_1543_divide__self__if,axiom,
    ! [A: real] :
      ( ( ( A = zero_zero_real )
       => ( ( divide_divide_real @ A @ A )
          = zero_zero_real ) )
      & ( ( A != zero_zero_real )
       => ( ( divide_divide_real @ A @ A )
          = one_one_real ) ) ) ).

% divide_self_if
thf(fact_1544_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
thf(fact_1545_divide__eq__eq__1,axiom,
    ! [B2: real,A: real] :
      ( ( ( divide_divide_real @ B2 @ A )
        = one_one_real )
      = ( ( A != zero_zero_real )
        & ( A = B2 ) ) ) ).

% divide_eq_eq_1
thf(fact_1546_divide__eq__eq__1,axiom,
    ! [B2: rat,A: rat] :
      ( ( ( divide_divide_rat @ B2 @ A )
        = one_one_rat )
      = ( ( A != zero_zero_rat )
        & ( A = B2 ) ) ) ).

% divide_eq_eq_1
thf(fact_1547_eq__divide__eq__1,axiom,
    ! [B2: real,A: real] :
      ( ( one_one_real
        = ( divide_divide_real @ B2 @ A ) )
      = ( ( A != zero_zero_real )
        & ( A = B2 ) ) ) ).

% eq_divide_eq_1
thf(fact_1548_eq__divide__eq__1,axiom,
    ! [B2: rat,A: rat] :
      ( ( one_one_rat
        = ( divide_divide_rat @ B2 @ A ) )
      = ( ( A != zero_zero_rat )
        & ( A = B2 ) ) ) ).

% eq_divide_eq_1
thf(fact_1549_one__divide__eq__0__iff,axiom,
    ! [A: real] :
      ( ( ( divide_divide_real @ one_one_real @ A )
        = zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% one_divide_eq_0_iff
thf(fact_1550_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
thf(fact_1551_zero__eq__1__divide__iff,axiom,
    ! [A: real] :
      ( ( zero_zero_real
        = ( divide_divide_real @ one_one_real @ A ) )
      = ( A = zero_zero_real ) ) ).

% zero_eq_1_divide_iff
thf(fact_1552_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
thf(fact_1553_dvd__times__right__cancel__iff,axiom,
    ! [A: code_integer,B2: code_integer,C: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ B2 @ A ) @ ( times_3573771949741848930nteger @ C @ A ) )
        = ( dvd_dvd_Code_integer @ B2 @ C ) ) ) ).

% dvd_times_right_cancel_iff
thf(fact_1554_dvd__times__right__cancel__iff,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( A != zero_zero_nat )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ B2 @ A ) @ ( times_times_nat @ C @ A ) )
        = ( dvd_dvd_nat @ B2 @ C ) ) ) ).

% dvd_times_right_cancel_iff
thf(fact_1555_dvd__times__right__cancel__iff,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( A != zero_zero_int )
     => ( ( dvd_dvd_int @ ( times_times_int @ B2 @ A ) @ ( times_times_int @ C @ A ) )
        = ( dvd_dvd_int @ B2 @ C ) ) ) ).

% dvd_times_right_cancel_iff
thf(fact_1556_dvd__times__left__cancel__iff,axiom,
    ! [A: code_integer,B2: code_integer,C: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B2 ) @ ( times_3573771949741848930nteger @ A @ C ) )
        = ( dvd_dvd_Code_integer @ B2 @ C ) ) ) ).

% dvd_times_left_cancel_iff
thf(fact_1557_dvd__times__left__cancel__iff,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( A != zero_zero_nat )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B2 ) @ ( times_times_nat @ A @ C ) )
        = ( dvd_dvd_nat @ B2 @ C ) ) ) ).

% dvd_times_left_cancel_iff
thf(fact_1558_dvd__times__left__cancel__iff,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( A != zero_zero_int )
     => ( ( dvd_dvd_int @ ( times_times_int @ A @ B2 ) @ ( times_times_int @ A @ C ) )
        = ( dvd_dvd_int @ B2 @ C ) ) ) ).

% dvd_times_left_cancel_iff
thf(fact_1559_dvd__mult__cancel__right,axiom,
    ! [A: code_integer,C: code_integer,B2: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B2 @ C ) )
      = ( ( C = zero_z3403309356797280102nteger )
        | ( dvd_dvd_Code_integer @ A @ B2 ) ) ) ).

% dvd_mult_cancel_right
thf(fact_1560_dvd__mult__cancel__right,axiom,
    ! [A: real,C: real,B2: real] :
      ( ( dvd_dvd_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) )
      = ( ( C = zero_zero_real )
        | ( dvd_dvd_real @ A @ B2 ) ) ) ).

% dvd_mult_cancel_right
thf(fact_1561_dvd__mult__cancel__right,axiom,
    ! [A: rat,C: rat,B2: rat] :
      ( ( dvd_dvd_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ C ) )
      = ( ( C = zero_zero_rat )
        | ( dvd_dvd_rat @ A @ B2 ) ) ) ).

% dvd_mult_cancel_right
thf(fact_1562_dvd__mult__cancel__right,axiom,
    ! [A: int,C: int,B2: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) )
      = ( ( C = zero_zero_int )
        | ( dvd_dvd_int @ A @ B2 ) ) ) ).

% dvd_mult_cancel_right
thf(fact_1563_dvd__mult__cancel__left,axiom,
    ! [C: code_integer,A: code_integer,B2: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B2 ) )
      = ( ( C = zero_z3403309356797280102nteger )
        | ( dvd_dvd_Code_integer @ A @ B2 ) ) ) ).

% dvd_mult_cancel_left
thf(fact_1564_dvd__mult__cancel__left,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( dvd_dvd_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
      = ( ( C = zero_zero_real )
        | ( dvd_dvd_real @ A @ B2 ) ) ) ).

% dvd_mult_cancel_left
thf(fact_1565_dvd__mult__cancel__left,axiom,
    ! [C: rat,A: rat,B2: rat] :
      ( ( dvd_dvd_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) )
      = ( ( C = zero_zero_rat )
        | ( dvd_dvd_rat @ A @ B2 ) ) ) ).

% dvd_mult_cancel_left
thf(fact_1566_dvd__mult__cancel__left,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
      = ( ( C = zero_zero_int )
        | ( dvd_dvd_int @ A @ B2 ) ) ) ).

% dvd_mult_cancel_left
thf(fact_1567_power__0__Suc,axiom,
    ! [N: nat] :
      ( ( power_power_rat @ zero_zero_rat @ ( suc @ N ) )
      = zero_zero_rat ) ).

% power_0_Suc
thf(fact_1568_power__0__Suc,axiom,
    ! [N: nat] :
      ( ( power_power_nat @ zero_zero_nat @ ( suc @ N ) )
      = zero_zero_nat ) ).

% power_0_Suc
thf(fact_1569_power__0__Suc,axiom,
    ! [N: nat] :
      ( ( power_power_real @ zero_zero_real @ ( suc @ N ) )
      = zero_zero_real ) ).

% power_0_Suc
thf(fact_1570_power__0__Suc,axiom,
    ! [N: nat] :
      ( ( power_power_int @ zero_zero_int @ ( suc @ N ) )
      = zero_zero_int ) ).

% power_0_Suc
thf(fact_1571_power__0__Suc,axiom,
    ! [N: nat] :
      ( ( power_power_complex @ zero_zero_complex @ ( suc @ N ) )
      = zero_zero_complex ) ).

% power_0_Suc
thf(fact_1572_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_power_rat @ zero_zero_rat @ ( numeral_numeral_nat @ K ) )
      = zero_zero_rat ) ).

% power_zero_numeral
thf(fact_1573_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ K ) )
      = zero_zero_nat ) ).

% power_zero_numeral
thf(fact_1574_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ K ) )
      = zero_zero_real ) ).

% power_zero_numeral
thf(fact_1575_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ K ) )
      = zero_zero_int ) ).

% power_zero_numeral
thf(fact_1576_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ K ) )
      = zero_zero_complex ) ).

% power_zero_numeral
thf(fact_1577_power__eq__0__iff,axiom,
    ! [A: rat,N: nat] :
      ( ( ( power_power_rat @ A @ N )
        = zero_zero_rat )
      = ( ( A = zero_zero_rat )
        & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% power_eq_0_iff
thf(fact_1578_power__eq__0__iff,axiom,
    ! [A: nat,N: nat] :
      ( ( ( power_power_nat @ A @ N )
        = zero_zero_nat )
      = ( ( A = zero_zero_nat )
        & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% power_eq_0_iff
thf(fact_1579_power__eq__0__iff,axiom,
    ! [A: real,N: nat] :
      ( ( ( power_power_real @ A @ N )
        = zero_zero_real )
      = ( ( A = zero_zero_real )
        & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% power_eq_0_iff
thf(fact_1580_power__eq__0__iff,axiom,
    ! [A: int,N: nat] :
      ( ( ( power_power_int @ A @ N )
        = zero_zero_int )
      = ( ( A = zero_zero_int )
        & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% power_eq_0_iff
thf(fact_1581_power__eq__0__iff,axiom,
    ! [A: complex,N: nat] :
      ( ( ( power_power_complex @ A @ N )
        = zero_zero_complex )
      = ( ( A = zero_zero_complex )
        & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% power_eq_0_iff
thf(fact_1582_unit__prod,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
       => ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B2 ) @ one_one_Code_integer ) ) ) ).

% unit_prod
thf(fact_1583_unit__prod,axiom,
    ! [A: nat,B2: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( dvd_dvd_nat @ B2 @ one_one_nat )
       => ( dvd_dvd_nat @ ( times_times_nat @ A @ B2 ) @ one_one_nat ) ) ) ).

% unit_prod
thf(fact_1584_unit__prod,axiom,
    ! [A: int,B2: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( dvd_dvd_int @ B2 @ one_one_int )
       => ( dvd_dvd_int @ ( times_times_int @ A @ B2 ) @ one_one_int ) ) ) ).

% unit_prod
thf(fact_1585_dvd__add__times__triv__left__iff,axiom,
    ! [A: code_integer,C: code_integer,B2: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ C @ A ) @ B2 ) )
      = ( dvd_dvd_Code_integer @ A @ B2 ) ) ).

% dvd_add_times_triv_left_iff
thf(fact_1586_dvd__add__times__triv__left__iff,axiom,
    ! [A: real,C: real,B2: real] :
      ( ( dvd_dvd_real @ A @ ( plus_plus_real @ ( times_times_real @ C @ A ) @ B2 ) )
      = ( dvd_dvd_real @ A @ B2 ) ) ).

% dvd_add_times_triv_left_iff
thf(fact_1587_dvd__add__times__triv__left__iff,axiom,
    ! [A: rat,C: rat,B2: rat] :
      ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ ( times_times_rat @ C @ A ) @ B2 ) )
      = ( dvd_dvd_rat @ A @ B2 ) ) ).

% dvd_add_times_triv_left_iff
thf(fact_1588_dvd__add__times__triv__left__iff,axiom,
    ! [A: nat,C: nat,B2: nat] :
      ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ ( times_times_nat @ C @ A ) @ B2 ) )
      = ( dvd_dvd_nat @ A @ B2 ) ) ).

% dvd_add_times_triv_left_iff
thf(fact_1589_dvd__add__times__triv__left__iff,axiom,
    ! [A: int,C: int,B2: int] :
      ( ( dvd_dvd_int @ A @ ( plus_plus_int @ ( times_times_int @ C @ A ) @ B2 ) )
      = ( dvd_dvd_int @ A @ B2 ) ) ).

% dvd_add_times_triv_left_iff
thf(fact_1590_dvd__add__times__triv__right__iff,axiom,
    ! [A: code_integer,B2: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B2 @ ( times_3573771949741848930nteger @ C @ A ) ) )
      = ( dvd_dvd_Code_integer @ A @ B2 ) ) ).

% dvd_add_times_triv_right_iff
thf(fact_1591_dvd__add__times__triv__right__iff,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B2 @ ( times_times_real @ C @ A ) ) )
      = ( dvd_dvd_real @ A @ B2 ) ) ).

% dvd_add_times_triv_right_iff
thf(fact_1592_dvd__add__times__triv__right__iff,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B2 @ ( times_times_rat @ C @ A ) ) )
      = ( dvd_dvd_rat @ A @ B2 ) ) ).

% dvd_add_times_triv_right_iff
thf(fact_1593_dvd__add__times__triv__right__iff,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B2 @ ( times_times_nat @ C @ A ) ) )
      = ( dvd_dvd_nat @ A @ B2 ) ) ).

% dvd_add_times_triv_right_iff
thf(fact_1594_dvd__add__times__triv__right__iff,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B2 @ ( times_times_int @ C @ A ) ) )
      = ( dvd_dvd_int @ A @ B2 ) ) ).

% dvd_add_times_triv_right_iff
thf(fact_1595_mod__mult__self1__is__0,axiom,
    ! [B2: nat,A: nat] :
      ( ( modulo_modulo_nat @ ( times_times_nat @ B2 @ A ) @ B2 )
      = zero_zero_nat ) ).

% mod_mult_self1_is_0
thf(fact_1596_mod__mult__self1__is__0,axiom,
    ! [B2: int,A: int] :
      ( ( modulo_modulo_int @ ( times_times_int @ B2 @ A ) @ B2 )
      = zero_zero_int ) ).

% mod_mult_self1_is_0
thf(fact_1597_mod__mult__self1__is__0,axiom,
    ! [B2: code_integer,A: code_integer] :
      ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ B2 @ A ) @ B2 )
      = zero_z3403309356797280102nteger ) ).

% mod_mult_self1_is_0
thf(fact_1598_mod__mult__self2__is__0,axiom,
    ! [A: nat,B2: nat] :
      ( ( modulo_modulo_nat @ ( times_times_nat @ A @ B2 ) @ B2 )
      = zero_zero_nat ) ).

% mod_mult_self2_is_0
thf(fact_1599_mod__mult__self2__is__0,axiom,
    ! [A: int,B2: int] :
      ( ( modulo_modulo_int @ ( times_times_int @ A @ B2 ) @ B2 )
      = zero_zero_int ) ).

% mod_mult_self2_is_0
thf(fact_1600_mod__mult__self2__is__0,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B2 ) @ B2 )
      = zero_z3403309356797280102nteger ) ).

% mod_mult_self2_is_0
thf(fact_1601_dvd__div__mult__self,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ B2 )
     => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ B2 @ A ) @ A )
        = B2 ) ) ).

% dvd_div_mult_self
thf(fact_1602_dvd__div__mult__self,axiom,
    ! [A: nat,B2: nat] :
      ( ( dvd_dvd_nat @ A @ B2 )
     => ( ( times_times_nat @ ( divide_divide_nat @ B2 @ A ) @ A )
        = B2 ) ) ).

% dvd_div_mult_self
thf(fact_1603_dvd__div__mult__self,axiom,
    ! [A: int,B2: int] :
      ( ( dvd_dvd_int @ A @ B2 )
     => ( ( times_times_int @ ( divide_divide_int @ B2 @ A ) @ A )
        = B2 ) ) ).

% dvd_div_mult_self
thf(fact_1604_dvd__mult__div__cancel,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ B2 )
     => ( ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B2 @ A ) )
        = B2 ) ) ).

% dvd_mult_div_cancel
thf(fact_1605_dvd__mult__div__cancel,axiom,
    ! [A: nat,B2: nat] :
      ( ( dvd_dvd_nat @ A @ B2 )
     => ( ( times_times_nat @ A @ ( divide_divide_nat @ B2 @ A ) )
        = B2 ) ) ).

% dvd_mult_div_cancel
thf(fact_1606_dvd__mult__div__cancel,axiom,
    ! [A: int,B2: int] :
      ( ( dvd_dvd_int @ A @ B2 )
     => ( ( times_times_int @ A @ ( divide_divide_int @ B2 @ A ) )
        = B2 ) ) ).

% dvd_mult_div_cancel
thf(fact_1607_mod__by__1,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ A @ one_one_nat )
      = zero_zero_nat ) ).

% mod_by_1
thf(fact_1608_mod__by__1,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ A @ one_one_int )
      = zero_zero_int ) ).

% mod_by_1
thf(fact_1609_mod__by__1,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger @ A @ one_one_Code_integer )
      = zero_z3403309356797280102nteger ) ).

% mod_by_1
thf(fact_1610_bits__mod__by__1,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ A @ one_one_nat )
      = zero_zero_nat ) ).

% bits_mod_by_1
thf(fact_1611_bits__mod__by__1,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ A @ one_one_int )
      = zero_zero_int ) ).

% bits_mod_by_1
thf(fact_1612_bits__mod__by__1,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger @ A @ one_one_Code_integer )
      = zero_z3403309356797280102nteger ) ).

% bits_mod_by_1
thf(fact_1613_unit__div,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
       => ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A @ B2 ) @ one_one_Code_integer ) ) ) ).

% unit_div
thf(fact_1614_unit__div,axiom,
    ! [A: nat,B2: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( dvd_dvd_nat @ B2 @ one_one_nat )
       => ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B2 ) @ one_one_nat ) ) ) ).

% unit_div
thf(fact_1615_unit__div,axiom,
    ! [A: int,B2: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( dvd_dvd_int @ B2 @ one_one_int )
       => ( dvd_dvd_int @ ( divide_divide_int @ A @ B2 ) @ one_one_int ) ) ) ).

% unit_div
thf(fact_1616_unit__div__1__unit,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ one_one_Code_integer @ A ) @ one_one_Code_integer ) ) ).

% unit_div_1_unit
thf(fact_1617_unit__div__1__unit,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( dvd_dvd_nat @ ( divide_divide_nat @ one_one_nat @ A ) @ one_one_nat ) ) ).

% unit_div_1_unit
thf(fact_1618_unit__div__1__unit,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( dvd_dvd_int @ ( divide_divide_int @ one_one_int @ A ) @ one_one_int ) ) ).

% unit_div_1_unit
thf(fact_1619_unit__div__1__div__1,axiom,
    ! [A: code_integer] :
      ( ( 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
thf(fact_1620_unit__div__1__div__1,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( divide_divide_nat @ one_one_nat @ ( divide_divide_nat @ one_one_nat @ A ) )
        = A ) ) ).

% unit_div_1_div_1
thf(fact_1621_unit__div__1__div__1,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( divide_divide_int @ one_one_int @ ( divide_divide_int @ one_one_int @ A ) )
        = A ) ) ).

% unit_div_1_div_1
thf(fact_1622_div__add,axiom,
    ! [C: code_integer,A: code_integer,B2: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ A )
     => ( ( dvd_dvd_Code_integer @ C @ B2 )
       => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B2 ) @ C )
          = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B2 @ C ) ) ) ) ) ).

% div_add
thf(fact_1623_div__add,axiom,
    ! [C: nat,A: nat,B2: nat] :
      ( ( dvd_dvd_nat @ C @ A )
     => ( ( dvd_dvd_nat @ C @ B2 )
       => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B2 ) @ C )
          = ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B2 @ C ) ) ) ) ) ).

% div_add
thf(fact_1624_div__add,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( dvd_dvd_int @ C @ A )
     => ( ( dvd_dvd_int @ C @ B2 )
       => ( ( divide_divide_int @ ( plus_plus_int @ A @ B2 ) @ C )
          = ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B2 @ C ) ) ) ) ) ).

% div_add
thf(fact_1625_div__diff,axiom,
    ! [C: code_integer,A: code_integer,B2: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ A )
     => ( ( dvd_dvd_Code_integer @ C @ B2 )
       => ( ( divide6298287555418463151nteger @ ( minus_8373710615458151222nteger @ A @ B2 ) @ C )
          = ( minus_8373710615458151222nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B2 @ C ) ) ) ) ) ).

% div_diff
thf(fact_1626_div__diff,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( dvd_dvd_int @ C @ A )
     => ( ( dvd_dvd_int @ C @ B2 )
       => ( ( divide_divide_int @ ( minus_minus_int @ A @ B2 ) @ C )
          = ( minus_minus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B2 @ C ) ) ) ) ) ).

% div_diff
thf(fact_1627_mod__div__trivial,axiom,
    ! [A: nat,B2: nat] :
      ( ( divide_divide_nat @ ( modulo_modulo_nat @ A @ B2 ) @ B2 )
      = zero_zero_nat ) ).

% mod_div_trivial
thf(fact_1628_mod__div__trivial,axiom,
    ! [A: int,B2: int] :
      ( ( divide_divide_int @ ( modulo_modulo_int @ A @ B2 ) @ B2 )
      = zero_zero_int ) ).

% mod_div_trivial
thf(fact_1629_mod__div__trivial,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ B2 ) @ B2 )
      = zero_z3403309356797280102nteger ) ).

% mod_div_trivial
thf(fact_1630_bits__mod__div__trivial,axiom,
    ! [A: nat,B2: nat] :
      ( ( divide_divide_nat @ ( modulo_modulo_nat @ A @ B2 ) @ B2 )
      = zero_zero_nat ) ).

% bits_mod_div_trivial
thf(fact_1631_bits__mod__div__trivial,axiom,
    ! [A: int,B2: int] :
      ( ( divide_divide_int @ ( modulo_modulo_int @ A @ B2 ) @ B2 )
      = zero_zero_int ) ).

% bits_mod_div_trivial
thf(fact_1632_bits__mod__div__trivial,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ B2 ) @ B2 )
      = zero_z3403309356797280102nteger ) ).

% bits_mod_div_trivial
thf(fact_1633_zmod__numeral__Bit0,axiom,
    ! [V: num,W: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) ) ).

% zmod_numeral_Bit0
thf(fact_1634_dvd__imp__mod__0,axiom,
    ! [A: nat,B2: nat] :
      ( ( dvd_dvd_nat @ A @ B2 )
     => ( ( modulo_modulo_nat @ B2 @ A )
        = zero_zero_nat ) ) ).

% dvd_imp_mod_0
thf(fact_1635_dvd__imp__mod__0,axiom,
    ! [A: int,B2: int] :
      ( ( dvd_dvd_int @ A @ B2 )
     => ( ( modulo_modulo_int @ B2 @ A )
        = zero_zero_int ) ) ).

% dvd_imp_mod_0
thf(fact_1636_dvd__imp__mod__0,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ B2 )
     => ( ( modulo364778990260209775nteger @ B2 @ A )
        = zero_z3403309356797280102nteger ) ) ).

% dvd_imp_mod_0
thf(fact_1637_Suc__pred,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
        = N ) ) ).

% Suc_pred
thf(fact_1638_one__le__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) )
      = ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M )
        & ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N ) ) ) ).

% one_le_mult_iff
thf(fact_1639_nat__mult__le__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ord_less_eq_nat @ M @ N ) ) ) ).

% nat_mult_le_cancel_disj
thf(fact_1640_mult__le__cancel2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ord_less_eq_nat @ M @ N ) ) ) ).

% mult_le_cancel2
thf(fact_1641_div__mult__self1__is__m,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( divide_divide_nat @ ( times_times_nat @ N @ M ) @ N )
        = M ) ) ).

% div_mult_self1_is_m
thf(fact_1642_div__mult__self__is__m,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( divide_divide_nat @ ( times_times_nat @ M @ N ) @ N )
        = M ) ) ).

% div_mult_self_is_m
thf(fact_1643_signed__take__bit__Suc__1,axiom,
    ! [N: nat] :
      ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ one_one_int )
      = one_one_int ) ).

% signed_take_bit_Suc_1
thf(fact_1644_signed__take__bit__numeral__of__1,axiom,
    ! [K: num] :
      ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ K ) @ one_one_int )
      = one_one_int ) ).

% signed_take_bit_numeral_of_1
thf(fact_1645_dbl__simps_I5_J,axiom,
    ! [K: num] :
      ( ( neg_nu7009210354673126013omplex @ ( numera6690914467698888265omplex @ K ) )
      = ( numera6690914467698888265omplex @ ( bit0 @ K ) ) ) ).

% dbl_simps(5)
thf(fact_1646_dbl__simps_I5_J,axiom,
    ! [K: num] :
      ( ( neg_numeral_dbl_real @ ( numeral_numeral_real @ K ) )
      = ( numeral_numeral_real @ ( bit0 @ K ) ) ) ).

% dbl_simps(5)
thf(fact_1647_dbl__simps_I5_J,axiom,
    ! [K: num] :
      ( ( neg_numeral_dbl_rat @ ( numeral_numeral_rat @ K ) )
      = ( numeral_numeral_rat @ ( bit0 @ K ) ) ) ).

% dbl_simps(5)
thf(fact_1648_dbl__simps_I5_J,axiom,
    ! [K: num] :
      ( ( neg_numeral_dbl_int @ ( numeral_numeral_int @ K ) )
      = ( numeral_numeral_int @ ( bit0 @ K ) ) ) ).

% dbl_simps(5)
thf(fact_1649_zero__le__divide__1__iff,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A ) )
      = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).

% zero_le_divide_1_iff
thf(fact_1650_zero__le__divide__1__iff,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ one_one_rat @ A ) )
      = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).

% zero_le_divide_1_iff
thf(fact_1651_divide__le__0__1__iff,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ A ) @ zero_zero_real )
      = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).

% divide_le_0_1_iff
thf(fact_1652_divide__le__0__1__iff,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ zero_zero_rat )
      = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).

% divide_le_0_1_iff
thf(fact_1653_zero__less__divide__1__iff,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A ) )
      = ( ord_less_real @ zero_zero_real @ A ) ) ).

% zero_less_divide_1_iff
thf(fact_1654_zero__less__divide__1__iff,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ one_one_rat @ A ) )
      = ( ord_less_rat @ zero_zero_rat @ A ) ) ).

% zero_less_divide_1_iff
thf(fact_1655_less__divide__eq__1__pos,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B2 @ A ) )
        = ( ord_less_real @ A @ B2 ) ) ) ).

% less_divide_eq_1_pos
thf(fact_1656_less__divide__eq__1__pos,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B2 @ A ) )
        = ( ord_less_rat @ A @ B2 ) ) ) ).

% less_divide_eq_1_pos
thf(fact_1657_less__divide__eq__1__neg,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B2 @ A ) )
        = ( ord_less_real @ B2 @ A ) ) ) ).

% less_divide_eq_1_neg
thf(fact_1658_less__divide__eq__1__neg,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B2 @ A ) )
        = ( ord_less_rat @ B2 @ A ) ) ) ).

% less_divide_eq_1_neg
thf(fact_1659_divide__less__eq__1__pos,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ ( divide_divide_real @ B2 @ A ) @ one_one_real )
        = ( ord_less_real @ B2 @ A ) ) ) ).

% divide_less_eq_1_pos
thf(fact_1660_divide__less__eq__1__pos,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ ( divide_divide_rat @ B2 @ A ) @ one_one_rat )
        = ( ord_less_rat @ B2 @ A ) ) ) ).

% divide_less_eq_1_pos
thf(fact_1661_divide__less__eq__1__neg,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( ord_less_real @ ( divide_divide_real @ B2 @ A ) @ one_one_real )
        = ( ord_less_real @ A @ B2 ) ) ) ).

% divide_less_eq_1_neg
thf(fact_1662_divide__less__eq__1__neg,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( ord_less_rat @ ( divide_divide_rat @ B2 @ A ) @ one_one_rat )
        = ( ord_less_rat @ A @ B2 ) ) ) ).

% divide_less_eq_1_neg
thf(fact_1663_divide__less__0__1__iff,axiom,
    ! [A: real] :
      ( ( ord_less_real @ ( divide_divide_real @ one_one_real @ A ) @ zero_zero_real )
      = ( ord_less_real @ A @ zero_zero_real ) ) ).

% divide_less_0_1_iff
thf(fact_1664_divide__less__0__1__iff,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ zero_zero_rat )
      = ( ord_less_rat @ A @ zero_zero_rat ) ) ).

% divide_less_0_1_iff
thf(fact_1665_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B2: complex,W: num,A: complex] :
      ( ( ( divide1717551699836669952omplex @ B2 @ ( numera6690914467698888265omplex @ W ) )
        = A )
      = ( ( ( ( numera6690914467698888265omplex @ W )
           != zero_zero_complex )
         => ( B2
            = ( times_times_complex @ A @ ( numera6690914467698888265omplex @ W ) ) ) )
        & ( ( ( numera6690914467698888265omplex @ W )
            = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% divide_eq_eq_numeral1(1)
thf(fact_1666_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B2: real,W: num,A: real] :
      ( ( ( divide_divide_real @ B2 @ ( numeral_numeral_real @ W ) )
        = A )
      = ( ( ( ( numeral_numeral_real @ W )
           != zero_zero_real )
         => ( B2
            = ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) ) )
        & ( ( ( numeral_numeral_real @ W )
            = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% divide_eq_eq_numeral1(1)
thf(fact_1667_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B2: rat,W: num,A: rat] :
      ( ( ( divide_divide_rat @ B2 @ ( numeral_numeral_rat @ W ) )
        = A )
      = ( ( ( ( numeral_numeral_rat @ W )
           != zero_zero_rat )
         => ( B2
            = ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) ) )
        & ( ( ( numeral_numeral_rat @ W )
            = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% divide_eq_eq_numeral1(1)
thf(fact_1668_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: complex,B2: complex,W: num] :
      ( ( A
        = ( divide1717551699836669952omplex @ B2 @ ( numera6690914467698888265omplex @ W ) ) )
      = ( ( ( ( numera6690914467698888265omplex @ W )
           != zero_zero_complex )
         => ( ( times_times_complex @ A @ ( numera6690914467698888265omplex @ W ) )
            = B2 ) )
        & ( ( ( numera6690914467698888265omplex @ W )
            = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% eq_divide_eq_numeral1(1)
thf(fact_1669_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: real,B2: real,W: num] :
      ( ( A
        = ( divide_divide_real @ B2 @ ( numeral_numeral_real @ W ) ) )
      = ( ( ( ( numeral_numeral_real @ W )
           != zero_zero_real )
         => ( ( times_times_real @ A @ ( numeral_numeral_real @ W ) )
            = B2 ) )
        & ( ( ( numeral_numeral_real @ W )
            = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral1(1)
thf(fact_1670_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B2: rat,W: num] :
      ( ( A
        = ( divide_divide_rat @ B2 @ ( numeral_numeral_rat @ W ) ) )
      = ( ( ( ( numeral_numeral_rat @ W )
           != zero_zero_rat )
         => ( ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) )
            = B2 ) )
        & ( ( ( numeral_numeral_rat @ W )
            = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral1(1)
thf(fact_1671_nonzero__divide__mult__cancel__left,axiom,
    ! [A: complex,B2: complex] :
      ( ( A != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ A @ ( times_times_complex @ A @ B2 ) )
        = ( divide1717551699836669952omplex @ one_one_complex @ B2 ) ) ) ).

% nonzero_divide_mult_cancel_left
thf(fact_1672_nonzero__divide__mult__cancel__left,axiom,
    ! [A: real,B2: real] :
      ( ( A != zero_zero_real )
     => ( ( divide_divide_real @ A @ ( times_times_real @ A @ B2 ) )
        = ( divide_divide_real @ one_one_real @ B2 ) ) ) ).

% nonzero_divide_mult_cancel_left
thf(fact_1673_nonzero__divide__mult__cancel__left,axiom,
    ! [A: rat,B2: rat] :
      ( ( A != zero_zero_rat )
     => ( ( divide_divide_rat @ A @ ( times_times_rat @ A @ B2 ) )
        = ( divide_divide_rat @ one_one_rat @ B2 ) ) ) ).

% nonzero_divide_mult_cancel_left
thf(fact_1674_nonzero__divide__mult__cancel__right,axiom,
    ! [B2: complex,A: complex] :
      ( ( B2 != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ B2 @ ( times_times_complex @ A @ B2 ) )
        = ( divide1717551699836669952omplex @ one_one_complex @ A ) ) ) ).

% nonzero_divide_mult_cancel_right
thf(fact_1675_nonzero__divide__mult__cancel__right,axiom,
    ! [B2: real,A: real] :
      ( ( B2 != zero_zero_real )
     => ( ( divide_divide_real @ B2 @ ( times_times_real @ A @ B2 ) )
        = ( divide_divide_real @ one_one_real @ A ) ) ) ).

% nonzero_divide_mult_cancel_right
thf(fact_1676_nonzero__divide__mult__cancel__right,axiom,
    ! [B2: rat,A: rat] :
      ( ( B2 != zero_zero_rat )
     => ( ( divide_divide_rat @ B2 @ ( times_times_rat @ A @ B2 ) )
        = ( divide_divide_rat @ one_one_rat @ A ) ) ) ).

% nonzero_divide_mult_cancel_right
thf(fact_1677_div__mult__self1,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( ( B2 != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C @ B2 ) ) @ B2 )
        = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B2 ) ) ) ) ).

% div_mult_self1
thf(fact_1678_div__mult__self1,axiom,
    ! [B2: int,A: int,C: int] :
      ( ( B2 != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ C @ B2 ) ) @ B2 )
        = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B2 ) ) ) ) ).

% div_mult_self1
thf(fact_1679_div__mult__self2,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( ( B2 != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B2 @ C ) ) @ B2 )
        = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B2 ) ) ) ) ).

% div_mult_self2
thf(fact_1680_div__mult__self2,axiom,
    ! [B2: int,A: int,C: int] :
      ( ( B2 != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ B2 @ C ) ) @ B2 )
        = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B2 ) ) ) ) ).

% div_mult_self2
thf(fact_1681_div__mult__self3,axiom,
    ! [B2: nat,C: nat,A: nat] :
      ( ( B2 != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B2 ) @ A ) @ B2 )
        = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B2 ) ) ) ) ).

% div_mult_self3
thf(fact_1682_div__mult__self3,axiom,
    ! [B2: int,C: int,A: int] :
      ( ( B2 != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ C @ B2 ) @ A ) @ B2 )
        = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B2 ) ) ) ) ).

% div_mult_self3
thf(fact_1683_div__mult__self4,axiom,
    ! [B2: nat,C: nat,A: nat] :
      ( ( B2 != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ B2 @ C ) @ A ) @ B2 )
        = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B2 ) ) ) ) ).

% div_mult_self4
thf(fact_1684_div__mult__self4,axiom,
    ! [B2: int,C: int,A: int] :
      ( ( B2 != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ B2 @ C ) @ A ) @ B2 )
        = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B2 ) ) ) ) ).

% div_mult_self4
thf(fact_1685_power__mono__iff,axiom,
    ! [A: real,B2: real,N: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
       => ( ( ord_less_nat @ zero_zero_nat @ N )
         => ( ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B2 @ N ) )
            = ( ord_less_eq_real @ A @ B2 ) ) ) ) ) ).

% power_mono_iff
thf(fact_1686_power__mono__iff,axiom,
    ! [A: rat,B2: rat,N: nat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
       => ( ( ord_less_nat @ zero_zero_nat @ N )
         => ( ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B2 @ N ) )
            = ( ord_less_eq_rat @ A @ B2 ) ) ) ) ) ).

% power_mono_iff
thf(fact_1687_power__mono__iff,axiom,
    ! [A: nat,B2: nat,N: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
       => ( ( ord_less_nat @ zero_zero_nat @ N )
         => ( ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B2 @ N ) )
            = ( ord_less_eq_nat @ A @ B2 ) ) ) ) ) ).

% power_mono_iff
thf(fact_1688_power__mono__iff,axiom,
    ! [A: int,B2: int,N: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
       => ( ( ord_less_nat @ zero_zero_nat @ N )
         => ( ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B2 @ N ) )
            = ( ord_less_eq_int @ A @ B2 ) ) ) ) ) ).

% power_mono_iff
thf(fact_1689_unit__mult__div__div,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ( ( times_3573771949741848930nteger @ B2 @ ( divide6298287555418463151nteger @ one_one_Code_integer @ A ) )
        = ( divide6298287555418463151nteger @ B2 @ A ) ) ) ).

% unit_mult_div_div
thf(fact_1690_unit__mult__div__div,axiom,
    ! [A: nat,B2: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( times_times_nat @ B2 @ ( divide_divide_nat @ one_one_nat @ A ) )
        = ( divide_divide_nat @ B2 @ A ) ) ) ).

% unit_mult_div_div
thf(fact_1691_unit__mult__div__div,axiom,
    ! [A: int,B2: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( times_times_int @ B2 @ ( divide_divide_int @ one_one_int @ A ) )
        = ( divide_divide_int @ B2 @ A ) ) ) ).

% unit_mult_div_div
thf(fact_1692_unit__div__mult__self,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ B2 @ A ) @ A )
        = B2 ) ) ).

% unit_div_mult_self
thf(fact_1693_unit__div__mult__self,axiom,
    ! [A: nat,B2: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( times_times_nat @ ( divide_divide_nat @ B2 @ A ) @ A )
        = B2 ) ) ).

% unit_div_mult_self
thf(fact_1694_unit__div__mult__self,axiom,
    ! [A: int,B2: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( times_times_int @ ( divide_divide_int @ B2 @ A ) @ A )
        = B2 ) ) ).

% unit_div_mult_self
thf(fact_1695_signed__take__bit__Suc__bit0,axiom,
    ! [N: nat,K: num] :
      ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
      = ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% signed_take_bit_Suc_bit0
thf(fact_1696_Suc__diff__1,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
        = N ) ) ).

% Suc_diff_1
thf(fact_1697_half__negative__int__iff,axiom,
    ! [K: int] :
      ( ( ord_less_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ zero_zero_int )
      = ( ord_less_int @ K @ zero_zero_int ) ) ).

% half_negative_int_iff
thf(fact_1698_half__nonnegative__int__iff,axiom,
    ! [K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
      = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).

% half_nonnegative_int_iff
thf(fact_1699_le__divide__eq__1__pos,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B2 @ A ) )
        = ( ord_less_eq_real @ A @ B2 ) ) ) ).

% le_divide_eq_1_pos
thf(fact_1700_le__divide__eq__1__pos,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B2 @ A ) )
        = ( ord_less_eq_rat @ A @ B2 ) ) ) ).

% le_divide_eq_1_pos
thf(fact_1701_le__divide__eq__1__neg,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B2 @ A ) )
        = ( ord_less_eq_real @ B2 @ A ) ) ) ).

% le_divide_eq_1_neg
thf(fact_1702_le__divide__eq__1__neg,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B2 @ A ) )
        = ( ord_less_eq_rat @ B2 @ A ) ) ) ).

% le_divide_eq_1_neg
thf(fact_1703_divide__le__eq__1__pos,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ ( divide_divide_real @ B2 @ A ) @ one_one_real )
        = ( ord_less_eq_real @ B2 @ A ) ) ) ).

% divide_le_eq_1_pos
thf(fact_1704_divide__le__eq__1__pos,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B2 @ A ) @ one_one_rat )
        = ( ord_less_eq_rat @ B2 @ A ) ) ) ).

% divide_le_eq_1_pos
thf(fact_1705_divide__le__eq__1__neg,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( ord_less_eq_real @ ( divide_divide_real @ B2 @ A ) @ one_one_real )
        = ( ord_less_eq_real @ A @ B2 ) ) ) ).

% divide_le_eq_1_neg
thf(fact_1706_divide__le__eq__1__neg,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B2 @ A ) @ one_one_rat )
        = ( ord_less_eq_rat @ A @ B2 ) ) ) ).

% divide_le_eq_1_neg
thf(fact_1707_power__strict__decreasing__iff,axiom,
    ! [B2: real,M: nat,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ B2 )
     => ( ( ord_less_real @ B2 @ one_one_real )
       => ( ( ord_less_real @ ( power_power_real @ B2 @ M ) @ ( power_power_real @ B2 @ N ) )
          = ( ord_less_nat @ N @ M ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_1708_power__strict__decreasing__iff,axiom,
    ! [B2: rat,M: nat,N: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ B2 )
     => ( ( ord_less_rat @ B2 @ one_one_rat )
       => ( ( ord_less_rat @ ( power_power_rat @ B2 @ M ) @ ( power_power_rat @ B2 @ N ) )
          = ( ord_less_nat @ N @ M ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_1709_power__strict__decreasing__iff,axiom,
    ! [B2: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B2 )
     => ( ( ord_less_nat @ B2 @ one_one_nat )
       => ( ( ord_less_nat @ ( power_power_nat @ B2 @ M ) @ ( power_power_nat @ B2 @ N ) )
          = ( ord_less_nat @ N @ M ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_1710_power__strict__decreasing__iff,axiom,
    ! [B2: int,M: nat,N: nat] :
      ( ( ord_less_int @ zero_zero_int @ B2 )
     => ( ( ord_less_int @ B2 @ one_one_int )
       => ( ( ord_less_int @ ( power_power_int @ B2 @ M ) @ ( power_power_int @ B2 @ N ) )
          = ( ord_less_nat @ N @ M ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_1711_even__mult__iff,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( times_3573771949741848930nteger @ A @ B2 ) )
      = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
        | ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) ) ) ).

% even_mult_iff
thf(fact_1712_even__mult__iff,axiom,
    ! [A: nat,B2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ A @ B2 ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
        | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) ) ).

% even_mult_iff
thf(fact_1713_even__mult__iff,axiom,
    ! [A: int,B2: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( times_times_int @ A @ B2 ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
        | ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) ) ).

% even_mult_iff
thf(fact_1714_zero__eq__power2,axiom,
    ! [A: rat] :
      ( ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = zero_zero_rat )
      = ( A = zero_zero_rat ) ) ).

% zero_eq_power2
thf(fact_1715_zero__eq__power2,axiom,
    ! [A: nat] :
      ( ( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = zero_zero_nat )
      = ( A = zero_zero_nat ) ) ).

% zero_eq_power2
thf(fact_1716_zero__eq__power2,axiom,
    ! [A: real] :
      ( ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% zero_eq_power2
thf(fact_1717_zero__eq__power2,axiom,
    ! [A: int] :
      ( ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = zero_zero_int )
      = ( A = zero_zero_int ) ) ).

% zero_eq_power2
thf(fact_1718_zero__eq__power2,axiom,
    ! [A: complex] :
      ( ( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = zero_zero_complex )
      = ( A = zero_zero_complex ) ) ).

% zero_eq_power2
thf(fact_1719_odd__add,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B2 ) ) )
      = ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) )
       != ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) ) ) ) ).

% odd_add
thf(fact_1720_odd__add,axiom,
    ! [A: nat,B2: nat] :
      ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B2 ) ) )
      = ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
       != ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) ) ) ).

% odd_add
thf(fact_1721_odd__add,axiom,
    ! [A: int,B2: int] :
      ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B2 ) ) )
      = ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
       != ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) ) ) ).

% odd_add
thf(fact_1722_even__add,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B2 ) )
      = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
        = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) ) ) ).

% even_add
thf(fact_1723_even__add,axiom,
    ! [A: nat,B2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B2 ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
        = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) ) ).

% even_add
thf(fact_1724_even__add,axiom,
    ! [A: int,B2: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B2 ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
        = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) ) ).

% even_add
thf(fact_1725_even__mod__2__iff,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ).

% even_mod_2_iff
thf(fact_1726_even__mod__2__iff,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
      = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ).

% even_mod_2_iff
thf(fact_1727_even__mod__2__iff,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) )
      = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ).

% even_mod_2_iff
thf(fact_1728_even__Suc,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N ) )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% even_Suc
thf(fact_1729_even__Suc__Suc__iff,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ N ) ) )
      = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).

% even_Suc_Suc_iff
thf(fact_1730_not__mod2__eq__Suc__0__eq__0,axiom,
    ! [N: nat] :
      ( ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
       != ( suc @ zero_zero_nat ) )
      = ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = zero_zero_nat ) ) ).

% not_mod2_eq_Suc_0_eq_0
thf(fact_1731_add__self__mod__2,axiom,
    ! [M: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ M @ M ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = zero_zero_nat ) ).

% add_self_mod_2
thf(fact_1732_one__div__two__eq__zero,axiom,
    ( ( divide_divide_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_nat ) ).

% one_div_two_eq_zero
thf(fact_1733_one__div__two__eq__zero,axiom,
    ( ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
    = zero_zero_int ) ).

% one_div_two_eq_zero
thf(fact_1734_bits__1__div__2,axiom,
    ( ( divide_divide_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_nat ) ).

% bits_1_div_2
thf(fact_1735_bits__1__div__2,axiom,
    ( ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
    = zero_zero_int ) ).

% bits_1_div_2
thf(fact_1736_power__decreasing__iff,axiom,
    ! [B2: real,M: nat,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ B2 )
     => ( ( ord_less_real @ B2 @ one_one_real )
       => ( ( ord_less_eq_real @ ( power_power_real @ B2 @ M ) @ ( power_power_real @ B2 @ N ) )
          = ( ord_less_eq_nat @ N @ M ) ) ) ) ).

% power_decreasing_iff
thf(fact_1737_power__decreasing__iff,axiom,
    ! [B2: rat,M: nat,N: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ B2 )
     => ( ( ord_less_rat @ B2 @ one_one_rat )
       => ( ( ord_less_eq_rat @ ( power_power_rat @ B2 @ M ) @ ( power_power_rat @ B2 @ N ) )
          = ( ord_less_eq_nat @ N @ M ) ) ) ) ).

% power_decreasing_iff
thf(fact_1738_power__decreasing__iff,axiom,
    ! [B2: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B2 )
     => ( ( ord_less_nat @ B2 @ one_one_nat )
       => ( ( ord_less_eq_nat @ ( power_power_nat @ B2 @ M ) @ ( power_power_nat @ B2 @ N ) )
          = ( ord_less_eq_nat @ N @ M ) ) ) ) ).

% power_decreasing_iff
thf(fact_1739_power__decreasing__iff,axiom,
    ! [B2: int,M: nat,N: nat] :
      ( ( ord_less_int @ zero_zero_int @ B2 )
     => ( ( ord_less_int @ B2 @ one_one_int )
       => ( ( ord_less_eq_int @ ( power_power_int @ B2 @ M ) @ ( power_power_int @ B2 @ N ) )
          = ( ord_less_eq_nat @ N @ M ) ) ) ) ).

% power_decreasing_iff
thf(fact_1740_power2__less__eq__zero__iff,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% power2_less_eq_zero_iff
thf(fact_1741_power2__less__eq__zero__iff,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_rat )
      = ( A = zero_zero_rat ) ) ).

% power2_less_eq_zero_iff
thf(fact_1742_power2__less__eq__zero__iff,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_int )
      = ( A = zero_zero_int ) ) ).

% power2_less_eq_zero_iff
thf(fact_1743_power2__eq__iff__nonneg,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
       => ( ( ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          = ( X4 = Y3 ) ) ) ) ).

% power2_eq_iff_nonneg
thf(fact_1744_power2__eq__iff__nonneg,axiom,
    ! [X4: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X4 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
       => ( ( ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          = ( X4 = Y3 ) ) ) ) ).

% power2_eq_iff_nonneg
thf(fact_1745_power2__eq__iff__nonneg,axiom,
    ! [X4: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ X4 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ Y3 )
       => ( ( ( power_power_nat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = ( power_power_nat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          = ( X4 = Y3 ) ) ) ) ).

% power2_eq_iff_nonneg
thf(fact_1746_power2__eq__iff__nonneg,axiom,
    ! [X4: int,Y3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X4 )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
       => ( ( ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          = ( X4 = Y3 ) ) ) ) ).

% power2_eq_iff_nonneg
thf(fact_1747_zero__less__power2,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( A != zero_zero_real ) ) ).

% zero_less_power2
thf(fact_1748_zero__less__power2,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( A != zero_zero_rat ) ) ).

% zero_less_power2
thf(fact_1749_zero__less__power2,axiom,
    ! [A: int] :
      ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( A != zero_zero_int ) ) ).

% zero_less_power2
thf(fact_1750_sum__power2__eq__zero__iff,axiom,
    ! [X4: rat,Y3: rat] :
      ( ( ( plus_plus_rat @ ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = zero_zero_rat )
      = ( ( X4 = zero_zero_rat )
        & ( Y3 = zero_zero_rat ) ) ) ).

% sum_power2_eq_zero_iff
thf(fact_1751_sum__power2__eq__zero__iff,axiom,
    ! [X4: real,Y3: real] :
      ( ( ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = zero_zero_real )
      = ( ( X4 = zero_zero_real )
        & ( Y3 = zero_zero_real ) ) ) ).

% sum_power2_eq_zero_iff
thf(fact_1752_sum__power2__eq__zero__iff,axiom,
    ! [X4: int,Y3: int] :
      ( ( ( plus_plus_int @ ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = zero_zero_int )
      = ( ( X4 = zero_zero_int )
        & ( Y3 = zero_zero_int ) ) ) ).

% sum_power2_eq_zero_iff
thf(fact_1753_even__plus__one__iff,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) )
      = ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).

% even_plus_one_iff
thf(fact_1754_even__plus__one__iff,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ one_one_nat ) )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).

% even_plus_one_iff
thf(fact_1755_even__plus__one__iff,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ one_one_int ) )
      = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).

% even_plus_one_iff
thf(fact_1756_even__diff,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_8373710615458151222nteger @ A @ B2 ) )
      = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B2 ) ) ) ).

% even_diff
thf(fact_1757_even__diff,axiom,
    ! [A: int,B2: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ A @ B2 ) )
      = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B2 ) ) ) ).

% even_diff
thf(fact_1758_not__mod__2__eq__0__eq__1,axiom,
    ! [A: nat] :
      ( ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
       != zero_zero_nat )
      = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_nat ) ) ).

% not_mod_2_eq_0_eq_1
thf(fact_1759_not__mod__2__eq__0__eq__1,axiom,
    ! [A: int] :
      ( ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
       != zero_zero_int )
      = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = one_one_int ) ) ).

% not_mod_2_eq_0_eq_1
thf(fact_1760_not__mod__2__eq__0__eq__1,axiom,
    ! [A: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
       != zero_z3403309356797280102nteger )
      = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = one_one_Code_integer ) ) ).

% not_mod_2_eq_0_eq_1
thf(fact_1761_not__mod__2__eq__1__eq__0,axiom,
    ! [A: nat] :
      ( ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
       != one_one_nat )
      = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = zero_zero_nat ) ) ).

% not_mod_2_eq_1_eq_0
thf(fact_1762_not__mod__2__eq__1__eq__0,axiom,
    ! [A: int] :
      ( ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
       != one_one_int )
      = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = zero_zero_int ) ) ).

% not_mod_2_eq_1_eq_0
thf(fact_1763_not__mod__2__eq__1__eq__0,axiom,
    ! [A: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
       != one_one_Code_integer )
      = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = zero_z3403309356797280102nteger ) ) ).

% not_mod_2_eq_1_eq_0
thf(fact_1764_even__power,axiom,
    ! [A: code_integer,N: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( power_8256067586552552935nteger @ A @ N ) )
      = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
        & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% even_power
thf(fact_1765_even__power,axiom,
    ! [A: nat,N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ A @ N ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
        & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% even_power
thf(fact_1766_even__power,axiom,
    ! [A: int,N: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( power_power_int @ A @ N ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
        & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% even_power
thf(fact_1767_odd__Suc__minus__one,axiom,
    ! [N: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
        = N ) ) ).

% odd_Suc_minus_one
thf(fact_1768_odd__Suc__div__two,axiom,
    ! [N: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( divide_divide_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% odd_Suc_div_two
thf(fact_1769_even__Suc__div__two,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( divide_divide_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% even_Suc_div_two
thf(fact_1770_mod2__gr__0,axiom,
    ! [M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_nat ) ) ).

% mod2_gr_0
thf(fact_1771_unset__bit__0,axiom,
    ! [A: int] :
      ( ( bit_se4203085406695923979it_int @ zero_zero_nat @ A )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).

% unset_bit_0
thf(fact_1772_unset__bit__0,axiom,
    ! [A: nat] :
      ( ( bit_se4205575877204974255it_nat @ zero_zero_nat @ A )
      = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% unset_bit_0
thf(fact_1773_odd__succ__div__two,axiom,
    ! [A: code_integer] :
      ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ one_one_Code_integer ) ) ) ).

% odd_succ_div_two
thf(fact_1774_odd__succ__div__two,axiom,
    ! [A: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( plus_plus_nat @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ).

% odd_succ_div_two
thf(fact_1775_odd__succ__div__two,axiom,
    ! [A: int] :
      ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ( ( divide_divide_int @ ( plus_plus_int @ A @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = ( plus_plus_int @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ) ).

% odd_succ_div_two
thf(fact_1776_even__succ__div__two,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).

% even_succ_div_two
thf(fact_1777_even__succ__div__two,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% even_succ_div_two
thf(fact_1778_even__succ__div__two,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ( ( divide_divide_int @ ( plus_plus_int @ A @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).

% even_succ_div_two
thf(fact_1779_even__succ__div__2,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).

% even_succ_div_2
thf(fact_1780_even__succ__div__2,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% even_succ_div_2
thf(fact_1781_even__succ__div__2,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ A ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).

% even_succ_div_2
thf(fact_1782_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) @ one_one_Code_integer ) )
      = ( N = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
thf(fact_1783_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) )
      = ( N = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
thf(fact_1784_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int ) )
      = ( N = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
thf(fact_1785_zero__le__power__eq__numeral,axiom,
    ! [A: real,W: num] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
          & ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ).

% zero_le_power_eq_numeral
thf(fact_1786_zero__le__power__eq__numeral,axiom,
    ! [A: rat,W: num] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
          & ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ).

% zero_le_power_eq_numeral
thf(fact_1787_zero__le__power__eq__numeral,axiom,
    ! [A: int,W: num] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
          & ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ) ).

% zero_le_power_eq_numeral
thf(fact_1788_zero__less__power__eq__numeral,axiom,
    ! [A: real,W: num] :
      ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) )
      = ( ( ( numeral_numeral_nat @ W )
          = zero_zero_nat )
        | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
          & ( A != zero_zero_real ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
          & ( ord_less_real @ zero_zero_real @ A ) ) ) ) ).

% zero_less_power_eq_numeral
thf(fact_1789_zero__less__power__eq__numeral,axiom,
    ! [A: rat,W: num] :
      ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) )
      = ( ( ( numeral_numeral_nat @ W )
          = zero_zero_nat )
        | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
          & ( A != zero_zero_rat ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
          & ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ).

% zero_less_power_eq_numeral
thf(fact_1790_zero__less__power__eq__numeral,axiom,
    ! [A: int,W: num] :
      ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) )
      = ( ( ( numeral_numeral_nat @ W )
          = zero_zero_nat )
        | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
          & ( A != zero_zero_int ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
          & ( ord_less_int @ zero_zero_int @ A ) ) ) ) ).

% zero_less_power_eq_numeral
thf(fact_1791_power__less__zero__eq,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_real @ ( power_power_real @ A @ N ) @ zero_zero_real )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
        & ( ord_less_real @ A @ zero_zero_real ) ) ) ).

% power_less_zero_eq
thf(fact_1792_power__less__zero__eq,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_rat @ ( power_power_rat @ A @ N ) @ zero_zero_rat )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
        & ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).

% power_less_zero_eq
thf(fact_1793_power__less__zero__eq,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_int @ ( power_power_int @ A @ N ) @ zero_zero_int )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
        & ( ord_less_int @ A @ zero_zero_int ) ) ) ).

% power_less_zero_eq
thf(fact_1794_power__less__zero__eq__numeral,axiom,
    ! [A: real,W: num] :
      ( ( ord_less_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_real )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
        & ( ord_less_real @ A @ zero_zero_real ) ) ) ).

% power_less_zero_eq_numeral
thf(fact_1795_power__less__zero__eq__numeral,axiom,
    ! [A: rat,W: num] :
      ( ( ord_less_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_rat )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
        & ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).

% power_less_zero_eq_numeral
thf(fact_1796_power__less__zero__eq__numeral,axiom,
    ! [A: int,W: num] :
      ( ( ord_less_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_int )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
        & ( ord_less_int @ A @ zero_zero_int ) ) ) ).

% power_less_zero_eq_numeral
thf(fact_1797_even__diff__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) )
      = ( ( ord_less_nat @ M @ N )
        | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) ) ) ) ).

% even_diff_nat
thf(fact_1798_odd__two__times__div__two__succ,axiom,
    ! [A: code_integer] :
      ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ one_one_Code_integer )
        = A ) ) ).

% odd_two_times_div_two_succ
thf(fact_1799_odd__two__times__div__two__succ,axiom,
    ! [A: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ( ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_nat )
        = A ) ) ).

% odd_two_times_div_two_succ
thf(fact_1800_odd__two__times__div__two__succ,axiom,
    ! [A: int] :
      ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ( ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ one_one_int )
        = A ) ) ).

% odd_two_times_div_two_succ
thf(fact_1801_power__le__zero__eq__numeral,axiom,
    ! [A: real,W: num] :
      ( ( ord_less_eq_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_real )
      = ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W ) )
        & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
            & ( ord_less_eq_real @ A @ zero_zero_real ) )
          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
            & ( A = zero_zero_real ) ) ) ) ) ).

% power_le_zero_eq_numeral
thf(fact_1802_power__le__zero__eq__numeral,axiom,
    ! [A: rat,W: num] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_rat )
      = ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W ) )
        & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
            & ( ord_less_eq_rat @ A @ zero_zero_rat ) )
          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
            & ( A = zero_zero_rat ) ) ) ) ) ).

% power_le_zero_eq_numeral
thf(fact_1803_power__le__zero__eq__numeral,axiom,
    ! [A: int,W: num] :
      ( ( ord_less_eq_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_int )
      = ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W ) )
        & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
            & ( ord_less_eq_int @ A @ zero_zero_int ) )
          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
            & ( A = zero_zero_int ) ) ) ) ) ).

% power_le_zero_eq_numeral
thf(fact_1804_set__bit__0,axiom,
    ! [A: int] :
      ( ( bit_se7879613467334960850it_int @ zero_zero_nat @ A )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ).

% set_bit_0
thf(fact_1805_set__bit__0,axiom,
    ! [A: nat] :
      ( ( bit_se7882103937844011126it_nat @ zero_zero_nat @ A )
      = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% set_bit_0
thf(fact_1806_even__succ__div__exp,axiom,
    ! [A: code_integer,N: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
          = ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).

% even_succ_div_exp
thf(fact_1807_even__succ__div__exp,axiom,
    ! [A: nat,N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( divide_divide_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
          = ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).

% even_succ_div_exp
thf(fact_1808_even__succ__div__exp,axiom,
    ! [A: int,N: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
          = ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).

% even_succ_div_exp
thf(fact_1809_even__succ__mod__exp,axiom,
    ! [A: nat,N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( modulo_modulo_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
          = ( plus_plus_nat @ one_one_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).

% even_succ_mod_exp
thf(fact_1810_even__succ__mod__exp,axiom,
    ! [A: int,N: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
          = ( plus_plus_int @ one_one_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).

% even_succ_mod_exp
thf(fact_1811_even__succ__mod__exp,axiom,
    ! [A: code_integer,N: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
          = ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).

% even_succ_mod_exp
thf(fact_1812_split__zdiv,axiom,
    ! [P: int > $o,N: int,K: int] :
      ( ( P @ ( divide_divide_int @ N @ K ) )
      = ( ( ( K = zero_zero_int )
         => ( P @ zero_zero_int ) )
        & ( ( ord_less_int @ zero_zero_int @ K )
         => ! [I2: int,J3: int] :
              ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
                & ( ord_less_int @ J3 @ K )
                & ( N
                  = ( plus_plus_int @ ( times_times_int @ K @ I2 ) @ J3 ) ) )
             => ( P @ I2 ) ) )
        & ( ( ord_less_int @ K @ zero_zero_int )
         => ! [I2: int,J3: int] :
              ( ( ( ord_less_int @ K @ J3 )
                & ( ord_less_eq_int @ J3 @ zero_zero_int )
                & ( N
                  = ( plus_plus_int @ ( times_times_int @ K @ I2 ) @ J3 ) ) )
             => ( P @ I2 ) ) ) ) ) ).

% split_zdiv
thf(fact_1813_div__pos__geq,axiom,
    ! [L2: int,K: int] :
      ( ( ord_less_int @ zero_zero_int @ L2 )
     => ( ( ord_less_eq_int @ L2 @ K )
       => ( ( divide_divide_int @ K @ L2 )
          = ( plus_plus_int @ ( divide_divide_int @ ( minus_minus_int @ K @ L2 ) @ L2 ) @ one_one_int ) ) ) ) ).

% div_pos_geq
thf(fact_1814_int__div__neg__eq,axiom,
    ! [A: int,B2: int,Q2: int,R2: int] :
      ( ( A
        = ( plus_plus_int @ ( times_times_int @ B2 @ Q2 ) @ R2 ) )
     => ( ( ord_less_eq_int @ R2 @ zero_zero_int )
       => ( ( ord_less_int @ B2 @ R2 )
         => ( ( divide_divide_int @ A @ B2 )
            = Q2 ) ) ) ) ).

% int_div_neg_eq
thf(fact_1815_int__div__pos__eq,axiom,
    ! [A: int,B2: int,Q2: int,R2: int] :
      ( ( A
        = ( plus_plus_int @ ( times_times_int @ B2 @ Q2 ) @ R2 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
       => ( ( ord_less_int @ R2 @ B2 )
         => ( ( divide_divide_int @ A @ B2 )
            = Q2 ) ) ) ) ).

% int_div_pos_eq
thf(fact_1816_div__mod__decomp__int,axiom,
    ! [A3: int,N: int] :
      ( A3
      = ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A3 @ N ) @ N ) @ ( modulo_modulo_int @ A3 @ N ) ) ) ).

% div_mod_decomp_int
thf(fact_1817_split__neg__lemma,axiom,
    ! [K: int,P: int > int > $o,N: int] :
      ( ( ord_less_int @ K @ zero_zero_int )
     => ( ( P @ ( divide_divide_int @ N @ K ) @ ( modulo_modulo_int @ N @ K ) )
        = ( ! [I2: int,J3: int] :
              ( ( ( ord_less_int @ K @ J3 )
                & ( ord_less_eq_int @ J3 @ zero_zero_int )
                & ( N
                  = ( plus_plus_int @ ( times_times_int @ K @ I2 ) @ J3 ) ) )
             => ( P @ I2 @ J3 ) ) ) ) ) ).

% split_neg_lemma
thf(fact_1818_split__pos__lemma,axiom,
    ! [K: int,P: int > int > $o,N: int] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ( ( P @ ( divide_divide_int @ N @ K ) @ ( modulo_modulo_int @ N @ K ) )
        = ( ! [I2: int,J3: int] :
              ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
                & ( ord_less_int @ J3 @ K )
                & ( N
                  = ( plus_plus_int @ ( times_times_int @ K @ I2 ) @ J3 ) ) )
             => ( P @ I2 @ J3 ) ) ) ) ) ).

% split_pos_lemma
thf(fact_1819_int__div__less__self,axiom,
    ! [X4: int,K: int] :
      ( ( ord_less_int @ zero_zero_int @ X4 )
     => ( ( ord_less_int @ one_one_int @ K )
       => ( ord_less_int @ ( divide_divide_int @ X4 @ K ) @ X4 ) ) ) ).

% int_div_less_self
thf(fact_1820_verit__le__mono__div__int,axiom,
    ! [A3: int,B4: int,N: int] :
      ( ( ord_less_int @ A3 @ B4 )
     => ( ( ord_less_int @ zero_zero_int @ N )
       => ( ord_less_eq_int
          @ ( plus_plus_int @ ( divide_divide_int @ A3 @ N )
            @ ( if_int
              @ ( ( modulo_modulo_int @ B4 @ N )
                = zero_zero_int )
              @ one_one_int
              @ zero_zero_int ) )
          @ ( divide_divide_int @ B4 @ N ) ) ) ) ).

% verit_le_mono_div_int
thf(fact_1821_zdiv__zmult2__eq,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ C )
     => ( ( divide_divide_int @ A @ ( times_times_int @ B2 @ C ) )
        = ( divide_divide_int @ ( divide_divide_int @ A @ B2 ) @ C ) ) ) ).

% zdiv_zmult2_eq
thf(fact_1822_zmod__zmult2__eq,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ C )
     => ( ( modulo_modulo_int @ A @ ( times_times_int @ B2 @ C ) )
        = ( plus_plus_int @ ( times_times_int @ B2 @ ( modulo_modulo_int @ ( divide_divide_int @ A @ B2 ) @ C ) ) @ ( modulo_modulo_int @ A @ B2 ) ) ) ) ).

% zmod_zmult2_eq
thf(fact_1823_nonneg1__imp__zdiv__pos__iff,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A @ B2 ) )
        = ( ( ord_less_eq_int @ B2 @ A )
          & ( ord_less_int @ zero_zero_int @ B2 ) ) ) ) ).

% nonneg1_imp_zdiv_pos_iff
thf(fact_1824_pos__imp__zdiv__nonneg__iff,axiom,
    ! [B2: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B2 ) )
        = ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).

% pos_imp_zdiv_nonneg_iff
thf(fact_1825_neg__imp__zdiv__nonneg__iff,axiom,
    ! [B2: int,A: int] :
      ( ( ord_less_int @ B2 @ zero_zero_int )
     => ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B2 ) )
        = ( ord_less_eq_int @ A @ zero_zero_int ) ) ) ).

% neg_imp_zdiv_nonneg_iff
thf(fact_1826_pos__imp__zdiv__pos__iff,axiom,
    ! [K: int,I: int] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ I @ K ) )
        = ( ord_less_eq_int @ K @ I ) ) ) ).

% pos_imp_zdiv_pos_iff
thf(fact_1827_pos__imp__zdiv__neg__iff,axiom,
    ! [B2: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B2 )
     => ( ( ord_less_int @ ( divide_divide_int @ A @ B2 ) @ zero_zero_int )
        = ( ord_less_int @ A @ zero_zero_int ) ) ) ).

% pos_imp_zdiv_neg_iff
thf(fact_1828_neg__imp__zdiv__neg__iff,axiom,
    ! [B2: int,A: int] :
      ( ( ord_less_int @ B2 @ zero_zero_int )
     => ( ( ord_less_int @ ( divide_divide_int @ A @ B2 ) @ zero_zero_int )
        = ( ord_less_int @ zero_zero_int @ A ) ) ) ).

% neg_imp_zdiv_neg_iff
thf(fact_1829_div__nonpos__pos__le0,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ zero_zero_int @ B2 )
       => ( ord_less_eq_int @ ( divide_divide_int @ A @ B2 ) @ zero_zero_int ) ) ) ).

% div_nonpos_pos_le0
thf(fact_1830_div__nonneg__neg__le0,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ B2 @ zero_zero_int )
       => ( ord_less_eq_int @ ( divide_divide_int @ A @ B2 ) @ zero_zero_int ) ) ) ).

% div_nonneg_neg_le0
thf(fact_1831_div__neg__pos__less0,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ zero_zero_int @ B2 )
       => ( ord_less_int @ ( divide_divide_int @ A @ B2 ) @ zero_zero_int ) ) ) ).

% div_neg_pos_less0
thf(fact_1832_div__positive__int,axiom,
    ! [L2: int,K: int] :
      ( ( ord_less_eq_int @ L2 @ K )
     => ( ( ord_less_int @ zero_zero_int @ L2 )
       => ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ K @ L2 ) ) ) ) ).

% div_positive_int
thf(fact_1833_div__int__pos__iff,axiom,
    ! [K: int,L2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ L2 ) )
      = ( ( K = zero_zero_int )
        | ( L2 = zero_zero_int )
        | ( ( ord_less_eq_int @ zero_zero_int @ K )
          & ( ord_less_eq_int @ zero_zero_int @ L2 ) )
        | ( ( ord_less_int @ K @ zero_zero_int )
          & ( ord_less_int @ L2 @ zero_zero_int ) ) ) ) ).

% div_int_pos_iff
thf(fact_1834_zdiv__mono2__neg,axiom,
    ! [A: int,B: int,B2: int] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ( ord_less_eq_int @ B @ B2 )
         => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A @ B2 ) ) ) ) ) ).

% zdiv_mono2_neg
thf(fact_1835_zdiv__mono1__neg,axiom,
    ! [A: int,A2: int,B2: int] :
      ( ( ord_less_eq_int @ A @ A2 )
     => ( ( ord_less_int @ B2 @ zero_zero_int )
       => ( ord_less_eq_int @ ( divide_divide_int @ A2 @ B2 ) @ ( divide_divide_int @ A @ B2 ) ) ) ) ).

% zdiv_mono1_neg
thf(fact_1836_zdiv__eq__0__iff,axiom,
    ! [I: int,K: int] :
      ( ( ( divide_divide_int @ I @ K )
        = zero_zero_int )
      = ( ( K = zero_zero_int )
        | ( ( ord_less_eq_int @ zero_zero_int @ I )
          & ( ord_less_int @ I @ K ) )
        | ( ( ord_less_eq_int @ I @ zero_zero_int )
          & ( ord_less_int @ K @ I ) ) ) ) ).

% zdiv_eq_0_iff
thf(fact_1837_zdiv__mono__strict,axiom,
    ! [A3: int,B4: int,N: int] :
      ( ( ord_less_int @ A3 @ B4 )
     => ( ( ord_less_int @ zero_zero_int @ N )
       => ( ( ( modulo_modulo_int @ A3 @ N )
            = zero_zero_int )
         => ( ( ( modulo_modulo_int @ B4 @ N )
              = zero_zero_int )
           => ( ord_less_int @ ( divide_divide_int @ A3 @ N ) @ ( divide_divide_int @ B4 @ N ) ) ) ) ) ) ).

% zdiv_mono_strict
thf(fact_1838_zdiv__mono2,axiom,
    ! [A: int,B: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ( ord_less_eq_int @ B @ B2 )
         => ( ord_less_eq_int @ ( divide_divide_int @ A @ B2 ) @ ( divide_divide_int @ A @ B ) ) ) ) ) ).

% zdiv_mono2
thf(fact_1839_zdiv__mono1,axiom,
    ! [A: int,A2: int,B2: int] :
      ( ( ord_less_eq_int @ A @ A2 )
     => ( ( ord_less_int @ zero_zero_int @ B2 )
       => ( ord_less_eq_int @ ( divide_divide_int @ A @ B2 ) @ ( divide_divide_int @ A2 @ B2 ) ) ) ) ).

% zdiv_mono1
thf(fact_1840_dvd__field__iff,axiom,
    ( dvd_dvd_real
    = ( ^ [A4: real,B3: real] :
          ( ( A4 = zero_zero_real )
         => ( B3 = zero_zero_real ) ) ) ) ).

% dvd_field_iff
thf(fact_1841_dvd__field__iff,axiom,
    ( dvd_dvd_rat
    = ( ^ [A4: rat,B3: rat] :
          ( ( A4 = zero_zero_rat )
         => ( B3 = zero_zero_rat ) ) ) ) ).

% dvd_field_iff
thf(fact_1842_dvd__refl,axiom,
    ! [A: nat] : ( dvd_dvd_nat @ A @ A ) ).

% dvd_refl
thf(fact_1843_dvd__refl,axiom,
    ! [A: int] : ( dvd_dvd_int @ A @ A ) ).

% dvd_refl
thf(fact_1844_dvd__refl,axiom,
    ! [A: code_integer] : ( dvd_dvd_Code_integer @ A @ A ) ).

% dvd_refl
thf(fact_1845_dvd__trans,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ B2 )
     => ( ( dvd_dvd_nat @ B2 @ C )
       => ( dvd_dvd_nat @ A @ C ) ) ) ).

% dvd_trans
thf(fact_1846_dvd__trans,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( dvd_dvd_int @ A @ B2 )
     => ( ( dvd_dvd_int @ B2 @ C )
       => ( dvd_dvd_int @ A @ C ) ) ) ).

% dvd_trans
thf(fact_1847_dvd__trans,axiom,
    ! [A: code_integer,B2: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ B2 )
     => ( ( dvd_dvd_Code_integer @ B2 @ C )
       => ( dvd_dvd_Code_integer @ A @ C ) ) ) ).

% dvd_trans
thf(fact_1848_dvd__0__left,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ A )
     => ( A = zero_z3403309356797280102nteger ) ) ).

% dvd_0_left
thf(fact_1849_dvd__0__left,axiom,
    ! [A: real] :
      ( ( dvd_dvd_real @ zero_zero_real @ A )
     => ( A = zero_zero_real ) ) ).

% dvd_0_left
thf(fact_1850_dvd__0__left,axiom,
    ! [A: rat] :
      ( ( dvd_dvd_rat @ zero_zero_rat @ A )
     => ( A = zero_zero_rat ) ) ).

% dvd_0_left
thf(fact_1851_dvd__0__left,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ zero_zero_nat @ A )
     => ( A = zero_zero_nat ) ) ).

% dvd_0_left
thf(fact_1852_dvd__0__left,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ zero_zero_int @ A )
     => ( A = zero_zero_int ) ) ).

% dvd_0_left
thf(fact_1853_dvd__antisym,axiom,
    ! [M: nat,N: nat] :
      ( ( dvd_dvd_nat @ M @ N )
     => ( ( dvd_dvd_nat @ N @ M )
       => ( M = N ) ) ) ).

% dvd_antisym
thf(fact_1854_zero__reorient,axiom,
    ! [X4: literal] :
      ( ( zero_zero_literal = X4 )
      = ( X4 = zero_zero_literal ) ) ).

% zero_reorient
thf(fact_1855_zero__reorient,axiom,
    ! [X4: real] :
      ( ( zero_zero_real = X4 )
      = ( X4 = zero_zero_real ) ) ).

% zero_reorient
thf(fact_1856_zero__reorient,axiom,
    ! [X4: rat] :
      ( ( zero_zero_rat = X4 )
      = ( X4 = zero_zero_rat ) ) ).

% zero_reorient
thf(fact_1857_zero__reorient,axiom,
    ! [X4: nat] :
      ( ( zero_zero_nat = X4 )
      = ( X4 = zero_zero_nat ) ) ).

% zero_reorient
thf(fact_1858_zero__reorient,axiom,
    ! [X4: int] :
      ( ( zero_zero_int = X4 )
      = ( X4 = zero_zero_int ) ) ).

% zero_reorient
thf(fact_1859_not__is__unit__0,axiom,
    ~ ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ one_one_Code_integer ) ).

% not_is_unit_0
thf(fact_1860_not__is__unit__0,axiom,
    ~ ( dvd_dvd_nat @ zero_zero_nat @ one_one_nat ) ).

% not_is_unit_0
thf(fact_1861_not__is__unit__0,axiom,
    ~ ( dvd_dvd_int @ zero_zero_int @ one_one_int ) ).

% not_is_unit_0
thf(fact_1862_dvd__div__eq__0__iff,axiom,
    ! [B2: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ B2 @ A )
     => ( ( ( divide6298287555418463151nteger @ A @ B2 )
          = zero_z3403309356797280102nteger )
        = ( A = zero_z3403309356797280102nteger ) ) ) ).

% dvd_div_eq_0_iff
thf(fact_1863_dvd__div__eq__0__iff,axiom,
    ! [B2: complex,A: complex] :
      ( ( dvd_dvd_complex @ B2 @ A )
     => ( ( ( divide1717551699836669952omplex @ A @ B2 )
          = zero_zero_complex )
        = ( A = zero_zero_complex ) ) ) ).

% dvd_div_eq_0_iff
thf(fact_1864_dvd__div__eq__0__iff,axiom,
    ! [B2: real,A: real] :
      ( ( dvd_dvd_real @ B2 @ A )
     => ( ( ( divide_divide_real @ A @ B2 )
          = zero_zero_real )
        = ( A = zero_zero_real ) ) ) ).

% dvd_div_eq_0_iff
thf(fact_1865_dvd__div__eq__0__iff,axiom,
    ! [B2: rat,A: rat] :
      ( ( dvd_dvd_rat @ B2 @ A )
     => ( ( ( divide_divide_rat @ A @ B2 )
          = zero_zero_rat )
        = ( A = zero_zero_rat ) ) ) ).

% dvd_div_eq_0_iff
thf(fact_1866_dvd__div__eq__0__iff,axiom,
    ! [B2: nat,A: nat] :
      ( ( dvd_dvd_nat @ B2 @ A )
     => ( ( ( divide_divide_nat @ A @ B2 )
          = zero_zero_nat )
        = ( A = zero_zero_nat ) ) ) ).

% dvd_div_eq_0_iff
thf(fact_1867_dvd__div__eq__0__iff,axiom,
    ! [B2: int,A: int] :
      ( ( dvd_dvd_int @ B2 @ A )
     => ( ( ( divide_divide_int @ A @ B2 )
          = zero_zero_int )
        = ( A = zero_zero_int ) ) ) ).

% dvd_div_eq_0_iff
thf(fact_1868_nat__dvd__not__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_nat @ M @ N )
       => ~ ( dvd_dvd_nat @ N @ M ) ) ) ).

% nat_dvd_not_less
thf(fact_1869_mod__0__imp__dvd,axiom,
    ! [A: nat,B2: nat] :
      ( ( ( modulo_modulo_nat @ A @ B2 )
        = zero_zero_nat )
     => ( dvd_dvd_nat @ B2 @ A ) ) ).

% mod_0_imp_dvd
thf(fact_1870_mod__0__imp__dvd,axiom,
    ! [A: int,B2: int] :
      ( ( ( modulo_modulo_int @ A @ B2 )
        = zero_zero_int )
     => ( dvd_dvd_int @ B2 @ A ) ) ).

% mod_0_imp_dvd
thf(fact_1871_mod__0__imp__dvd,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ B2 )
        = zero_z3403309356797280102nteger )
     => ( dvd_dvd_Code_integer @ B2 @ A ) ) ).

% mod_0_imp_dvd
thf(fact_1872_dvd__eq__mod__eq__0,axiom,
    ( dvd_dvd_nat
    = ( ^ [A4: nat,B3: nat] :
          ( ( modulo_modulo_nat @ B3 @ A4 )
          = zero_zero_nat ) ) ) ).

% dvd_eq_mod_eq_0
thf(fact_1873_dvd__eq__mod__eq__0,axiom,
    ( dvd_dvd_int
    = ( ^ [A4: int,B3: int] :
          ( ( modulo_modulo_int @ B3 @ A4 )
          = zero_zero_int ) ) ) ).

% dvd_eq_mod_eq_0
thf(fact_1874_dvd__eq__mod__eq__0,axiom,
    ( dvd_dvd_Code_integer
    = ( ^ [A4: code_integer,B3: code_integer] :
          ( ( modulo364778990260209775nteger @ B3 @ A4 )
          = zero_z3403309356797280102nteger ) ) ) ).

% dvd_eq_mod_eq_0
thf(fact_1875_mod__eq__0__iff__dvd,axiom,
    ! [A: nat,B2: nat] :
      ( ( ( modulo_modulo_nat @ A @ B2 )
        = zero_zero_nat )
      = ( dvd_dvd_nat @ B2 @ A ) ) ).

% mod_eq_0_iff_dvd
thf(fact_1876_mod__eq__0__iff__dvd,axiom,
    ! [A: int,B2: int] :
      ( ( ( modulo_modulo_int @ A @ B2 )
        = zero_zero_int )
      = ( dvd_dvd_int @ B2 @ A ) ) ).

% mod_eq_0_iff_dvd
thf(fact_1877_mod__eq__0__iff__dvd,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ B2 )
        = zero_z3403309356797280102nteger )
      = ( dvd_dvd_Code_integer @ B2 @ A ) ) ).

% mod_eq_0_iff_dvd
thf(fact_1878_list__decode_Ocases,axiom,
    ! [X4: nat] :
      ( ( X4 != zero_zero_nat )
     => ~ ! [N4: nat] :
            ( X4
           != ( suc @ N4 ) ) ) ).

% list_decode.cases
thf(fact_1879_pos__zmod__mult__2,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
        = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ B2 @ A ) ) ) ) ) ).

% pos_zmod_mult_2
thf(fact_1880_neg__zmod__mult__2,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
        = ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ ( plus_plus_int @ B2 @ one_one_int ) @ A ) ) @ one_one_int ) ) ) ).

% neg_zmod_mult_2
thf(fact_1881_unit__dvdE,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ~ ( ( A != zero_z3403309356797280102nteger )
         => ! [C3: code_integer] :
              ( B2
             != ( times_3573771949741848930nteger @ A @ C3 ) ) ) ) ).

% unit_dvdE
thf(fact_1882_unit__dvdE,axiom,
    ! [A: nat,B2: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ~ ( ( A != zero_zero_nat )
         => ! [C3: nat] :
              ( B2
             != ( times_times_nat @ A @ C3 ) ) ) ) ).

% unit_dvdE
thf(fact_1883_unit__dvdE,axiom,
    ! [A: int,B2: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ~ ( ( A != zero_zero_int )
         => ! [C3: int] :
              ( B2
             != ( times_times_int @ A @ C3 ) ) ) ) ).

% unit_dvdE
thf(fact_1884_dvd__div__eq__mult,axiom,
    ! [A: code_integer,B2: code_integer,C: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ A @ B2 )
       => ( ( ( divide6298287555418463151nteger @ B2 @ A )
            = C )
          = ( B2
            = ( times_3573771949741848930nteger @ C @ A ) ) ) ) ) ).

% dvd_div_eq_mult
thf(fact_1885_dvd__div__eq__mult,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( A != zero_zero_nat )
     => ( ( dvd_dvd_nat @ A @ B2 )
       => ( ( ( divide_divide_nat @ B2 @ A )
            = C )
          = ( B2
            = ( times_times_nat @ C @ A ) ) ) ) ) ).

% dvd_div_eq_mult
thf(fact_1886_dvd__div__eq__mult,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( A != zero_zero_int )
     => ( ( dvd_dvd_int @ A @ B2 )
       => ( ( ( divide_divide_int @ B2 @ A )
            = C )
          = ( B2
            = ( times_times_int @ C @ A ) ) ) ) ) ).

% dvd_div_eq_mult
thf(fact_1887_div__dvd__iff__mult,axiom,
    ! [B2: code_integer,A: code_integer,C: code_integer] :
      ( ( B2 != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ B2 @ A )
       => ( ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A @ B2 ) @ C )
          = ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ C @ B2 ) ) ) ) ) ).

% div_dvd_iff_mult
thf(fact_1888_div__dvd__iff__mult,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( ( B2 != zero_zero_nat )
     => ( ( dvd_dvd_nat @ B2 @ A )
       => ( ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B2 ) @ C )
          = ( dvd_dvd_nat @ A @ ( times_times_nat @ C @ B2 ) ) ) ) ) ).

% div_dvd_iff_mult
thf(fact_1889_div__dvd__iff__mult,axiom,
    ! [B2: int,A: int,C: int] :
      ( ( B2 != zero_zero_int )
     => ( ( dvd_dvd_int @ B2 @ A )
       => ( ( dvd_dvd_int @ ( divide_divide_int @ A @ B2 ) @ C )
          = ( dvd_dvd_int @ A @ ( times_times_int @ C @ B2 ) ) ) ) ) ).

% div_dvd_iff_mult
thf(fact_1890_dvd__div__iff__mult,axiom,
    ! [C: code_integer,B2: code_integer,A: code_integer] :
      ( ( C != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ C @ B2 )
       => ( ( dvd_dvd_Code_integer @ A @ ( divide6298287555418463151nteger @ B2 @ C ) )
          = ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ B2 ) ) ) ) ).

% dvd_div_iff_mult
thf(fact_1891_dvd__div__iff__mult,axiom,
    ! [C: nat,B2: nat,A: nat] :
      ( ( C != zero_zero_nat )
     => ( ( dvd_dvd_nat @ C @ B2 )
       => ( ( dvd_dvd_nat @ A @ ( divide_divide_nat @ B2 @ C ) )
          = ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ B2 ) ) ) ) ).

% dvd_div_iff_mult
thf(fact_1892_dvd__div__iff__mult,axiom,
    ! [C: int,B2: int,A: int] :
      ( ( C != zero_zero_int )
     => ( ( dvd_dvd_int @ C @ B2 )
       => ( ( dvd_dvd_int @ A @ ( divide_divide_int @ B2 @ C ) )
          = ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ B2 ) ) ) ) ).

% dvd_div_iff_mult
thf(fact_1893_dvd__div__div__eq__mult,axiom,
    ! [A: code_integer,C: code_integer,B2: code_integer,D: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( C != zero_z3403309356797280102nteger )
       => ( ( dvd_dvd_Code_integer @ A @ B2 )
         => ( ( dvd_dvd_Code_integer @ C @ D )
           => ( ( ( divide6298287555418463151nteger @ B2 @ A )
                = ( divide6298287555418463151nteger @ D @ C ) )
              = ( ( times_3573771949741848930nteger @ B2 @ C )
                = ( times_3573771949741848930nteger @ A @ D ) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
thf(fact_1894_dvd__div__div__eq__mult,axiom,
    ! [A: nat,C: nat,B2: nat,D: nat] :
      ( ( A != zero_zero_nat )
     => ( ( C != zero_zero_nat )
       => ( ( dvd_dvd_nat @ A @ B2 )
         => ( ( dvd_dvd_nat @ C @ D )
           => ( ( ( divide_divide_nat @ B2 @ A )
                = ( divide_divide_nat @ D @ C ) )
              = ( ( times_times_nat @ B2 @ C )
                = ( times_times_nat @ A @ D ) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
thf(fact_1895_dvd__div__div__eq__mult,axiom,
    ! [A: int,C: int,B2: int,D: int] :
      ( ( A != zero_zero_int )
     => ( ( C != zero_zero_int )
       => ( ( dvd_dvd_int @ A @ B2 )
         => ( ( dvd_dvd_int @ C @ D )
           => ( ( ( divide_divide_int @ B2 @ A )
                = ( divide_divide_int @ D @ C ) )
              = ( ( times_times_int @ B2 @ C )
                = ( times_times_int @ A @ D ) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
thf(fact_1896_unit__div__eq__0__iff,axiom,
    ! [B2: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
     => ( ( ( divide6298287555418463151nteger @ A @ B2 )
          = zero_z3403309356797280102nteger )
        = ( A = zero_z3403309356797280102nteger ) ) ) ).

% unit_div_eq_0_iff
thf(fact_1897_unit__div__eq__0__iff,axiom,
    ! [B2: nat,A: nat] :
      ( ( dvd_dvd_nat @ B2 @ one_one_nat )
     => ( ( ( divide_divide_nat @ A @ B2 )
          = zero_zero_nat )
        = ( A = zero_zero_nat ) ) ) ).

% unit_div_eq_0_iff
thf(fact_1898_unit__div__eq__0__iff,axiom,
    ! [B2: int,A: int] :
      ( ( dvd_dvd_int @ B2 @ one_one_int )
     => ( ( ( divide_divide_int @ A @ B2 )
          = zero_zero_int )
        = ( A = zero_zero_int ) ) ) ).

% unit_div_eq_0_iff
thf(fact_1899_is__unit__power__iff,axiom,
    ! [A: code_integer,N: nat] :
      ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) @ one_one_Code_integer )
      = ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
        | ( N = zero_zero_nat ) ) ) ).

% is_unit_power_iff
thf(fact_1900_is__unit__power__iff,axiom,
    ! [A: nat,N: nat] :
      ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ one_one_nat )
      = ( ( dvd_dvd_nat @ A @ one_one_nat )
        | ( N = zero_zero_nat ) ) ) ).

% is_unit_power_iff
thf(fact_1901_is__unit__power__iff,axiom,
    ! [A: int,N: nat] :
      ( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ one_one_int )
      = ( ( dvd_dvd_int @ A @ one_one_int )
        | ( N = zero_zero_nat ) ) ) ).

% is_unit_power_iff
thf(fact_1902_unit__imp__mod__eq__0,axiom,
    ! [B2: nat,A: nat] :
      ( ( dvd_dvd_nat @ B2 @ one_one_nat )
     => ( ( modulo_modulo_nat @ A @ B2 )
        = zero_zero_nat ) ) ).

% unit_imp_mod_eq_0
thf(fact_1903_unit__imp__mod__eq__0,axiom,
    ! [B2: int,A: int] :
      ( ( dvd_dvd_int @ B2 @ one_one_int )
     => ( ( modulo_modulo_int @ A @ B2 )
        = zero_zero_int ) ) ).

% unit_imp_mod_eq_0
thf(fact_1904_unit__imp__mod__eq__0,axiom,
    ! [B2: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
     => ( ( modulo364778990260209775nteger @ A @ B2 )
        = zero_z3403309356797280102nteger ) ) ).

% unit_imp_mod_eq_0
thf(fact_1905_dvd__imp__le,axiom,
    ! [K: nat,N: nat] :
      ( ( dvd_dvd_nat @ K @ N )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_eq_nat @ K @ N ) ) ) ).

% dvd_imp_le
thf(fact_1906_nat__mult__dvd__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
        = ( dvd_dvd_nat @ M @ N ) ) ) ).

% nat_mult_dvd_cancel1
thf(fact_1907_dvd__mult__cancel,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( dvd_dvd_nat @ M @ N ) ) ) ).

% dvd_mult_cancel
thf(fact_1908_power__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( power_power_rat @ zero_zero_rat @ N )
          = one_one_rat ) )
      & ( ( N != zero_zero_nat )
       => ( ( power_power_rat @ zero_zero_rat @ N )
          = zero_zero_rat ) ) ) ).

% power_0_left
thf(fact_1909_power__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( power_power_nat @ zero_zero_nat @ N )
          = one_one_nat ) )
      & ( ( N != zero_zero_nat )
       => ( ( power_power_nat @ zero_zero_nat @ N )
          = zero_zero_nat ) ) ) ).

% power_0_left
thf(fact_1910_power__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( power_power_real @ zero_zero_real @ N )
          = one_one_real ) )
      & ( ( N != zero_zero_nat )
       => ( ( power_power_real @ zero_zero_real @ N )
          = zero_zero_real ) ) ) ).

% power_0_left
thf(fact_1911_power__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( power_power_int @ zero_zero_int @ N )
          = one_one_int ) )
      & ( ( N != zero_zero_nat )
       => ( ( power_power_int @ zero_zero_int @ N )
          = zero_zero_int ) ) ) ).

% power_0_left
thf(fact_1912_power__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( power_power_complex @ zero_zero_complex @ N )
          = one_one_complex ) )
      & ( ( N != zero_zero_nat )
       => ( ( power_power_complex @ zero_zero_complex @ N )
          = zero_zero_complex ) ) ) ).

% power_0_left
thf(fact_1913_zero__power,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( power_power_rat @ zero_zero_rat @ N )
        = zero_zero_rat ) ) ).

% zero_power
thf(fact_1914_zero__power,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( power_power_nat @ zero_zero_nat @ N )
        = zero_zero_nat ) ) ).

% zero_power
thf(fact_1915_zero__power,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( power_power_real @ zero_zero_real @ N )
        = zero_zero_real ) ) ).

% zero_power
thf(fact_1916_zero__power,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( power_power_int @ zero_zero_int @ N )
        = zero_zero_int ) ) ).

% zero_power
thf(fact_1917_zero__power,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( power_power_complex @ zero_zero_complex @ N )
        = zero_zero_complex ) ) ).

% zero_power
thf(fact_1918_mod__greater__zero__iff__not__dvd,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( modulo_modulo_nat @ M @ N ) )
      = ( ~ ( dvd_dvd_nat @ N @ M ) ) ) ).

% mod_greater_zero_iff_not_dvd
thf(fact_1919_even__set__bit__iff,axiom,
    ! [M: nat,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se2793503036327961859nteger @ M @ A ) )
      = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
        & ( M != zero_zero_nat ) ) ) ).

% even_set_bit_iff
thf(fact_1920_even__set__bit__iff,axiom,
    ! [M: nat,A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se7879613467334960850it_int @ M @ A ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
        & ( M != zero_zero_nat ) ) ) ).

% even_set_bit_iff
thf(fact_1921_even__set__bit__iff,axiom,
    ! [M: nat,A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se7882103937844011126it_nat @ M @ A ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
        & ( M != zero_zero_nat ) ) ) ).

% even_set_bit_iff
thf(fact_1922_even__flip__bit__iff,axiom,
    ! [M: nat,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1345352211410354436nteger @ M @ A ) )
      = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
       != ( M = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
thf(fact_1923_even__flip__bit__iff,axiom,
    ! [M: nat,A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2159334234014336723it_int @ M @ A ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
       != ( M = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
thf(fact_1924_even__flip__bit__iff,axiom,
    ! [M: nat,A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2161824704523386999it_nat @ M @ A ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
       != ( M = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
thf(fact_1925_even__unset__bit__iff,axiom,
    ! [M: nat,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se8260200283734997820nteger @ M @ A ) )
      = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
        | ( M = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
thf(fact_1926_even__unset__bit__iff,axiom,
    ! [M: nat,A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se4203085406695923979it_int @ M @ A ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
        | ( M = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
thf(fact_1927_even__unset__bit__iff,axiom,
    ! [M: nat,A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se4205575877204974255it_nat @ M @ A ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
        | ( M = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
thf(fact_1928_dvd__triv__right,axiom,
    ! [A: code_integer,B2: code_integer] : ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B2 @ A ) ) ).

% dvd_triv_right
thf(fact_1929_dvd__triv__right,axiom,
    ! [A: real,B2: real] : ( dvd_dvd_real @ A @ ( times_times_real @ B2 @ A ) ) ).

% dvd_triv_right
thf(fact_1930_dvd__triv__right,axiom,
    ! [A: rat,B2: rat] : ( dvd_dvd_rat @ A @ ( times_times_rat @ B2 @ A ) ) ).

% dvd_triv_right
thf(fact_1931_dvd__triv__right,axiom,
    ! [A: nat,B2: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ B2 @ A ) ) ).

% dvd_triv_right
thf(fact_1932_dvd__triv__right,axiom,
    ! [A: int,B2: int] : ( dvd_dvd_int @ A @ ( times_times_int @ B2 @ A ) ) ).

% dvd_triv_right
thf(fact_1933_dvd__mult__right,axiom,
    ! [A: code_integer,B2: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B2 ) @ C )
     => ( dvd_dvd_Code_integer @ B2 @ C ) ) ).

% dvd_mult_right
thf(fact_1934_dvd__mult__right,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( dvd_dvd_real @ ( times_times_real @ A @ B2 ) @ C )
     => ( dvd_dvd_real @ B2 @ C ) ) ).

% dvd_mult_right
thf(fact_1935_dvd__mult__right,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( dvd_dvd_rat @ ( times_times_rat @ A @ B2 ) @ C )
     => ( dvd_dvd_rat @ B2 @ C ) ) ).

% dvd_mult_right
thf(fact_1936_dvd__mult__right,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B2 ) @ C )
     => ( dvd_dvd_nat @ B2 @ C ) ) ).

% dvd_mult_right
thf(fact_1937_dvd__mult__right,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ A @ B2 ) @ C )
     => ( dvd_dvd_int @ B2 @ C ) ) ).

% dvd_mult_right
thf(fact_1938_mult__dvd__mono,axiom,
    ! [A: code_integer,B2: code_integer,C: code_integer,D: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ B2 )
     => ( ( dvd_dvd_Code_integer @ C @ D )
       => ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B2 @ D ) ) ) ) ).

% mult_dvd_mono
thf(fact_1939_mult__dvd__mono,axiom,
    ! [A: real,B2: real,C: real,D: real] :
      ( ( dvd_dvd_real @ A @ B2 )
     => ( ( dvd_dvd_real @ C @ D )
       => ( dvd_dvd_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ D ) ) ) ) ).

% mult_dvd_mono
thf(fact_1940_mult__dvd__mono,axiom,
    ! [A: rat,B2: rat,C: rat,D: rat] :
      ( ( dvd_dvd_rat @ A @ B2 )
     => ( ( dvd_dvd_rat @ C @ D )
       => ( dvd_dvd_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ D ) ) ) ) ).

% mult_dvd_mono
thf(fact_1941_mult__dvd__mono,axiom,
    ! [A: nat,B2: nat,C: nat,D: nat] :
      ( ( dvd_dvd_nat @ A @ B2 )
     => ( ( dvd_dvd_nat @ C @ D )
       => ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ).

% mult_dvd_mono
thf(fact_1942_mult__dvd__mono,axiom,
    ! [A: int,B2: int,C: int,D: int] :
      ( ( dvd_dvd_int @ A @ B2 )
     => ( ( dvd_dvd_int @ C @ D )
       => ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ D ) ) ) ) ).

% mult_dvd_mono
thf(fact_1943_dvd__triv__left,axiom,
    ! [A: code_integer,B2: code_integer] : ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ A @ B2 ) ) ).

% dvd_triv_left
thf(fact_1944_dvd__triv__left,axiom,
    ! [A: real,B2: real] : ( dvd_dvd_real @ A @ ( times_times_real @ A @ B2 ) ) ).

% dvd_triv_left
thf(fact_1945_dvd__triv__left,axiom,
    ! [A: rat,B2: rat] : ( dvd_dvd_rat @ A @ ( times_times_rat @ A @ B2 ) ) ).

% dvd_triv_left
thf(fact_1946_dvd__triv__left,axiom,
    ! [A: nat,B2: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ A @ B2 ) ) ).

% dvd_triv_left
thf(fact_1947_dvd__triv__left,axiom,
    ! [A: int,B2: int] : ( dvd_dvd_int @ A @ ( times_times_int @ A @ B2 ) ) ).

% dvd_triv_left
thf(fact_1948_dvd__mult__left,axiom,
    ! [A: code_integer,B2: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B2 ) @ C )
     => ( dvd_dvd_Code_integer @ A @ C ) ) ).

% dvd_mult_left
thf(fact_1949_dvd__mult__left,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( dvd_dvd_real @ ( times_times_real @ A @ B2 ) @ C )
     => ( dvd_dvd_real @ A @ C ) ) ).

% dvd_mult_left
thf(fact_1950_dvd__mult__left,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( dvd_dvd_rat @ ( times_times_rat @ A @ B2 ) @ C )
     => ( dvd_dvd_rat @ A @ C ) ) ).

% dvd_mult_left
thf(fact_1951_dvd__mult__left,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B2 ) @ C )
     => ( dvd_dvd_nat @ A @ C ) ) ).

% dvd_mult_left
thf(fact_1952_dvd__mult__left,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ A @ B2 ) @ C )
     => ( dvd_dvd_int @ A @ C ) ) ).

% dvd_mult_left
thf(fact_1953_dvd__mult2,axiom,
    ! [A: code_integer,B2: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ B2 )
     => ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B2 @ C ) ) ) ).

% dvd_mult2
thf(fact_1954_dvd__mult2,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( dvd_dvd_real @ A @ B2 )
     => ( dvd_dvd_real @ A @ ( times_times_real @ B2 @ C ) ) ) ).

% dvd_mult2
thf(fact_1955_dvd__mult2,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( dvd_dvd_rat @ A @ B2 )
     => ( dvd_dvd_rat @ A @ ( times_times_rat @ B2 @ C ) ) ) ).

% dvd_mult2
thf(fact_1956_dvd__mult2,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ B2 )
     => ( dvd_dvd_nat @ A @ ( times_times_nat @ B2 @ C ) ) ) ).

% dvd_mult2
thf(fact_1957_dvd__mult2,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( dvd_dvd_int @ A @ B2 )
     => ( dvd_dvd_int @ A @ ( times_times_int @ B2 @ C ) ) ) ).

% dvd_mult2
thf(fact_1958_dvd__mult,axiom,
    ! [A: code_integer,C: code_integer,B2: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ C )
     => ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B2 @ C ) ) ) ).

% dvd_mult
thf(fact_1959_dvd__mult,axiom,
    ! [A: real,C: real,B2: real] :
      ( ( dvd_dvd_real @ A @ C )
     => ( dvd_dvd_real @ A @ ( times_times_real @ B2 @ C ) ) ) ).

% dvd_mult
thf(fact_1960_dvd__mult,axiom,
    ! [A: rat,C: rat,B2: rat] :
      ( ( dvd_dvd_rat @ A @ C )
     => ( dvd_dvd_rat @ A @ ( times_times_rat @ B2 @ C ) ) ) ).

% dvd_mult
thf(fact_1961_dvd__mult,axiom,
    ! [A: nat,C: nat,B2: nat] :
      ( ( dvd_dvd_nat @ A @ C )
     => ( dvd_dvd_nat @ A @ ( times_times_nat @ B2 @ C ) ) ) ).

% dvd_mult
thf(fact_1962_dvd__mult,axiom,
    ! [A: int,C: int,B2: int] :
      ( ( dvd_dvd_int @ A @ C )
     => ( dvd_dvd_int @ A @ ( times_times_int @ B2 @ C ) ) ) ).

% dvd_mult
thf(fact_1963_dvd__def,axiom,
    ( dvd_dvd_Code_integer
    = ( ^ [B3: code_integer,A4: code_integer] :
        ? [K3: code_integer] :
          ( A4
          = ( times_3573771949741848930nteger @ B3 @ K3 ) ) ) ) ).

% dvd_def
thf(fact_1964_dvd__def,axiom,
    ( dvd_dvd_real
    = ( ^ [B3: real,A4: real] :
        ? [K3: real] :
          ( A4
          = ( times_times_real @ B3 @ K3 ) ) ) ) ).

% dvd_def
thf(fact_1965_dvd__def,axiom,
    ( dvd_dvd_rat
    = ( ^ [B3: rat,A4: rat] :
        ? [K3: rat] :
          ( A4
          = ( times_times_rat @ B3 @ K3 ) ) ) ) ).

% dvd_def
thf(fact_1966_dvd__def,axiom,
    ( dvd_dvd_nat
    = ( ^ [B3: nat,A4: nat] :
        ? [K3: nat] :
          ( A4
          = ( times_times_nat @ B3 @ K3 ) ) ) ) ).

% dvd_def
thf(fact_1967_dvd__def,axiom,
    ( dvd_dvd_int
    = ( ^ [B3: int,A4: int] :
        ? [K3: int] :
          ( A4
          = ( times_times_int @ B3 @ K3 ) ) ) ) ).

% dvd_def
thf(fact_1968_dvdI,axiom,
    ! [A: code_integer,B2: code_integer,K: code_integer] :
      ( ( A
        = ( times_3573771949741848930nteger @ B2 @ K ) )
     => ( dvd_dvd_Code_integer @ B2 @ A ) ) ).

% dvdI
thf(fact_1969_dvdI,axiom,
    ! [A: real,B2: real,K: real] :
      ( ( A
        = ( times_times_real @ B2 @ K ) )
     => ( dvd_dvd_real @ B2 @ A ) ) ).

% dvdI
thf(fact_1970_dvdI,axiom,
    ! [A: rat,B2: rat,K: rat] :
      ( ( A
        = ( times_times_rat @ B2 @ K ) )
     => ( dvd_dvd_rat @ B2 @ A ) ) ).

% dvdI
thf(fact_1971_dvdI,axiom,
    ! [A: nat,B2: nat,K: nat] :
      ( ( A
        = ( times_times_nat @ B2 @ K ) )
     => ( dvd_dvd_nat @ B2 @ A ) ) ).

% dvdI
thf(fact_1972_dvdI,axiom,
    ! [A: int,B2: int,K: int] :
      ( ( A
        = ( times_times_int @ B2 @ K ) )
     => ( dvd_dvd_int @ B2 @ A ) ) ).

% dvdI
thf(fact_1973_dvdE,axiom,
    ! [B2: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ B2 @ A )
     => ~ ! [K2: code_integer] :
            ( A
           != ( times_3573771949741848930nteger @ B2 @ K2 ) ) ) ).

% dvdE
thf(fact_1974_dvdE,axiom,
    ! [B2: real,A: real] :
      ( ( dvd_dvd_real @ B2 @ A )
     => ~ ! [K2: real] :
            ( A
           != ( times_times_real @ B2 @ K2 ) ) ) ).

% dvdE
thf(fact_1975_dvdE,axiom,
    ! [B2: rat,A: rat] :
      ( ( dvd_dvd_rat @ B2 @ A )
     => ~ ! [K2: rat] :
            ( A
           != ( times_times_rat @ B2 @ K2 ) ) ) ).

% dvdE
thf(fact_1976_dvdE,axiom,
    ! [B2: nat,A: nat] :
      ( ( dvd_dvd_nat @ B2 @ A )
     => ~ ! [K2: nat] :
            ( A
           != ( times_times_nat @ B2 @ K2 ) ) ) ).

% dvdE
thf(fact_1977_dvdE,axiom,
    ! [B2: int,A: int] :
      ( ( dvd_dvd_int @ B2 @ A )
     => ~ ! [K2: int] :
            ( A
           != ( times_times_int @ B2 @ K2 ) ) ) ).

% dvdE
thf(fact_1978_dvd__unit__imp__unit,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ B2 )
     => ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
       => ( dvd_dvd_Code_integer @ A @ one_one_Code_integer ) ) ) ).

% dvd_unit_imp_unit
thf(fact_1979_dvd__unit__imp__unit,axiom,
    ! [A: nat,B2: nat] :
      ( ( dvd_dvd_nat @ A @ B2 )
     => ( ( dvd_dvd_nat @ B2 @ one_one_nat )
       => ( dvd_dvd_nat @ A @ one_one_nat ) ) ) ).

% dvd_unit_imp_unit
thf(fact_1980_dvd__unit__imp__unit,axiom,
    ! [A: int,B2: int] :
      ( ( dvd_dvd_int @ A @ B2 )
     => ( ( dvd_dvd_int @ B2 @ one_one_int )
       => ( dvd_dvd_int @ A @ one_one_int ) ) ) ).

% dvd_unit_imp_unit
thf(fact_1981_unit__imp__dvd,axiom,
    ! [B2: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
     => ( dvd_dvd_Code_integer @ B2 @ A ) ) ).

% unit_imp_dvd
thf(fact_1982_unit__imp__dvd,axiom,
    ! [B2: nat,A: nat] :
      ( ( dvd_dvd_nat @ B2 @ one_one_nat )
     => ( dvd_dvd_nat @ B2 @ A ) ) ).

% unit_imp_dvd
thf(fact_1983_unit__imp__dvd,axiom,
    ! [B2: int,A: int] :
      ( ( dvd_dvd_int @ B2 @ one_one_int )
     => ( dvd_dvd_int @ B2 @ A ) ) ).

% unit_imp_dvd
thf(fact_1984_one__dvd,axiom,
    ! [A: code_integer] : ( dvd_dvd_Code_integer @ one_one_Code_integer @ A ) ).

% one_dvd
thf(fact_1985_one__dvd,axiom,
    ! [A: complex] : ( dvd_dvd_complex @ one_one_complex @ A ) ).

% one_dvd
thf(fact_1986_one__dvd,axiom,
    ! [A: real] : ( dvd_dvd_real @ one_one_real @ A ) ).

% one_dvd
thf(fact_1987_one__dvd,axiom,
    ! [A: rat] : ( dvd_dvd_rat @ one_one_rat @ A ) ).

% one_dvd
thf(fact_1988_one__dvd,axiom,
    ! [A: nat] : ( dvd_dvd_nat @ one_one_nat @ A ) ).

% one_dvd
thf(fact_1989_one__dvd,axiom,
    ! [A: int] : ( dvd_dvd_int @ one_one_int @ A ) ).

% one_dvd
thf(fact_1990_dvd__add__right__iff,axiom,
    ! [A: code_integer,B2: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ B2 )
     => ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B2 @ C ) )
        = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).

% dvd_add_right_iff
thf(fact_1991_dvd__add__right__iff,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( dvd_dvd_real @ A @ B2 )
     => ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B2 @ C ) )
        = ( dvd_dvd_real @ A @ C ) ) ) ).

% dvd_add_right_iff
thf(fact_1992_dvd__add__right__iff,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( dvd_dvd_rat @ A @ B2 )
     => ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B2 @ C ) )
        = ( dvd_dvd_rat @ A @ C ) ) ) ).

% dvd_add_right_iff
thf(fact_1993_dvd__add__right__iff,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ B2 )
     => ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B2 @ C ) )
        = ( dvd_dvd_nat @ A @ C ) ) ) ).

% dvd_add_right_iff
thf(fact_1994_dvd__add__right__iff,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( dvd_dvd_int @ A @ B2 )
     => ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B2 @ C ) )
        = ( dvd_dvd_int @ A @ C ) ) ) ).

% dvd_add_right_iff
thf(fact_1995_dvd__add__left__iff,axiom,
    ! [A: code_integer,C: code_integer,B2: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ C )
     => ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B2 @ C ) )
        = ( dvd_dvd_Code_integer @ A @ B2 ) ) ) ).

% dvd_add_left_iff
thf(fact_1996_dvd__add__left__iff,axiom,
    ! [A: real,C: real,B2: real] :
      ( ( dvd_dvd_real @ A @ C )
     => ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B2 @ C ) )
        = ( dvd_dvd_real @ A @ B2 ) ) ) ).

% dvd_add_left_iff
thf(fact_1997_dvd__add__left__iff,axiom,
    ! [A: rat,C: rat,B2: rat] :
      ( ( dvd_dvd_rat @ A @ C )
     => ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B2 @ C ) )
        = ( dvd_dvd_rat @ A @ B2 ) ) ) ).

% dvd_add_left_iff
thf(fact_1998_dvd__add__left__iff,axiom,
    ! [A: nat,C: nat,B2: nat] :
      ( ( dvd_dvd_nat @ A @ C )
     => ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B2 @ C ) )
        = ( dvd_dvd_nat @ A @ B2 ) ) ) ).

% dvd_add_left_iff
thf(fact_1999_dvd__add__left__iff,axiom,
    ! [A: int,C: int,B2: int] :
      ( ( dvd_dvd_int @ A @ C )
     => ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B2 @ C ) )
        = ( dvd_dvd_int @ A @ B2 ) ) ) ).

% dvd_add_left_iff
thf(fact_2000_dvd__add,axiom,
    ! [A: code_integer,B2: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ B2 )
     => ( ( dvd_dvd_Code_integer @ A @ C )
       => ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B2 @ C ) ) ) ) ).

% dvd_add
thf(fact_2001_dvd__add,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( dvd_dvd_real @ A @ B2 )
     => ( ( dvd_dvd_real @ A @ C )
       => ( dvd_dvd_real @ A @ ( plus_plus_real @ B2 @ C ) ) ) ) ).

% dvd_add
thf(fact_2002_dvd__add,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( dvd_dvd_rat @ A @ B2 )
     => ( ( dvd_dvd_rat @ A @ C )
       => ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B2 @ C ) ) ) ) ).

% dvd_add
thf(fact_2003_dvd__add,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ B2 )
     => ( ( dvd_dvd_nat @ A @ C )
       => ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B2 @ C ) ) ) ) ).

% dvd_add
thf(fact_2004_dvd__add,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( dvd_dvd_int @ A @ B2 )
     => ( ( dvd_dvd_int @ A @ C )
       => ( dvd_dvd_int @ A @ ( plus_plus_int @ B2 @ C ) ) ) ) ).

% dvd_add
thf(fact_2005_dvd__diff__commute,axiom,
    ! [A: code_integer,C: code_integer,B2: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ ( minus_8373710615458151222nteger @ C @ B2 ) )
      = ( dvd_dvd_Code_integer @ A @ ( minus_8373710615458151222nteger @ B2 @ C ) ) ) ).

% dvd_diff_commute
thf(fact_2006_dvd__diff__commute,axiom,
    ! [A: int,C: int,B2: int] :
      ( ( dvd_dvd_int @ A @ ( minus_minus_int @ C @ B2 ) )
      = ( dvd_dvd_int @ A @ ( minus_minus_int @ B2 @ C ) ) ) ).

% dvd_diff_commute
thf(fact_2007_dvd__diff,axiom,
    ! [X4: code_integer,Y3: code_integer,Z: code_integer] :
      ( ( dvd_dvd_Code_integer @ X4 @ Y3 )
     => ( ( dvd_dvd_Code_integer @ X4 @ Z )
       => ( dvd_dvd_Code_integer @ X4 @ ( minus_8373710615458151222nteger @ Y3 @ Z ) ) ) ) ).

% dvd_diff
thf(fact_2008_dvd__diff,axiom,
    ! [X4: real,Y3: real,Z: real] :
      ( ( dvd_dvd_real @ X4 @ Y3 )
     => ( ( dvd_dvd_real @ X4 @ Z )
       => ( dvd_dvd_real @ X4 @ ( minus_minus_real @ Y3 @ Z ) ) ) ) ).

% dvd_diff
thf(fact_2009_dvd__diff,axiom,
    ! [X4: rat,Y3: rat,Z: rat] :
      ( ( dvd_dvd_rat @ X4 @ Y3 )
     => ( ( dvd_dvd_rat @ X4 @ Z )
       => ( dvd_dvd_rat @ X4 @ ( minus_minus_rat @ Y3 @ Z ) ) ) ) ).

% dvd_diff
thf(fact_2010_dvd__diff,axiom,
    ! [X4: int,Y3: int,Z: int] :
      ( ( dvd_dvd_int @ X4 @ Y3 )
     => ( ( dvd_dvd_int @ X4 @ Z )
       => ( dvd_dvd_int @ X4 @ ( minus_minus_int @ Y3 @ Z ) ) ) ) ).

% dvd_diff
thf(fact_2011_dvd__div__eq__iff,axiom,
    ! [C: code_integer,A: code_integer,B2: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ A )
     => ( ( dvd_dvd_Code_integer @ C @ B2 )
       => ( ( ( divide6298287555418463151nteger @ A @ C )
            = ( divide6298287555418463151nteger @ B2 @ C ) )
          = ( A = B2 ) ) ) ) ).

% dvd_div_eq_iff
thf(fact_2012_dvd__div__eq__iff,axiom,
    ! [C: complex,A: complex,B2: complex] :
      ( ( dvd_dvd_complex @ C @ A )
     => ( ( dvd_dvd_complex @ C @ B2 )
       => ( ( ( divide1717551699836669952omplex @ A @ C )
            = ( divide1717551699836669952omplex @ B2 @ C ) )
          = ( A = B2 ) ) ) ) ).

% dvd_div_eq_iff
thf(fact_2013_dvd__div__eq__iff,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( dvd_dvd_real @ C @ A )
     => ( ( dvd_dvd_real @ C @ B2 )
       => ( ( ( divide_divide_real @ A @ C )
            = ( divide_divide_real @ B2 @ C ) )
          = ( A = B2 ) ) ) ) ).

% dvd_div_eq_iff
thf(fact_2014_dvd__div__eq__iff,axiom,
    ! [C: rat,A: rat,B2: rat] :
      ( ( dvd_dvd_rat @ C @ A )
     => ( ( dvd_dvd_rat @ C @ B2 )
       => ( ( ( divide_divide_rat @ A @ C )
            = ( divide_divide_rat @ B2 @ C ) )
          = ( A = B2 ) ) ) ) ).

% dvd_div_eq_iff
thf(fact_2015_dvd__div__eq__iff,axiom,
    ! [C: nat,A: nat,B2: nat] :
      ( ( dvd_dvd_nat @ C @ A )
     => ( ( dvd_dvd_nat @ C @ B2 )
       => ( ( ( divide_divide_nat @ A @ C )
            = ( divide_divide_nat @ B2 @ C ) )
          = ( A = B2 ) ) ) ) ).

% dvd_div_eq_iff
thf(fact_2016_dvd__div__eq__iff,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( dvd_dvd_int @ C @ A )
     => ( ( dvd_dvd_int @ C @ B2 )
       => ( ( ( divide_divide_int @ A @ C )
            = ( divide_divide_int @ B2 @ C ) )
          = ( A = B2 ) ) ) ) ).

% dvd_div_eq_iff
thf(fact_2017_dvd__div__eq__cancel,axiom,
    ! [A: code_integer,C: code_integer,B2: code_integer] :
      ( ( ( divide6298287555418463151nteger @ A @ C )
        = ( divide6298287555418463151nteger @ B2 @ C ) )
     => ( ( dvd_dvd_Code_integer @ C @ A )
       => ( ( dvd_dvd_Code_integer @ C @ B2 )
         => ( A = B2 ) ) ) ) ).

% dvd_div_eq_cancel
thf(fact_2018_dvd__div__eq__cancel,axiom,
    ! [A: complex,C: complex,B2: complex] :
      ( ( ( divide1717551699836669952omplex @ A @ C )
        = ( divide1717551699836669952omplex @ B2 @ C ) )
     => ( ( dvd_dvd_complex @ C @ A )
       => ( ( dvd_dvd_complex @ C @ B2 )
         => ( A = B2 ) ) ) ) ).

% dvd_div_eq_cancel
thf(fact_2019_dvd__div__eq__cancel,axiom,
    ! [A: real,C: real,B2: real] :
      ( ( ( divide_divide_real @ A @ C )
        = ( divide_divide_real @ B2 @ C ) )
     => ( ( dvd_dvd_real @ C @ A )
       => ( ( dvd_dvd_real @ C @ B2 )
         => ( A = B2 ) ) ) ) ).

% dvd_div_eq_cancel
thf(fact_2020_dvd__div__eq__cancel,axiom,
    ! [A: rat,C: rat,B2: rat] :
      ( ( ( divide_divide_rat @ A @ C )
        = ( divide_divide_rat @ B2 @ C ) )
     => ( ( dvd_dvd_rat @ C @ A )
       => ( ( dvd_dvd_rat @ C @ B2 )
         => ( A = B2 ) ) ) ) ).

% dvd_div_eq_cancel
thf(fact_2021_dvd__div__eq__cancel,axiom,
    ! [A: nat,C: nat,B2: nat] :
      ( ( ( divide_divide_nat @ A @ C )
        = ( divide_divide_nat @ B2 @ C ) )
     => ( ( dvd_dvd_nat @ C @ A )
       => ( ( dvd_dvd_nat @ C @ B2 )
         => ( A = B2 ) ) ) ) ).

% dvd_div_eq_cancel
thf(fact_2022_dvd__div__eq__cancel,axiom,
    ! [A: int,C: int,B2: int] :
      ( ( ( divide_divide_int @ A @ C )
        = ( divide_divide_int @ B2 @ C ) )
     => ( ( dvd_dvd_int @ C @ A )
       => ( ( dvd_dvd_int @ C @ B2 )
         => ( A = B2 ) ) ) ) ).

% dvd_div_eq_cancel
thf(fact_2023_div__div__div__same,axiom,
    ! [D: code_integer,B2: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ D @ B2 )
     => ( ( dvd_dvd_Code_integer @ B2 @ A )
       => ( ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ D ) @ ( divide6298287555418463151nteger @ B2 @ D ) )
          = ( divide6298287555418463151nteger @ A @ B2 ) ) ) ) ).

% div_div_div_same
thf(fact_2024_div__div__div__same,axiom,
    ! [D: nat,B2: nat,A: nat] :
      ( ( dvd_dvd_nat @ D @ B2 )
     => ( ( dvd_dvd_nat @ B2 @ A )
       => ( ( divide_divide_nat @ ( divide_divide_nat @ A @ D ) @ ( divide_divide_nat @ B2 @ D ) )
          = ( divide_divide_nat @ A @ B2 ) ) ) ) ).

% div_div_div_same
thf(fact_2025_div__div__div__same,axiom,
    ! [D: int,B2: int,A: int] :
      ( ( dvd_dvd_int @ D @ B2 )
     => ( ( dvd_dvd_int @ B2 @ A )
       => ( ( divide_divide_int @ ( divide_divide_int @ A @ D ) @ ( divide_divide_int @ B2 @ D ) )
          = ( divide_divide_int @ A @ B2 ) ) ) ) ).

% div_div_div_same
thf(fact_2026_dvd__power__same,axiom,
    ! [X4: code_integer,Y3: code_integer,N: nat] :
      ( ( dvd_dvd_Code_integer @ X4 @ Y3 )
     => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X4 @ N ) @ ( power_8256067586552552935nteger @ Y3 @ N ) ) ) ).

% dvd_power_same
thf(fact_2027_dvd__power__same,axiom,
    ! [X4: nat,Y3: nat,N: nat] :
      ( ( dvd_dvd_nat @ X4 @ Y3 )
     => ( dvd_dvd_nat @ ( power_power_nat @ X4 @ N ) @ ( power_power_nat @ Y3 @ N ) ) ) ).

% dvd_power_same
thf(fact_2028_dvd__power__same,axiom,
    ! [X4: real,Y3: real,N: nat] :
      ( ( dvd_dvd_real @ X4 @ Y3 )
     => ( dvd_dvd_real @ ( power_power_real @ X4 @ N ) @ ( power_power_real @ Y3 @ N ) ) ) ).

% dvd_power_same
thf(fact_2029_dvd__power__same,axiom,
    ! [X4: int,Y3: int,N: nat] :
      ( ( dvd_dvd_int @ X4 @ Y3 )
     => ( dvd_dvd_int @ ( power_power_int @ X4 @ N ) @ ( power_power_int @ Y3 @ N ) ) ) ).

% dvd_power_same
thf(fact_2030_dvd__power__same,axiom,
    ! [X4: complex,Y3: complex,N: nat] :
      ( ( dvd_dvd_complex @ X4 @ Y3 )
     => ( dvd_dvd_complex @ ( power_power_complex @ X4 @ N ) @ ( power_power_complex @ Y3 @ N ) ) ) ).

% dvd_power_same
thf(fact_2031_dvd__mod__iff,axiom,
    ! [C: nat,B2: nat,A: nat] :
      ( ( dvd_dvd_nat @ C @ B2 )
     => ( ( dvd_dvd_nat @ C @ ( modulo_modulo_nat @ A @ B2 ) )
        = ( dvd_dvd_nat @ C @ A ) ) ) ).

% dvd_mod_iff
thf(fact_2032_dvd__mod__iff,axiom,
    ! [C: int,B2: int,A: int] :
      ( ( dvd_dvd_int @ C @ B2 )
     => ( ( dvd_dvd_int @ C @ ( modulo_modulo_int @ A @ B2 ) )
        = ( dvd_dvd_int @ C @ A ) ) ) ).

% dvd_mod_iff
thf(fact_2033_dvd__mod__iff,axiom,
    ! [C: code_integer,B2: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ B2 )
     => ( ( dvd_dvd_Code_integer @ C @ ( modulo364778990260209775nteger @ A @ B2 ) )
        = ( dvd_dvd_Code_integer @ C @ A ) ) ) ).

% dvd_mod_iff
thf(fact_2034_dvd__mod__imp__dvd,axiom,
    ! [C: nat,A: nat,B2: nat] :
      ( ( dvd_dvd_nat @ C @ ( modulo_modulo_nat @ A @ B2 ) )
     => ( ( dvd_dvd_nat @ C @ B2 )
       => ( dvd_dvd_nat @ C @ A ) ) ) ).

% dvd_mod_imp_dvd
thf(fact_2035_dvd__mod__imp__dvd,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( dvd_dvd_int @ C @ ( modulo_modulo_int @ A @ B2 ) )
     => ( ( dvd_dvd_int @ C @ B2 )
       => ( dvd_dvd_int @ C @ A ) ) ) ).

% dvd_mod_imp_dvd
thf(fact_2036_dvd__mod__imp__dvd,axiom,
    ! [C: code_integer,A: code_integer,B2: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ ( modulo364778990260209775nteger @ A @ B2 ) )
     => ( ( dvd_dvd_Code_integer @ C @ B2 )
       => ( dvd_dvd_Code_integer @ C @ A ) ) ) ).

% dvd_mod_imp_dvd
thf(fact_2037_dvd__mod,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( dvd_dvd_nat @ K @ M )
     => ( ( dvd_dvd_nat @ K @ N )
       => ( dvd_dvd_nat @ K @ ( modulo_modulo_nat @ M @ N ) ) ) ) ).

% dvd_mod
thf(fact_2038_dvd__mod,axiom,
    ! [K: int,M: int,N: int] :
      ( ( dvd_dvd_int @ K @ M )
     => ( ( dvd_dvd_int @ K @ N )
       => ( dvd_dvd_int @ K @ ( modulo_modulo_int @ M @ N ) ) ) ) ).

% dvd_mod
thf(fact_2039_dvd__mod,axiom,
    ! [K: code_integer,M: code_integer,N: code_integer] :
      ( ( dvd_dvd_Code_integer @ K @ M )
     => ( ( dvd_dvd_Code_integer @ K @ N )
       => ( dvd_dvd_Code_integer @ K @ ( modulo364778990260209775nteger @ M @ N ) ) ) ) ).

% dvd_mod
thf(fact_2040_mod__mod__cancel,axiom,
    ! [C: nat,B2: nat,A: nat] :
      ( ( dvd_dvd_nat @ C @ B2 )
     => ( ( modulo_modulo_nat @ ( modulo_modulo_nat @ A @ B2 ) @ C )
        = ( modulo_modulo_nat @ A @ C ) ) ) ).

% mod_mod_cancel
thf(fact_2041_mod__mod__cancel,axiom,
    ! [C: int,B2: int,A: int] :
      ( ( dvd_dvd_int @ C @ B2 )
     => ( ( modulo_modulo_int @ ( modulo_modulo_int @ A @ B2 ) @ C )
        = ( modulo_modulo_int @ A @ C ) ) ) ).

% mod_mod_cancel
thf(fact_2042_mod__mod__cancel,axiom,
    ! [C: code_integer,B2: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ B2 )
     => ( ( modulo364778990260209775nteger @ ( modulo364778990260209775nteger @ A @ B2 ) @ C )
        = ( modulo364778990260209775nteger @ A @ C ) ) ) ).

% mod_mod_cancel
thf(fact_2043_dvd__diff__nat,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( dvd_dvd_nat @ K @ M )
     => ( ( dvd_dvd_nat @ K @ N )
       => ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) ) ) ) ).

% dvd_diff_nat
thf(fact_2044_zero__le,axiom,
    ! [X4: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X4 ) ).

% zero_le
thf(fact_2045_le__numeral__extra_I3_J,axiom,
    ord_less_eq_real @ zero_zero_real @ zero_zero_real ).

% le_numeral_extra(3)
thf(fact_2046_le__numeral__extra_I3_J,axiom,
    ord_less_eq_rat @ zero_zero_rat @ zero_zero_rat ).

% le_numeral_extra(3)
thf(fact_2047_le__numeral__extra_I3_J,axiom,
    ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).

% le_numeral_extra(3)
thf(fact_2048_le__numeral__extra_I3_J,axiom,
    ord_less_eq_int @ zero_zero_int @ zero_zero_int ).

% le_numeral_extra(3)
thf(fact_2049_zero__less__iff__neq__zero,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
      = ( N != zero_zero_nat ) ) ).

% zero_less_iff_neq_zero
thf(fact_2050_gr__implies__not__zero,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( N != zero_zero_nat ) ) ).

% gr_implies_not_zero
thf(fact_2051_not__less__zero,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% not_less_zero
thf(fact_2052_gr__zeroI,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% gr_zeroI
thf(fact_2053_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).

% less_numeral_extra(3)
thf(fact_2054_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_rat @ zero_zero_rat @ zero_zero_rat ) ).

% less_numeral_extra(3)
thf(fact_2055_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).

% less_numeral_extra(3)
thf(fact_2056_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).

% less_numeral_extra(3)
thf(fact_2057_field__lbound__gt__zero,axiom,
    ! [D1: real,D22: real] :
      ( ( ord_less_real @ zero_zero_real @ D1 )
     => ( ( ord_less_real @ zero_zero_real @ D22 )
       => ? [E2: real] :
            ( ( ord_less_real @ zero_zero_real @ E2 )
            & ( ord_less_real @ E2 @ D1 )
            & ( ord_less_real @ E2 @ D22 ) ) ) ) ).

% field_lbound_gt_zero
thf(fact_2058_field__lbound__gt__zero,axiom,
    ! [D1: rat,D22: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ D1 )
     => ( ( ord_less_rat @ zero_zero_rat @ D22 )
       => ? [E2: rat] :
            ( ( ord_less_rat @ zero_zero_rat @ E2 )
            & ( ord_less_rat @ E2 @ D1 )
            & ( ord_less_rat @ E2 @ D22 ) ) ) ) ).

% field_lbound_gt_zero
thf(fact_2059_zero__neq__numeral,axiom,
    ! [N: num] :
      ( zero_zero_complex
     != ( numera6690914467698888265omplex @ N ) ) ).

% zero_neq_numeral
thf(fact_2060_zero__neq__numeral,axiom,
    ! [N: num] :
      ( zero_zero_real
     != ( numeral_numeral_real @ N ) ) ).

% zero_neq_numeral
thf(fact_2061_zero__neq__numeral,axiom,
    ! [N: num] :
      ( zero_zero_rat
     != ( numeral_numeral_rat @ N ) ) ).

% zero_neq_numeral
thf(fact_2062_zero__neq__numeral,axiom,
    ! [N: num] :
      ( zero_zero_nat
     != ( numeral_numeral_nat @ N ) ) ).

% zero_neq_numeral
thf(fact_2063_zero__neq__numeral,axiom,
    ! [N: num] :
      ( zero_zero_int
     != ( numeral_numeral_int @ N ) ) ).

% zero_neq_numeral
thf(fact_2064_mult__right__cancel,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( C != zero_zero_real )
     => ( ( ( times_times_real @ A @ C )
          = ( times_times_real @ B2 @ C ) )
        = ( A = B2 ) ) ) ).

% mult_right_cancel
thf(fact_2065_mult__right__cancel,axiom,
    ! [C: rat,A: rat,B2: rat] :
      ( ( C != zero_zero_rat )
     => ( ( ( times_times_rat @ A @ C )
          = ( times_times_rat @ B2 @ C ) )
        = ( A = B2 ) ) ) ).

% mult_right_cancel
thf(fact_2066_mult__right__cancel,axiom,
    ! [C: nat,A: nat,B2: nat] :
      ( ( C != zero_zero_nat )
     => ( ( ( times_times_nat @ A @ C )
          = ( times_times_nat @ B2 @ C ) )
        = ( A = B2 ) ) ) ).

% mult_right_cancel
thf(fact_2067_mult__right__cancel,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( C != zero_zero_int )
     => ( ( ( times_times_int @ A @ C )
          = ( times_times_int @ B2 @ C ) )
        = ( A = B2 ) ) ) ).

% mult_right_cancel
thf(fact_2068_mult__left__cancel,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( C != zero_zero_real )
     => ( ( ( times_times_real @ C @ A )
          = ( times_times_real @ C @ B2 ) )
        = ( A = B2 ) ) ) ).

% mult_left_cancel
thf(fact_2069_mult__left__cancel,axiom,
    ! [C: rat,A: rat,B2: rat] :
      ( ( C != zero_zero_rat )
     => ( ( ( times_times_rat @ C @ A )
          = ( times_times_rat @ C @ B2 ) )
        = ( A = B2 ) ) ) ).

% mult_left_cancel
thf(fact_2070_mult__left__cancel,axiom,
    ! [C: nat,A: nat,B2: nat] :
      ( ( C != zero_zero_nat )
     => ( ( ( times_times_nat @ C @ A )
          = ( times_times_nat @ C @ B2 ) )
        = ( A = B2 ) ) ) ).

% mult_left_cancel
thf(fact_2071_mult__left__cancel,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( C != zero_zero_int )
     => ( ( ( times_times_int @ C @ A )
          = ( times_times_int @ C @ B2 ) )
        = ( A = B2 ) ) ) ).

% mult_left_cancel
thf(fact_2072_no__zero__divisors,axiom,
    ! [A: real,B2: real] :
      ( ( A != zero_zero_real )
     => ( ( B2 != zero_zero_real )
       => ( ( times_times_real @ A @ B2 )
         != zero_zero_real ) ) ) ).

% no_zero_divisors
thf(fact_2073_no__zero__divisors,axiom,
    ! [A: rat,B2: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B2 != zero_zero_rat )
       => ( ( times_times_rat @ A @ B2 )
         != zero_zero_rat ) ) ) ).

% no_zero_divisors
thf(fact_2074_no__zero__divisors,axiom,
    ! [A: nat,B2: nat] :
      ( ( A != zero_zero_nat )
     => ( ( B2 != zero_zero_nat )
       => ( ( times_times_nat @ A @ B2 )
         != zero_zero_nat ) ) ) ).

% no_zero_divisors
thf(fact_2075_no__zero__divisors,axiom,
    ! [A: int,B2: int] :
      ( ( A != zero_zero_int )
     => ( ( B2 != zero_zero_int )
       => ( ( times_times_int @ A @ B2 )
         != zero_zero_int ) ) ) ).

% no_zero_divisors
thf(fact_2076_divisors__zero,axiom,
    ! [A: real,B2: real] :
      ( ( ( times_times_real @ A @ B2 )
        = zero_zero_real )
     => ( ( A = zero_zero_real )
        | ( B2 = zero_zero_real ) ) ) ).

% divisors_zero
thf(fact_2077_divisors__zero,axiom,
    ! [A: rat,B2: rat] :
      ( ( ( times_times_rat @ A @ B2 )
        = zero_zero_rat )
     => ( ( A = zero_zero_rat )
        | ( B2 = zero_zero_rat ) ) ) ).

% divisors_zero
thf(fact_2078_divisors__zero,axiom,
    ! [A: nat,B2: nat] :
      ( ( ( times_times_nat @ A @ B2 )
        = zero_zero_nat )
     => ( ( A = zero_zero_nat )
        | ( B2 = zero_zero_nat ) ) ) ).

% divisors_zero
thf(fact_2079_divisors__zero,axiom,
    ! [A: int,B2: int] :
      ( ( ( times_times_int @ A @ B2 )
        = zero_zero_int )
     => ( ( A = zero_zero_int )
        | ( B2 = zero_zero_int ) ) ) ).

% divisors_zero
thf(fact_2080_mult__not__zero,axiom,
    ! [A: real,B2: real] :
      ( ( ( times_times_real @ A @ B2 )
       != zero_zero_real )
     => ( ( A != zero_zero_real )
        & ( B2 != zero_zero_real ) ) ) ).

% mult_not_zero
thf(fact_2081_mult__not__zero,axiom,
    ! [A: rat,B2: rat] :
      ( ( ( times_times_rat @ A @ B2 )
       != zero_zero_rat )
     => ( ( A != zero_zero_rat )
        & ( B2 != zero_zero_rat ) ) ) ).

% mult_not_zero
thf(fact_2082_mult__not__zero,axiom,
    ! [A: nat,B2: nat] :
      ( ( ( times_times_nat @ A @ B2 )
       != zero_zero_nat )
     => ( ( A != zero_zero_nat )
        & ( B2 != zero_zero_nat ) ) ) ).

% mult_not_zero
thf(fact_2083_mult__not__zero,axiom,
    ! [A: int,B2: int] :
      ( ( ( times_times_int @ A @ B2 )
       != zero_zero_int )
     => ( ( A != zero_zero_int )
        & ( B2 != zero_zero_int ) ) ) ).

% mult_not_zero
thf(fact_2084_zero__neq__one,axiom,
    zero_zero_complex != one_one_complex ).

% zero_neq_one
thf(fact_2085_zero__neq__one,axiom,
    zero_zero_real != one_one_real ).

% zero_neq_one
thf(fact_2086_zero__neq__one,axiom,
    zero_zero_rat != one_one_rat ).

% zero_neq_one
thf(fact_2087_zero__neq__one,axiom,
    zero_zero_nat != one_one_nat ).

% zero_neq_one
thf(fact_2088_zero__neq__one,axiom,
    zero_zero_int != one_one_int ).

% zero_neq_one
thf(fact_2089_add_Ogroup__left__neutral,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ zero_zero_real @ A )
      = A ) ).

% add.group_left_neutral
thf(fact_2090_add_Ogroup__left__neutral,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ zero_zero_rat @ A )
      = A ) ).

% add.group_left_neutral
thf(fact_2091_add_Ogroup__left__neutral,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ zero_zero_int @ A )
      = A ) ).

% add.group_left_neutral
thf(fact_2092_add_Ocomm__neutral,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ A @ zero_zero_real )
      = A ) ).

% add.comm_neutral
thf(fact_2093_add_Ocomm__neutral,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ A @ zero_zero_rat )
      = A ) ).

% add.comm_neutral
thf(fact_2094_add_Ocomm__neutral,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% add.comm_neutral
thf(fact_2095_add_Ocomm__neutral,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ zero_zero_int )
      = A ) ).

% add.comm_neutral
thf(fact_2096_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ zero_zero_real @ A )
      = A ) ).

% comm_monoid_add_class.add_0
thf(fact_2097_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ zero_zero_rat @ A )
      = A ) ).

% comm_monoid_add_class.add_0
thf(fact_2098_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ A )
      = A ) ).

% comm_monoid_add_class.add_0
thf(fact_2099_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ zero_zero_int @ A )
      = A ) ).

% comm_monoid_add_class.add_0
thf(fact_2100_verit__sum__simplify,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ A @ zero_zero_real )
      = A ) ).

% verit_sum_simplify
thf(fact_2101_verit__sum__simplify,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ A @ zero_zero_rat )
      = A ) ).

% verit_sum_simplify
thf(fact_2102_verit__sum__simplify,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% verit_sum_simplify
thf(fact_2103_verit__sum__simplify,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ zero_zero_int )
      = A ) ).

% verit_sum_simplify
thf(fact_2104_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y6: real,Z4: real] : Y6 = Z4 )
    = ( ^ [A4: real,B3: real] :
          ( ( minus_minus_real @ A4 @ B3 )
          = zero_zero_real ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_2105_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y6: rat,Z4: rat] : Y6 = Z4 )
    = ( ^ [A4: rat,B3: rat] :
          ( ( minus_minus_rat @ A4 @ B3 )
          = zero_zero_rat ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_2106_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y6: int,Z4: int] : Y6 = Z4 )
    = ( ^ [A4: int,B3: int] :
          ( ( minus_minus_int @ A4 @ B3 )
          = zero_zero_int ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_2107_power__not__zero,axiom,
    ! [A: rat,N: nat] :
      ( ( A != zero_zero_rat )
     => ( ( power_power_rat @ A @ N )
       != zero_zero_rat ) ) ).

% power_not_zero
thf(fact_2108_power__not__zero,axiom,
    ! [A: nat,N: nat] :
      ( ( A != zero_zero_nat )
     => ( ( power_power_nat @ A @ N )
       != zero_zero_nat ) ) ).

% power_not_zero
thf(fact_2109_power__not__zero,axiom,
    ! [A: real,N: nat] :
      ( ( A != zero_zero_real )
     => ( ( power_power_real @ A @ N )
       != zero_zero_real ) ) ).

% power_not_zero
thf(fact_2110_power__not__zero,axiom,
    ! [A: int,N: nat] :
      ( ( A != zero_zero_int )
     => ( ( power_power_int @ A @ N )
       != zero_zero_int ) ) ).

% power_not_zero
thf(fact_2111_power__not__zero,axiom,
    ! [A: complex,N: nat] :
      ( ( A != zero_zero_complex )
     => ( ( power_power_complex @ A @ N )
       != zero_zero_complex ) ) ).

% power_not_zero
thf(fact_2112_not0__implies__Suc,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ? [M4: nat] :
          ( N
          = ( suc @ M4 ) ) ) ).

% not0_implies_Suc
thf(fact_2113_Zero__not__Suc,axiom,
    ! [M: nat] :
      ( zero_zero_nat
     != ( suc @ M ) ) ).

% Zero_not_Suc
thf(fact_2114_Zero__neq__Suc,axiom,
    ! [M: nat] :
      ( zero_zero_nat
     != ( suc @ M ) ) ).

% Zero_neq_Suc
thf(fact_2115_Suc__neq__Zero,axiom,
    ! [M: nat] :
      ( ( suc @ M )
     != zero_zero_nat ) ).

% Suc_neq_Zero
thf(fact_2116_zero__induct,axiom,
    ! [P: nat > $o,K: nat] :
      ( ( P @ K )
     => ( ! [N4: nat] :
            ( ( P @ ( suc @ N4 ) )
           => ( P @ N4 ) )
       => ( P @ zero_zero_nat ) ) ) ).

% zero_induct
thf(fact_2117_diff__induct,axiom,
    ! [P: nat > nat > $o,M: nat,N: nat] :
      ( ! [X3: nat] : ( P @ X3 @ zero_zero_nat )
     => ( ! [Y4: nat] : ( P @ zero_zero_nat @ ( suc @ Y4 ) )
       => ( ! [X3: nat,Y4: nat] :
              ( ( P @ X3 @ Y4 )
             => ( P @ ( suc @ X3 ) @ ( suc @ Y4 ) ) )
         => ( P @ M @ N ) ) ) ) ).

% diff_induct
thf(fact_2118_nat__induct,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N4: nat] :
            ( ( P @ N4 )
           => ( P @ ( suc @ N4 ) ) )
       => ( P @ N ) ) ) ).

% nat_induct
thf(fact_2119_old_Onat_Oexhaust,axiom,
    ! [Y3: nat] :
      ( ( Y3 != zero_zero_nat )
     => ~ ! [Nat3: nat] :
            ( Y3
           != ( suc @ Nat3 ) ) ) ).

% old.nat.exhaust
thf(fact_2120_nat_OdiscI,axiom,
    ! [Nat: nat,X2: nat] :
      ( ( Nat
        = ( suc @ X2 ) )
     => ( Nat != zero_zero_nat ) ) ).

% nat.discI
thf(fact_2121_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] :
      ( zero_zero_nat
     != ( suc @ Nat2 ) ) ).

% old.nat.distinct(1)
thf(fact_2122_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] :
      ( ( suc @ Nat2 )
     != zero_zero_nat ) ).

% old.nat.distinct(2)
thf(fact_2123_nat_Odistinct_I1_J,axiom,
    ! [X2: nat] :
      ( zero_zero_nat
     != ( suc @ X2 ) ) ).

% nat.distinct(1)
thf(fact_2124_infinite__descent0,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N4: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N4 )
           => ( ~ ( P @ N4 )
             => ? [M5: nat] :
                  ( ( ord_less_nat @ M5 @ N4 )
                  & ~ ( P @ M5 ) ) ) )
       => ( P @ N ) ) ) ).

% infinite_descent0
thf(fact_2125_gr__implies__not0,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( N != zero_zero_nat ) ) ).

% gr_implies_not0
thf(fact_2126_less__zeroE,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% less_zeroE
thf(fact_2127_not__less0,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% not_less0
thf(fact_2128_not__gr0,axiom,
    ! [N: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
      = ( N = zero_zero_nat ) ) ).

% not_gr0
thf(fact_2129_gr0I,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% gr0I
thf(fact_2130_bot__nat__0_Oextremum__strict,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ zero_zero_nat ) ).

% bot_nat_0.extremum_strict
thf(fact_2131_le__0__eq,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ N @ zero_zero_nat )
      = ( N = zero_zero_nat ) ) ).

% le_0_eq
thf(fact_2132_bot__nat__0_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( A = zero_zero_nat ) ) ).

% bot_nat_0.extremum_uniqueI
thf(fact_2133_bot__nat__0_Oextremum__unique,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
      = ( A = zero_zero_nat ) ) ).

% bot_nat_0.extremum_unique
thf(fact_2134_less__eq__nat_Osimps_I1_J,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).

% less_eq_nat.simps(1)
thf(fact_2135_add__eq__self__zero,axiom,
    ! [M: nat,N: nat] :
      ( ( ( plus_plus_nat @ M @ N )
        = M )
     => ( N = zero_zero_nat ) ) ).

% add_eq_self_zero
thf(fact_2136_plus__nat_Oadd__0,axiom,
    ! [N: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ N )
      = N ) ).

% plus_nat.add_0
thf(fact_2137_diffs0__imp__equal,axiom,
    ! [M: nat,N: nat] :
      ( ( ( minus_minus_nat @ M @ N )
        = zero_zero_nat )
     => ( ( ( minus_minus_nat @ N @ M )
          = zero_zero_nat )
       => ( M = N ) ) ) ).

% diffs0_imp_equal
thf(fact_2138_minus__nat_Odiff__0,axiom,
    ! [M: nat] :
      ( ( minus_minus_nat @ M @ zero_zero_nat )
      = M ) ).

% minus_nat.diff_0
thf(fact_2139_nat__mult__eq__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ( times_times_nat @ K @ M )
        = ( times_times_nat @ K @ N ) )
      = ( ( K = zero_zero_nat )
        | ( M = N ) ) ) ).

% nat_mult_eq_cancel_disj
thf(fact_2140_mult__0,axiom,
    ! [N: nat] :
      ( ( times_times_nat @ zero_zero_nat @ N )
      = zero_zero_nat ) ).

% mult_0
thf(fact_2141_even__zero,axiom,
    dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ zero_z3403309356797280102nteger ).

% even_zero
thf(fact_2142_even__zero,axiom,
    dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ zero_zero_nat ).

% even_zero
thf(fact_2143_even__zero,axiom,
    dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ zero_zero_int ).

% even_zero
thf(fact_2144_is__unitE,axiom,
    ! [A: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ~ ( ( A != zero_z3403309356797280102nteger )
         => ! [B5: code_integer] :
              ( ( B5 != zero_z3403309356797280102nteger )
             => ( ( dvd_dvd_Code_integer @ B5 @ one_one_Code_integer )
               => ( ( ( divide6298287555418463151nteger @ one_one_Code_integer @ A )
                    = B5 )
                 => ( ( ( divide6298287555418463151nteger @ one_one_Code_integer @ B5 )
                      = A )
                   => ( ( ( times_3573771949741848930nteger @ A @ B5 )
                        = one_one_Code_integer )
                     => ( ( divide6298287555418463151nteger @ C @ A )
                       != ( times_3573771949741848930nteger @ C @ B5 ) ) ) ) ) ) ) ) ) ).

% is_unitE
thf(fact_2145_is__unitE,axiom,
    ! [A: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ~ ( ( A != zero_zero_nat )
         => ! [B5: nat] :
              ( ( B5 != zero_zero_nat )
             => ( ( dvd_dvd_nat @ B5 @ one_one_nat )
               => ( ( ( divide_divide_nat @ one_one_nat @ A )
                    = B5 )
                 => ( ( ( divide_divide_nat @ one_one_nat @ B5 )
                      = A )
                   => ( ( ( times_times_nat @ A @ B5 )
                        = one_one_nat )
                     => ( ( divide_divide_nat @ C @ A )
                       != ( times_times_nat @ C @ B5 ) ) ) ) ) ) ) ) ) ).

% is_unitE
thf(fact_2146_is__unitE,axiom,
    ! [A: int,C: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ~ ( ( A != zero_zero_int )
         => ! [B5: int] :
              ( ( B5 != zero_zero_int )
             => ( ( dvd_dvd_int @ B5 @ one_one_int )
               => ( ( ( divide_divide_int @ one_one_int @ A )
                    = B5 )
                 => ( ( ( divide_divide_int @ one_one_int @ B5 )
                      = A )
                   => ( ( ( times_times_int @ A @ B5 )
                        = one_one_int )
                     => ( ( divide_divide_int @ C @ A )
                       != ( times_times_int @ C @ B5 ) ) ) ) ) ) ) ) ) ).

% is_unitE
thf(fact_2147_is__unit__div__mult__cancel__left,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
       => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ A @ B2 ) )
          = ( divide6298287555418463151nteger @ one_one_Code_integer @ B2 ) ) ) ) ).

% is_unit_div_mult_cancel_left
thf(fact_2148_is__unit__div__mult__cancel__left,axiom,
    ! [A: nat,B2: nat] :
      ( ( A != zero_zero_nat )
     => ( ( dvd_dvd_nat @ B2 @ one_one_nat )
       => ( ( divide_divide_nat @ A @ ( times_times_nat @ A @ B2 ) )
          = ( divide_divide_nat @ one_one_nat @ B2 ) ) ) ) ).

% is_unit_div_mult_cancel_left
thf(fact_2149_is__unit__div__mult__cancel__left,axiom,
    ! [A: int,B2: int] :
      ( ( A != zero_zero_int )
     => ( ( dvd_dvd_int @ B2 @ one_one_int )
       => ( ( divide_divide_int @ A @ ( times_times_int @ A @ B2 ) )
          = ( divide_divide_int @ one_one_int @ B2 ) ) ) ) ).

% is_unit_div_mult_cancel_left
thf(fact_2150_is__unit__div__mult__cancel__right,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
       => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B2 @ A ) )
          = ( divide6298287555418463151nteger @ one_one_Code_integer @ B2 ) ) ) ) ).

% is_unit_div_mult_cancel_right
thf(fact_2151_is__unit__div__mult__cancel__right,axiom,
    ! [A: nat,B2: nat] :
      ( ( A != zero_zero_nat )
     => ( ( dvd_dvd_nat @ B2 @ one_one_nat )
       => ( ( divide_divide_nat @ A @ ( times_times_nat @ B2 @ A ) )
          = ( divide_divide_nat @ one_one_nat @ B2 ) ) ) ) ).

% is_unit_div_mult_cancel_right
thf(fact_2152_is__unit__div__mult__cancel__right,axiom,
    ! [A: int,B2: int] :
      ( ( A != zero_zero_int )
     => ( ( dvd_dvd_int @ B2 @ one_one_int )
       => ( ( divide_divide_int @ A @ ( times_times_int @ B2 @ A ) )
          = ( divide_divide_int @ one_one_int @ B2 ) ) ) ) ).

% is_unit_div_mult_cancel_right
thf(fact_2153_dvd__power__iff,axiom,
    ! [X4: code_integer,M: nat,N: nat] :
      ( ( X4 != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X4 @ M ) @ ( power_8256067586552552935nteger @ X4 @ N ) )
        = ( ( dvd_dvd_Code_integer @ X4 @ one_one_Code_integer )
          | ( ord_less_eq_nat @ M @ N ) ) ) ) ).

% dvd_power_iff
thf(fact_2154_dvd__power__iff,axiom,
    ! [X4: nat,M: nat,N: nat] :
      ( ( X4 != zero_zero_nat )
     => ( ( dvd_dvd_nat @ ( power_power_nat @ X4 @ M ) @ ( power_power_nat @ X4 @ N ) )
        = ( ( dvd_dvd_nat @ X4 @ one_one_nat )
          | ( ord_less_eq_nat @ M @ N ) ) ) ) ).

% dvd_power_iff
thf(fact_2155_dvd__power__iff,axiom,
    ! [X4: int,M: nat,N: nat] :
      ( ( X4 != zero_zero_int )
     => ( ( dvd_dvd_int @ ( power_power_int @ X4 @ M ) @ ( power_power_int @ X4 @ N ) )
        = ( ( dvd_dvd_int @ X4 @ one_one_int )
          | ( ord_less_eq_nat @ M @ N ) ) ) ) ).

% dvd_power_iff
thf(fact_2156_dvd__power,axiom,
    ! [N: nat,X4: code_integer] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N )
        | ( X4 = one_one_Code_integer ) )
     => ( dvd_dvd_Code_integer @ X4 @ ( power_8256067586552552935nteger @ X4 @ N ) ) ) ).

% dvd_power
thf(fact_2157_dvd__power,axiom,
    ! [N: nat,X4: rat] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N )
        | ( X4 = one_one_rat ) )
     => ( dvd_dvd_rat @ X4 @ ( power_power_rat @ X4 @ N ) ) ) ).

% dvd_power
thf(fact_2158_dvd__power,axiom,
    ! [N: nat,X4: nat] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N )
        | ( X4 = one_one_nat ) )
     => ( dvd_dvd_nat @ X4 @ ( power_power_nat @ X4 @ N ) ) ) ).

% dvd_power
thf(fact_2159_dvd__power,axiom,
    ! [N: nat,X4: real] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N )
        | ( X4 = one_one_real ) )
     => ( dvd_dvd_real @ X4 @ ( power_power_real @ X4 @ N ) ) ) ).

% dvd_power
thf(fact_2160_dvd__power,axiom,
    ! [N: nat,X4: int] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N )
        | ( X4 = one_one_int ) )
     => ( dvd_dvd_int @ X4 @ ( power_power_int @ X4 @ N ) ) ) ).

% dvd_power
thf(fact_2161_dvd__power,axiom,
    ! [N: nat,X4: complex] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N )
        | ( X4 = one_one_complex ) )
     => ( dvd_dvd_complex @ X4 @ ( power_power_complex @ X4 @ N ) ) ) ).

% dvd_power
thf(fact_2162_enat__0__less__mult__iff,axiom,
    ! [M: extended_enat,N: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( times_7803423173614009249d_enat @ M @ N ) )
      = ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ M )
        & ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N ) ) ) ).

% enat_0_less_mult_iff
thf(fact_2163_not__iless0,axiom,
    ! [N: extended_enat] :
      ~ ( ord_le72135733267957522d_enat @ N @ zero_z5237406670263579293d_enat ) ).

% not_iless0
thf(fact_2164_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
thf(fact_2165_ile0__eq,axiom,
    ! [N: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ N @ zero_z5237406670263579293d_enat )
      = ( N = zero_z5237406670263579293d_enat ) ) ).

% ile0_eq
thf(fact_2166_i0__lb,axiom,
    ! [N: extended_enat] : ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ N ) ).

% i0_lb
thf(fact_2167_dvd__mult__cancel1,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ M @ N ) @ M )
        = ( N = one_one_nat ) ) ) ).

% dvd_mult_cancel1
thf(fact_2168_dvd__mult__cancel2,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ N @ M ) @ M )
        = ( N = one_one_nat ) ) ) ).

% dvd_mult_cancel2
thf(fact_2169_power__eq__imp__eq__base,axiom,
    ! [A: real,N: nat,B2: real] :
      ( ( ( power_power_real @ A @ N )
        = ( power_power_real @ B2 @ N ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
         => ( ( ord_less_nat @ zero_zero_nat @ N )
           => ( A = B2 ) ) ) ) ) ).

% power_eq_imp_eq_base
thf(fact_2170_power__eq__imp__eq__base,axiom,
    ! [A: rat,N: nat,B2: rat] :
      ( ( ( power_power_rat @ A @ N )
        = ( power_power_rat @ B2 @ N ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
         => ( ( ord_less_nat @ zero_zero_nat @ N )
           => ( A = B2 ) ) ) ) ) ).

% power_eq_imp_eq_base
thf(fact_2171_power__eq__imp__eq__base,axiom,
    ! [A: nat,N: nat,B2: nat] :
      ( ( ( power_power_nat @ A @ N )
        = ( power_power_nat @ B2 @ N ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
         => ( ( ord_less_nat @ zero_zero_nat @ N )
           => ( A = B2 ) ) ) ) ) ).

% power_eq_imp_eq_base
thf(fact_2172_power__eq__imp__eq__base,axiom,
    ! [A: int,N: nat,B2: int] :
      ( ( ( power_power_int @ A @ N )
        = ( power_power_int @ B2 @ N ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
         => ( ( ord_less_nat @ zero_zero_nat @ N )
           => ( A = B2 ) ) ) ) ) ).

% power_eq_imp_eq_base
thf(fact_2173_power__eq__iff__eq__base,axiom,
    ! [N: nat,A: real,B2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
         => ( ( ( power_power_real @ A @ N )
              = ( power_power_real @ B2 @ N ) )
            = ( A = B2 ) ) ) ) ) ).

% power_eq_iff_eq_base
thf(fact_2174_power__eq__iff__eq__base,axiom,
    ! [N: nat,A: rat,B2: rat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
         => ( ( ( power_power_rat @ A @ N )
              = ( power_power_rat @ B2 @ N ) )
            = ( A = B2 ) ) ) ) ) ).

% power_eq_iff_eq_base
thf(fact_2175_power__eq__iff__eq__base,axiom,
    ! [N: nat,A: nat,B2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
         => ( ( ( power_power_nat @ A @ N )
              = ( power_power_nat @ B2 @ N ) )
            = ( A = B2 ) ) ) ) ) ).

% power_eq_iff_eq_base
thf(fact_2176_power__eq__iff__eq__base,axiom,
    ! [N: nat,A: int,B2: int] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
         => ( ( ( power_power_int @ A @ N )
              = ( power_power_int @ B2 @ N ) )
            = ( A = B2 ) ) ) ) ) ).

% power_eq_iff_eq_base
thf(fact_2177_int__power__div__base,axiom,
    ! [M: nat,K: int] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_int @ zero_zero_int @ K )
       => ( ( divide_divide_int @ ( power_power_int @ K @ M ) @ K )
          = ( power_power_int @ K @ ( minus_minus_nat @ M @ ( suc @ zero_zero_nat ) ) ) ) ) ) ).

% int_power_div_base
thf(fact_2178_zero__less__power__eq,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ N ) )
      = ( ( N = zero_zero_nat )
        | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
          & ( A != zero_zero_real ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
          & ( ord_less_real @ zero_zero_real @ A ) ) ) ) ).

% zero_less_power_eq
thf(fact_2179_zero__less__power__eq,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) )
      = ( ( N = zero_zero_nat )
        | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
          & ( A != zero_zero_rat ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
          & ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ).

% zero_less_power_eq
thf(fact_2180_zero__less__power__eq,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N ) )
      = ( ( N = zero_zero_nat )
        | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
          & ( A != zero_zero_int ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
          & ( ord_less_int @ zero_zero_int @ A ) ) ) ) ).

% zero_less_power_eq
thf(fact_2181_even__signed__take__bit__iff,axiom,
    ! [M: nat,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ M @ A ) )
      = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ).

% even_signed_take_bit_iff
thf(fact_2182_even__signed__take__bit__iff,axiom,
    ! [M: nat,A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ M @ A ) )
      = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ).

% even_signed_take_bit_iff
thf(fact_2183_realpow__pos__nth,axiom,
    ! [N: nat,A: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ? [R3: real] :
            ( ( ord_less_real @ zero_zero_real @ R3 )
            & ( ( power_power_real @ R3 @ N )
              = A ) ) ) ) ).

% realpow_pos_nth
thf(fact_2184_realpow__pos__nth__unique,axiom,
    ! [N: nat,A: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ? [X3: real] :
            ( ( ord_less_real @ zero_zero_real @ X3 )
            & ( ( power_power_real @ X3 @ N )
              = A )
            & ! [Y5: real] :
                ( ( ( ord_less_real @ zero_zero_real @ Y5 )
                  & ( ( power_power_real @ Y5 @ N )
                    = A ) )
               => ( Y5 = X3 ) ) ) ) ) ).

% realpow_pos_nth_unique
thf(fact_2185_power__le__zero__eq,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ zero_zero_real )
      = ( ( ord_less_nat @ zero_zero_nat @ N )
        & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
            & ( ord_less_eq_real @ A @ zero_zero_real ) )
          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
            & ( A = zero_zero_real ) ) ) ) ) ).

% power_le_zero_eq
thf(fact_2186_power__le__zero__eq,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ zero_zero_rat )
      = ( ( ord_less_nat @ zero_zero_nat @ N )
        & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
            & ( ord_less_eq_rat @ A @ zero_zero_rat ) )
          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
            & ( A = zero_zero_rat ) ) ) ) ) ).

% power_le_zero_eq
thf(fact_2187_power__le__zero__eq,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ zero_zero_int )
      = ( ( ord_less_nat @ zero_zero_nat @ N )
        & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
            & ( ord_less_eq_int @ A @ zero_zero_int ) )
          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
            & ( A = zero_zero_int ) ) ) ) ) ).

% power_le_zero_eq
thf(fact_2188_even__iff__mod__2__eq__zero,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
      = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = zero_zero_nat ) ) ).

% even_iff_mod_2_eq_zero
thf(fact_2189_even__iff__mod__2__eq__zero,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
      = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = zero_zero_int ) ) ).

% even_iff_mod_2_eq_zero
thf(fact_2190_even__iff__mod__2__eq__zero,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
      = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = zero_z3403309356797280102nteger ) ) ).

% even_iff_mod_2_eq_zero
thf(fact_2191_odd__pos,axiom,
    ! [N: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% odd_pos
thf(fact_2192_signed__take__bit__int__less__self__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ K )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K ) ) ).

% signed_take_bit_int_less_self_iff
thf(fact_2193_signed__take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,N: nat] :
      ( ( ord_less_eq_int @ K @ ( bit_ri631733984087533419it_int @ N @ K ) )
      = ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).

% signed_take_bit_int_greater_eq_self_iff
thf(fact_2194_signed__take__bit__int__less__exp,axiom,
    ! [N: nat,K: int] : ( ord_less_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).

% signed_take_bit_int_less_exp
thf(fact_2195_is__unit__mult__iff,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B2 ) @ one_one_Code_integer )
      = ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
        & ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer ) ) ) ).

% is_unit_mult_iff
thf(fact_2196_is__unit__mult__iff,axiom,
    ! [A: nat,B2: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B2 ) @ one_one_nat )
      = ( ( dvd_dvd_nat @ A @ one_one_nat )
        & ( dvd_dvd_nat @ B2 @ one_one_nat ) ) ) ).

% is_unit_mult_iff
thf(fact_2197_is__unit__mult__iff,axiom,
    ! [A: int,B2: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ A @ B2 ) @ one_one_int )
      = ( ( dvd_dvd_int @ A @ one_one_int )
        & ( dvd_dvd_int @ B2 @ one_one_int ) ) ) ).

% is_unit_mult_iff
thf(fact_2198_dvd__mult__unit__iff,axiom,
    ! [B2: code_integer,A: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ C @ B2 ) )
        = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).

% dvd_mult_unit_iff
thf(fact_2199_dvd__mult__unit__iff,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( ( dvd_dvd_nat @ B2 @ one_one_nat )
     => ( ( dvd_dvd_nat @ A @ ( times_times_nat @ C @ B2 ) )
        = ( dvd_dvd_nat @ A @ C ) ) ) ).

% dvd_mult_unit_iff
thf(fact_2200_dvd__mult__unit__iff,axiom,
    ! [B2: int,A: int,C: int] :
      ( ( dvd_dvd_int @ B2 @ one_one_int )
     => ( ( dvd_dvd_int @ A @ ( times_times_int @ C @ B2 ) )
        = ( dvd_dvd_int @ A @ C ) ) ) ).

% dvd_mult_unit_iff
thf(fact_2201_mult__unit__dvd__iff,axiom,
    ! [B2: code_integer,A: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B2 ) @ C )
        = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).

% mult_unit_dvd_iff
thf(fact_2202_mult__unit__dvd__iff,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( ( dvd_dvd_nat @ B2 @ one_one_nat )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B2 ) @ C )
        = ( dvd_dvd_nat @ A @ C ) ) ) ).

% mult_unit_dvd_iff
thf(fact_2203_mult__unit__dvd__iff,axiom,
    ! [B2: int,A: int,C: int] :
      ( ( dvd_dvd_int @ B2 @ one_one_int )
     => ( ( dvd_dvd_int @ ( times_times_int @ A @ B2 ) @ C )
        = ( dvd_dvd_int @ A @ C ) ) ) ).

% mult_unit_dvd_iff
thf(fact_2204_dvd__mult__unit__iff_H,axiom,
    ! [B2: code_integer,A: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B2 @ C ) )
        = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).

% dvd_mult_unit_iff'
thf(fact_2205_dvd__mult__unit__iff_H,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( ( dvd_dvd_nat @ B2 @ one_one_nat )
     => ( ( dvd_dvd_nat @ A @ ( times_times_nat @ B2 @ C ) )
        = ( dvd_dvd_nat @ A @ C ) ) ) ).

% dvd_mult_unit_iff'
thf(fact_2206_dvd__mult__unit__iff_H,axiom,
    ! [B2: int,A: int,C: int] :
      ( ( dvd_dvd_int @ B2 @ one_one_int )
     => ( ( dvd_dvd_int @ A @ ( times_times_int @ B2 @ C ) )
        = ( dvd_dvd_int @ A @ C ) ) ) ).

% dvd_mult_unit_iff'
thf(fact_2207_mult__unit__dvd__iff_H,axiom,
    ! [A: code_integer,B2: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B2 ) @ C )
        = ( dvd_dvd_Code_integer @ B2 @ C ) ) ) ).

% mult_unit_dvd_iff'
thf(fact_2208_mult__unit__dvd__iff_H,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B2 ) @ C )
        = ( dvd_dvd_nat @ B2 @ C ) ) ) ).

% mult_unit_dvd_iff'
thf(fact_2209_mult__unit__dvd__iff_H,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( dvd_dvd_int @ ( times_times_int @ A @ B2 ) @ C )
        = ( dvd_dvd_int @ B2 @ C ) ) ) ).

% mult_unit_dvd_iff'
thf(fact_2210_unit__mult__left__cancel,axiom,
    ! [A: code_integer,B2: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ( ( ( times_3573771949741848930nteger @ A @ B2 )
          = ( times_3573771949741848930nteger @ A @ C ) )
        = ( B2 = C ) ) ) ).

% unit_mult_left_cancel
thf(fact_2211_unit__mult__left__cancel,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( ( times_times_nat @ A @ B2 )
          = ( times_times_nat @ A @ C ) )
        = ( B2 = C ) ) ) ).

% unit_mult_left_cancel
thf(fact_2212_unit__mult__left__cancel,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( ( times_times_int @ A @ B2 )
          = ( times_times_int @ A @ C ) )
        = ( B2 = C ) ) ) ).

% unit_mult_left_cancel
thf(fact_2213_unit__mult__right__cancel,axiom,
    ! [A: code_integer,B2: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ( ( ( times_3573771949741848930nteger @ B2 @ A )
          = ( times_3573771949741848930nteger @ C @ A ) )
        = ( B2 = C ) ) ) ).

% unit_mult_right_cancel
thf(fact_2214_unit__mult__right__cancel,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( ( times_times_nat @ B2 @ A )
          = ( times_times_nat @ C @ A ) )
        = ( B2 = C ) ) ) ).

% unit_mult_right_cancel
thf(fact_2215_unit__mult__right__cancel,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( ( times_times_int @ B2 @ A )
          = ( times_times_int @ C @ A ) )
        = ( B2 = C ) ) ) ).

% unit_mult_right_cancel
thf(fact_2216_dvd__div__mult,axiom,
    ! [C: code_integer,B2: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ B2 )
     => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ B2 @ C ) @ A )
        = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ B2 @ A ) @ C ) ) ) ).

% dvd_div_mult
thf(fact_2217_dvd__div__mult,axiom,
    ! [C: nat,B2: nat,A: nat] :
      ( ( dvd_dvd_nat @ C @ B2 )
     => ( ( times_times_nat @ ( divide_divide_nat @ B2 @ C ) @ A )
        = ( divide_divide_nat @ ( times_times_nat @ B2 @ A ) @ C ) ) ) ).

% dvd_div_mult
thf(fact_2218_dvd__div__mult,axiom,
    ! [C: int,B2: int,A: int] :
      ( ( dvd_dvd_int @ C @ B2 )
     => ( ( times_times_int @ ( divide_divide_int @ B2 @ C ) @ A )
        = ( divide_divide_int @ ( times_times_int @ B2 @ A ) @ C ) ) ) ).

% dvd_div_mult
thf(fact_2219_div__mult__swap,axiom,
    ! [C: code_integer,B2: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ B2 )
     => ( ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B2 @ C ) )
        = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B2 ) @ C ) ) ) ).

% div_mult_swap
thf(fact_2220_div__mult__swap,axiom,
    ! [C: nat,B2: nat,A: nat] :
      ( ( dvd_dvd_nat @ C @ B2 )
     => ( ( times_times_nat @ A @ ( divide_divide_nat @ B2 @ C ) )
        = ( divide_divide_nat @ ( times_times_nat @ A @ B2 ) @ C ) ) ) ).

% div_mult_swap
thf(fact_2221_div__mult__swap,axiom,
    ! [C: int,B2: int,A: int] :
      ( ( dvd_dvd_int @ C @ B2 )
     => ( ( times_times_int @ A @ ( divide_divide_int @ B2 @ C ) )
        = ( divide_divide_int @ ( times_times_int @ A @ B2 ) @ C ) ) ) ).

% div_mult_swap
thf(fact_2222_div__div__eq__right,axiom,
    ! [C: code_integer,B2: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ B2 )
     => ( ( dvd_dvd_Code_integer @ B2 @ A )
       => ( ( divide6298287555418463151nteger @ A @ ( divide6298287555418463151nteger @ B2 @ C ) )
          = ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B2 ) @ C ) ) ) ) ).

% div_div_eq_right
thf(fact_2223_div__div__eq__right,axiom,
    ! [C: nat,B2: nat,A: nat] :
      ( ( dvd_dvd_nat @ C @ B2 )
     => ( ( dvd_dvd_nat @ B2 @ A )
       => ( ( divide_divide_nat @ A @ ( divide_divide_nat @ B2 @ C ) )
          = ( times_times_nat @ ( divide_divide_nat @ A @ B2 ) @ C ) ) ) ) ).

% div_div_eq_right
thf(fact_2224_div__div__eq__right,axiom,
    ! [C: int,B2: int,A: int] :
      ( ( dvd_dvd_int @ C @ B2 )
     => ( ( dvd_dvd_int @ B2 @ A )
       => ( ( divide_divide_int @ A @ ( divide_divide_int @ B2 @ C ) )
          = ( times_times_int @ ( divide_divide_int @ A @ B2 ) @ C ) ) ) ) ).

% div_div_eq_right
thf(fact_2225_dvd__div__mult2__eq,axiom,
    ! [B2: code_integer,C: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ B2 @ C ) @ A )
     => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B2 @ C ) )
        = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ B2 ) @ C ) ) ) ).

% dvd_div_mult2_eq
thf(fact_2226_dvd__div__mult2__eq,axiom,
    ! [B2: nat,C: nat,A: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ B2 @ C ) @ A )
     => ( ( divide_divide_nat @ A @ ( times_times_nat @ B2 @ C ) )
        = ( divide_divide_nat @ ( divide_divide_nat @ A @ B2 ) @ C ) ) ) ).

% dvd_div_mult2_eq
thf(fact_2227_dvd__div__mult2__eq,axiom,
    ! [B2: int,C: int,A: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ B2 @ C ) @ A )
     => ( ( divide_divide_int @ A @ ( times_times_int @ B2 @ C ) )
        = ( divide_divide_int @ ( divide_divide_int @ A @ B2 ) @ C ) ) ) ).

% dvd_div_mult2_eq
thf(fact_2228_dvd__mult__imp__div,axiom,
    ! [A: code_integer,C: code_integer,B2: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ B2 )
     => ( dvd_dvd_Code_integer @ A @ ( divide6298287555418463151nteger @ B2 @ C ) ) ) ).

% dvd_mult_imp_div
thf(fact_2229_dvd__mult__imp__div,axiom,
    ! [A: nat,C: nat,B2: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ B2 )
     => ( dvd_dvd_nat @ A @ ( divide_divide_nat @ B2 @ C ) ) ) ).

% dvd_mult_imp_div
thf(fact_2230_dvd__mult__imp__div,axiom,
    ! [A: int,C: int,B2: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ B2 )
     => ( dvd_dvd_int @ A @ ( divide_divide_int @ B2 @ C ) ) ) ).

% dvd_mult_imp_div
thf(fact_2231_div__mult__div__if__dvd,axiom,
    ! [B2: code_integer,A: code_integer,D: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ B2 @ A )
     => ( ( dvd_dvd_Code_integer @ D @ C )
       => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B2 ) @ ( divide6298287555418463151nteger @ C @ D ) )
          = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B2 @ D ) ) ) ) ) ).

% div_mult_div_if_dvd
thf(fact_2232_div__mult__div__if__dvd,axiom,
    ! [B2: nat,A: nat,D: nat,C: nat] :
      ( ( dvd_dvd_nat @ B2 @ A )
     => ( ( dvd_dvd_nat @ D @ C )
       => ( ( times_times_nat @ ( divide_divide_nat @ A @ B2 ) @ ( divide_divide_nat @ C @ D ) )
          = ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ).

% div_mult_div_if_dvd
thf(fact_2233_div__mult__div__if__dvd,axiom,
    ! [B2: int,A: int,D: int,C: int] :
      ( ( dvd_dvd_int @ B2 @ A )
     => ( ( dvd_dvd_int @ D @ C )
       => ( ( times_times_int @ ( divide_divide_int @ A @ B2 ) @ ( divide_divide_int @ C @ D ) )
          = ( divide_divide_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ).

% div_mult_div_if_dvd
thf(fact_2234_unit__div__cancel,axiom,
    ! [A: code_integer,B2: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ( ( ( divide6298287555418463151nteger @ B2 @ A )
          = ( divide6298287555418463151nteger @ C @ A ) )
        = ( B2 = C ) ) ) ).

% unit_div_cancel
thf(fact_2235_unit__div__cancel,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( ( divide_divide_nat @ B2 @ A )
          = ( divide_divide_nat @ C @ A ) )
        = ( B2 = C ) ) ) ).

% unit_div_cancel
thf(fact_2236_unit__div__cancel,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( ( divide_divide_int @ B2 @ A )
          = ( divide_divide_int @ C @ A ) )
        = ( B2 = C ) ) ) ).

% unit_div_cancel
thf(fact_2237_div__unit__dvd__iff,axiom,
    ! [B2: code_integer,A: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A @ B2 ) @ C )
        = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).

% div_unit_dvd_iff
thf(fact_2238_div__unit__dvd__iff,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( ( dvd_dvd_nat @ B2 @ one_one_nat )
     => ( ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B2 ) @ C )
        = ( dvd_dvd_nat @ A @ C ) ) ) ).

% div_unit_dvd_iff
thf(fact_2239_div__unit__dvd__iff,axiom,
    ! [B2: int,A: int,C: int] :
      ( ( dvd_dvd_int @ B2 @ one_one_int )
     => ( ( dvd_dvd_int @ ( divide_divide_int @ A @ B2 ) @ C )
        = ( dvd_dvd_int @ A @ C ) ) ) ).

% div_unit_dvd_iff
thf(fact_2240_dvd__div__unit__iff,axiom,
    ! [B2: code_integer,A: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ A @ ( divide6298287555418463151nteger @ C @ B2 ) )
        = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).

% dvd_div_unit_iff
thf(fact_2241_dvd__div__unit__iff,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( ( dvd_dvd_nat @ B2 @ one_one_nat )
     => ( ( dvd_dvd_nat @ A @ ( divide_divide_nat @ C @ B2 ) )
        = ( dvd_dvd_nat @ A @ C ) ) ) ).

% dvd_div_unit_iff
thf(fact_2242_dvd__div__unit__iff,axiom,
    ! [B2: int,A: int,C: int] :
      ( ( dvd_dvd_int @ B2 @ one_one_int )
     => ( ( dvd_dvd_int @ A @ ( divide_divide_int @ C @ B2 ) )
        = ( dvd_dvd_int @ A @ C ) ) ) ).

% dvd_div_unit_iff
thf(fact_2243_div__plus__div__distrib__dvd__right,axiom,
    ! [C: code_integer,B2: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ B2 )
     => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B2 ) @ C )
        = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B2 @ C ) ) ) ) ).

% div_plus_div_distrib_dvd_right
thf(fact_2244_div__plus__div__distrib__dvd__right,axiom,
    ! [C: nat,B2: nat,A: nat] :
      ( ( dvd_dvd_nat @ C @ B2 )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B2 ) @ C )
        = ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B2 @ C ) ) ) ) ).

% div_plus_div_distrib_dvd_right
thf(fact_2245_div__plus__div__distrib__dvd__right,axiom,
    ! [C: int,B2: int,A: int] :
      ( ( dvd_dvd_int @ C @ B2 )
     => ( ( divide_divide_int @ ( plus_plus_int @ A @ B2 ) @ C )
        = ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B2 @ C ) ) ) ) ).

% div_plus_div_distrib_dvd_right
thf(fact_2246_div__plus__div__distrib__dvd__left,axiom,
    ! [C: code_integer,A: code_integer,B2: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ A )
     => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B2 ) @ C )
        = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B2 @ C ) ) ) ) ).

% div_plus_div_distrib_dvd_left
thf(fact_2247_div__plus__div__distrib__dvd__left,axiom,
    ! [C: nat,A: nat,B2: nat] :
      ( ( dvd_dvd_nat @ C @ A )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B2 ) @ C )
        = ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B2 @ C ) ) ) ) ).

% div_plus_div_distrib_dvd_left
thf(fact_2248_div__plus__div__distrib__dvd__left,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( dvd_dvd_int @ C @ A )
     => ( ( divide_divide_int @ ( plus_plus_int @ A @ B2 ) @ C )
        = ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B2 @ C ) ) ) ) ).

% div_plus_div_distrib_dvd_left
thf(fact_2249_div__power,axiom,
    ! [B2: code_integer,A: code_integer,N: nat] :
      ( ( dvd_dvd_Code_integer @ B2 @ A )
     => ( ( power_8256067586552552935nteger @ ( divide6298287555418463151nteger @ A @ B2 ) @ N )
        = ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ A @ N ) @ ( power_8256067586552552935nteger @ B2 @ N ) ) ) ) ).

% div_power
thf(fact_2250_div__power,axiom,
    ! [B2: nat,A: nat,N: nat] :
      ( ( dvd_dvd_nat @ B2 @ A )
     => ( ( power_power_nat @ ( divide_divide_nat @ A @ B2 ) @ N )
        = ( divide_divide_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B2 @ N ) ) ) ) ).

% div_power
thf(fact_2251_div__power,axiom,
    ! [B2: int,A: int,N: nat] :
      ( ( dvd_dvd_int @ B2 @ A )
     => ( ( power_power_int @ ( divide_divide_int @ A @ B2 ) @ N )
        = ( divide_divide_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B2 @ N ) ) ) ) ).

% div_power
thf(fact_2252_power__strict__mono,axiom,
    ! [A: real,B2: real,N: nat] :
      ( ( ord_less_real @ A @ B2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ( ord_less_nat @ zero_zero_nat @ N )
         => ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B2 @ N ) ) ) ) ) ).

% power_strict_mono
thf(fact_2253_power__strict__mono,axiom,
    ! [A: rat,B2: rat,N: nat] :
      ( ( ord_less_rat @ A @ B2 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
       => ( ( ord_less_nat @ zero_zero_nat @ N )
         => ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B2 @ N ) ) ) ) ) ).

% power_strict_mono
thf(fact_2254_power__strict__mono,axiom,
    ! [A: nat,B2: nat,N: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ( ord_less_nat @ zero_zero_nat @ N )
         => ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B2 @ N ) ) ) ) ) ).

% power_strict_mono
thf(fact_2255_power__strict__mono,axiom,
    ! [A: int,B2: int,N: nat] :
      ( ( ord_less_int @ A @ B2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ( ord_less_nat @ zero_zero_nat @ N )
         => ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B2 @ N ) ) ) ) ) ).

% power_strict_mono
thf(fact_2256_le__imp__power__dvd,axiom,
    ! [M: nat,N: nat,A: code_integer] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ M ) @ ( power_8256067586552552935nteger @ A @ N ) ) ) ).

% le_imp_power_dvd
thf(fact_2257_le__imp__power__dvd,axiom,
    ! [M: nat,N: nat,A: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( dvd_dvd_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) ) ) ).

% le_imp_power_dvd
thf(fact_2258_le__imp__power__dvd,axiom,
    ! [M: nat,N: nat,A: real] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( dvd_dvd_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) ) ) ).

% le_imp_power_dvd
thf(fact_2259_le__imp__power__dvd,axiom,
    ! [M: nat,N: nat,A: int] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( dvd_dvd_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) ) ) ).

% le_imp_power_dvd
thf(fact_2260_le__imp__power__dvd,axiom,
    ! [M: nat,N: nat,A: complex] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( dvd_dvd_complex @ ( power_power_complex @ A @ M ) @ ( power_power_complex @ A @ N ) ) ) ).

% le_imp_power_dvd
thf(fact_2261_power__le__dvd,axiom,
    ! [A: code_integer,N: nat,B2: code_integer,M: nat] :
      ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) @ B2 )
     => ( ( ord_less_eq_nat @ M @ N )
       => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ M ) @ B2 ) ) ) ).

% power_le_dvd
thf(fact_2262_power__le__dvd,axiom,
    ! [A: nat,N: nat,B2: nat,M: nat] :
      ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ B2 )
     => ( ( ord_less_eq_nat @ M @ N )
       => ( dvd_dvd_nat @ ( power_power_nat @ A @ M ) @ B2 ) ) ) ).

% power_le_dvd
thf(fact_2263_power__le__dvd,axiom,
    ! [A: real,N: nat,B2: real,M: nat] :
      ( ( dvd_dvd_real @ ( power_power_real @ A @ N ) @ B2 )
     => ( ( ord_less_eq_nat @ M @ N )
       => ( dvd_dvd_real @ ( power_power_real @ A @ M ) @ B2 ) ) ) ).

% power_le_dvd
thf(fact_2264_power__le__dvd,axiom,
    ! [A: int,N: nat,B2: int,M: nat] :
      ( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ B2 )
     => ( ( ord_less_eq_nat @ M @ N )
       => ( dvd_dvd_int @ ( power_power_int @ A @ M ) @ B2 ) ) ) ).

% power_le_dvd
thf(fact_2265_power__le__dvd,axiom,
    ! [A: complex,N: nat,B2: complex,M: nat] :
      ( ( dvd_dvd_complex @ ( power_power_complex @ A @ N ) @ B2 )
     => ( ( ord_less_eq_nat @ M @ N )
       => ( dvd_dvd_complex @ ( power_power_complex @ A @ M ) @ B2 ) ) ) ).

% power_le_dvd
thf(fact_2266_dvd__power__le,axiom,
    ! [X4: code_integer,Y3: code_integer,N: nat,M: nat] :
      ( ( dvd_dvd_Code_integer @ X4 @ Y3 )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X4 @ N ) @ ( power_8256067586552552935nteger @ Y3 @ M ) ) ) ) ).

% dvd_power_le
thf(fact_2267_dvd__power__le,axiom,
    ! [X4: nat,Y3: nat,N: nat,M: nat] :
      ( ( dvd_dvd_nat @ X4 @ Y3 )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( dvd_dvd_nat @ ( power_power_nat @ X4 @ N ) @ ( power_power_nat @ Y3 @ M ) ) ) ) ).

% dvd_power_le
thf(fact_2268_dvd__power__le,axiom,
    ! [X4: real,Y3: real,N: nat,M: nat] :
      ( ( dvd_dvd_real @ X4 @ Y3 )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( dvd_dvd_real @ ( power_power_real @ X4 @ N ) @ ( power_power_real @ Y3 @ M ) ) ) ) ).

% dvd_power_le
thf(fact_2269_dvd__power__le,axiom,
    ! [X4: int,Y3: int,N: nat,M: nat] :
      ( ( dvd_dvd_int @ X4 @ Y3 )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( dvd_dvd_int @ ( power_power_int @ X4 @ N ) @ ( power_power_int @ Y3 @ M ) ) ) ) ).

% dvd_power_le
thf(fact_2270_dvd__power__le,axiom,
    ! [X4: complex,Y3: complex,N: nat,M: nat] :
      ( ( dvd_dvd_complex @ X4 @ Y3 )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( dvd_dvd_complex @ ( power_power_complex @ X4 @ N ) @ ( power_power_complex @ Y3 @ M ) ) ) ) ).

% dvd_power_le
thf(fact_2271_mod__eq__dvd__iff,axiom,
    ! [A: int,C: int,B2: int] :
      ( ( ( modulo_modulo_int @ A @ C )
        = ( modulo_modulo_int @ B2 @ C ) )
      = ( dvd_dvd_int @ C @ ( minus_minus_int @ A @ B2 ) ) ) ).

% mod_eq_dvd_iff
thf(fact_2272_mod__eq__dvd__iff,axiom,
    ! [A: code_integer,C: code_integer,B2: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ C )
        = ( modulo364778990260209775nteger @ B2 @ C ) )
      = ( dvd_dvd_Code_integer @ C @ ( minus_8373710615458151222nteger @ A @ B2 ) ) ) ).

% mod_eq_dvd_iff
thf(fact_2273_dvd__minus__mod,axiom,
    ! [B2: nat,A: nat] : ( dvd_dvd_nat @ B2 @ ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B2 ) ) ) ).

% dvd_minus_mod
thf(fact_2274_dvd__minus__mod,axiom,
    ! [B2: int,A: int] : ( dvd_dvd_int @ B2 @ ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B2 ) ) ) ).

% dvd_minus_mod
thf(fact_2275_dvd__minus__mod,axiom,
    ! [B2: code_integer,A: code_integer] : ( dvd_dvd_Code_integer @ B2 @ ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ A @ B2 ) ) ) ).

% dvd_minus_mod
thf(fact_2276_dvd__minus__self,axiom,
    ! [M: nat,N: nat] :
      ( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ M ) )
      = ( ( ord_less_nat @ N @ M )
        | ( dvd_dvd_nat @ M @ N ) ) ) ).

% dvd_minus_self
thf(fact_2277_dvd__diffD,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) )
     => ( ( dvd_dvd_nat @ K @ N )
       => ( ( ord_less_eq_nat @ N @ M )
         => ( dvd_dvd_nat @ K @ M ) ) ) ) ).

% dvd_diffD
thf(fact_2278_dvd__diffD1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) )
     => ( ( dvd_dvd_nat @ K @ M )
       => ( ( ord_less_eq_nat @ N @ M )
         => ( dvd_dvd_nat @ K @ N ) ) ) ) ).

% dvd_diffD1
thf(fact_2279_less__eq__dvd__minus,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( dvd_dvd_nat @ M @ N )
        = ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ M ) ) ) ) ).

% less_eq_dvd_minus
thf(fact_2280_dbl__def,axiom,
    ( neg_numeral_dbl_real
    = ( ^ [X: real] : ( plus_plus_real @ X @ X ) ) ) ).

% dbl_def
thf(fact_2281_dbl__def,axiom,
    ( neg_numeral_dbl_rat
    = ( ^ [X: rat] : ( plus_plus_rat @ X @ X ) ) ) ).

% dbl_def
thf(fact_2282_dbl__def,axiom,
    ( neg_numeral_dbl_int
    = ( ^ [X: int] : ( plus_plus_int @ X @ X ) ) ) ).

% dbl_def
thf(fact_2283_zero__le__numeral,axiom,
    ! [N: num] : ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ N ) ) ).

% zero_le_numeral
thf(fact_2284_zero__le__numeral,axiom,
    ! [N: num] : ( ord_less_eq_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N ) ) ).

% zero_le_numeral
thf(fact_2285_zero__le__numeral,axiom,
    ! [N: num] : ( ord_less_eq_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).

% zero_le_numeral
thf(fact_2286_zero__le__numeral,axiom,
    ! [N: num] : ( ord_less_eq_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).

% zero_le_numeral
thf(fact_2287_not__numeral__le__zero,axiom,
    ! [N: num] :
      ~ ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ zero_zero_real ) ).

% not_numeral_le_zero
thf(fact_2288_not__numeral__le__zero,axiom,
    ! [N: num] :
      ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ N ) @ zero_zero_rat ) ).

% not_numeral_le_zero
thf(fact_2289_not__numeral__le__zero,axiom,
    ! [N: num] :
      ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).

% not_numeral_le_zero
thf(fact_2290_not__numeral__le__zero,axiom,
    ! [N: num] :
      ~ ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).

% not_numeral_le_zero
thf(fact_2291_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ord_less_eq_real @ A @ B2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_2292_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B2 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_2293_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
       => ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B2 ) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_2294_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( ord_less_eq_int @ A @ B2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ C )
       => ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_2295_zero__le__mult__iff,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
          & ( ord_less_eq_real @ zero_zero_real @ B2 ) )
        | ( ( ord_less_eq_real @ A @ zero_zero_real )
          & ( ord_less_eq_real @ B2 @ zero_zero_real ) ) ) ) ).

% zero_le_mult_iff
thf(fact_2296_zero__le__mult__iff,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B2 ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
          & ( ord_less_eq_rat @ zero_zero_rat @ B2 ) )
        | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
          & ( ord_less_eq_rat @ B2 @ zero_zero_rat ) ) ) ) ).

% zero_le_mult_iff
thf(fact_2297_zero__le__mult__iff,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B2 ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ A )
          & ( ord_less_eq_int @ zero_zero_int @ B2 ) )
        | ( ( ord_less_eq_int @ A @ zero_zero_int )
          & ( ord_less_eq_int @ B2 @ zero_zero_int ) ) ) ) ).

% zero_le_mult_iff
thf(fact_2298_mult__nonneg__nonpos2,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ B2 @ zero_zero_real )
       => ( ord_less_eq_real @ ( times_times_real @ B2 @ A ) @ zero_zero_real ) ) ) ).

% mult_nonneg_nonpos2
thf(fact_2299_mult__nonneg__nonpos2,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ B2 @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( times_times_rat @ B2 @ A ) @ zero_zero_rat ) ) ) ).

% mult_nonneg_nonpos2
thf(fact_2300_mult__nonneg__nonpos2,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
       => ( ord_less_eq_nat @ ( times_times_nat @ B2 @ A ) @ zero_zero_nat ) ) ) ).

% mult_nonneg_nonpos2
thf(fact_2301_mult__nonneg__nonpos2,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ B2 @ zero_zero_int )
       => ( ord_less_eq_int @ ( times_times_int @ B2 @ A ) @ zero_zero_int ) ) ) ).

% mult_nonneg_nonpos2
thf(fact_2302_mult__nonpos__nonneg,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ A @ zero_zero_real )
     => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
       => ( ord_less_eq_real @ ( times_times_real @ A @ B2 ) @ zero_zero_real ) ) ) ).

% mult_nonpos_nonneg
thf(fact_2303_mult__nonpos__nonneg,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_eq_rat @ A @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
       => ( ord_less_eq_rat @ ( times_times_rat @ A @ B2 ) @ zero_zero_rat ) ) ) ).

% mult_nonpos_nonneg
thf(fact_2304_mult__nonpos__nonneg,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
       => ( ord_less_eq_nat @ ( times_times_nat @ A @ B2 ) @ zero_zero_nat ) ) ) ).

% mult_nonpos_nonneg
thf(fact_2305_mult__nonpos__nonneg,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
       => ( ord_less_eq_int @ ( times_times_int @ A @ B2 ) @ zero_zero_int ) ) ) ).

% mult_nonpos_nonneg
thf(fact_2306_mult__nonneg__nonpos,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ B2 @ zero_zero_real )
       => ( ord_less_eq_real @ ( times_times_real @ A @ B2 ) @ zero_zero_real ) ) ) ).

% mult_nonneg_nonpos
thf(fact_2307_mult__nonneg__nonpos,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ B2 @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( times_times_rat @ A @ B2 ) @ zero_zero_rat ) ) ) ).

% mult_nonneg_nonpos
thf(fact_2308_mult__nonneg__nonpos,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
       => ( ord_less_eq_nat @ ( times_times_nat @ A @ B2 ) @ zero_zero_nat ) ) ) ).

% mult_nonneg_nonpos
thf(fact_2309_mult__nonneg__nonpos,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ B2 @ zero_zero_int )
       => ( ord_less_eq_int @ ( times_times_int @ A @ B2 ) @ zero_zero_int ) ) ) ).

% mult_nonneg_nonpos
thf(fact_2310_mult__nonneg__nonneg,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
       => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) ) ) ) ).

% mult_nonneg_nonneg
thf(fact_2311_mult__nonneg__nonneg,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B2 ) ) ) ) ).

% mult_nonneg_nonneg
thf(fact_2312_mult__nonneg__nonneg,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
       => ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A @ B2 ) ) ) ) ).

% mult_nonneg_nonneg
thf(fact_2313_mult__nonneg__nonneg,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
       => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B2 ) ) ) ) ).

% mult_nonneg_nonneg
thf(fact_2314_split__mult__neg__le,axiom,
    ! [A: real,B2: real] :
      ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
          & ( ord_less_eq_real @ B2 @ zero_zero_real ) )
        | ( ( ord_less_eq_real @ A @ zero_zero_real )
          & ( ord_less_eq_real @ zero_zero_real @ B2 ) ) )
     => ( ord_less_eq_real @ ( times_times_real @ A @ B2 ) @ zero_zero_real ) ) ).

% split_mult_neg_le
thf(fact_2315_split__mult__neg__le,axiom,
    ! [A: rat,B2: rat] :
      ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
          & ( ord_less_eq_rat @ B2 @ zero_zero_rat ) )
        | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
          & ( ord_less_eq_rat @ zero_zero_rat @ B2 ) ) )
     => ( ord_less_eq_rat @ ( times_times_rat @ A @ B2 ) @ zero_zero_rat ) ) ).

% split_mult_neg_le
thf(fact_2316_split__mult__neg__le,axiom,
    ! [A: nat,B2: nat] :
      ( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A )
          & ( ord_less_eq_nat @ B2 @ zero_zero_nat ) )
        | ( ( ord_less_eq_nat @ A @ zero_zero_nat )
          & ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) )
     => ( ord_less_eq_nat @ ( times_times_nat @ A @ B2 ) @ zero_zero_nat ) ) ).

% split_mult_neg_le
thf(fact_2317_split__mult__neg__le,axiom,
    ! [A: int,B2: int] :
      ( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
          & ( ord_less_eq_int @ B2 @ zero_zero_int ) )
        | ( ( ord_less_eq_int @ A @ zero_zero_int )
          & ( ord_less_eq_int @ zero_zero_int @ B2 ) ) )
     => ( ord_less_eq_int @ ( times_times_int @ A @ B2 ) @ zero_zero_int ) ) ).

% split_mult_neg_le
thf(fact_2318_mult__le__0__iff,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ A @ B2 ) @ zero_zero_real )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
          & ( ord_less_eq_real @ B2 @ zero_zero_real ) )
        | ( ( ord_less_eq_real @ A @ zero_zero_real )
          & ( ord_less_eq_real @ zero_zero_real @ B2 ) ) ) ) ).

% mult_le_0_iff
thf(fact_2319_mult__le__0__iff,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ A @ B2 ) @ zero_zero_rat )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
          & ( ord_less_eq_rat @ B2 @ zero_zero_rat ) )
        | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
          & ( ord_less_eq_rat @ zero_zero_rat @ B2 ) ) ) ) ).

% mult_le_0_iff
thf(fact_2320_mult__le__0__iff,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ A @ B2 ) @ zero_zero_int )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ A )
          & ( ord_less_eq_int @ B2 @ zero_zero_int ) )
        | ( ( ord_less_eq_int @ A @ zero_zero_int )
          & ( ord_less_eq_int @ zero_zero_int @ B2 ) ) ) ) ).

% mult_le_0_iff
thf(fact_2321_mult__right__mono,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ord_less_eq_real @ A @ B2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) ) ) ) ).

% mult_right_mono
thf(fact_2322_mult__right__mono,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B2 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ C ) ) ) ) ).

% mult_right_mono
thf(fact_2323_mult__right__mono,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
       => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ C ) ) ) ) ).

% mult_right_mono
thf(fact_2324_mult__right__mono,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( ord_less_eq_int @ A @ B2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ C )
       => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) ) ) ) ).

% mult_right_mono
thf(fact_2325_mult__right__mono__neg,axiom,
    ! [B2: real,A: real,C: real] :
      ( ( ord_less_eq_real @ B2 @ A )
     => ( ( ord_less_eq_real @ C @ zero_zero_real )
       => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) ) ) ) ).

% mult_right_mono_neg
thf(fact_2326_mult__right__mono__neg,axiom,
    ! [B2: rat,A: rat,C: rat] :
      ( ( ord_less_eq_rat @ B2 @ A )
     => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ C ) ) ) ) ).

% mult_right_mono_neg
thf(fact_2327_mult__right__mono__neg,axiom,
    ! [B2: int,A: int,C: int] :
      ( ( ord_less_eq_int @ B2 @ A )
     => ( ( ord_less_eq_int @ C @ zero_zero_int )
       => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) ) ) ) ).

% mult_right_mono_neg
thf(fact_2328_mult__left__mono,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ord_less_eq_real @ A @ B2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) ) ) ) ).

% mult_left_mono
thf(fact_2329_mult__left__mono,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B2 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) ) ) ) ).

% mult_left_mono
thf(fact_2330_mult__left__mono,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
       => ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B2 ) ) ) ) ).

% mult_left_mono
thf(fact_2331_mult__left__mono,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( ord_less_eq_int @ A @ B2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ C )
       => ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) ) ) ) ).

% mult_left_mono
thf(fact_2332_mult__nonpos__nonpos,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ A @ zero_zero_real )
     => ( ( ord_less_eq_real @ B2 @ zero_zero_real )
       => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) ) ) ) ).

% mult_nonpos_nonpos
thf(fact_2333_mult__nonpos__nonpos,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_eq_rat @ A @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ B2 @ zero_zero_rat )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B2 ) ) ) ) ).

% mult_nonpos_nonpos
thf(fact_2334_mult__nonpos__nonpos,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( ord_less_eq_int @ B2 @ zero_zero_int )
       => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B2 ) ) ) ) ).

% mult_nonpos_nonpos
thf(fact_2335_mult__left__mono__neg,axiom,
    ! [B2: real,A: real,C: real] :
      ( ( ord_less_eq_real @ B2 @ A )
     => ( ( ord_less_eq_real @ C @ zero_zero_real )
       => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) ) ) ) ).

% mult_left_mono_neg
thf(fact_2336_mult__left__mono__neg,axiom,
    ! [B2: rat,A: rat,C: rat] :
      ( ( ord_less_eq_rat @ B2 @ A )
     => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) ) ) ) ).

% mult_left_mono_neg
thf(fact_2337_mult__left__mono__neg,axiom,
    ! [B2: int,A: int,C: int] :
      ( ( ord_less_eq_int @ B2 @ A )
     => ( ( ord_less_eq_int @ C @ zero_zero_int )
       => ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) ) ) ) ).

% mult_left_mono_neg
thf(fact_2338_split__mult__pos__le,axiom,
    ! [A: real,B2: real] :
      ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
          & ( ord_less_eq_real @ zero_zero_real @ B2 ) )
        | ( ( ord_less_eq_real @ A @ zero_zero_real )
          & ( ord_less_eq_real @ B2 @ zero_zero_real ) ) )
     => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) ) ) ).

% split_mult_pos_le
thf(fact_2339_split__mult__pos__le,axiom,
    ! [A: rat,B2: rat] :
      ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
          & ( ord_less_eq_rat @ zero_zero_rat @ B2 ) )
        | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
          & ( ord_less_eq_rat @ B2 @ zero_zero_rat ) ) )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B2 ) ) ) ).

% split_mult_pos_le
thf(fact_2340_split__mult__pos__le,axiom,
    ! [A: int,B2: int] :
      ( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
          & ( ord_less_eq_int @ zero_zero_int @ B2 ) )
        | ( ( ord_less_eq_int @ A @ zero_zero_int )
          & ( ord_less_eq_int @ B2 @ zero_zero_int ) ) )
     => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B2 ) ) ) ).

% split_mult_pos_le
thf(fact_2341_zero__le__square,axiom,
    ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ A ) ) ).

% zero_le_square
thf(fact_2342_zero__le__square,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ A ) ) ).

% zero_le_square
thf(fact_2343_zero__le__square,axiom,
    ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ A ) ) ).

% zero_le_square
thf(fact_2344_mult__mono_H,axiom,
    ! [A: real,B2: real,C: real,D: real] :
      ( ( ord_less_eq_real @ A @ B2 )
     => ( ( ord_less_eq_real @ C @ D )
       => ( ( ord_less_eq_real @ zero_zero_real @ A )
         => ( ( ord_less_eq_real @ zero_zero_real @ C )
           => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ D ) ) ) ) ) ) ).

% mult_mono'
thf(fact_2345_mult__mono_H,axiom,
    ! [A: rat,B2: rat,C: rat,D: rat] :
      ( ( ord_less_eq_rat @ A @ B2 )
     => ( ( ord_less_eq_rat @ C @ D )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
           => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ D ) ) ) ) ) ) ).

% mult_mono'
thf(fact_2346_mult__mono_H,axiom,
    ! [A: nat,B2: nat,C: nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( ord_less_eq_nat @ C @ D )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
           => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ) ).

% mult_mono'
thf(fact_2347_mult__mono_H,axiom,
    ! [A: int,B2: int,C: int,D: int] :
      ( ( ord_less_eq_int @ A @ B2 )
     => ( ( ord_less_eq_int @ C @ D )
       => ( ( ord_less_eq_int @ zero_zero_int @ A )
         => ( ( ord_less_eq_int @ zero_zero_int @ C )
           => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ) ).

% mult_mono'
thf(fact_2348_mult__mono,axiom,
    ! [A: real,B2: real,C: real,D: real] :
      ( ( ord_less_eq_real @ A @ B2 )
     => ( ( ord_less_eq_real @ C @ D )
       => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
         => ( ( ord_less_eq_real @ zero_zero_real @ C )
           => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ D ) ) ) ) ) ) ).

% mult_mono
thf(fact_2349_mult__mono,axiom,
    ! [A: rat,B2: rat,C: rat,D: rat] :
      ( ( ord_less_eq_rat @ A @ B2 )
     => ( ( ord_less_eq_rat @ C @ D )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
           => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ D ) ) ) ) ) ) ).

% mult_mono
thf(fact_2350_mult__mono,axiom,
    ! [A: nat,B2: nat,C: nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( ord_less_eq_nat @ C @ D )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
           => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ) ).

% mult_mono
thf(fact_2351_mult__mono,axiom,
    ! [A: int,B2: int,C: int,D: int] :
      ( ( ord_less_eq_int @ A @ B2 )
     => ( ( ord_less_eq_int @ C @ D )
       => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
         => ( ( ord_less_eq_int @ zero_zero_int @ C )
           => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ) ).

% mult_mono
thf(fact_2352_not__numeral__less__zero,axiom,
    ! [N: num] :
      ~ ( ord_less_real @ ( numeral_numeral_real @ N ) @ zero_zero_real ) ).

% not_numeral_less_zero
thf(fact_2353_not__numeral__less__zero,axiom,
    ! [N: num] :
      ~ ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ zero_zero_rat ) ).

% not_numeral_less_zero
thf(fact_2354_not__numeral__less__zero,axiom,
    ! [N: num] :
      ~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).

% not_numeral_less_zero
thf(fact_2355_not__numeral__less__zero,axiom,
    ! [N: num] :
      ~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).

% not_numeral_less_zero
thf(fact_2356_zero__less__numeral,axiom,
    ! [N: num] : ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ N ) ) ).

% zero_less_numeral
thf(fact_2357_zero__less__numeral,axiom,
    ! [N: num] : ( ord_less_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N ) ) ).

% zero_less_numeral
thf(fact_2358_zero__less__numeral,axiom,
    ! [N: num] : ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).

% zero_less_numeral
thf(fact_2359_zero__less__numeral,axiom,
    ! [N: num] : ( ord_less_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).

% zero_less_numeral
thf(fact_2360_zero__less__one__class_Ozero__le__one,axiom,
    ord_less_eq_real @ zero_zero_real @ one_one_real ).

% zero_less_one_class.zero_le_one
thf(fact_2361_zero__less__one__class_Ozero__le__one,axiom,
    ord_less_eq_rat @ zero_zero_rat @ one_one_rat ).

% zero_less_one_class.zero_le_one
thf(fact_2362_zero__less__one__class_Ozero__le__one,axiom,
    ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).

% zero_less_one_class.zero_le_one
thf(fact_2363_zero__less__one__class_Ozero__le__one,axiom,
    ord_less_eq_int @ zero_zero_int @ one_one_int ).

% zero_less_one_class.zero_le_one
thf(fact_2364_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    ord_less_eq_real @ zero_zero_real @ one_one_real ).

% linordered_nonzero_semiring_class.zero_le_one
thf(fact_2365_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    ord_less_eq_rat @ zero_zero_rat @ one_one_rat ).

% linordered_nonzero_semiring_class.zero_le_one
thf(fact_2366_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).

% linordered_nonzero_semiring_class.zero_le_one
thf(fact_2367_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    ord_less_eq_int @ zero_zero_int @ one_one_int ).

% linordered_nonzero_semiring_class.zero_le_one
thf(fact_2368_not__one__le__zero,axiom,
    ~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).

% not_one_le_zero
thf(fact_2369_not__one__le__zero,axiom,
    ~ ( ord_less_eq_rat @ one_one_rat @ zero_zero_rat ) ).

% not_one_le_zero
thf(fact_2370_not__one__le__zero,axiom,
    ~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).

% not_one_le_zero
thf(fact_2371_not__one__le__zero,axiom,
    ~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).

% not_one_le_zero
thf(fact_2372_add__decreasing,axiom,
    ! [A: real,C: real,B2: real] :
      ( ( ord_less_eq_real @ A @ zero_zero_real )
     => ( ( ord_less_eq_real @ C @ B2 )
       => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B2 ) ) ) ).

% add_decreasing
thf(fact_2373_add__decreasing,axiom,
    ! [A: rat,C: rat,B2: rat] :
      ( ( ord_less_eq_rat @ A @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ C @ B2 )
       => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ B2 ) ) ) ).

% add_decreasing
thf(fact_2374_add__decreasing,axiom,
    ! [A: nat,C: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ C @ B2 )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B2 ) ) ) ).

% add_decreasing
thf(fact_2375_add__decreasing,axiom,
    ! [A: int,C: int,B2: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( ord_less_eq_int @ C @ B2 )
       => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B2 ) ) ) ).

% add_decreasing
thf(fact_2376_add__increasing,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ B2 @ C )
       => ( ord_less_eq_real @ B2 @ ( plus_plus_real @ A @ C ) ) ) ) ).

% add_increasing
thf(fact_2377_add__increasing,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ B2 @ C )
       => ( ord_less_eq_rat @ B2 @ ( plus_plus_rat @ A @ C ) ) ) ) ).

% add_increasing
thf(fact_2378_add__increasing,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ord_less_eq_nat @ B2 @ ( plus_plus_nat @ A @ C ) ) ) ) ).

% add_increasing
thf(fact_2379_add__increasing,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ B2 @ C )
       => ( ord_less_eq_int @ B2 @ ( plus_plus_int @ A @ C ) ) ) ) ).

% add_increasing
thf(fact_2380_add__decreasing2,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( ord_less_eq_real @ C @ zero_zero_real )
     => ( ( ord_less_eq_real @ A @ B2 )
       => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B2 ) ) ) ).

% add_decreasing2
thf(fact_2381_add__decreasing2,axiom,
    ! [C: rat,A: rat,B2: rat] :
      ( ( ord_less_eq_rat @ C @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ A @ B2 )
       => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ B2 ) ) ) ).

% add_decreasing2
thf(fact_2382_add__decreasing2,axiom,
    ! [C: nat,A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ C @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ A @ B2 )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B2 ) ) ) ).

% add_decreasing2
thf(fact_2383_add__decreasing2,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( ord_less_eq_int @ C @ zero_zero_int )
     => ( ( ord_less_eq_int @ A @ B2 )
       => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B2 ) ) ) ).

% add_decreasing2
thf(fact_2384_add__increasing2,axiom,
    ! [C: real,B2: real,A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ C )
     => ( ( ord_less_eq_real @ B2 @ A )
       => ( ord_less_eq_real @ B2 @ ( plus_plus_real @ A @ C ) ) ) ) ).

% add_increasing2
thf(fact_2385_add__increasing2,axiom,
    ! [C: rat,B2: rat,A: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ C )
     => ( ( ord_less_eq_rat @ B2 @ A )
       => ( ord_less_eq_rat @ B2 @ ( plus_plus_rat @ A @ C ) ) ) ) ).

% add_increasing2
thf(fact_2386_add__increasing2,axiom,
    ! [C: nat,B2: nat,A: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ C )
     => ( ( ord_less_eq_nat @ B2 @ A )
       => ( ord_less_eq_nat @ B2 @ ( plus_plus_nat @ A @ C ) ) ) ) ).

% add_increasing2
thf(fact_2387_add__increasing2,axiom,
    ! [C: int,B2: int,A: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ C )
     => ( ( ord_less_eq_int @ B2 @ A )
       => ( ord_less_eq_int @ B2 @ ( plus_plus_int @ A @ C ) ) ) ) ).

% add_increasing2
thf(fact_2388_add__nonneg__nonneg,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
       => ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ B2 ) ) ) ) ).

% add_nonneg_nonneg
thf(fact_2389_add__nonneg__nonneg,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B2 ) ) ) ) ).

% add_nonneg_nonneg
thf(fact_2390_add__nonneg__nonneg,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
       => ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B2 ) ) ) ) ).

% add_nonneg_nonneg
thf(fact_2391_add__nonneg__nonneg,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
       => ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ B2 ) ) ) ) ).

% add_nonneg_nonneg
thf(fact_2392_add__nonpos__nonpos,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ A @ zero_zero_real )
     => ( ( ord_less_eq_real @ B2 @ zero_zero_real )
       => ( ord_less_eq_real @ ( plus_plus_real @ A @ B2 ) @ zero_zero_real ) ) ) ).

% add_nonpos_nonpos
thf(fact_2393_add__nonpos__nonpos,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_eq_rat @ A @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ B2 @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B2 ) @ zero_zero_rat ) ) ) ).

% add_nonpos_nonpos
thf(fact_2394_add__nonpos__nonpos,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B2 ) @ zero_zero_nat ) ) ) ).

% add_nonpos_nonpos
thf(fact_2395_add__nonpos__nonpos,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( ord_less_eq_int @ B2 @ zero_zero_int )
       => ( ord_less_eq_int @ ( plus_plus_int @ A @ B2 ) @ zero_zero_int ) ) ) ).

% add_nonpos_nonpos
thf(fact_2396_add__nonneg__eq__0__iff,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
       => ( ( ( plus_plus_real @ X4 @ Y3 )
            = zero_zero_real )
          = ( ( X4 = zero_zero_real )
            & ( Y3 = zero_zero_real ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_2397_add__nonneg__eq__0__iff,axiom,
    ! [X4: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X4 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
       => ( ( ( plus_plus_rat @ X4 @ Y3 )
            = zero_zero_rat )
          = ( ( X4 = zero_zero_rat )
            & ( Y3 = zero_zero_rat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_2398_add__nonneg__eq__0__iff,axiom,
    ! [X4: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ X4 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ Y3 )
       => ( ( ( plus_plus_nat @ X4 @ Y3 )
            = zero_zero_nat )
          = ( ( X4 = zero_zero_nat )
            & ( Y3 = zero_zero_nat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_2399_add__nonneg__eq__0__iff,axiom,
    ! [X4: int,Y3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X4 )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
       => ( ( ( plus_plus_int @ X4 @ Y3 )
            = zero_zero_int )
          = ( ( X4 = zero_zero_int )
            & ( Y3 = zero_zero_int ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_2400_add__nonpos__eq__0__iff,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_eq_real @ X4 @ zero_zero_real )
     => ( ( ord_less_eq_real @ Y3 @ zero_zero_real )
       => ( ( ( plus_plus_real @ X4 @ Y3 )
            = zero_zero_real )
          = ( ( X4 = zero_zero_real )
            & ( Y3 = zero_zero_real ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_2401_add__nonpos__eq__0__iff,axiom,
    ! [X4: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ X4 @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ Y3 @ zero_zero_rat )
       => ( ( ( plus_plus_rat @ X4 @ Y3 )
            = zero_zero_rat )
          = ( ( X4 = zero_zero_rat )
            & ( Y3 = zero_zero_rat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_2402_add__nonpos__eq__0__iff,axiom,
    ! [X4: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ X4 @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ Y3 @ zero_zero_nat )
       => ( ( ( plus_plus_nat @ X4 @ Y3 )
            = zero_zero_nat )
          = ( ( X4 = zero_zero_nat )
            & ( Y3 = zero_zero_nat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_2403_add__nonpos__eq__0__iff,axiom,
    ! [X4: int,Y3: int] :
      ( ( ord_less_eq_int @ X4 @ zero_zero_int )
     => ( ( ord_less_eq_int @ Y3 @ zero_zero_int )
       => ( ( ( plus_plus_int @ X4 @ Y3 )
            = zero_zero_int )
          = ( ( X4 = zero_zero_int )
            & ( Y3 = zero_zero_int ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_2404_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ( ord_less_real @ zero_zero_real @ C )
       => ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_2405_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( ord_less_rat @ A @ B2 )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_2406_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( ( ord_less_nat @ zero_zero_nat @ C )
       => ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B2 ) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_2407_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( ord_less_int @ A @ B2 )
     => ( ( ord_less_int @ zero_zero_int @ C )
       => ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_2408_mult__less__cancel__right__disj,axiom,
    ! [A: real,C: real,B2: real] :
      ( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
          & ( ord_less_real @ A @ B2 ) )
        | ( ( ord_less_real @ C @ zero_zero_real )
          & ( ord_less_real @ B2 @ A ) ) ) ) ).

% mult_less_cancel_right_disj
thf(fact_2409_mult__less__cancel__right__disj,axiom,
    ! [A: rat,C: rat,B2: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
          & ( ord_less_rat @ A @ B2 ) )
        | ( ( ord_less_rat @ C @ zero_zero_rat )
          & ( ord_less_rat @ B2 @ A ) ) ) ) ).

% mult_less_cancel_right_disj
thf(fact_2410_mult__less__cancel__right__disj,axiom,
    ! [A: int,C: int,B2: int] :
      ( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
          & ( ord_less_int @ A @ B2 ) )
        | ( ( ord_less_int @ C @ zero_zero_int )
          & ( ord_less_int @ B2 @ A ) ) ) ) ).

% mult_less_cancel_right_disj
thf(fact_2411_mult__strict__right__mono,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ( ord_less_real @ zero_zero_real @ C )
       => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) ) ) ) ).

% mult_strict_right_mono
thf(fact_2412_mult__strict__right__mono,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( ord_less_rat @ A @ B2 )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ C ) ) ) ) ).

% mult_strict_right_mono
thf(fact_2413_mult__strict__right__mono,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( ( ord_less_nat @ zero_zero_nat @ C )
       => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ C ) ) ) ) ).

% mult_strict_right_mono
thf(fact_2414_mult__strict__right__mono,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( ord_less_int @ A @ B2 )
     => ( ( ord_less_int @ zero_zero_int @ C )
       => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) ) ) ) ).

% mult_strict_right_mono
thf(fact_2415_mult__strict__right__mono__neg,axiom,
    ! [B2: real,A: real,C: real] :
      ( ( ord_less_real @ B2 @ A )
     => ( ( ord_less_real @ C @ zero_zero_real )
       => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) ) ) ) ).

% mult_strict_right_mono_neg
thf(fact_2416_mult__strict__right__mono__neg,axiom,
    ! [B2: rat,A: rat,C: rat] :
      ( ( ord_less_rat @ B2 @ A )
     => ( ( ord_less_rat @ C @ zero_zero_rat )
       => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ C ) ) ) ) ).

% mult_strict_right_mono_neg
thf(fact_2417_mult__strict__right__mono__neg,axiom,
    ! [B2: int,A: int,C: int] :
      ( ( ord_less_int @ B2 @ A )
     => ( ( ord_less_int @ C @ zero_zero_int )
       => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) ) ) ) ).

% mult_strict_right_mono_neg
thf(fact_2418_mult__less__cancel__left__disj,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
          & ( ord_less_real @ A @ B2 ) )
        | ( ( ord_less_real @ C @ zero_zero_real )
          & ( ord_less_real @ B2 @ A ) ) ) ) ).

% mult_less_cancel_left_disj
thf(fact_2419_mult__less__cancel__left__disj,axiom,
    ! [C: rat,A: rat,B2: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
          & ( ord_less_rat @ A @ B2 ) )
        | ( ( ord_less_rat @ C @ zero_zero_rat )
          & ( ord_less_rat @ B2 @ A ) ) ) ) ).

% mult_less_cancel_left_disj
thf(fact_2420_mult__less__cancel__left__disj,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
          & ( ord_less_int @ A @ B2 ) )
        | ( ( ord_less_int @ C @ zero_zero_int )
          & ( ord_less_int @ B2 @ A ) ) ) ) ).

% mult_less_cancel_left_disj
thf(fact_2421_mult__strict__left__mono,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ( ord_less_real @ zero_zero_real @ C )
       => ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) ) ) ) ).

% mult_strict_left_mono
thf(fact_2422_mult__strict__left__mono,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( ord_less_rat @ A @ B2 )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) ) ) ) ).

% mult_strict_left_mono
thf(fact_2423_mult__strict__left__mono,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( ( ord_less_nat @ zero_zero_nat @ C )
       => ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B2 ) ) ) ) ).

% mult_strict_left_mono
thf(fact_2424_mult__strict__left__mono,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( ord_less_int @ A @ B2 )
     => ( ( ord_less_int @ zero_zero_int @ C )
       => ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) ) ) ) ).

% mult_strict_left_mono
thf(fact_2425_mult__strict__left__mono__neg,axiom,
    ! [B2: real,A: real,C: real] :
      ( ( ord_less_real @ B2 @ A )
     => ( ( ord_less_real @ C @ zero_zero_real )
       => ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) ) ) ) ).

% mult_strict_left_mono_neg
thf(fact_2426_mult__strict__left__mono__neg,axiom,
    ! [B2: rat,A: rat,C: rat] :
      ( ( ord_less_rat @ B2 @ A )
     => ( ( ord_less_rat @ C @ zero_zero_rat )
       => ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) ) ) ) ).

% mult_strict_left_mono_neg
thf(fact_2427_mult__strict__left__mono__neg,axiom,
    ! [B2: int,A: int,C: int] :
      ( ( ord_less_int @ B2 @ A )
     => ( ( ord_less_int @ C @ zero_zero_int )
       => ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) ) ) ) ).

% mult_strict_left_mono_neg
thf(fact_2428_mult__less__cancel__left__pos,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
        = ( ord_less_real @ A @ B2 ) ) ) ).

% mult_less_cancel_left_pos
thf(fact_2429_mult__less__cancel__left__pos,axiom,
    ! [C: rat,A: rat,B2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) )
        = ( ord_less_rat @ A @ B2 ) ) ) ).

% mult_less_cancel_left_pos
thf(fact_2430_mult__less__cancel__left__pos,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( ord_less_int @ zero_zero_int @ C )
     => ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
        = ( ord_less_int @ A @ B2 ) ) ) ).

% mult_less_cancel_left_pos
thf(fact_2431_mult__less__cancel__left__neg,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
        = ( ord_less_real @ B2 @ A ) ) ) ).

% mult_less_cancel_left_neg
thf(fact_2432_mult__less__cancel__left__neg,axiom,
    ! [C: rat,A: rat,B2: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) )
        = ( ord_less_rat @ B2 @ A ) ) ) ).

% mult_less_cancel_left_neg
thf(fact_2433_mult__less__cancel__left__neg,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( ord_less_int @ C @ zero_zero_int )
     => ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
        = ( ord_less_int @ B2 @ A ) ) ) ).

% mult_less_cancel_left_neg
thf(fact_2434_zero__less__mult__pos2,axiom,
    ! [B2: real,A: real] :
      ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ B2 @ A ) )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ( ord_less_real @ zero_zero_real @ B2 ) ) ) ).

% zero_less_mult_pos2
thf(fact_2435_zero__less__mult__pos2,axiom,
    ! [B2: rat,A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ B2 @ A ) )
     => ( ( ord_less_rat @ zero_zero_rat @ A )
       => ( ord_less_rat @ zero_zero_rat @ B2 ) ) ) ).

% zero_less_mult_pos2
thf(fact_2436_zero__less__mult__pos2,axiom,
    ! [B2: nat,A: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B2 @ A ) )
     => ( ( ord_less_nat @ zero_zero_nat @ A )
       => ( ord_less_nat @ zero_zero_nat @ B2 ) ) ) ).

% zero_less_mult_pos2
thf(fact_2437_zero__less__mult__pos2,axiom,
    ! [B2: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B2 @ A ) )
     => ( ( ord_less_int @ zero_zero_int @ A )
       => ( ord_less_int @ zero_zero_int @ B2 ) ) ) ).

% zero_less_mult_pos2
thf(fact_2438_zero__less__mult__pos,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ( ord_less_real @ zero_zero_real @ B2 ) ) ) ).

% zero_less_mult_pos
thf(fact_2439_zero__less__mult__pos,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B2 ) )
     => ( ( ord_less_rat @ zero_zero_rat @ A )
       => ( ord_less_rat @ zero_zero_rat @ B2 ) ) ) ).

% zero_less_mult_pos
thf(fact_2440_zero__less__mult__pos,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B2 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ A )
       => ( ord_less_nat @ zero_zero_nat @ B2 ) ) ) ).

% zero_less_mult_pos
thf(fact_2441_zero__less__mult__pos,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B2 ) )
     => ( ( ord_less_int @ zero_zero_int @ A )
       => ( ord_less_int @ zero_zero_int @ B2 ) ) ) ).

% zero_less_mult_pos
thf(fact_2442_zero__less__mult__iff,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) )
      = ( ( ( ord_less_real @ zero_zero_real @ A )
          & ( ord_less_real @ zero_zero_real @ B2 ) )
        | ( ( ord_less_real @ A @ zero_zero_real )
          & ( ord_less_real @ B2 @ zero_zero_real ) ) ) ) ).

% zero_less_mult_iff
thf(fact_2443_zero__less__mult__iff,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B2 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A )
          & ( ord_less_rat @ zero_zero_rat @ B2 ) )
        | ( ( ord_less_rat @ A @ zero_zero_rat )
          & ( ord_less_rat @ B2 @ zero_zero_rat ) ) ) ) ).

% zero_less_mult_iff
thf(fact_2444_zero__less__mult__iff,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B2 ) )
      = ( ( ( ord_less_int @ zero_zero_int @ A )
          & ( ord_less_int @ zero_zero_int @ B2 ) )
        | ( ( ord_less_int @ A @ zero_zero_int )
          & ( ord_less_int @ B2 @ zero_zero_int ) ) ) ) ).

% zero_less_mult_iff
thf(fact_2445_mult__pos__neg2,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ B2 @ zero_zero_real )
       => ( ord_less_real @ ( times_times_real @ B2 @ A ) @ zero_zero_real ) ) ) ).

% mult_pos_neg2
thf(fact_2446_mult__pos__neg2,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ B2 @ zero_zero_rat )
       => ( ord_less_rat @ ( times_times_rat @ B2 @ A ) @ zero_zero_rat ) ) ) ).

% mult_pos_neg2
thf(fact_2447_mult__pos__neg2,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ B2 @ zero_zero_nat )
       => ( ord_less_nat @ ( times_times_nat @ B2 @ A ) @ zero_zero_nat ) ) ) ).

% mult_pos_neg2
thf(fact_2448_mult__pos__neg2,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ B2 @ zero_zero_int )
       => ( ord_less_int @ ( times_times_int @ B2 @ A ) @ zero_zero_int ) ) ) ).

% mult_pos_neg2
thf(fact_2449_mult__pos__pos,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ B2 )
       => ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) ) ) ) ).

% mult_pos_pos
thf(fact_2450_mult__pos__pos,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ zero_zero_rat @ B2 )
       => ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B2 ) ) ) ) ).

% mult_pos_pos
thf(fact_2451_mult__pos__pos,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ B2 )
       => ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B2 ) ) ) ) ).

% mult_pos_pos
thf(fact_2452_mult__pos__pos,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ zero_zero_int @ B2 )
       => ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B2 ) ) ) ) ).

% mult_pos_pos
thf(fact_2453_mult__pos__neg,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ B2 @ zero_zero_real )
       => ( ord_less_real @ ( times_times_real @ A @ B2 ) @ zero_zero_real ) ) ) ).

% mult_pos_neg
thf(fact_2454_mult__pos__neg,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ B2 @ zero_zero_rat )
       => ( ord_less_rat @ ( times_times_rat @ A @ B2 ) @ zero_zero_rat ) ) ) ).

% mult_pos_neg
thf(fact_2455_mult__pos__neg,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ B2 @ zero_zero_nat )
       => ( ord_less_nat @ ( times_times_nat @ A @ B2 ) @ zero_zero_nat ) ) ) ).

% mult_pos_neg
thf(fact_2456_mult__pos__neg,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ B2 @ zero_zero_int )
       => ( ord_less_int @ ( times_times_int @ A @ B2 ) @ zero_zero_int ) ) ) ).

% mult_pos_neg
thf(fact_2457_mult__neg__pos,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( ord_less_real @ zero_zero_real @ B2 )
       => ( ord_less_real @ ( times_times_real @ A @ B2 ) @ zero_zero_real ) ) ) ).

% mult_neg_pos
thf(fact_2458_mult__neg__pos,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( ord_less_rat @ zero_zero_rat @ B2 )
       => ( ord_less_rat @ ( times_times_rat @ A @ B2 ) @ zero_zero_rat ) ) ) ).

% mult_neg_pos
thf(fact_2459_mult__neg__pos,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_nat @ A @ zero_zero_nat )
     => ( ( ord_less_nat @ zero_zero_nat @ B2 )
       => ( ord_less_nat @ ( times_times_nat @ A @ B2 ) @ zero_zero_nat ) ) ) ).

% mult_neg_pos
thf(fact_2460_mult__neg__pos,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ zero_zero_int @ B2 )
       => ( ord_less_int @ ( times_times_int @ A @ B2 ) @ zero_zero_int ) ) ) ).

% mult_neg_pos
thf(fact_2461_mult__less__0__iff,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ ( times_times_real @ A @ B2 ) @ zero_zero_real )
      = ( ( ( ord_less_real @ zero_zero_real @ A )
          & ( ord_less_real @ B2 @ zero_zero_real ) )
        | ( ( ord_less_real @ A @ zero_zero_real )
          & ( ord_less_real @ zero_zero_real @ B2 ) ) ) ) ).

% mult_less_0_iff
thf(fact_2462_mult__less__0__iff,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ A @ B2 ) @ zero_zero_rat )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A )
          & ( ord_less_rat @ B2 @ zero_zero_rat ) )
        | ( ( ord_less_rat @ A @ zero_zero_rat )
          & ( ord_less_rat @ zero_zero_rat @ B2 ) ) ) ) ).

% mult_less_0_iff
thf(fact_2463_mult__less__0__iff,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_int @ ( times_times_int @ A @ B2 ) @ zero_zero_int )
      = ( ( ( ord_less_int @ zero_zero_int @ A )
          & ( ord_less_int @ B2 @ zero_zero_int ) )
        | ( ( ord_less_int @ A @ zero_zero_int )
          & ( ord_less_int @ zero_zero_int @ B2 ) ) ) ) ).

% mult_less_0_iff
thf(fact_2464_not__square__less__zero,axiom,
    ! [A: real] :
      ~ ( ord_less_real @ ( times_times_real @ A @ A ) @ zero_zero_real ) ).

% not_square_less_zero
thf(fact_2465_not__square__less__zero,axiom,
    ! [A: rat] :
      ~ ( ord_less_rat @ ( times_times_rat @ A @ A ) @ zero_zero_rat ) ).

% not_square_less_zero
thf(fact_2466_not__square__less__zero,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ ( times_times_int @ A @ A ) @ zero_zero_int ) ).

% not_square_less_zero
thf(fact_2467_mult__neg__neg,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( ord_less_real @ B2 @ zero_zero_real )
       => ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) ) ) ) ).

% mult_neg_neg
thf(fact_2468_mult__neg__neg,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( ord_less_rat @ B2 @ zero_zero_rat )
       => ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B2 ) ) ) ) ).

% mult_neg_neg
thf(fact_2469_mult__neg__neg,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ B2 @ zero_zero_int )
       => ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B2 ) ) ) ) ).

% mult_neg_neg
thf(fact_2470_le__iff__diff__le__0,axiom,
    ( ord_less_eq_real
    = ( ^ [A4: real,B3: real] : ( ord_less_eq_real @ ( minus_minus_real @ A4 @ B3 ) @ zero_zero_real ) ) ) ).

% le_iff_diff_le_0
thf(fact_2471_le__iff__diff__le__0,axiom,
    ( ord_less_eq_rat
    = ( ^ [A4: rat,B3: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ A4 @ B3 ) @ zero_zero_rat ) ) ) ).

% le_iff_diff_le_0
thf(fact_2472_le__iff__diff__le__0,axiom,
    ( ord_less_eq_int
    = ( ^ [A4: int,B3: int] : ( ord_less_eq_int @ ( minus_minus_int @ A4 @ B3 ) @ zero_zero_int ) ) ) ).

% le_iff_diff_le_0
thf(fact_2473_less__numeral__extra_I1_J,axiom,
    ord_less_real @ zero_zero_real @ one_one_real ).

% less_numeral_extra(1)
thf(fact_2474_less__numeral__extra_I1_J,axiom,
    ord_less_rat @ zero_zero_rat @ one_one_rat ).

% less_numeral_extra(1)
thf(fact_2475_less__numeral__extra_I1_J,axiom,
    ord_less_nat @ zero_zero_nat @ one_one_nat ).

% less_numeral_extra(1)
thf(fact_2476_less__numeral__extra_I1_J,axiom,
    ord_less_int @ zero_zero_int @ one_one_int ).

% less_numeral_extra(1)
thf(fact_2477_zero__less__one,axiom,
    ord_less_real @ zero_zero_real @ one_one_real ).

% zero_less_one
thf(fact_2478_zero__less__one,axiom,
    ord_less_rat @ zero_zero_rat @ one_one_rat ).

% zero_less_one
thf(fact_2479_zero__less__one,axiom,
    ord_less_nat @ zero_zero_nat @ one_one_nat ).

% zero_less_one
thf(fact_2480_zero__less__one,axiom,
    ord_less_int @ zero_zero_int @ one_one_int ).

% zero_less_one
thf(fact_2481_not__one__less__zero,axiom,
    ~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).

% not_one_less_zero
thf(fact_2482_not__one__less__zero,axiom,
    ~ ( ord_less_rat @ one_one_rat @ zero_zero_rat ) ).

% not_one_less_zero
thf(fact_2483_not__one__less__zero,axiom,
    ~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).

% not_one_less_zero
thf(fact_2484_not__one__less__zero,axiom,
    ~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).

% not_one_less_zero
thf(fact_2485_add__neg__neg,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( ord_less_real @ B2 @ zero_zero_real )
       => ( ord_less_real @ ( plus_plus_real @ A @ B2 ) @ zero_zero_real ) ) ) ).

% add_neg_neg
thf(fact_2486_add__neg__neg,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( ord_less_rat @ B2 @ zero_zero_rat )
       => ( ord_less_rat @ ( plus_plus_rat @ A @ B2 ) @ zero_zero_rat ) ) ) ).

% add_neg_neg
thf(fact_2487_add__neg__neg,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_nat @ A @ zero_zero_nat )
     => ( ( ord_less_nat @ B2 @ zero_zero_nat )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ B2 ) @ zero_zero_nat ) ) ) ).

% add_neg_neg
thf(fact_2488_add__neg__neg,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ B2 @ zero_zero_int )
       => ( ord_less_int @ ( plus_plus_int @ A @ B2 ) @ zero_zero_int ) ) ) ).

% add_neg_neg
thf(fact_2489_add__pos__pos,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ B2 )
       => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B2 ) ) ) ) ).

% add_pos_pos
thf(fact_2490_add__pos__pos,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ zero_zero_rat @ B2 )
       => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B2 ) ) ) ) ).

% add_pos_pos
thf(fact_2491_add__pos__pos,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ B2 )
       => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B2 ) ) ) ) ).

% add_pos_pos
thf(fact_2492_add__pos__pos,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ zero_zero_int @ B2 )
       => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B2 ) ) ) ) ).

% add_pos_pos
thf(fact_2493_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ~ ! [C3: nat] :
            ( ( B2
              = ( plus_plus_nat @ A @ C3 ) )
           => ( C3 = zero_zero_nat ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
thf(fact_2494_pos__add__strict,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ B2 @ C )
       => ( ord_less_real @ B2 @ ( plus_plus_real @ A @ C ) ) ) ) ).

% pos_add_strict
thf(fact_2495_pos__add__strict,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ B2 @ C )
       => ( ord_less_rat @ B2 @ ( plus_plus_rat @ A @ C ) ) ) ) ).

% pos_add_strict
thf(fact_2496_pos__add__strict,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ B2 @ C )
       => ( ord_less_nat @ B2 @ ( plus_plus_nat @ A @ C ) ) ) ) ).

% pos_add_strict
thf(fact_2497_pos__add__strict,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ B2 @ C )
       => ( ord_less_int @ B2 @ ( plus_plus_int @ A @ C ) ) ) ) ).

% pos_add_strict
thf(fact_2498_add__less__zeroD,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_real @ ( plus_plus_real @ X4 @ Y3 ) @ zero_zero_real )
     => ( ( ord_less_real @ X4 @ zero_zero_real )
        | ( ord_less_real @ Y3 @ zero_zero_real ) ) ) ).

% add_less_zeroD
thf(fact_2499_add__less__zeroD,axiom,
    ! [X4: rat,Y3: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ X4 @ Y3 ) @ zero_zero_rat )
     => ( ( ord_less_rat @ X4 @ zero_zero_rat )
        | ( ord_less_rat @ Y3 @ zero_zero_rat ) ) ) ).

% add_less_zeroD
thf(fact_2500_add__less__zeroD,axiom,
    ! [X4: int,Y3: int] :
      ( ( ord_less_int @ ( plus_plus_int @ X4 @ Y3 ) @ zero_zero_int )
     => ( ( ord_less_int @ X4 @ zero_zero_int )
        | ( ord_less_int @ Y3 @ zero_zero_int ) ) ) ).

% add_less_zeroD
thf(fact_2501_divide__le__0__iff,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ A @ B2 ) @ zero_zero_real )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
          & ( ord_less_eq_real @ B2 @ zero_zero_real ) )
        | ( ( ord_less_eq_real @ A @ zero_zero_real )
          & ( ord_less_eq_real @ zero_zero_real @ B2 ) ) ) ) ).

% divide_le_0_iff
thf(fact_2502_divide__le__0__iff,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ B2 ) @ zero_zero_rat )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
          & ( ord_less_eq_rat @ B2 @ zero_zero_rat ) )
        | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
          & ( ord_less_eq_rat @ zero_zero_rat @ B2 ) ) ) ) ).

% divide_le_0_iff
thf(fact_2503_divide__right__mono,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ord_less_eq_real @ A @ B2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ord_less_eq_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B2 @ C ) ) ) ) ).

% divide_right_mono
thf(fact_2504_divide__right__mono,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B2 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B2 @ C ) ) ) ) ).

% divide_right_mono
thf(fact_2505_zero__le__divide__iff,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A @ B2 ) )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
          & ( ord_less_eq_real @ zero_zero_real @ B2 ) )
        | ( ( ord_less_eq_real @ A @ zero_zero_real )
          & ( ord_less_eq_real @ B2 @ zero_zero_real ) ) ) ) ).

% zero_le_divide_iff
thf(fact_2506_zero__le__divide__iff,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ B2 ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
          & ( ord_less_eq_rat @ zero_zero_rat @ B2 ) )
        | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
          & ( ord_less_eq_rat @ B2 @ zero_zero_rat ) ) ) ) ).

% zero_le_divide_iff
thf(fact_2507_divide__nonneg__nonneg,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
       => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X4 @ Y3 ) ) ) ) ).

% divide_nonneg_nonneg
thf(fact_2508_divide__nonneg__nonneg,axiom,
    ! [X4: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X4 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X4 @ Y3 ) ) ) ) ).

% divide_nonneg_nonneg
thf(fact_2509_divide__nonneg__nonpos,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_eq_real @ Y3 @ zero_zero_real )
       => ( ord_less_eq_real @ ( divide_divide_real @ X4 @ Y3 ) @ zero_zero_real ) ) ) ).

% divide_nonneg_nonpos
thf(fact_2510_divide__nonneg__nonpos,axiom,
    ! [X4: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X4 )
     => ( ( ord_less_eq_rat @ Y3 @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ X4 @ Y3 ) @ zero_zero_rat ) ) ) ).

% divide_nonneg_nonpos
thf(fact_2511_divide__nonpos__nonneg,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_eq_real @ X4 @ zero_zero_real )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
       => ( ord_less_eq_real @ ( divide_divide_real @ X4 @ Y3 ) @ zero_zero_real ) ) ) ).

% divide_nonpos_nonneg
thf(fact_2512_divide__nonpos__nonneg,axiom,
    ! [X4: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ X4 @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ X4 @ Y3 ) @ zero_zero_rat ) ) ) ).

% divide_nonpos_nonneg
thf(fact_2513_divide__nonpos__nonpos,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_eq_real @ X4 @ zero_zero_real )
     => ( ( ord_less_eq_real @ Y3 @ zero_zero_real )
       => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X4 @ Y3 ) ) ) ) ).

% divide_nonpos_nonpos
thf(fact_2514_divide__nonpos__nonpos,axiom,
    ! [X4: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ X4 @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ Y3 @ zero_zero_rat )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X4 @ Y3 ) ) ) ) ).

% divide_nonpos_nonpos
thf(fact_2515_divide__right__mono__neg,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ord_less_eq_real @ A @ B2 )
     => ( ( ord_less_eq_real @ C @ zero_zero_real )
       => ( ord_less_eq_real @ ( divide_divide_real @ B2 @ C ) @ ( divide_divide_real @ A @ C ) ) ) ) ).

% divide_right_mono_neg
thf(fact_2516_divide__right__mono__neg,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B2 )
     => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ B2 @ C ) @ ( divide_divide_rat @ A @ C ) ) ) ) ).

% divide_right_mono_neg
thf(fact_2517_less__iff__diff__less__0,axiom,
    ( ord_less_real
    = ( ^ [A4: real,B3: real] : ( ord_less_real @ ( minus_minus_real @ A4 @ B3 ) @ zero_zero_real ) ) ) ).

% less_iff_diff_less_0
thf(fact_2518_less__iff__diff__less__0,axiom,
    ( ord_less_rat
    = ( ^ [A4: rat,B3: rat] : ( ord_less_rat @ ( minus_minus_rat @ A4 @ B3 ) @ zero_zero_rat ) ) ) ).

% less_iff_diff_less_0
thf(fact_2519_less__iff__diff__less__0,axiom,
    ( ord_less_int
    = ( ^ [A4: int,B3: int] : ( ord_less_int @ ( minus_minus_int @ A4 @ B3 ) @ zero_zero_int ) ) ) ).

% less_iff_diff_less_0
thf(fact_2520_divide__strict__right__mono__neg,axiom,
    ! [B2: real,A: real,C: real] :
      ( ( ord_less_real @ B2 @ A )
     => ( ( ord_less_real @ C @ zero_zero_real )
       => ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B2 @ C ) ) ) ) ).

% divide_strict_right_mono_neg
thf(fact_2521_divide__strict__right__mono__neg,axiom,
    ! [B2: rat,A: rat,C: rat] :
      ( ( ord_less_rat @ B2 @ A )
     => ( ( ord_less_rat @ C @ zero_zero_rat )
       => ( ord_less_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B2 @ C ) ) ) ) ).

% divide_strict_right_mono_neg
thf(fact_2522_divide__strict__right__mono,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ( ord_less_real @ zero_zero_real @ C )
       => ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B2 @ C ) ) ) ) ).

% divide_strict_right_mono
thf(fact_2523_divide__strict__right__mono,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( ord_less_rat @ A @ B2 )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ord_less_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B2 @ C ) ) ) ) ).

% divide_strict_right_mono
thf(fact_2524_zero__less__divide__iff,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ B2 ) )
      = ( ( ( ord_less_real @ zero_zero_real @ A )
          & ( ord_less_real @ zero_zero_real @ B2 ) )
        | ( ( ord_less_real @ A @ zero_zero_real )
          & ( ord_less_real @ B2 @ zero_zero_real ) ) ) ) ).

% zero_less_divide_iff
thf(fact_2525_zero__less__divide__iff,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ B2 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A )
          & ( ord_less_rat @ zero_zero_rat @ B2 ) )
        | ( ( ord_less_rat @ A @ zero_zero_rat )
          & ( ord_less_rat @ B2 @ zero_zero_rat ) ) ) ) ).

% zero_less_divide_iff
thf(fact_2526_divide__less__cancel,axiom,
    ! [A: real,C: real,B2: real] :
      ( ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B2 @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ A @ B2 ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_real @ B2 @ A ) )
        & ( C != zero_zero_real ) ) ) ).

% divide_less_cancel
thf(fact_2527_divide__less__cancel,axiom,
    ! [A: rat,C: rat,B2: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B2 @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ A @ B2 ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ B2 @ A ) )
        & ( C != zero_zero_rat ) ) ) ).

% divide_less_cancel
thf(fact_2528_divide__less__0__iff,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ ( divide_divide_real @ A @ B2 ) @ zero_zero_real )
      = ( ( ( ord_less_real @ zero_zero_real @ A )
          & ( ord_less_real @ B2 @ zero_zero_real ) )
        | ( ( ord_less_real @ A @ zero_zero_real )
          & ( ord_less_real @ zero_zero_real @ B2 ) ) ) ) ).

% divide_less_0_iff
thf(fact_2529_divide__less__0__iff,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ A @ B2 ) @ zero_zero_rat )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A )
          & ( ord_less_rat @ B2 @ zero_zero_rat ) )
        | ( ( ord_less_rat @ A @ zero_zero_rat )
          & ( ord_less_rat @ zero_zero_rat @ B2 ) ) ) ) ).

% divide_less_0_iff
thf(fact_2530_divide__pos__pos,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( ord_less_real @ zero_zero_real @ Y3 )
       => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X4 @ Y3 ) ) ) ) ).

% divide_pos_pos
thf(fact_2531_divide__pos__pos,axiom,
    ! [X4: rat,Y3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ X4 )
     => ( ( ord_less_rat @ zero_zero_rat @ Y3 )
       => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ X4 @ Y3 ) ) ) ) ).

% divide_pos_pos
thf(fact_2532_divide__pos__neg,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( ord_less_real @ Y3 @ zero_zero_real )
       => ( ord_less_real @ ( divide_divide_real @ X4 @ Y3 ) @ zero_zero_real ) ) ) ).

% divide_pos_neg
thf(fact_2533_divide__pos__neg,axiom,
    ! [X4: rat,Y3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ X4 )
     => ( ( ord_less_rat @ Y3 @ zero_zero_rat )
       => ( ord_less_rat @ ( divide_divide_rat @ X4 @ Y3 ) @ zero_zero_rat ) ) ) ).

% divide_pos_neg
thf(fact_2534_divide__neg__pos,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_real @ X4 @ zero_zero_real )
     => ( ( ord_less_real @ zero_zero_real @ Y3 )
       => ( ord_less_real @ ( divide_divide_real @ X4 @ Y3 ) @ zero_zero_real ) ) ) ).

% divide_neg_pos
thf(fact_2535_divide__neg__pos,axiom,
    ! [X4: rat,Y3: rat] :
      ( ( ord_less_rat @ X4 @ zero_zero_rat )
     => ( ( ord_less_rat @ zero_zero_rat @ Y3 )
       => ( ord_less_rat @ ( divide_divide_rat @ X4 @ Y3 ) @ zero_zero_rat ) ) ) ).

% divide_neg_pos
thf(fact_2536_divide__neg__neg,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_real @ X4 @ zero_zero_real )
     => ( ( ord_less_real @ Y3 @ zero_zero_real )
       => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X4 @ Y3 ) ) ) ) ).

% divide_neg_neg
thf(fact_2537_divide__neg__neg,axiom,
    ! [X4: rat,Y3: rat] :
      ( ( ord_less_rat @ X4 @ zero_zero_rat )
     => ( ( ord_less_rat @ Y3 @ zero_zero_rat )
       => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ X4 @ Y3 ) ) ) ) ).

% divide_neg_neg
thf(fact_2538_zero__le__power,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).

% zero_le_power
thf(fact_2539_zero__le__power,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) ) ) ).

% zero_le_power
thf(fact_2540_zero__le__power,axiom,
    ! [A: nat,N: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).

% zero_le_power
thf(fact_2541_zero__le__power,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).

% zero_le_power
thf(fact_2542_power__mono,axiom,
    ! [A: real,B2: real,N: nat] :
      ( ( ord_less_eq_real @ A @ B2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B2 @ N ) ) ) ) ).

% power_mono
thf(fact_2543_power__mono,axiom,
    ! [A: rat,B2: rat,N: nat] :
      ( ( ord_less_eq_rat @ A @ B2 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
       => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B2 @ N ) ) ) ) ).

% power_mono
thf(fact_2544_power__mono,axiom,
    ! [A: nat,B2: nat,N: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B2 @ N ) ) ) ) ).

% power_mono
thf(fact_2545_power__mono,axiom,
    ! [A: int,B2: int,N: nat] :
      ( ( ord_less_eq_int @ A @ B2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B2 @ N ) ) ) ) ).

% power_mono
thf(fact_2546_zero__less__power,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).

% zero_less_power
thf(fact_2547_zero__less__power,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) ) ) ).

% zero_less_power
thf(fact_2548_zero__less__power,axiom,
    ! [A: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).

% zero_less_power
thf(fact_2549_zero__less__power,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).

% zero_less_power
thf(fact_2550_frac__eq__eq,axiom,
    ! [Y3: complex,Z: complex,X4: complex,W: complex] :
      ( ( Y3 != zero_zero_complex )
     => ( ( Z != zero_zero_complex )
       => ( ( ( divide1717551699836669952omplex @ X4 @ Y3 )
            = ( divide1717551699836669952omplex @ W @ Z ) )
          = ( ( times_times_complex @ X4 @ Z )
            = ( times_times_complex @ W @ Y3 ) ) ) ) ) ).

% frac_eq_eq
thf(fact_2551_frac__eq__eq,axiom,
    ! [Y3: real,Z: real,X4: real,W: real] :
      ( ( Y3 != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( ( ( divide_divide_real @ X4 @ Y3 )
            = ( divide_divide_real @ W @ Z ) )
          = ( ( times_times_real @ X4 @ Z )
            = ( times_times_real @ W @ Y3 ) ) ) ) ) ).

% frac_eq_eq
thf(fact_2552_frac__eq__eq,axiom,
    ! [Y3: rat,Z: rat,X4: rat,W: rat] :
      ( ( Y3 != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( ( ( divide_divide_rat @ X4 @ Y3 )
            = ( divide_divide_rat @ W @ Z ) )
          = ( ( times_times_rat @ X4 @ Z )
            = ( times_times_rat @ W @ Y3 ) ) ) ) ) ).

% frac_eq_eq
thf(fact_2553_divide__eq__eq,axiom,
    ! [B2: complex,C: complex,A: complex] :
      ( ( ( divide1717551699836669952omplex @ B2 @ C )
        = A )
      = ( ( ( C != zero_zero_complex )
         => ( B2
            = ( times_times_complex @ A @ C ) ) )
        & ( ( C = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% divide_eq_eq
thf(fact_2554_divide__eq__eq,axiom,
    ! [B2: real,C: real,A: real] :
      ( ( ( divide_divide_real @ B2 @ C )
        = A )
      = ( ( ( C != zero_zero_real )
         => ( B2
            = ( times_times_real @ A @ C ) ) )
        & ( ( C = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% divide_eq_eq
thf(fact_2555_divide__eq__eq,axiom,
    ! [B2: rat,C: rat,A: rat] :
      ( ( ( divide_divide_rat @ B2 @ C )
        = A )
      = ( ( ( C != zero_zero_rat )
         => ( B2
            = ( times_times_rat @ A @ C ) ) )
        & ( ( C = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% divide_eq_eq
thf(fact_2556_eq__divide__eq,axiom,
    ! [A: complex,B2: complex,C: complex] :
      ( ( A
        = ( divide1717551699836669952omplex @ B2 @ C ) )
      = ( ( ( C != zero_zero_complex )
         => ( ( times_times_complex @ A @ C )
            = B2 ) )
        & ( ( C = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% eq_divide_eq
thf(fact_2557_eq__divide__eq,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( A
        = ( divide_divide_real @ B2 @ C ) )
      = ( ( ( C != zero_zero_real )
         => ( ( times_times_real @ A @ C )
            = B2 ) )
        & ( ( C = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% eq_divide_eq
thf(fact_2558_eq__divide__eq,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( A
        = ( divide_divide_rat @ B2 @ C ) )
      = ( ( ( C != zero_zero_rat )
         => ( ( times_times_rat @ A @ C )
            = B2 ) )
        & ( ( C = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% eq_divide_eq
thf(fact_2559_divide__eq__imp,axiom,
    ! [C: complex,B2: complex,A: complex] :
      ( ( C != zero_zero_complex )
     => ( ( B2
          = ( times_times_complex @ A @ C ) )
       => ( ( divide1717551699836669952omplex @ B2 @ C )
          = A ) ) ) ).

% divide_eq_imp
thf(fact_2560_divide__eq__imp,axiom,
    ! [C: real,B2: real,A: real] :
      ( ( C != zero_zero_real )
     => ( ( B2
          = ( times_times_real @ A @ C ) )
       => ( ( divide_divide_real @ B2 @ C )
          = A ) ) ) ).

% divide_eq_imp
thf(fact_2561_divide__eq__imp,axiom,
    ! [C: rat,B2: rat,A: rat] :
      ( ( C != zero_zero_rat )
     => ( ( B2
          = ( times_times_rat @ A @ C ) )
       => ( ( divide_divide_rat @ B2 @ C )
          = A ) ) ) ).

% divide_eq_imp
thf(fact_2562_eq__divide__imp,axiom,
    ! [C: complex,A: complex,B2: complex] :
      ( ( C != zero_zero_complex )
     => ( ( ( times_times_complex @ A @ C )
          = B2 )
       => ( A
          = ( divide1717551699836669952omplex @ B2 @ C ) ) ) ) ).

% eq_divide_imp
thf(fact_2563_eq__divide__imp,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( C != zero_zero_real )
     => ( ( ( times_times_real @ A @ C )
          = B2 )
       => ( A
          = ( divide_divide_real @ B2 @ C ) ) ) ) ).

% eq_divide_imp
thf(fact_2564_eq__divide__imp,axiom,
    ! [C: rat,A: rat,B2: rat] :
      ( ( C != zero_zero_rat )
     => ( ( ( times_times_rat @ A @ C )
          = B2 )
       => ( A
          = ( divide_divide_rat @ B2 @ C ) ) ) ) ).

% eq_divide_imp
thf(fact_2565_nonzero__divide__eq__eq,axiom,
    ! [C: complex,B2: complex,A: complex] :
      ( ( C != zero_zero_complex )
     => ( ( ( divide1717551699836669952omplex @ B2 @ C )
          = A )
        = ( B2
          = ( times_times_complex @ A @ C ) ) ) ) ).

% nonzero_divide_eq_eq
thf(fact_2566_nonzero__divide__eq__eq,axiom,
    ! [C: real,B2: real,A: real] :
      ( ( C != zero_zero_real )
     => ( ( ( divide_divide_real @ B2 @ C )
          = A )
        = ( B2
          = ( times_times_real @ A @ C ) ) ) ) ).

% nonzero_divide_eq_eq
thf(fact_2567_nonzero__divide__eq__eq,axiom,
    ! [C: rat,B2: rat,A: rat] :
      ( ( C != zero_zero_rat )
     => ( ( ( divide_divide_rat @ B2 @ C )
          = A )
        = ( B2
          = ( times_times_rat @ A @ C ) ) ) ) ).

% nonzero_divide_eq_eq
thf(fact_2568_nonzero__eq__divide__eq,axiom,
    ! [C: complex,A: complex,B2: complex] :
      ( ( C != zero_zero_complex )
     => ( ( A
          = ( divide1717551699836669952omplex @ B2 @ C ) )
        = ( ( times_times_complex @ A @ C )
          = B2 ) ) ) ).

% nonzero_eq_divide_eq
thf(fact_2569_nonzero__eq__divide__eq,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( C != zero_zero_real )
     => ( ( A
          = ( divide_divide_real @ B2 @ C ) )
        = ( ( times_times_real @ A @ C )
          = B2 ) ) ) ).

% nonzero_eq_divide_eq
thf(fact_2570_nonzero__eq__divide__eq,axiom,
    ! [C: rat,A: rat,B2: rat] :
      ( ( C != zero_zero_rat )
     => ( ( A
          = ( divide_divide_rat @ B2 @ C ) )
        = ( ( times_times_rat @ A @ C )
          = B2 ) ) ) ).

% nonzero_eq_divide_eq
thf(fact_2571_right__inverse__eq,axiom,
    ! [B2: complex,A: complex] :
      ( ( B2 != zero_zero_complex )
     => ( ( ( divide1717551699836669952omplex @ A @ B2 )
          = one_one_complex )
        = ( A = B2 ) ) ) ).

% right_inverse_eq
thf(fact_2572_right__inverse__eq,axiom,
    ! [B2: real,A: real] :
      ( ( B2 != zero_zero_real )
     => ( ( ( divide_divide_real @ A @ B2 )
          = one_one_real )
        = ( A = B2 ) ) ) ).

% right_inverse_eq
thf(fact_2573_right__inverse__eq,axiom,
    ! [B2: rat,A: rat] :
      ( ( B2 != zero_zero_rat )
     => ( ( ( divide_divide_rat @ A @ B2 )
          = one_one_rat )
        = ( A = B2 ) ) ) ).

% right_inverse_eq
thf(fact_2574_not__exp__less__eq__0__int,axiom,
    ! [N: nat] :
      ~ ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ zero_zero_int ) ).

% not_exp_less_eq_0_int
thf(fact_2575_pos__zdiv__mult__2,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
        = ( divide_divide_int @ B2 @ A ) ) ) ).

% pos_zdiv_mult_2
thf(fact_2576_neg__zdiv__mult__2,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
        = ( divide_divide_int @ ( plus_plus_int @ B2 @ one_one_int ) @ A ) ) ) ).

% neg_zdiv_mult_2
thf(fact_2577_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ord_le3102999989581377725nteger @ ( modulo364778990260209775nteger @ A @ B2 ) @ A ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
thf(fact_2578_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ord_less_eq_nat @ ( modulo_modulo_nat @ A @ B2 ) @ A ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
thf(fact_2579_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ord_less_eq_int @ ( modulo_modulo_int @ A @ B2 ) @ A ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
thf(fact_2580_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B2: nat,A: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B2 )
     => ( ord_less_nat @ ( modulo_modulo_nat @ A @ B2 ) @ B2 ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
thf(fact_2581_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B2: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B2 )
     => ( ord_less_int @ ( modulo_modulo_int @ A @ B2 ) @ B2 ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
thf(fact_2582_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B2: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B2 )
     => ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ B2 ) @ B2 ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
thf(fact_2583_power__0,axiom,
    ! [A: rat] :
      ( ( power_power_rat @ A @ zero_zero_nat )
      = one_one_rat ) ).

% power_0
thf(fact_2584_power__0,axiom,
    ! [A: nat] :
      ( ( power_power_nat @ A @ zero_zero_nat )
      = one_one_nat ) ).

% power_0
thf(fact_2585_power__0,axiom,
    ! [A: real] :
      ( ( power_power_real @ A @ zero_zero_nat )
      = one_one_real ) ).

% power_0
thf(fact_2586_power__0,axiom,
    ! [A: int] :
      ( ( power_power_int @ A @ zero_zero_nat )
      = one_one_int ) ).

% power_0
thf(fact_2587_power__0,axiom,
    ! [A: complex] :
      ( ( power_power_complex @ A @ zero_zero_nat )
      = one_one_complex ) ).

% power_0
thf(fact_2588_mod__eq__self__iff__div__eq__0,axiom,
    ! [A: nat,B2: nat] :
      ( ( ( modulo_modulo_nat @ A @ B2 )
        = A )
      = ( ( divide_divide_nat @ A @ B2 )
        = zero_zero_nat ) ) ).

% mod_eq_self_iff_div_eq_0
thf(fact_2589_mod__eq__self__iff__div__eq__0,axiom,
    ! [A: int,B2: int] :
      ( ( ( modulo_modulo_int @ A @ B2 )
        = A )
      = ( ( divide_divide_int @ A @ B2 )
        = zero_zero_int ) ) ).

% mod_eq_self_iff_div_eq_0
thf(fact_2590_mod__eq__self__iff__div__eq__0,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ B2 )
        = A )
      = ( ( divide6298287555418463151nteger @ A @ B2 )
        = zero_z3403309356797280102nteger ) ) ).

% mod_eq_self_iff_div_eq_0
thf(fact_2591_less__Suc__eq__0__disj,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N ) )
      = ( ( M = zero_zero_nat )
        | ? [J3: nat] :
            ( ( M
              = ( suc @ J3 ) )
            & ( ord_less_nat @ J3 @ N ) ) ) ) ).

% less_Suc_eq_0_disj
thf(fact_2592_gr0__implies__Suc,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ? [M4: nat] :
          ( N
          = ( suc @ M4 ) ) ) ).

% gr0_implies_Suc
thf(fact_2593_All__less__Suc2,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( suc @ N ) )
           => ( P @ I2 ) ) )
      = ( ( P @ zero_zero_nat )
        & ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ N )
           => ( P @ ( suc @ I2 ) ) ) ) ) ).

% All_less_Suc2
thf(fact_2594_gr0__conv__Suc,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
      = ( ? [M6: nat] :
            ( N
            = ( suc @ M6 ) ) ) ) ).

% gr0_conv_Suc
thf(fact_2595_Ex__less__Suc2,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ? [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( suc @ N ) )
            & ( P @ I2 ) ) )
      = ( ( P @ zero_zero_nat )
        | ? [I2: nat] :
            ( ( ord_less_nat @ I2 @ N )
            & ( P @ ( suc @ I2 ) ) ) ) ) ).

% Ex_less_Suc2
thf(fact_2596_add__is__1,axiom,
    ! [M: nat,N: nat] :
      ( ( ( plus_plus_nat @ M @ N )
        = ( suc @ zero_zero_nat ) )
      = ( ( ( M
            = ( suc @ zero_zero_nat ) )
          & ( N = zero_zero_nat ) )
        | ( ( M = zero_zero_nat )
          & ( N
            = ( suc @ zero_zero_nat ) ) ) ) ) ).

% add_is_1
thf(fact_2597_one__is__add,axiom,
    ! [M: nat,N: nat] :
      ( ( ( suc @ zero_zero_nat )
        = ( plus_plus_nat @ M @ N ) )
      = ( ( ( M
            = ( suc @ zero_zero_nat ) )
          & ( N = zero_zero_nat ) )
        | ( ( M = zero_zero_nat )
          & ( N
            = ( suc @ zero_zero_nat ) ) ) ) ) ).

% one_is_add
thf(fact_2598_ex__least__nat__le,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ N )
     => ( ~ ( P @ zero_zero_nat )
       => ? [K2: nat] :
            ( ( ord_less_eq_nat @ K2 @ N )
            & ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ K2 )
               => ~ ( P @ I4 ) )
            & ( P @ K2 ) ) ) ) ).

% ex_least_nat_le
thf(fact_2599_less__imp__add__positive,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_nat @ I @ J )
     => ? [K2: nat] :
          ( ( ord_less_nat @ zero_zero_nat @ K2 )
          & ( ( plus_plus_nat @ I @ K2 )
            = J ) ) ) ).

% less_imp_add_positive
thf(fact_2600_diff__less,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_nat @ zero_zero_nat @ M )
       => ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).

% diff_less
thf(fact_2601_nat__mult__less__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
        = ( ord_less_nat @ M @ N ) ) ) ).

% nat_mult_less_cancel1
thf(fact_2602_nat__mult__eq__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( ( times_times_nat @ K @ M )
          = ( times_times_nat @ K @ N ) )
        = ( M = N ) ) ) ).

% nat_mult_eq_cancel1
thf(fact_2603_mult__less__mono2,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ord_less_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ) ).

% mult_less_mono2
thf(fact_2604_mult__less__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ord_less_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).

% mult_less_mono1
thf(fact_2605_One__nat__def,axiom,
    ( one_one_nat
    = ( suc @ zero_zero_nat ) ) ).

% One_nat_def
thf(fact_2606_Euclidean__Division_Odiv__eq__0__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ( divide_divide_nat @ M @ N )
        = zero_zero_nat )
      = ( ( ord_less_nat @ M @ N )
        | ( N = zero_zero_nat ) ) ) ).

% Euclidean_Division.div_eq_0_iff
thf(fact_2607_diff__add__0,axiom,
    ! [N: nat,M: nat] :
      ( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
      = zero_zero_nat ) ).

% diff_add_0
thf(fact_2608_nat__power__less__imp__less,axiom,
    ! [I: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ I )
     => ( ( ord_less_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N ) )
       => ( ord_less_nat @ M @ N ) ) ) ).

% nat_power_less_imp_less
thf(fact_2609_mult__eq__self__implies__10,axiom,
    ! [M: nat,N: nat] :
      ( ( M
        = ( times_times_nat @ M @ N ) )
     => ( ( N = one_one_nat )
        | ( M = zero_zero_nat ) ) ) ).

% mult_eq_self_implies_10
thf(fact_2610_mod__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( ( ( suc @ ( modulo_modulo_nat @ M @ N ) )
          = N )
       => ( ( modulo_modulo_nat @ ( suc @ M ) @ N )
          = zero_zero_nat ) )
      & ( ( ( suc @ ( modulo_modulo_nat @ M @ N ) )
         != N )
       => ( ( modulo_modulo_nat @ ( suc @ M ) @ N )
          = ( suc @ ( modulo_modulo_nat @ M @ N ) ) ) ) ) ).

% mod_Suc
thf(fact_2611_mod__less__divisor,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ord_less_nat @ ( modulo_modulo_nat @ M @ N ) @ N ) ) ).

% mod_less_divisor
thf(fact_2612_mod2__eq__if,axiom,
    ! [A: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
       => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
          = zero_zero_nat ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
       => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
          = one_one_nat ) ) ) ).

% mod2_eq_if
thf(fact_2613_mod2__eq__if,axiom,
    ! [A: int] :
      ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
       => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
          = zero_zero_int ) )
      & ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
       => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
          = one_one_int ) ) ) ).

% mod2_eq_if
thf(fact_2614_mod2__eq__if,axiom,
    ! [A: code_integer] :
      ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
       => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
          = zero_z3403309356797280102nteger ) )
      & ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
       => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
          = one_one_Code_integer ) ) ) ).

% mod2_eq_if
thf(fact_2615_parity__cases,axiom,
    ! [A: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
       => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
         != zero_zero_nat ) )
     => ~ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
         => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
           != one_one_nat ) ) ) ).

% parity_cases
thf(fact_2616_parity__cases,axiom,
    ! [A: int] :
      ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
       => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
         != zero_zero_int ) )
     => ~ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
         => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
           != one_one_int ) ) ) ).

% parity_cases
thf(fact_2617_parity__cases,axiom,
    ! [A: code_integer] :
      ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
       => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
         != zero_z3403309356797280102nteger ) )
     => ~ ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
         => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
           != one_one_Code_integer ) ) ) ).

% parity_cases
thf(fact_2618_pow_Osimps_I1_J,axiom,
    ! [X4: num] :
      ( ( pow @ X4 @ one )
      = X4 ) ).

% pow.simps(1)
thf(fact_2619_zero__le__power__eq,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
          & ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ).

% zero_le_power_eq
thf(fact_2620_zero__le__power__eq,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
          & ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ).

% zero_le_power_eq
thf(fact_2621_zero__le__power__eq,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
          & ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ) ).

% zero_le_power_eq
thf(fact_2622_zero__le__odd__power,axiom,
    ! [N: nat,A: real] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) )
        = ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ).

% zero_le_odd_power
thf(fact_2623_zero__le__odd__power,axiom,
    ! [N: nat,A: rat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) )
        = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ).

% zero_le_odd_power
thf(fact_2624_zero__le__odd__power,axiom,
    ! [N: nat,A: int] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) )
        = ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).

% zero_le_odd_power
thf(fact_2625_zero__le__even__power,axiom,
    ! [N: nat,A: real] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).

% zero_le_even_power
thf(fact_2626_zero__le__even__power,axiom,
    ! [N: nat,A: rat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) ) ) ).

% zero_le_even_power
thf(fact_2627_zero__le__even__power,axiom,
    ! [N: nat,A: int] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).

% zero_le_even_power
thf(fact_2628_mod__eq__0D,axiom,
    ! [M: nat,D: nat] :
      ( ( ( modulo_modulo_nat @ M @ D )
        = zero_zero_nat )
     => ? [Q3: nat] :
          ( M
          = ( times_times_nat @ D @ Q3 ) ) ) ).

% mod_eq_0D
thf(fact_2629_div2__even__ext__nat,axiom,
    ! [X4: nat,Y3: nat] :
      ( ( ( divide_divide_nat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( divide_divide_nat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X4 )
          = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Y3 ) )
       => ( X4 = Y3 ) ) ) ).

% div2_even_ext_nat
thf(fact_2630_even__numeral,axiom,
    ! [N: num] : ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) ) ).

% even_numeral
thf(fact_2631_even__numeral,axiom,
    ! [N: num] : ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bit0 @ N ) ) ) ).

% even_numeral
thf(fact_2632_even__numeral,axiom,
    ! [N: num] : ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) ).

% even_numeral
thf(fact_2633_signed__take__bit__int__less__eq,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K )
     => ( ord_less_eq_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( minus_minus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N ) ) ) ) ) ).

% signed_take_bit_int_less_eq
thf(fact_2634_unit__eq__div1,axiom,
    ! [B2: code_integer,A: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
     => ( ( ( divide6298287555418463151nteger @ A @ B2 )
          = C )
        = ( A
          = ( times_3573771949741848930nteger @ C @ B2 ) ) ) ) ).

% unit_eq_div1
thf(fact_2635_unit__eq__div1,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( ( dvd_dvd_nat @ B2 @ one_one_nat )
     => ( ( ( divide_divide_nat @ A @ B2 )
          = C )
        = ( A
          = ( times_times_nat @ C @ B2 ) ) ) ) ).

% unit_eq_div1
thf(fact_2636_unit__eq__div1,axiom,
    ! [B2: int,A: int,C: int] :
      ( ( dvd_dvd_int @ B2 @ one_one_int )
     => ( ( ( divide_divide_int @ A @ B2 )
          = C )
        = ( A
          = ( times_times_int @ C @ B2 ) ) ) ) ).

% unit_eq_div1
thf(fact_2637_unit__eq__div2,axiom,
    ! [B2: code_integer,A: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
     => ( ( A
          = ( divide6298287555418463151nteger @ C @ B2 ) )
        = ( ( times_3573771949741848930nteger @ A @ B2 )
          = C ) ) ) ).

% unit_eq_div2
thf(fact_2638_unit__eq__div2,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( ( dvd_dvd_nat @ B2 @ one_one_nat )
     => ( ( A
          = ( divide_divide_nat @ C @ B2 ) )
        = ( ( times_times_nat @ A @ B2 )
          = C ) ) ) ).

% unit_eq_div2
thf(fact_2639_unit__eq__div2,axiom,
    ! [B2: int,A: int,C: int] :
      ( ( dvd_dvd_int @ B2 @ one_one_int )
     => ( ( A
          = ( divide_divide_int @ C @ B2 ) )
        = ( ( times_times_int @ A @ B2 )
          = C ) ) ) ).

% unit_eq_div2
thf(fact_2640_div__mult__unit2,axiom,
    ! [C: code_integer,B2: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ B2 @ A )
       => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B2 @ C ) )
          = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ B2 ) @ C ) ) ) ) ).

% div_mult_unit2
thf(fact_2641_div__mult__unit2,axiom,
    ! [C: nat,B2: nat,A: nat] :
      ( ( dvd_dvd_nat @ C @ one_one_nat )
     => ( ( dvd_dvd_nat @ B2 @ A )
       => ( ( divide_divide_nat @ A @ ( times_times_nat @ B2 @ C ) )
          = ( divide_divide_nat @ ( divide_divide_nat @ A @ B2 ) @ C ) ) ) ) ).

% div_mult_unit2
thf(fact_2642_div__mult__unit2,axiom,
    ! [C: int,B2: int,A: int] :
      ( ( dvd_dvd_int @ C @ one_one_int )
     => ( ( dvd_dvd_int @ B2 @ A )
       => ( ( divide_divide_int @ A @ ( times_times_int @ B2 @ C ) )
          = ( divide_divide_int @ ( divide_divide_int @ A @ B2 ) @ C ) ) ) ) ).

% div_mult_unit2
thf(fact_2643_unit__div__commute,axiom,
    ! [B2: code_integer,A: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
     => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B2 ) @ C )
        = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ C ) @ B2 ) ) ) ).

% unit_div_commute
thf(fact_2644_unit__div__commute,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( ( dvd_dvd_nat @ B2 @ one_one_nat )
     => ( ( times_times_nat @ ( divide_divide_nat @ A @ B2 ) @ C )
        = ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ B2 ) ) ) ).

% unit_div_commute
thf(fact_2645_unit__div__commute,axiom,
    ! [B2: int,A: int,C: int] :
      ( ( dvd_dvd_int @ B2 @ one_one_int )
     => ( ( times_times_int @ ( divide_divide_int @ A @ B2 ) @ C )
        = ( divide_divide_int @ ( times_times_int @ A @ C ) @ B2 ) ) ) ).

% unit_div_commute
thf(fact_2646_unit__div__mult__swap,axiom,
    ! [C: code_integer,A: code_integer,B2: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ one_one_Code_integer )
     => ( ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B2 @ C ) )
        = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B2 ) @ C ) ) ) ).

% unit_div_mult_swap
thf(fact_2647_unit__div__mult__swap,axiom,
    ! [C: nat,A: nat,B2: nat] :
      ( ( dvd_dvd_nat @ C @ one_one_nat )
     => ( ( times_times_nat @ A @ ( divide_divide_nat @ B2 @ C ) )
        = ( divide_divide_nat @ ( times_times_nat @ A @ B2 ) @ C ) ) ) ).

% unit_div_mult_swap
thf(fact_2648_unit__div__mult__swap,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( dvd_dvd_int @ C @ one_one_int )
     => ( ( times_times_int @ A @ ( divide_divide_int @ B2 @ C ) )
        = ( divide_divide_int @ ( times_times_int @ A @ B2 ) @ C ) ) ) ).

% unit_div_mult_swap
thf(fact_2649_is__unit__div__mult2__eq,axiom,
    ! [B2: code_integer,C: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ C @ one_one_Code_integer )
       => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B2 @ C ) )
          = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ B2 ) @ C ) ) ) ) ).

% is_unit_div_mult2_eq
thf(fact_2650_is__unit__div__mult2__eq,axiom,
    ! [B2: nat,C: nat,A: nat] :
      ( ( dvd_dvd_nat @ B2 @ one_one_nat )
     => ( ( dvd_dvd_nat @ C @ one_one_nat )
       => ( ( divide_divide_nat @ A @ ( times_times_nat @ B2 @ C ) )
          = ( divide_divide_nat @ ( divide_divide_nat @ A @ B2 ) @ C ) ) ) ) ).

% is_unit_div_mult2_eq
thf(fact_2651_is__unit__div__mult2__eq,axiom,
    ! [B2: int,C: int,A: int] :
      ( ( dvd_dvd_int @ B2 @ one_one_int )
     => ( ( dvd_dvd_int @ C @ one_one_int )
       => ( ( divide_divide_int @ A @ ( times_times_int @ B2 @ C ) )
          = ( divide_divide_int @ ( divide_divide_int @ A @ B2 ) @ C ) ) ) ) ).

% is_unit_div_mult2_eq
thf(fact_2652_mod__eq__dvd__iff__nat,axiom,
    ! [N: nat,M: nat,Q2: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( ( modulo_modulo_nat @ M @ Q2 )
          = ( modulo_modulo_nat @ N @ Q2 ) )
        = ( dvd_dvd_nat @ Q2 @ ( minus_minus_nat @ M @ N ) ) ) ) ).

% mod_eq_dvd_iff_nat
thf(fact_2653_mult__less__le__imp__less,axiom,
    ! [A: real,B2: real,C: real,D: real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ( ord_less_eq_real @ C @ D )
       => ( ( ord_less_eq_real @ zero_zero_real @ A )
         => ( ( ord_less_real @ zero_zero_real @ C )
           => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ D ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_2654_mult__less__le__imp__less,axiom,
    ! [A: rat,B2: rat,C: rat,D: rat] :
      ( ( ord_less_rat @ A @ B2 )
     => ( ( ord_less_eq_rat @ C @ D )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
         => ( ( ord_less_rat @ zero_zero_rat @ C )
           => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ D ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_2655_mult__less__le__imp__less,axiom,
    ! [A: nat,B2: nat,C: nat,D: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( ( ord_less_eq_nat @ C @ D )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
         => ( ( ord_less_nat @ zero_zero_nat @ C )
           => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_2656_mult__less__le__imp__less,axiom,
    ! [A: int,B2: int,C: int,D: int] :
      ( ( ord_less_int @ A @ B2 )
     => ( ( ord_less_eq_int @ C @ D )
       => ( ( ord_less_eq_int @ zero_zero_int @ A )
         => ( ( ord_less_int @ zero_zero_int @ C )
           => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_2657_mult__le__less__imp__less,axiom,
    ! [A: real,B2: real,C: real,D: real] :
      ( ( ord_less_eq_real @ A @ B2 )
     => ( ( ord_less_real @ C @ D )
       => ( ( ord_less_real @ zero_zero_real @ A )
         => ( ( ord_less_eq_real @ zero_zero_real @ C )
           => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ D ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_2658_mult__le__less__imp__less,axiom,
    ! [A: rat,B2: rat,C: rat,D: rat] :
      ( ( ord_less_eq_rat @ A @ B2 )
     => ( ( ord_less_rat @ C @ D )
       => ( ( ord_less_rat @ zero_zero_rat @ A )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
           => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ D ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_2659_mult__le__less__imp__less,axiom,
    ! [A: nat,B2: nat,C: nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( ord_less_nat @ C @ D )
       => ( ( ord_less_nat @ zero_zero_nat @ A )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
           => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_2660_mult__le__less__imp__less,axiom,
    ! [A: int,B2: int,C: int,D: int] :
      ( ( ord_less_eq_int @ A @ B2 )
     => ( ( ord_less_int @ C @ D )
       => ( ( ord_less_int @ zero_zero_int @ A )
         => ( ( ord_less_eq_int @ zero_zero_int @ C )
           => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_2661_mult__right__le__imp__le,axiom,
    ! [A: real,C: real,B2: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) )
     => ( ( ord_less_real @ zero_zero_real @ C )
       => ( ord_less_eq_real @ A @ B2 ) ) ) ).

% mult_right_le_imp_le
thf(fact_2662_mult__right__le__imp__le,axiom,
    ! [A: rat,C: rat,B2: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ C ) )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ord_less_eq_rat @ A @ B2 ) ) ) ).

% mult_right_le_imp_le
thf(fact_2663_mult__right__le__imp__le,axiom,
    ! [A: nat,C: nat,B2: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ C ) )
     => ( ( ord_less_nat @ zero_zero_nat @ C )
       => ( ord_less_eq_nat @ A @ B2 ) ) ) ).

% mult_right_le_imp_le
thf(fact_2664_mult__right__le__imp__le,axiom,
    ! [A: int,C: int,B2: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) )
     => ( ( ord_less_int @ zero_zero_int @ C )
       => ( ord_less_eq_int @ A @ B2 ) ) ) ).

% mult_right_le_imp_le
thf(fact_2665_mult__left__le__imp__le,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
     => ( ( ord_less_real @ zero_zero_real @ C )
       => ( ord_less_eq_real @ A @ B2 ) ) ) ).

% mult_left_le_imp_le
thf(fact_2666_mult__left__le__imp__le,axiom,
    ! [C: rat,A: rat,B2: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ord_less_eq_rat @ A @ B2 ) ) ) ).

% mult_left_le_imp_le
thf(fact_2667_mult__left__le__imp__le,axiom,
    ! [C: nat,A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B2 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ C )
       => ( ord_less_eq_nat @ A @ B2 ) ) ) ).

% mult_left_le_imp_le
thf(fact_2668_mult__left__le__imp__le,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
     => ( ( ord_less_int @ zero_zero_int @ C )
       => ( ord_less_eq_int @ A @ B2 ) ) ) ).

% mult_left_le_imp_le
thf(fact_2669_mult__le__cancel__left__pos,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
        = ( ord_less_eq_real @ A @ B2 ) ) ) ).

% mult_le_cancel_left_pos
thf(fact_2670_mult__le__cancel__left__pos,axiom,
    ! [C: rat,A: rat,B2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) )
        = ( ord_less_eq_rat @ A @ B2 ) ) ) ).

% mult_le_cancel_left_pos
thf(fact_2671_mult__le__cancel__left__pos,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( ord_less_int @ zero_zero_int @ C )
     => ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
        = ( ord_less_eq_int @ A @ B2 ) ) ) ).

% mult_le_cancel_left_pos
thf(fact_2672_mult__le__cancel__left__neg,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
        = ( ord_less_eq_real @ B2 @ A ) ) ) ).

% mult_le_cancel_left_neg
thf(fact_2673_mult__le__cancel__left__neg,axiom,
    ! [C: rat,A: rat,B2: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) )
        = ( ord_less_eq_rat @ B2 @ A ) ) ) ).

% mult_le_cancel_left_neg
thf(fact_2674_mult__le__cancel__left__neg,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( ord_less_int @ C @ zero_zero_int )
     => ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
        = ( ord_less_eq_int @ B2 @ A ) ) ) ).

% mult_le_cancel_left_neg
thf(fact_2675_mult__less__cancel__right,axiom,
    ! [A: real,C: real,B2: real] :
      ( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
         => ( ord_less_real @ A @ B2 ) )
        & ( ( ord_less_eq_real @ C @ zero_zero_real )
         => ( ord_less_real @ B2 @ A ) ) ) ) ).

% mult_less_cancel_right
thf(fact_2676_mult__less__cancel__right,axiom,
    ! [A: rat,C: rat,B2: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ C ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ A @ B2 ) )
        & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ B2 @ A ) ) ) ) ).

% mult_less_cancel_right
thf(fact_2677_mult__less__cancel__right,axiom,
    ! [A: int,C: int,B2: int] :
      ( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ A @ B2 ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ B2 @ A ) ) ) ) ).

% mult_less_cancel_right
thf(fact_2678_mult__strict__mono_H,axiom,
    ! [A: real,B2: real,C: real,D: real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ( ord_less_real @ C @ D )
       => ( ( ord_less_eq_real @ zero_zero_real @ A )
         => ( ( ord_less_eq_real @ zero_zero_real @ C )
           => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ D ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_2679_mult__strict__mono_H,axiom,
    ! [A: rat,B2: rat,C: rat,D: rat] :
      ( ( ord_less_rat @ A @ B2 )
     => ( ( ord_less_rat @ C @ D )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
           => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ D ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_2680_mult__strict__mono_H,axiom,
    ! [A: nat,B2: nat,C: nat,D: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( ( ord_less_nat @ C @ D )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
           => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_2681_mult__strict__mono_H,axiom,
    ! [A: int,B2: int,C: int,D: int] :
      ( ( ord_less_int @ A @ B2 )
     => ( ( ord_less_int @ C @ D )
       => ( ( ord_less_eq_int @ zero_zero_int @ A )
         => ( ( ord_less_eq_int @ zero_zero_int @ C )
           => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_2682_mult__right__less__imp__less,axiom,
    ! [A: real,C: real,B2: real] :
      ( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ord_less_real @ A @ B2 ) ) ) ).

% mult_right_less_imp_less
thf(fact_2683_mult__right__less__imp__less,axiom,
    ! [A: rat,C: rat,B2: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ C ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ord_less_rat @ A @ B2 ) ) ) ).

% mult_right_less_imp_less
thf(fact_2684_mult__right__less__imp__less,axiom,
    ! [A: nat,C: nat,B2: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ C ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
       => ( ord_less_nat @ A @ B2 ) ) ) ).

% mult_right_less_imp_less
thf(fact_2685_mult__right__less__imp__less,axiom,
    ! [A: int,C: int,B2: int] :
      ( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ C )
       => ( ord_less_int @ A @ B2 ) ) ) ).

% mult_right_less_imp_less
thf(fact_2686_mult__less__cancel__left,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
         => ( ord_less_real @ A @ B2 ) )
        & ( ( ord_less_eq_real @ C @ zero_zero_real )
         => ( ord_less_real @ B2 @ A ) ) ) ) ).

% mult_less_cancel_left
thf(fact_2687_mult__less__cancel__left,axiom,
    ! [C: rat,A: rat,B2: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ A @ B2 ) )
        & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ B2 @ A ) ) ) ) ).

% mult_less_cancel_left
thf(fact_2688_mult__less__cancel__left,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ A @ B2 ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ B2 @ A ) ) ) ) ).

% mult_less_cancel_left
thf(fact_2689_mult__strict__mono,axiom,
    ! [A: real,B2: real,C: real,D: real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ( ord_less_real @ C @ D )
       => ( ( ord_less_real @ zero_zero_real @ B2 )
         => ( ( ord_less_eq_real @ zero_zero_real @ C )
           => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ D ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_2690_mult__strict__mono,axiom,
    ! [A: rat,B2: rat,C: rat,D: rat] :
      ( ( ord_less_rat @ A @ B2 )
     => ( ( ord_less_rat @ C @ D )
       => ( ( ord_less_rat @ zero_zero_rat @ B2 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
           => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ D ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_2691_mult__strict__mono,axiom,
    ! [A: nat,B2: nat,C: nat,D: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( ( ord_less_nat @ C @ D )
       => ( ( ord_less_nat @ zero_zero_nat @ B2 )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
           => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_2692_mult__strict__mono,axiom,
    ! [A: int,B2: int,C: int,D: int] :
      ( ( ord_less_int @ A @ B2 )
     => ( ( ord_less_int @ C @ D )
       => ( ( ord_less_int @ zero_zero_int @ B2 )
         => ( ( ord_less_eq_int @ zero_zero_int @ C )
           => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_2693_mult__left__less__imp__less,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ord_less_real @ A @ B2 ) ) ) ).

% mult_left_less_imp_less
thf(fact_2694_mult__left__less__imp__less,axiom,
    ! [C: rat,A: rat,B2: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ord_less_rat @ A @ B2 ) ) ) ).

% mult_left_less_imp_less
thf(fact_2695_mult__left__less__imp__less,axiom,
    ! [C: nat,A: nat,B2: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B2 ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
       => ( ord_less_nat @ A @ B2 ) ) ) ).

% mult_left_less_imp_less
thf(fact_2696_mult__left__less__imp__less,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ C )
       => ( ord_less_int @ A @ B2 ) ) ) ).

% mult_left_less_imp_less
thf(fact_2697_mult__le__cancel__right,axiom,
    ! [A: real,C: real,B2: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ A @ B2 ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_eq_real @ B2 @ A ) ) ) ) ).

% mult_le_cancel_right
thf(fact_2698_mult__le__cancel__right,axiom,
    ! [A: rat,C: rat,B2: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ A @ B2 ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ B2 @ A ) ) ) ) ).

% mult_le_cancel_right
thf(fact_2699_mult__le__cancel__right,axiom,
    ! [A: int,C: int,B2: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ A @ B2 ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ B2 @ A ) ) ) ) ).

% mult_le_cancel_right
thf(fact_2700_mult__le__cancel__left,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ A @ B2 ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_eq_real @ B2 @ A ) ) ) ) ).

% mult_le_cancel_left
thf(fact_2701_mult__le__cancel__left,axiom,
    ! [C: rat,A: rat,B2: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ A @ B2 ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ B2 @ A ) ) ) ) ).

% mult_le_cancel_left
thf(fact_2702_mult__le__cancel__left,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ A @ B2 ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ B2 @ A ) ) ) ) ).

% mult_le_cancel_left
thf(fact_2703_add__strict__increasing2,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ B2 @ C )
       => ( ord_less_real @ B2 @ ( plus_plus_real @ A @ C ) ) ) ) ).

% add_strict_increasing2
thf(fact_2704_add__strict__increasing2,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ B2 @ C )
       => ( ord_less_rat @ B2 @ ( plus_plus_rat @ A @ C ) ) ) ) ).

% add_strict_increasing2
thf(fact_2705_add__strict__increasing2,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ B2 @ C )
       => ( ord_less_nat @ B2 @ ( plus_plus_nat @ A @ C ) ) ) ) ).

% add_strict_increasing2
thf(fact_2706_add__strict__increasing2,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ B2 @ C )
       => ( ord_less_int @ B2 @ ( plus_plus_int @ A @ C ) ) ) ) ).

% add_strict_increasing2
thf(fact_2707_add__strict__increasing,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ B2 @ C )
       => ( ord_less_real @ B2 @ ( plus_plus_real @ A @ C ) ) ) ) ).

% add_strict_increasing
thf(fact_2708_add__strict__increasing,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ B2 @ C )
       => ( ord_less_rat @ B2 @ ( plus_plus_rat @ A @ C ) ) ) ) ).

% add_strict_increasing
thf(fact_2709_add__strict__increasing,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ord_less_nat @ B2 @ ( plus_plus_nat @ A @ C ) ) ) ) ).

% add_strict_increasing
thf(fact_2710_add__strict__increasing,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ B2 @ C )
       => ( ord_less_int @ B2 @ ( plus_plus_int @ A @ C ) ) ) ) ).

% add_strict_increasing
thf(fact_2711_add__pos__nonneg,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
       => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B2 ) ) ) ) ).

% add_pos_nonneg
thf(fact_2712_add__pos__nonneg,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
       => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B2 ) ) ) ) ).

% add_pos_nonneg
thf(fact_2713_add__pos__nonneg,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
       => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B2 ) ) ) ) ).

% add_pos_nonneg
thf(fact_2714_add__pos__nonneg,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
       => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B2 ) ) ) ) ).

% add_pos_nonneg
thf(fact_2715_add__nonpos__neg,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ A @ zero_zero_real )
     => ( ( ord_less_real @ B2 @ zero_zero_real )
       => ( ord_less_real @ ( plus_plus_real @ A @ B2 ) @ zero_zero_real ) ) ) ).

% add_nonpos_neg
thf(fact_2716_add__nonpos__neg,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_eq_rat @ A @ zero_zero_rat )
     => ( ( ord_less_rat @ B2 @ zero_zero_rat )
       => ( ord_less_rat @ ( plus_plus_rat @ A @ B2 ) @ zero_zero_rat ) ) ) ).

% add_nonpos_neg
thf(fact_2717_add__nonpos__neg,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( ( ord_less_nat @ B2 @ zero_zero_nat )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ B2 ) @ zero_zero_nat ) ) ) ).

% add_nonpos_neg
thf(fact_2718_add__nonpos__neg,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ B2 @ zero_zero_int )
       => ( ord_less_int @ ( plus_plus_int @ A @ B2 ) @ zero_zero_int ) ) ) ).

% add_nonpos_neg
thf(fact_2719_add__nonneg__pos,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ B2 )
       => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B2 ) ) ) ) ).

% add_nonneg_pos
thf(fact_2720_add__nonneg__pos,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ zero_zero_rat @ B2 )
       => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B2 ) ) ) ) ).

% add_nonneg_pos
thf(fact_2721_add__nonneg__pos,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ B2 )
       => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B2 ) ) ) ) ).

% add_nonneg_pos
thf(fact_2722_add__nonneg__pos,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ zero_zero_int @ B2 )
       => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B2 ) ) ) ) ).

% add_nonneg_pos
thf(fact_2723_add__neg__nonpos,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( ord_less_eq_real @ B2 @ zero_zero_real )
       => ( ord_less_real @ ( plus_plus_real @ A @ B2 ) @ zero_zero_real ) ) ) ).

% add_neg_nonpos
thf(fact_2724_add__neg__nonpos,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ B2 @ zero_zero_rat )
       => ( ord_less_rat @ ( plus_plus_rat @ A @ B2 ) @ zero_zero_rat ) ) ) ).

% add_neg_nonpos
thf(fact_2725_add__neg__nonpos,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_nat @ A @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ B2 ) @ zero_zero_nat ) ) ) ).

% add_neg_nonpos
thf(fact_2726_add__neg__nonpos,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ( ord_less_eq_int @ B2 @ zero_zero_int )
       => ( ord_less_int @ ( plus_plus_int @ A @ B2 ) @ zero_zero_int ) ) ) ).

% add_neg_nonpos
thf(fact_2727_field__le__epsilon,axiom,
    ! [X4: real,Y3: real] :
      ( ! [E2: real] :
          ( ( ord_less_real @ zero_zero_real @ E2 )
         => ( ord_less_eq_real @ X4 @ ( plus_plus_real @ Y3 @ E2 ) ) )
     => ( ord_less_eq_real @ X4 @ Y3 ) ) ).

% field_le_epsilon
thf(fact_2728_field__le__epsilon,axiom,
    ! [X4: rat,Y3: rat] :
      ( ! [E2: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ E2 )
         => ( ord_less_eq_rat @ X4 @ ( plus_plus_rat @ Y3 @ E2 ) ) )
     => ( ord_less_eq_rat @ X4 @ Y3 ) ) ).

% field_le_epsilon
thf(fact_2729_div__positive,axiom,
    ! [B2: nat,A: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B2 )
     => ( ( ord_less_eq_nat @ B2 @ A )
       => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ A @ B2 ) ) ) ) ).

% div_positive
thf(fact_2730_div__positive,axiom,
    ! [B2: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B2 )
     => ( ( ord_less_eq_int @ B2 @ A )
       => ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A @ B2 ) ) ) ) ).

% div_positive
thf(fact_2731_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ A @ B2 )
       => ( ( divide_divide_nat @ A @ B2 )
          = zero_zero_nat ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
thf(fact_2732_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ A @ B2 )
       => ( ( divide_divide_int @ A @ B2 )
          = zero_zero_int ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
thf(fact_2733_divide__nonpos__pos,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_eq_real @ X4 @ zero_zero_real )
     => ( ( ord_less_real @ zero_zero_real @ Y3 )
       => ( ord_less_eq_real @ ( divide_divide_real @ X4 @ Y3 ) @ zero_zero_real ) ) ) ).

% divide_nonpos_pos
thf(fact_2734_divide__nonpos__pos,axiom,
    ! [X4: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ X4 @ zero_zero_rat )
     => ( ( ord_less_rat @ zero_zero_rat @ Y3 )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ X4 @ Y3 ) @ zero_zero_rat ) ) ) ).

% divide_nonpos_pos
thf(fact_2735_divide__nonpos__neg,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_eq_real @ X4 @ zero_zero_real )
     => ( ( ord_less_real @ Y3 @ zero_zero_real )
       => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X4 @ Y3 ) ) ) ) ).

% divide_nonpos_neg
thf(fact_2736_divide__nonpos__neg,axiom,
    ! [X4: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ X4 @ zero_zero_rat )
     => ( ( ord_less_rat @ Y3 @ zero_zero_rat )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X4 @ Y3 ) ) ) ) ).

% divide_nonpos_neg
thf(fact_2737_divide__nonneg__pos,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_real @ zero_zero_real @ Y3 )
       => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X4 @ Y3 ) ) ) ) ).

% divide_nonneg_pos
thf(fact_2738_divide__nonneg__pos,axiom,
    ! [X4: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X4 )
     => ( ( ord_less_rat @ zero_zero_rat @ Y3 )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X4 @ Y3 ) ) ) ) ).

% divide_nonneg_pos
thf(fact_2739_divide__nonneg__neg,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_real @ Y3 @ zero_zero_real )
       => ( ord_less_eq_real @ ( divide_divide_real @ X4 @ Y3 ) @ zero_zero_real ) ) ) ).

% divide_nonneg_neg
thf(fact_2740_divide__nonneg__neg,axiom,
    ! [X4: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X4 )
     => ( ( ord_less_rat @ Y3 @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ X4 @ Y3 ) @ zero_zero_rat ) ) ) ).

% divide_nonneg_neg
thf(fact_2741_divide__le__cancel,axiom,
    ! [A: real,C: real,B2: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B2 @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ A @ B2 ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_eq_real @ B2 @ A ) ) ) ) ).

% divide_le_cancel
thf(fact_2742_divide__le__cancel,axiom,
    ! [A: rat,C: rat,B2: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B2 @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ A @ B2 ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ B2 @ A ) ) ) ) ).

% divide_le_cancel
thf(fact_2743_frac__less2,axiom,
    ! [X4: real,Y3: real,W: real,Z: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( ord_less_eq_real @ X4 @ Y3 )
       => ( ( ord_less_real @ zero_zero_real @ W )
         => ( ( ord_less_real @ W @ Z )
           => ( ord_less_real @ ( divide_divide_real @ X4 @ Z ) @ ( divide_divide_real @ Y3 @ W ) ) ) ) ) ) ).

% frac_less2
thf(fact_2744_frac__less2,axiom,
    ! [X4: rat,Y3: rat,W: rat,Z: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ X4 )
     => ( ( ord_less_eq_rat @ X4 @ Y3 )
       => ( ( ord_less_rat @ zero_zero_rat @ W )
         => ( ( ord_less_rat @ W @ Z )
           => ( ord_less_rat @ ( divide_divide_rat @ X4 @ Z ) @ ( divide_divide_rat @ Y3 @ W ) ) ) ) ) ) ).

% frac_less2
thf(fact_2745_frac__less,axiom,
    ! [X4: real,Y3: real,W: real,Z: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_real @ X4 @ Y3 )
       => ( ( ord_less_real @ zero_zero_real @ W )
         => ( ( ord_less_eq_real @ W @ Z )
           => ( ord_less_real @ ( divide_divide_real @ X4 @ Z ) @ ( divide_divide_real @ Y3 @ W ) ) ) ) ) ) ).

% frac_less
thf(fact_2746_frac__less,axiom,
    ! [X4: rat,Y3: rat,W: rat,Z: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X4 )
     => ( ( ord_less_rat @ X4 @ Y3 )
       => ( ( ord_less_rat @ zero_zero_rat @ W )
         => ( ( ord_less_eq_rat @ W @ Z )
           => ( ord_less_rat @ ( divide_divide_rat @ X4 @ Z ) @ ( divide_divide_rat @ Y3 @ W ) ) ) ) ) ) ).

% frac_less
thf(fact_2747_frac__le,axiom,
    ! [Y3: real,X4: real,W: real,Z: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
     => ( ( ord_less_eq_real @ X4 @ Y3 )
       => ( ( ord_less_real @ zero_zero_real @ W )
         => ( ( ord_less_eq_real @ W @ Z )
           => ( ord_less_eq_real @ ( divide_divide_real @ X4 @ Z ) @ ( divide_divide_real @ Y3 @ W ) ) ) ) ) ) ).

% frac_le
thf(fact_2748_frac__le,axiom,
    ! [Y3: rat,X4: rat,W: rat,Z: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
     => ( ( ord_less_eq_rat @ X4 @ Y3 )
       => ( ( ord_less_rat @ zero_zero_rat @ W )
         => ( ( ord_less_eq_rat @ W @ Z )
           => ( ord_less_eq_rat @ ( divide_divide_rat @ X4 @ Z ) @ ( divide_divide_rat @ Y3 @ W ) ) ) ) ) ) ).

% frac_le
thf(fact_2749_mult__left__le__one__le,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
       => ( ( ord_less_eq_real @ Y3 @ one_one_real )
         => ( ord_less_eq_real @ ( times_times_real @ Y3 @ X4 ) @ X4 ) ) ) ) ).

% mult_left_le_one_le
thf(fact_2750_mult__left__le__one__le,axiom,
    ! [X4: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X4 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
       => ( ( ord_less_eq_rat @ Y3 @ one_one_rat )
         => ( ord_less_eq_rat @ ( times_times_rat @ Y3 @ X4 ) @ X4 ) ) ) ) ).

% mult_left_le_one_le
thf(fact_2751_mult__left__le__one__le,axiom,
    ! [X4: int,Y3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X4 )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
       => ( ( ord_less_eq_int @ Y3 @ one_one_int )
         => ( ord_less_eq_int @ ( times_times_int @ Y3 @ X4 ) @ X4 ) ) ) ) ).

% mult_left_le_one_le
thf(fact_2752_mult__right__le__one__le,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
       => ( ( ord_less_eq_real @ Y3 @ one_one_real )
         => ( ord_less_eq_real @ ( times_times_real @ X4 @ Y3 ) @ X4 ) ) ) ) ).

% mult_right_le_one_le
thf(fact_2753_mult__right__le__one__le,axiom,
    ! [X4: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X4 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
       => ( ( ord_less_eq_rat @ Y3 @ one_one_rat )
         => ( ord_less_eq_rat @ ( times_times_rat @ X4 @ Y3 ) @ X4 ) ) ) ) ).

% mult_right_le_one_le
thf(fact_2754_mult__right__le__one__le,axiom,
    ! [X4: int,Y3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X4 )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
       => ( ( ord_less_eq_int @ Y3 @ one_one_int )
         => ( ord_less_eq_int @ ( times_times_int @ X4 @ Y3 ) @ X4 ) ) ) ) ).

% mult_right_le_one_le
thf(fact_2755_mult__le__one,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ A @ one_one_real )
     => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
       => ( ( ord_less_eq_real @ B2 @ one_one_real )
         => ( ord_less_eq_real @ ( times_times_real @ A @ B2 ) @ one_one_real ) ) ) ) ).

% mult_le_one
thf(fact_2756_mult__le__one,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_eq_rat @ A @ one_one_rat )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
       => ( ( ord_less_eq_rat @ B2 @ one_one_rat )
         => ( ord_less_eq_rat @ ( times_times_rat @ A @ B2 ) @ one_one_rat ) ) ) ) ).

% mult_le_one
thf(fact_2757_mult__le__one,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A @ one_one_nat )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
       => ( ( ord_less_eq_nat @ B2 @ one_one_nat )
         => ( ord_less_eq_nat @ ( times_times_nat @ A @ B2 ) @ one_one_nat ) ) ) ) ).

% mult_le_one
thf(fact_2758_mult__le__one,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ A @ one_one_int )
     => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
       => ( ( ord_less_eq_int @ B2 @ one_one_int )
         => ( ord_less_eq_int @ ( times_times_int @ A @ B2 ) @ one_one_int ) ) ) ) ).

% mult_le_one
thf(fact_2759_mult__left__le,axiom,
    ! [C: real,A: real] :
      ( ( ord_less_eq_real @ C @ one_one_real )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ A ) ) ) ).

% mult_left_le
thf(fact_2760_mult__left__le,axiom,
    ! [C: rat,A: rat] :
      ( ( ord_less_eq_rat @ C @ one_one_rat )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
       => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ A ) ) ) ).

% mult_left_le
thf(fact_2761_mult__left__le,axiom,
    ! [C: nat,A: nat] :
      ( ( ord_less_eq_nat @ C @ one_one_nat )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ A ) ) ) ).

% mult_left_le
thf(fact_2762_mult__left__le,axiom,
    ! [C: int,A: int] :
      ( ( ord_less_eq_int @ C @ one_one_int )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ A ) ) ) ).

% mult_left_le
thf(fact_2763_sum__squares__le__zero__iff,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ X4 @ X4 ) @ ( times_times_real @ Y3 @ Y3 ) ) @ zero_zero_real )
      = ( ( X4 = zero_zero_real )
        & ( Y3 = zero_zero_real ) ) ) ).

% sum_squares_le_zero_iff
thf(fact_2764_sum__squares__le__zero__iff,axiom,
    ! [X4: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ X4 @ X4 ) @ ( times_times_rat @ Y3 @ Y3 ) ) @ zero_zero_rat )
      = ( ( X4 = zero_zero_rat )
        & ( Y3 = zero_zero_rat ) ) ) ).

% sum_squares_le_zero_iff
thf(fact_2765_sum__squares__le__zero__iff,axiom,
    ! [X4: int,Y3: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ X4 @ X4 ) @ ( times_times_int @ Y3 @ Y3 ) ) @ zero_zero_int )
      = ( ( X4 = zero_zero_int )
        & ( Y3 = zero_zero_int ) ) ) ).

% sum_squares_le_zero_iff
thf(fact_2766_sum__squares__ge__zero,axiom,
    ! [X4: real,Y3: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X4 @ X4 ) @ ( times_times_real @ Y3 @ Y3 ) ) ) ).

% sum_squares_ge_zero
thf(fact_2767_sum__squares__ge__zero,axiom,
    ! [X4: rat,Y3: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X4 @ X4 ) @ ( times_times_rat @ Y3 @ Y3 ) ) ) ).

% sum_squares_ge_zero
thf(fact_2768_sum__squares__ge__zero,axiom,
    ! [X4: int,Y3: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X4 @ X4 ) @ ( times_times_int @ Y3 @ Y3 ) ) ) ).

% sum_squares_ge_zero
thf(fact_2769_power__less__imp__less__base,axiom,
    ! [A: real,N: nat,B2: real] :
      ( ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B2 @ N ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
       => ( ord_less_real @ A @ B2 ) ) ) ).

% power_less_imp_less_base
thf(fact_2770_power__less__imp__less__base,axiom,
    ! [A: rat,N: nat,B2: rat] :
      ( ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B2 @ N ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
       => ( ord_less_rat @ A @ B2 ) ) ) ).

% power_less_imp_less_base
thf(fact_2771_power__less__imp__less__base,axiom,
    ! [A: nat,N: nat,B2: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B2 @ N ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
       => ( ord_less_nat @ A @ B2 ) ) ) ).

% power_less_imp_less_base
thf(fact_2772_power__less__imp__less__base,axiom,
    ! [A: int,N: nat,B2: int] :
      ( ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B2 @ N ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
       => ( ord_less_int @ A @ B2 ) ) ) ).

% power_less_imp_less_base
thf(fact_2773_sum__squares__gt__zero__iff,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X4 @ X4 ) @ ( times_times_real @ Y3 @ Y3 ) ) )
      = ( ( X4 != zero_zero_real )
        | ( Y3 != zero_zero_real ) ) ) ).

% sum_squares_gt_zero_iff
thf(fact_2774_sum__squares__gt__zero__iff,axiom,
    ! [X4: rat,Y3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X4 @ X4 ) @ ( times_times_rat @ Y3 @ Y3 ) ) )
      = ( ( X4 != zero_zero_rat )
        | ( Y3 != zero_zero_rat ) ) ) ).

% sum_squares_gt_zero_iff
thf(fact_2775_sum__squares__gt__zero__iff,axiom,
    ! [X4: int,Y3: int] :
      ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X4 @ X4 ) @ ( times_times_int @ Y3 @ Y3 ) ) )
      = ( ( X4 != zero_zero_int )
        | ( Y3 != zero_zero_int ) ) ) ).

% sum_squares_gt_zero_iff
thf(fact_2776_not__sum__squares__lt__zero,axiom,
    ! [X4: real,Y3: real] :
      ~ ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ X4 @ X4 ) @ ( times_times_real @ Y3 @ Y3 ) ) @ zero_zero_real ) ).

% not_sum_squares_lt_zero
thf(fact_2777_not__sum__squares__lt__zero,axiom,
    ! [X4: rat,Y3: rat] :
      ~ ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ X4 @ X4 ) @ ( times_times_rat @ Y3 @ Y3 ) ) @ zero_zero_rat ) ).

% not_sum_squares_lt_zero
thf(fact_2778_not__sum__squares__lt__zero,axiom,
    ! [X4: int,Y3: int] :
      ~ ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ X4 @ X4 ) @ ( times_times_int @ Y3 @ Y3 ) ) @ zero_zero_int ) ).

% not_sum_squares_lt_zero
thf(fact_2779_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C: nat,A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ C )
     => ( ( divide_divide_nat @ A @ ( times_times_nat @ B2 @ C ) )
        = ( divide_divide_nat @ ( divide_divide_nat @ A @ B2 ) @ C ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
thf(fact_2780_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ C )
     => ( ( divide_divide_int @ A @ ( times_times_int @ B2 @ C ) )
        = ( divide_divide_int @ ( divide_divide_int @ A @ B2 ) @ C ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
thf(fact_2781_zero__less__two,axiom,
    ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ one_one_real ) ).

% zero_less_two
thf(fact_2782_zero__less__two,axiom,
    ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ).

% zero_less_two
thf(fact_2783_zero__less__two,axiom,
    ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).

% zero_less_two
thf(fact_2784_zero__less__two,axiom,
    ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).

% zero_less_two
thf(fact_2785_divide__strict__left__mono__neg,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ( ord_less_real @ C @ zero_zero_real )
       => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) )
         => ( ord_less_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B2 ) ) ) ) ) ).

% divide_strict_left_mono_neg
thf(fact_2786_divide__strict__left__mono__neg,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( ord_less_rat @ A @ B2 )
     => ( ( ord_less_rat @ C @ zero_zero_rat )
       => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B2 ) )
         => ( ord_less_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B2 ) ) ) ) ) ).

% divide_strict_left_mono_neg
thf(fact_2787_divide__strict__left__mono,axiom,
    ! [B2: real,A: real,C: real] :
      ( ( ord_less_real @ B2 @ A )
     => ( ( ord_less_real @ zero_zero_real @ C )
       => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) )
         => ( ord_less_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B2 ) ) ) ) ) ).

% divide_strict_left_mono
thf(fact_2788_divide__strict__left__mono,axiom,
    ! [B2: rat,A: rat,C: rat] :
      ( ( ord_less_rat @ B2 @ A )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B2 ) )
         => ( ord_less_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B2 ) ) ) ) ) ).

% divide_strict_left_mono
thf(fact_2789_mult__imp__less__div__pos,axiom,
    ! [Y3: real,Z: real,X4: real] :
      ( ( ord_less_real @ zero_zero_real @ Y3 )
     => ( ( ord_less_real @ ( times_times_real @ Z @ Y3 ) @ X4 )
       => ( ord_less_real @ Z @ ( divide_divide_real @ X4 @ Y3 ) ) ) ) ).

% mult_imp_less_div_pos
thf(fact_2790_mult__imp__less__div__pos,axiom,
    ! [Y3: rat,Z: rat,X4: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Y3 )
     => ( ( ord_less_rat @ ( times_times_rat @ Z @ Y3 ) @ X4 )
       => ( ord_less_rat @ Z @ ( divide_divide_rat @ X4 @ Y3 ) ) ) ) ).

% mult_imp_less_div_pos
thf(fact_2791_mult__imp__div__pos__less,axiom,
    ! [Y3: real,X4: real,Z: real] :
      ( ( ord_less_real @ zero_zero_real @ Y3 )
     => ( ( ord_less_real @ X4 @ ( times_times_real @ Z @ Y3 ) )
       => ( ord_less_real @ ( divide_divide_real @ X4 @ Y3 ) @ Z ) ) ) ).

% mult_imp_div_pos_less
thf(fact_2792_mult__imp__div__pos__less,axiom,
    ! [Y3: rat,X4: rat,Z: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Y3 )
     => ( ( ord_less_rat @ X4 @ ( times_times_rat @ Z @ Y3 ) )
       => ( ord_less_rat @ ( divide_divide_rat @ X4 @ Y3 ) @ Z ) ) ) ).

% mult_imp_div_pos_less
thf(fact_2793_pos__less__divide__eq,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_real @ A @ ( divide_divide_real @ B2 @ C ) )
        = ( ord_less_real @ ( times_times_real @ A @ C ) @ B2 ) ) ) ).

% pos_less_divide_eq
thf(fact_2794_pos__less__divide__eq,axiom,
    ! [C: rat,A: rat,B2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_rat @ A @ ( divide_divide_rat @ B2 @ C ) )
        = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B2 ) ) ) ).

% pos_less_divide_eq
thf(fact_2795_pos__divide__less__eq,axiom,
    ! [C: real,B2: real,A: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_real @ ( divide_divide_real @ B2 @ C ) @ A )
        = ( ord_less_real @ B2 @ ( times_times_real @ A @ C ) ) ) ) ).

% pos_divide_less_eq
thf(fact_2796_pos__divide__less__eq,axiom,
    ! [C: rat,B2: rat,A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_rat @ ( divide_divide_rat @ B2 @ C ) @ A )
        = ( ord_less_rat @ B2 @ ( times_times_rat @ A @ C ) ) ) ) ).

% pos_divide_less_eq
thf(fact_2797_neg__less__divide__eq,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_real @ A @ ( divide_divide_real @ B2 @ C ) )
        = ( ord_less_real @ B2 @ ( times_times_real @ A @ C ) ) ) ) ).

% neg_less_divide_eq
thf(fact_2798_neg__less__divide__eq,axiom,
    ! [C: rat,A: rat,B2: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_rat @ A @ ( divide_divide_rat @ B2 @ C ) )
        = ( ord_less_rat @ B2 @ ( times_times_rat @ A @ C ) ) ) ) ).

% neg_less_divide_eq
thf(fact_2799_neg__divide__less__eq,axiom,
    ! [C: real,B2: real,A: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_real @ ( divide_divide_real @ B2 @ C ) @ A )
        = ( ord_less_real @ ( times_times_real @ A @ C ) @ B2 ) ) ) ).

% neg_divide_less_eq
thf(fact_2800_neg__divide__less__eq,axiom,
    ! [C: rat,B2: rat,A: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_rat @ ( divide_divide_rat @ B2 @ C ) @ A )
        = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B2 ) ) ) ).

% neg_divide_less_eq
thf(fact_2801_less__divide__eq,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ord_less_real @ A @ ( divide_divide_real @ B2 @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ ( times_times_real @ A @ C ) @ B2 ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ B2 @ ( times_times_real @ A @ C ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ A @ zero_zero_real ) ) ) ) ) ) ).

% less_divide_eq
thf(fact_2802_less__divide__eq,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( ord_less_rat @ A @ ( divide_divide_rat @ B2 @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B2 ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ B2 @ ( times_times_rat @ A @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).

% less_divide_eq
thf(fact_2803_divide__less__eq,axiom,
    ! [B2: real,C: real,A: real] :
      ( ( ord_less_real @ ( divide_divide_real @ B2 @ C ) @ A )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ B2 @ ( times_times_real @ A @ C ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ ( times_times_real @ A @ C ) @ B2 ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ zero_zero_real @ A ) ) ) ) ) ) ).

% divide_less_eq
thf(fact_2804_divide__less__eq,axiom,
    ! [B2: rat,C: rat,A: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ B2 @ C ) @ A )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ B2 @ ( times_times_rat @ A @ C ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B2 ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).

% divide_less_eq
thf(fact_2805_less__divide__eq__1,axiom,
    ! [B2: real,A: real] :
      ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B2 @ A ) )
      = ( ( ( ord_less_real @ zero_zero_real @ A )
          & ( ord_less_real @ A @ B2 ) )
        | ( ( ord_less_real @ A @ zero_zero_real )
          & ( ord_less_real @ B2 @ A ) ) ) ) ).

% less_divide_eq_1
thf(fact_2806_less__divide__eq__1,axiom,
    ! [B2: rat,A: rat] :
      ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B2 @ A ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A )
          & ( ord_less_rat @ A @ B2 ) )
        | ( ( ord_less_rat @ A @ zero_zero_rat )
          & ( ord_less_rat @ B2 @ A ) ) ) ) ).

% less_divide_eq_1
thf(fact_2807_divide__less__eq__1,axiom,
    ! [B2: real,A: real] :
      ( ( ord_less_real @ ( divide_divide_real @ B2 @ A ) @ one_one_real )
      = ( ( ( ord_less_real @ zero_zero_real @ A )
          & ( ord_less_real @ B2 @ A ) )
        | ( ( ord_less_real @ A @ zero_zero_real )
          & ( ord_less_real @ A @ B2 ) )
        | ( A = zero_zero_real ) ) ) ).

% divide_less_eq_1
thf(fact_2808_divide__less__eq__1,axiom,
    ! [B2: rat,A: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ B2 @ A ) @ one_one_rat )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A )
          & ( ord_less_rat @ B2 @ A ) )
        | ( ( ord_less_rat @ A @ zero_zero_rat )
          & ( ord_less_rat @ A @ B2 ) )
        | ( A = zero_zero_rat ) ) ) ).

% divide_less_eq_1
thf(fact_2809_power__le__one,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ A @ one_one_real )
       => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ one_one_real ) ) ) ).

% power_le_one
thf(fact_2810_power__le__one,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ A @ one_one_rat )
       => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ one_one_rat ) ) ) ).

% power_le_one
thf(fact_2811_power__le__one,axiom,
    ! [A: nat,N: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ A @ one_one_nat )
       => ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ one_one_nat ) ) ) ).

% power_le_one
thf(fact_2812_power__le__one,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ A @ one_one_int )
       => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ one_one_int ) ) ) ).

% power_le_one
thf(fact_2813_divide__eq__eq__numeral_I1_J,axiom,
    ! [B2: complex,C: complex,W: num] :
      ( ( ( divide1717551699836669952omplex @ B2 @ C )
        = ( numera6690914467698888265omplex @ W ) )
      = ( ( ( C != zero_zero_complex )
         => ( B2
            = ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ C ) ) )
        & ( ( C = zero_zero_complex )
         => ( ( numera6690914467698888265omplex @ W )
            = zero_zero_complex ) ) ) ) ).

% divide_eq_eq_numeral(1)
thf(fact_2814_divide__eq__eq__numeral_I1_J,axiom,
    ! [B2: real,C: real,W: num] :
      ( ( ( divide_divide_real @ B2 @ C )
        = ( numeral_numeral_real @ W ) )
      = ( ( ( C != zero_zero_real )
         => ( B2
            = ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
        & ( ( C = zero_zero_real )
         => ( ( numeral_numeral_real @ W )
            = zero_zero_real ) ) ) ) ).

% divide_eq_eq_numeral(1)
thf(fact_2815_divide__eq__eq__numeral_I1_J,axiom,
    ! [B2: rat,C: rat,W: num] :
      ( ( ( divide_divide_rat @ B2 @ C )
        = ( numeral_numeral_rat @ W ) )
      = ( ( ( C != zero_zero_rat )
         => ( B2
            = ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
        & ( ( C = zero_zero_rat )
         => ( ( numeral_numeral_rat @ W )
            = zero_zero_rat ) ) ) ) ).

% divide_eq_eq_numeral(1)
thf(fact_2816_eq__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B2: complex,C: complex] :
      ( ( ( numera6690914467698888265omplex @ W )
        = ( divide1717551699836669952omplex @ B2 @ C ) )
      = ( ( ( C != zero_zero_complex )
         => ( ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ C )
            = B2 ) )
        & ( ( C = zero_zero_complex )
         => ( ( numera6690914467698888265omplex @ W )
            = zero_zero_complex ) ) ) ) ).

% eq_divide_eq_numeral(1)
thf(fact_2817_eq__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B2: real,C: real] :
      ( ( ( numeral_numeral_real @ W )
        = ( divide_divide_real @ B2 @ C ) )
      = ( ( ( C != zero_zero_real )
         => ( ( times_times_real @ ( numeral_numeral_real @ W ) @ C )
            = B2 ) )
        & ( ( C = zero_zero_real )
         => ( ( numeral_numeral_real @ W )
            = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral(1)
thf(fact_2818_eq__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B2: rat,C: rat] :
      ( ( ( numeral_numeral_rat @ W )
        = ( divide_divide_rat @ B2 @ C ) )
      = ( ( ( C != zero_zero_rat )
         => ( ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C )
            = B2 ) )
        & ( ( C = zero_zero_rat )
         => ( ( numeral_numeral_rat @ W )
            = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral(1)
thf(fact_2819_add__divide__eq__if__simps_I2_J,axiom,
    ! [Z: complex,A: complex,B2: complex] :
      ( ( ( Z = zero_zero_complex )
       => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ Z ) @ B2 )
          = B2 ) )
      & ( ( Z != zero_zero_complex )
       => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ Z ) @ B2 )
          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ ( times_times_complex @ B2 @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(2)
thf(fact_2820_add__divide__eq__if__simps_I2_J,axiom,
    ! [Z: real,A: real,B2: real] :
      ( ( ( Z = zero_zero_real )
       => ( ( plus_plus_real @ ( divide_divide_real @ A @ Z ) @ B2 )
          = B2 ) )
      & ( ( Z != zero_zero_real )
       => ( ( plus_plus_real @ ( divide_divide_real @ A @ Z ) @ B2 )
          = ( divide_divide_real @ ( plus_plus_real @ A @ ( times_times_real @ B2 @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(2)
thf(fact_2821_add__divide__eq__if__simps_I2_J,axiom,
    ! [Z: rat,A: rat,B2: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( ( plus_plus_rat @ ( divide_divide_rat @ A @ Z ) @ B2 )
          = B2 ) )
      & ( ( Z != zero_zero_rat )
       => ( ( plus_plus_rat @ ( divide_divide_rat @ A @ Z ) @ B2 )
          = ( divide_divide_rat @ ( plus_plus_rat @ A @ ( times_times_rat @ B2 @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(2)
thf(fact_2822_add__divide__eq__if__simps_I1_J,axiom,
    ! [Z: complex,A: complex,B2: complex] :
      ( ( ( Z = zero_zero_complex )
       => ( ( plus_plus_complex @ A @ ( divide1717551699836669952omplex @ B2 @ Z ) )
          = A ) )
      & ( ( Z != zero_zero_complex )
       => ( ( plus_plus_complex @ A @ ( divide1717551699836669952omplex @ B2 @ Z ) )
          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ A @ Z ) @ B2 ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(1)
thf(fact_2823_add__divide__eq__if__simps_I1_J,axiom,
    ! [Z: real,A: real,B2: real] :
      ( ( ( Z = zero_zero_real )
       => ( ( plus_plus_real @ A @ ( divide_divide_real @ B2 @ Z ) )
          = A ) )
      & ( ( Z != zero_zero_real )
       => ( ( plus_plus_real @ A @ ( divide_divide_real @ B2 @ Z ) )
          = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ A @ Z ) @ B2 ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(1)
thf(fact_2824_add__divide__eq__if__simps_I1_J,axiom,
    ! [Z: rat,A: rat,B2: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( ( plus_plus_rat @ A @ ( divide_divide_rat @ B2 @ Z ) )
          = A ) )
      & ( ( Z != zero_zero_rat )
       => ( ( plus_plus_rat @ A @ ( divide_divide_rat @ B2 @ Z ) )
          = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ Z ) @ B2 ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(1)
thf(fact_2825_add__frac__eq,axiom,
    ! [Y3: complex,Z: complex,X4: complex,W: complex] :
      ( ( Y3 != zero_zero_complex )
     => ( ( Z != zero_zero_complex )
       => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X4 @ Y3 ) @ ( divide1717551699836669952omplex @ W @ Z ) )
          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ X4 @ Z ) @ ( times_times_complex @ W @ Y3 ) ) @ ( times_times_complex @ Y3 @ Z ) ) ) ) ) ).

% add_frac_eq
thf(fact_2826_add__frac__eq,axiom,
    ! [Y3: real,Z: real,X4: real,W: real] :
      ( ( Y3 != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( ( plus_plus_real @ ( divide_divide_real @ X4 @ Y3 ) @ ( divide_divide_real @ W @ Z ) )
          = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X4 @ Z ) @ ( times_times_real @ W @ Y3 ) ) @ ( times_times_real @ Y3 @ Z ) ) ) ) ) ).

% add_frac_eq
thf(fact_2827_add__frac__eq,axiom,
    ! [Y3: rat,Z: rat,X4: rat,W: rat] :
      ( ( Y3 != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( ( plus_plus_rat @ ( divide_divide_rat @ X4 @ Y3 ) @ ( divide_divide_rat @ W @ Z ) )
          = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X4 @ Z ) @ ( times_times_rat @ W @ Y3 ) ) @ ( times_times_rat @ Y3 @ Z ) ) ) ) ) ).

% add_frac_eq
thf(fact_2828_add__frac__num,axiom,
    ! [Y3: complex,X4: complex,Z: complex] :
      ( ( Y3 != zero_zero_complex )
     => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X4 @ Y3 ) @ Z )
        = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X4 @ ( times_times_complex @ Z @ Y3 ) ) @ Y3 ) ) ) ).

% add_frac_num
thf(fact_2829_add__frac__num,axiom,
    ! [Y3: real,X4: real,Z: real] :
      ( ( Y3 != zero_zero_real )
     => ( ( plus_plus_real @ ( divide_divide_real @ X4 @ Y3 ) @ Z )
        = ( divide_divide_real @ ( plus_plus_real @ X4 @ ( times_times_real @ Z @ Y3 ) ) @ Y3 ) ) ) ).

% add_frac_num
thf(fact_2830_add__frac__num,axiom,
    ! [Y3: rat,X4: rat,Z: rat] :
      ( ( Y3 != zero_zero_rat )
     => ( ( plus_plus_rat @ ( divide_divide_rat @ X4 @ Y3 ) @ Z )
        = ( divide_divide_rat @ ( plus_plus_rat @ X4 @ ( times_times_rat @ Z @ Y3 ) ) @ Y3 ) ) ) ).

% add_frac_num
thf(fact_2831_add__num__frac,axiom,
    ! [Y3: complex,Z: complex,X4: complex] :
      ( ( Y3 != zero_zero_complex )
     => ( ( plus_plus_complex @ Z @ ( divide1717551699836669952omplex @ X4 @ Y3 ) )
        = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X4 @ ( times_times_complex @ Z @ Y3 ) ) @ Y3 ) ) ) ).

% add_num_frac
thf(fact_2832_add__num__frac,axiom,
    ! [Y3: real,Z: real,X4: real] :
      ( ( Y3 != zero_zero_real )
     => ( ( plus_plus_real @ Z @ ( divide_divide_real @ X4 @ Y3 ) )
        = ( divide_divide_real @ ( plus_plus_real @ X4 @ ( times_times_real @ Z @ Y3 ) ) @ Y3 ) ) ) ).

% add_num_frac
thf(fact_2833_add__num__frac,axiom,
    ! [Y3: rat,Z: rat,X4: rat] :
      ( ( Y3 != zero_zero_rat )
     => ( ( plus_plus_rat @ Z @ ( divide_divide_rat @ X4 @ Y3 ) )
        = ( divide_divide_rat @ ( plus_plus_rat @ X4 @ ( times_times_rat @ Z @ Y3 ) ) @ Y3 ) ) ) ).

% add_num_frac
thf(fact_2834_add__divide__eq__iff,axiom,
    ! [Z: complex,X4: complex,Y3: complex] :
      ( ( Z != zero_zero_complex )
     => ( ( plus_plus_complex @ X4 @ ( divide1717551699836669952omplex @ Y3 @ Z ) )
        = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ X4 @ Z ) @ Y3 ) @ Z ) ) ) ).

% add_divide_eq_iff
thf(fact_2835_add__divide__eq__iff,axiom,
    ! [Z: real,X4: real,Y3: real] :
      ( ( Z != zero_zero_real )
     => ( ( plus_plus_real @ X4 @ ( divide_divide_real @ Y3 @ Z ) )
        = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X4 @ Z ) @ Y3 ) @ Z ) ) ) ).

% add_divide_eq_iff
thf(fact_2836_add__divide__eq__iff,axiom,
    ! [Z: rat,X4: rat,Y3: rat] :
      ( ( Z != zero_zero_rat )
     => ( ( plus_plus_rat @ X4 @ ( divide_divide_rat @ Y3 @ Z ) )
        = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X4 @ Z ) @ Y3 ) @ Z ) ) ) ).

% add_divide_eq_iff
thf(fact_2837_divide__add__eq__iff,axiom,
    ! [Z: complex,X4: complex,Y3: complex] :
      ( ( Z != zero_zero_complex )
     => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X4 @ Z ) @ Y3 )
        = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X4 @ ( times_times_complex @ Y3 @ Z ) ) @ Z ) ) ) ).

% divide_add_eq_iff
thf(fact_2838_divide__add__eq__iff,axiom,
    ! [Z: real,X4: real,Y3: real] :
      ( ( Z != zero_zero_real )
     => ( ( plus_plus_real @ ( divide_divide_real @ X4 @ Z ) @ Y3 )
        = ( divide_divide_real @ ( plus_plus_real @ X4 @ ( times_times_real @ Y3 @ Z ) ) @ Z ) ) ) ).

% divide_add_eq_iff
thf(fact_2839_divide__add__eq__iff,axiom,
    ! [Z: rat,X4: rat,Y3: rat] :
      ( ( Z != zero_zero_rat )
     => ( ( plus_plus_rat @ ( divide_divide_rat @ X4 @ Z ) @ Y3 )
        = ( divide_divide_rat @ ( plus_plus_rat @ X4 @ ( times_times_rat @ Y3 @ Z ) ) @ Z ) ) ) ).

% divide_add_eq_iff
thf(fact_2840_power__le__imp__le__base,axiom,
    ! [A: real,N: nat,B2: real] :
      ( ( ord_less_eq_real @ ( power_power_real @ A @ ( suc @ N ) ) @ ( power_power_real @ B2 @ ( suc @ N ) ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
       => ( ord_less_eq_real @ A @ B2 ) ) ) ).

% power_le_imp_le_base
thf(fact_2841_power__le__imp__le__base,axiom,
    ! [A: rat,N: nat,B2: rat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( suc @ N ) ) @ ( power_power_rat @ B2 @ ( suc @ N ) ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
       => ( ord_less_eq_rat @ A @ B2 ) ) ) ).

% power_le_imp_le_base
thf(fact_2842_power__le__imp__le__base,axiom,
    ! [A: nat,N: nat,B2: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ ( power_power_nat @ B2 @ ( suc @ N ) ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
       => ( ord_less_eq_nat @ A @ B2 ) ) ) ).

% power_le_imp_le_base
thf(fact_2843_power__le__imp__le__base,axiom,
    ! [A: int,N: nat,B2: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N ) ) @ ( power_power_int @ B2 @ ( suc @ N ) ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
       => ( ord_less_eq_int @ A @ B2 ) ) ) ).

% power_le_imp_le_base
thf(fact_2844_power__inject__base,axiom,
    ! [A: real,N: nat,B2: real] :
      ( ( ( power_power_real @ A @ ( suc @ N ) )
        = ( power_power_real @ B2 @ ( suc @ N ) ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
         => ( A = B2 ) ) ) ) ).

% power_inject_base
thf(fact_2845_power__inject__base,axiom,
    ! [A: rat,N: nat,B2: rat] :
      ( ( ( power_power_rat @ A @ ( suc @ N ) )
        = ( power_power_rat @ B2 @ ( suc @ N ) ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
         => ( A = B2 ) ) ) ) ).

% power_inject_base
thf(fact_2846_power__inject__base,axiom,
    ! [A: nat,N: nat,B2: nat] :
      ( ( ( power_power_nat @ A @ ( suc @ N ) )
        = ( power_power_nat @ B2 @ ( suc @ N ) ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
         => ( A = B2 ) ) ) ) ).

% power_inject_base
thf(fact_2847_power__inject__base,axiom,
    ! [A: int,N: nat,B2: int] :
      ( ( ( power_power_int @ A @ ( suc @ N ) )
        = ( power_power_int @ B2 @ ( suc @ N ) ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
         => ( A = B2 ) ) ) ) ).

% power_inject_base
thf(fact_2848_div__add__self1,axiom,
    ! [B2: nat,A: nat] :
      ( ( B2 != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ B2 @ A ) @ B2 )
        = ( plus_plus_nat @ ( divide_divide_nat @ A @ B2 ) @ one_one_nat ) ) ) ).

% div_add_self1
thf(fact_2849_div__add__self1,axiom,
    ! [B2: int,A: int] :
      ( ( B2 != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ B2 @ A ) @ B2 )
        = ( plus_plus_int @ ( divide_divide_int @ A @ B2 ) @ one_one_int ) ) ) ).

% div_add_self1
thf(fact_2850_div__add__self2,axiom,
    ! [B2: nat,A: nat] :
      ( ( B2 != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B2 ) @ B2 )
        = ( plus_plus_nat @ ( divide_divide_nat @ A @ B2 ) @ one_one_nat ) ) ) ).

% div_add_self2
thf(fact_2851_div__add__self2,axiom,
    ! [B2: int,A: int] :
      ( ( B2 != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ A @ B2 ) @ B2 )
        = ( plus_plus_int @ ( divide_divide_int @ A @ B2 ) @ one_one_int ) ) ) ).

% div_add_self2
thf(fact_2852_add__divide__eq__if__simps_I4_J,axiom,
    ! [Z: complex,A: complex,B2: complex] :
      ( ( ( Z = zero_zero_complex )
       => ( ( minus_minus_complex @ A @ ( divide1717551699836669952omplex @ B2 @ Z ) )
          = A ) )
      & ( ( Z != zero_zero_complex )
       => ( ( minus_minus_complex @ A @ ( divide1717551699836669952omplex @ B2 @ Z ) )
          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ A @ Z ) @ B2 ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(4)
thf(fact_2853_add__divide__eq__if__simps_I4_J,axiom,
    ! [Z: real,A: real,B2: real] :
      ( ( ( Z = zero_zero_real )
       => ( ( minus_minus_real @ A @ ( divide_divide_real @ B2 @ Z ) )
          = A ) )
      & ( ( Z != zero_zero_real )
       => ( ( minus_minus_real @ A @ ( divide_divide_real @ B2 @ Z ) )
          = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ A @ Z ) @ B2 ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(4)
thf(fact_2854_add__divide__eq__if__simps_I4_J,axiom,
    ! [Z: rat,A: rat,B2: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( ( minus_minus_rat @ A @ ( divide_divide_rat @ B2 @ Z ) )
          = A ) )
      & ( ( Z != zero_zero_rat )
       => ( ( minus_minus_rat @ A @ ( divide_divide_rat @ B2 @ Z ) )
          = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ A @ Z ) @ B2 ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(4)
thf(fact_2855_diff__frac__eq,axiom,
    ! [Y3: complex,Z: complex,X4: complex,W: complex] :
      ( ( Y3 != zero_zero_complex )
     => ( ( Z != zero_zero_complex )
       => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ X4 @ Y3 ) @ ( divide1717551699836669952omplex @ W @ Z ) )
          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ X4 @ Z ) @ ( times_times_complex @ W @ Y3 ) ) @ ( times_times_complex @ Y3 @ Z ) ) ) ) ) ).

% diff_frac_eq
thf(fact_2856_diff__frac__eq,axiom,
    ! [Y3: real,Z: real,X4: real,W: real] :
      ( ( Y3 != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( ( minus_minus_real @ ( divide_divide_real @ X4 @ Y3 ) @ ( divide_divide_real @ W @ Z ) )
          = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X4 @ Z ) @ ( times_times_real @ W @ Y3 ) ) @ ( times_times_real @ Y3 @ Z ) ) ) ) ) ).

% diff_frac_eq
thf(fact_2857_diff__frac__eq,axiom,
    ! [Y3: rat,Z: rat,X4: rat,W: rat] :
      ( ( Y3 != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( ( minus_minus_rat @ ( divide_divide_rat @ X4 @ Y3 ) @ ( divide_divide_rat @ W @ Z ) )
          = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X4 @ Z ) @ ( times_times_rat @ W @ Y3 ) ) @ ( times_times_rat @ Y3 @ Z ) ) ) ) ) ).

% diff_frac_eq
thf(fact_2858_diff__divide__eq__iff,axiom,
    ! [Z: complex,X4: complex,Y3: complex] :
      ( ( Z != zero_zero_complex )
     => ( ( minus_minus_complex @ X4 @ ( divide1717551699836669952omplex @ Y3 @ Z ) )
        = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ X4 @ Z ) @ Y3 ) @ Z ) ) ) ).

% diff_divide_eq_iff
thf(fact_2859_diff__divide__eq__iff,axiom,
    ! [Z: real,X4: real,Y3: real] :
      ( ( Z != zero_zero_real )
     => ( ( minus_minus_real @ X4 @ ( divide_divide_real @ Y3 @ Z ) )
        = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X4 @ Z ) @ Y3 ) @ Z ) ) ) ).

% diff_divide_eq_iff
thf(fact_2860_diff__divide__eq__iff,axiom,
    ! [Z: rat,X4: rat,Y3: rat] :
      ( ( Z != zero_zero_rat )
     => ( ( minus_minus_rat @ X4 @ ( divide_divide_rat @ Y3 @ Z ) )
        = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X4 @ Z ) @ Y3 ) @ Z ) ) ) ).

% diff_divide_eq_iff
thf(fact_2861_divide__diff__eq__iff,axiom,
    ! [Z: complex,X4: complex,Y3: complex] :
      ( ( Z != zero_zero_complex )
     => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ X4 @ Z ) @ Y3 )
        = ( divide1717551699836669952omplex @ ( minus_minus_complex @ X4 @ ( times_times_complex @ Y3 @ Z ) ) @ Z ) ) ) ).

% divide_diff_eq_iff
thf(fact_2862_divide__diff__eq__iff,axiom,
    ! [Z: real,X4: real,Y3: real] :
      ( ( Z != zero_zero_real )
     => ( ( minus_minus_real @ ( divide_divide_real @ X4 @ Z ) @ Y3 )
        = ( divide_divide_real @ ( minus_minus_real @ X4 @ ( times_times_real @ Y3 @ Z ) ) @ Z ) ) ) ).

% divide_diff_eq_iff
thf(fact_2863_divide__diff__eq__iff,axiom,
    ! [Z: rat,X4: rat,Y3: rat] :
      ( ( Z != zero_zero_rat )
     => ( ( minus_minus_rat @ ( divide_divide_rat @ X4 @ Z ) @ Y3 )
        = ( divide_divide_rat @ ( minus_minus_rat @ X4 @ ( times_times_rat @ Y3 @ Z ) ) @ Z ) ) ) ).

% divide_diff_eq_iff
thf(fact_2864_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B2: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B2 )
     => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( modulo364778990260209775nteger @ A @ B2 ) ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
thf(fact_2865_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B2: nat,A: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B2 )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( modulo_modulo_nat @ A @ B2 ) ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
thf(fact_2866_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B2: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B2 )
     => ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ A @ B2 ) ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
thf(fact_2867_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le6747313008572928689nteger @ A @ B2 )
       => ( ( modulo364778990260209775nteger @ A @ B2 )
          = A ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
thf(fact_2868_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ A @ B2 )
       => ( ( modulo_modulo_nat @ A @ B2 )
          = A ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
thf(fact_2869_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ A @ B2 )
       => ( ( modulo_modulo_int @ A @ B2 )
          = A ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
thf(fact_2870_cong__exp__iff__simps_I2_J,axiom,
    ! [N: num,Q2: num] :
      ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
        = zero_zero_nat )
      = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q2 ) )
        = zero_zero_nat ) ) ).

% cong_exp_iff_simps(2)
thf(fact_2871_cong__exp__iff__simps_I2_J,axiom,
    ! [N: num,Q2: num] :
      ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
        = zero_zero_int )
      = ( ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q2 ) )
        = zero_zero_int ) ) ).

% cong_exp_iff_simps(2)
thf(fact_2872_cong__exp__iff__simps_I2_J,axiom,
    ! [N: num,Q2: num] :
      ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
        = zero_z3403309356797280102nteger )
      = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q2 ) )
        = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(2)
thf(fact_2873_cong__exp__iff__simps_I1_J,axiom,
    ! [N: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ one ) )
      = zero_zero_nat ) ).

% cong_exp_iff_simps(1)
thf(fact_2874_cong__exp__iff__simps_I1_J,axiom,
    ! [N: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ one ) )
      = zero_zero_int ) ).

% cong_exp_iff_simps(1)
thf(fact_2875_cong__exp__iff__simps_I1_J,axiom,
    ! [N: num] :
      ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ one ) )
      = zero_z3403309356797280102nteger ) ).

% cong_exp_iff_simps(1)
thf(fact_2876_numeral__1__eq__Suc__0,axiom,
    ( ( numeral_numeral_nat @ one )
    = ( suc @ zero_zero_nat ) ) ).

% numeral_1_eq_Suc_0
thf(fact_2877_ex__least__nat__less,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ N )
     => ( ~ ( P @ zero_zero_nat )
       => ? [K2: nat] :
            ( ( ord_less_nat @ K2 @ N )
            & ! [I4: nat] :
                ( ( ord_less_eq_nat @ I4 @ K2 )
               => ~ ( P @ I4 ) )
            & ( P @ ( suc @ K2 ) ) ) ) ) ).

% ex_least_nat_less
thf(fact_2878_diff__Suc__less,axiom,
    ! [N: nat,I: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I ) ) @ N ) ) ).

% diff_Suc_less
thf(fact_2879_n__less__n__mult__m,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
       => ( ord_less_nat @ N @ ( times_times_nat @ N @ M ) ) ) ) ).

% n_less_n_mult_m
thf(fact_2880_n__less__m__mult__n,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
       => ( ord_less_nat @ N @ ( times_times_nat @ M @ N ) ) ) ) ).

% n_less_m_mult_n
thf(fact_2881_one__less__mult,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
     => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
       => ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) ) ) ) ).

% one_less_mult
thf(fact_2882_realpow__pos__nth2,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ? [R3: real] :
          ( ( ord_less_real @ zero_zero_real @ R3 )
          & ( ( power_power_real @ R3 @ ( suc @ N ) )
            = A ) ) ) ).

% realpow_pos_nth2
thf(fact_2883_nat__induct__non__zero,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( P @ one_one_nat )
       => ( ! [N4: nat] :
              ( ( ord_less_nat @ zero_zero_nat @ N4 )
             => ( ( P @ N4 )
               => ( P @ ( suc @ N4 ) ) ) )
         => ( P @ N ) ) ) ) ).

% nat_induct_non_zero
thf(fact_2884_nat__mult__le__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
        = ( ord_less_eq_nat @ M @ N ) ) ) ).

% nat_mult_le_cancel1
thf(fact_2885_div__greater__zero__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M @ N ) )
      = ( ( ord_less_eq_nat @ N @ M )
        & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% div_greater_zero_iff
thf(fact_2886_div__le__mono2,axiom,
    ! [M: nat,N: nat,K: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_eq_nat @ M @ N )
       => ( ord_less_eq_nat @ ( divide_divide_nat @ K @ N ) @ ( divide_divide_nat @ K @ M ) ) ) ) ).

% div_le_mono2
thf(fact_2887_power__gt__expt,axiom,
    ! [N: nat,K: nat] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
     => ( ord_less_nat @ K @ ( power_power_nat @ N @ K ) ) ) ).

% power_gt_expt
thf(fact_2888_nat__diff__split,axiom,
    ! [P: nat > $o,A: nat,B2: nat] :
      ( ( P @ ( minus_minus_nat @ A @ B2 ) )
      = ( ( ( ord_less_nat @ A @ B2 )
         => ( P @ zero_zero_nat ) )
        & ! [D2: nat] :
            ( ( A
              = ( plus_plus_nat @ B2 @ D2 ) )
           => ( P @ D2 ) ) ) ) ).

% nat_diff_split
thf(fact_2889_nat__diff__split__asm,axiom,
    ! [P: nat > $o,A: nat,B2: nat] :
      ( ( P @ ( minus_minus_nat @ A @ B2 ) )
      = ( ~ ( ( ( ord_less_nat @ A @ B2 )
              & ~ ( P @ zero_zero_nat ) )
            | ? [D2: nat] :
                ( ( A
                  = ( plus_plus_nat @ B2 @ D2 ) )
                & ~ ( P @ D2 ) ) ) ) ) ).

% nat_diff_split_asm
thf(fact_2890_nat__one__le__power,axiom,
    ! [I: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ I )
     => ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( power_power_nat @ I @ N ) ) ) ).

% nat_one_le_power
thf(fact_2891_real__arch__pow__inv,axiom,
    ! [Y3: real,X4: real] :
      ( ( ord_less_real @ zero_zero_real @ Y3 )
     => ( ( ord_less_real @ X4 @ one_one_real )
       => ? [N4: nat] : ( ord_less_real @ ( power_power_real @ X4 @ N4 ) @ Y3 ) ) ) ).

% real_arch_pow_inv
thf(fact_2892_div__less__iff__less__mult,axiom,
    ! [Q2: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ Q2 )
     => ( ( ord_less_nat @ ( divide_divide_nat @ M @ Q2 ) @ N )
        = ( ord_less_nat @ M @ ( times_times_nat @ N @ Q2 ) ) ) ) ).

% div_less_iff_less_mult
thf(fact_2893_nat__mult__div__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
        = ( divide_divide_nat @ M @ N ) ) ) ).

% nat_mult_div_cancel1
thf(fact_2894_div__eq__dividend__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ( divide_divide_nat @ M @ N )
          = M )
        = ( N = one_one_nat ) ) ) ).

% div_eq_dividend_iff
thf(fact_2895_div__less__dividend,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ one_one_nat @ N )
     => ( ( ord_less_nat @ zero_zero_nat @ M )
       => ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ M ) ) ) ).

% div_less_dividend
thf(fact_2896_mod__le__divisor,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ N ) @ N ) ) ).

% mod_le_divisor
thf(fact_2897_div__less__mono,axiom,
    ! [A3: nat,B4: nat,N: nat] :
      ( ( ord_less_nat @ A3 @ B4 )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( ( modulo_modulo_nat @ A3 @ N )
            = zero_zero_nat )
         => ( ( ( modulo_modulo_nat @ B4 @ N )
              = zero_zero_nat )
           => ( ord_less_nat @ ( divide_divide_nat @ A3 @ N ) @ ( divide_divide_nat @ B4 @ N ) ) ) ) ) ) ).

% div_less_mono
thf(fact_2898_evenE,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ~ ! [B5: code_integer] :
            ( A
           != ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B5 ) ) ) ).

% evenE
thf(fact_2899_evenE,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ~ ! [B5: nat] :
            ( A
           != ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B5 ) ) ) ).

% evenE
thf(fact_2900_evenE,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ~ ! [B5: int] :
            ( A
           != ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B5 ) ) ) ).

% evenE
thf(fact_2901_odd__one,axiom,
    ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ one_one_Code_integer ) ).

% odd_one
thf(fact_2902_odd__one,axiom,
    ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ one_one_nat ) ).

% odd_one
thf(fact_2903_odd__one,axiom,
    ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ one_one_int ) ).

% odd_one
thf(fact_2904_odd__even__add,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 )
       => ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B2 ) ) ) ) ).

% odd_even_add
thf(fact_2905_odd__even__add,axiom,
    ! [A: nat,B2: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 )
       => ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B2 ) ) ) ) ).

% odd_even_add
thf(fact_2906_odd__even__add,axiom,
    ! [A: int,B2: int] :
      ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 )
       => ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B2 ) ) ) ) ).

% odd_even_add
thf(fact_2907_bit__eq__rec,axiom,
    ( ( ^ [Y6: code_integer,Z4: code_integer] : Y6 = Z4 )
    = ( ^ [A4: code_integer,B3: code_integer] :
          ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A4 )
            = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B3 ) )
          & ( ( divide6298287555418463151nteger @ A4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
            = ( divide6298287555418463151nteger @ B3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% bit_eq_rec
thf(fact_2908_bit__eq__rec,axiom,
    ( ( ^ [Y6: nat,Z4: nat] : Y6 = Z4 )
    = ( ^ [A4: nat,B3: nat] :
          ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A4 )
            = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 ) )
          & ( ( divide_divide_nat @ A4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = ( divide_divide_nat @ B3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% bit_eq_rec
thf(fact_2909_bit__eq__rec,axiom,
    ( ( ^ [Y6: int,Z4: int] : Y6 = Z4 )
    = ( ^ [A4: int,B3: int] :
          ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A4 )
            = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) )
          & ( ( divide_divide_int @ A4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
            = ( divide_divide_int @ B3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).

% bit_eq_rec
thf(fact_2910_even__mask__div__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ one_one_Code_integer ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) )
      = ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N )
          = zero_z3403309356797280102nteger )
        | ( ord_less_eq_nat @ M @ N ) ) ) ).

% even_mask_div_iff
thf(fact_2911_even__mask__div__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ one_one_nat ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
          = zero_zero_nat )
        | ( ord_less_eq_nat @ M @ N ) ) ) ).

% even_mask_div_iff
thf(fact_2912_even__mask__div__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
      = ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
          = zero_zero_int )
        | ( ord_less_eq_nat @ M @ N ) ) ) ).

% even_mask_div_iff
thf(fact_2913_dvd__minus__add,axiom,
    ! [Q2: nat,N: nat,R2: nat,M: nat] :
      ( ( ord_less_eq_nat @ Q2 @ N )
     => ( ( ord_less_eq_nat @ Q2 @ ( times_times_nat @ R2 @ M ) )
       => ( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ Q2 ) )
          = ( dvd_dvd_nat @ M @ ( plus_plus_nat @ N @ ( minus_minus_nat @ ( times_times_nat @ R2 @ M ) @ Q2 ) ) ) ) ) ) ).

% dvd_minus_add
thf(fact_2914_power__dvd__imp__le,axiom,
    ! [I: nat,M: nat,N: nat] :
      ( ( dvd_dvd_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N ) )
     => ( ( ord_less_nat @ one_one_nat @ I )
       => ( ord_less_eq_nat @ M @ N ) ) ) ).

% power_dvd_imp_le
thf(fact_2915_mod__nat__eqI,axiom,
    ! [R2: nat,N: nat,M: nat] :
      ( ( ord_less_nat @ R2 @ N )
     => ( ( ord_less_eq_nat @ R2 @ M )
       => ( ( dvd_dvd_nat @ N @ ( minus_minus_nat @ M @ R2 ) )
         => ( ( modulo_modulo_nat @ M @ N )
            = R2 ) ) ) ) ).

% mod_nat_eqI
thf(fact_2916_mult__less__cancel__right2,axiom,
    ! [A: real,C: real] :
      ( ( ord_less_real @ ( times_times_real @ A @ C ) @ C )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
         => ( ord_less_real @ A @ one_one_real ) )
        & ( ( ord_less_eq_real @ C @ zero_zero_real )
         => ( ord_less_real @ one_one_real @ A ) ) ) ) ).

% mult_less_cancel_right2
thf(fact_2917_mult__less__cancel__right2,axiom,
    ! [A: rat,C: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ C )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ A @ one_one_rat ) )
        & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ one_one_rat @ A ) ) ) ) ).

% mult_less_cancel_right2
thf(fact_2918_mult__less__cancel__right2,axiom,
    ! [A: int,C: int] :
      ( ( ord_less_int @ ( times_times_int @ A @ C ) @ C )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ A @ one_one_int ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ one_one_int @ A ) ) ) ) ).

% mult_less_cancel_right2
thf(fact_2919_mult__less__cancel__right1,axiom,
    ! [C: real,B2: real] :
      ( ( ord_less_real @ C @ ( times_times_real @ B2 @ C ) )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
         => ( ord_less_real @ one_one_real @ B2 ) )
        & ( ( ord_less_eq_real @ C @ zero_zero_real )
         => ( ord_less_real @ B2 @ one_one_real ) ) ) ) ).

% mult_less_cancel_right1
thf(fact_2920_mult__less__cancel__right1,axiom,
    ! [C: rat,B2: rat] :
      ( ( ord_less_rat @ C @ ( times_times_rat @ B2 @ C ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ one_one_rat @ B2 ) )
        & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ B2 @ one_one_rat ) ) ) ) ).

% mult_less_cancel_right1
thf(fact_2921_mult__less__cancel__right1,axiom,
    ! [C: int,B2: int] :
      ( ( ord_less_int @ C @ ( times_times_int @ B2 @ C ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ one_one_int @ B2 ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ B2 @ one_one_int ) ) ) ) ).

% mult_less_cancel_right1
thf(fact_2922_mult__less__cancel__left2,axiom,
    ! [C: real,A: real] :
      ( ( ord_less_real @ ( times_times_real @ C @ A ) @ C )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
         => ( ord_less_real @ A @ one_one_real ) )
        & ( ( ord_less_eq_real @ C @ zero_zero_real )
         => ( ord_less_real @ one_one_real @ A ) ) ) ) ).

% mult_less_cancel_left2
thf(fact_2923_mult__less__cancel__left2,axiom,
    ! [C: rat,A: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ C )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ A @ one_one_rat ) )
        & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ one_one_rat @ A ) ) ) ) ).

% mult_less_cancel_left2
thf(fact_2924_mult__less__cancel__left2,axiom,
    ! [C: int,A: int] :
      ( ( ord_less_int @ ( times_times_int @ C @ A ) @ C )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ A @ one_one_int ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ one_one_int @ A ) ) ) ) ).

% mult_less_cancel_left2
thf(fact_2925_mult__less__cancel__left1,axiom,
    ! [C: real,B2: real] :
      ( ( ord_less_real @ C @ ( times_times_real @ C @ B2 ) )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
         => ( ord_less_real @ one_one_real @ B2 ) )
        & ( ( ord_less_eq_real @ C @ zero_zero_real )
         => ( ord_less_real @ B2 @ one_one_real ) ) ) ) ).

% mult_less_cancel_left1
thf(fact_2926_mult__less__cancel__left1,axiom,
    ! [C: rat,B2: rat] :
      ( ( ord_less_rat @ C @ ( times_times_rat @ C @ B2 ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ one_one_rat @ B2 ) )
        & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ B2 @ one_one_rat ) ) ) ) ).

% mult_less_cancel_left1
thf(fact_2927_mult__less__cancel__left1,axiom,
    ! [C: int,B2: int] :
      ( ( ord_less_int @ C @ ( times_times_int @ C @ B2 ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ one_one_int @ B2 ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ B2 @ one_one_int ) ) ) ) ).

% mult_less_cancel_left1
thf(fact_2928_mult__le__cancel__right2,axiom,
    ! [A: real,C: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ C )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ A @ one_one_real ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_eq_real @ one_one_real @ A ) ) ) ) ).

% mult_le_cancel_right2
thf(fact_2929_mult__le__cancel__right2,axiom,
    ! [A: rat,C: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ C )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ A @ one_one_rat ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ one_one_rat @ A ) ) ) ) ).

% mult_le_cancel_right2
thf(fact_2930_mult__le__cancel__right2,axiom,
    ! [A: int,C: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ C )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ A @ one_one_int ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).

% mult_le_cancel_right2
thf(fact_2931_mult__le__cancel__right1,axiom,
    ! [C: real,B2: real] :
      ( ( ord_less_eq_real @ C @ ( times_times_real @ B2 @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ one_one_real @ B2 ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_eq_real @ B2 @ one_one_real ) ) ) ) ).

% mult_le_cancel_right1
thf(fact_2932_mult__le__cancel__right1,axiom,
    ! [C: rat,B2: rat] :
      ( ( ord_less_eq_rat @ C @ ( times_times_rat @ B2 @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ one_one_rat @ B2 ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ B2 @ one_one_rat ) ) ) ) ).

% mult_le_cancel_right1
thf(fact_2933_mult__le__cancel__right1,axiom,
    ! [C: int,B2: int] :
      ( ( ord_less_eq_int @ C @ ( times_times_int @ B2 @ C ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ one_one_int @ B2 ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ B2 @ one_one_int ) ) ) ) ).

% mult_le_cancel_right1
thf(fact_2934_mult__le__cancel__left2,axiom,
    ! [C: real,A: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ C )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ A @ one_one_real ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_eq_real @ one_one_real @ A ) ) ) ) ).

% mult_le_cancel_left2
thf(fact_2935_mult__le__cancel__left2,axiom,
    ! [C: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ C )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ A @ one_one_rat ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ one_one_rat @ A ) ) ) ) ).

% mult_le_cancel_left2
thf(fact_2936_mult__le__cancel__left2,axiom,
    ! [C: int,A: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ C )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ A @ one_one_int ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).

% mult_le_cancel_left2
thf(fact_2937_mult__le__cancel__left1,axiom,
    ! [C: real,B2: real] :
      ( ( ord_less_eq_real @ C @ ( times_times_real @ C @ B2 ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ one_one_real @ B2 ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_eq_real @ B2 @ one_one_real ) ) ) ) ).

% mult_le_cancel_left1
thf(fact_2938_mult__le__cancel__left1,axiom,
    ! [C: rat,B2: rat] :
      ( ( ord_less_eq_rat @ C @ ( times_times_rat @ C @ B2 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ one_one_rat @ B2 ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ B2 @ one_one_rat ) ) ) ) ).

% mult_le_cancel_left1
thf(fact_2939_mult__le__cancel__left1,axiom,
    ! [C: int,B2: int] :
      ( ( ord_less_eq_int @ C @ ( times_times_int @ C @ B2 ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ one_one_int @ B2 ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ B2 @ one_one_int ) ) ) ) ).

% mult_le_cancel_left1
thf(fact_2940_field__le__mult__one__interval,axiom,
    ! [X4: real,Y3: real] :
      ( ! [Z2: real] :
          ( ( ord_less_real @ zero_zero_real @ Z2 )
         => ( ( ord_less_real @ Z2 @ one_one_real )
           => ( ord_less_eq_real @ ( times_times_real @ Z2 @ X4 ) @ Y3 ) ) )
     => ( ord_less_eq_real @ X4 @ Y3 ) ) ).

% field_le_mult_one_interval
thf(fact_2941_field__le__mult__one__interval,axiom,
    ! [X4: rat,Y3: rat] :
      ( ! [Z2: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ Z2 )
         => ( ( ord_less_rat @ Z2 @ one_one_rat )
           => ( ord_less_eq_rat @ ( times_times_rat @ Z2 @ X4 ) @ Y3 ) ) )
     => ( ord_less_eq_rat @ X4 @ Y3 ) ) ).

% field_le_mult_one_interval
thf(fact_2942_divide__left__mono__neg,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ord_less_eq_real @ A @ B2 )
     => ( ( ord_less_eq_real @ C @ zero_zero_real )
       => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) )
         => ( ord_less_eq_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B2 ) ) ) ) ) ).

% divide_left_mono_neg
thf(fact_2943_divide__left__mono__neg,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B2 )
     => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
       => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B2 ) )
         => ( ord_less_eq_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B2 ) ) ) ) ) ).

% divide_left_mono_neg
thf(fact_2944_mult__imp__le__div__pos,axiom,
    ! [Y3: real,Z: real,X4: real] :
      ( ( ord_less_real @ zero_zero_real @ Y3 )
     => ( ( ord_less_eq_real @ ( times_times_real @ Z @ Y3 ) @ X4 )
       => ( ord_less_eq_real @ Z @ ( divide_divide_real @ X4 @ Y3 ) ) ) ) ).

% mult_imp_le_div_pos
thf(fact_2945_mult__imp__le__div__pos,axiom,
    ! [Y3: rat,Z: rat,X4: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Y3 )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ Z @ Y3 ) @ X4 )
       => ( ord_less_eq_rat @ Z @ ( divide_divide_rat @ X4 @ Y3 ) ) ) ) ).

% mult_imp_le_div_pos
thf(fact_2946_mult__imp__div__pos__le,axiom,
    ! [Y3: real,X4: real,Z: real] :
      ( ( ord_less_real @ zero_zero_real @ Y3 )
     => ( ( ord_less_eq_real @ X4 @ ( times_times_real @ Z @ Y3 ) )
       => ( ord_less_eq_real @ ( divide_divide_real @ X4 @ Y3 ) @ Z ) ) ) ).

% mult_imp_div_pos_le
thf(fact_2947_mult__imp__div__pos__le,axiom,
    ! [Y3: rat,X4: rat,Z: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Y3 )
     => ( ( ord_less_eq_rat @ X4 @ ( times_times_rat @ Z @ Y3 ) )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ X4 @ Y3 ) @ Z ) ) ) ).

% mult_imp_div_pos_le
thf(fact_2948_pos__le__divide__eq,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B2 @ C ) )
        = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B2 ) ) ) ).

% pos_le_divide_eq
thf(fact_2949_pos__le__divide__eq,axiom,
    ! [C: rat,A: rat,B2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B2 @ C ) )
        = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B2 ) ) ) ).

% pos_le_divide_eq
thf(fact_2950_pos__divide__le__eq,axiom,
    ! [C: real,B2: real,A: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_eq_real @ ( divide_divide_real @ B2 @ C ) @ A )
        = ( ord_less_eq_real @ B2 @ ( times_times_real @ A @ C ) ) ) ) ).

% pos_divide_le_eq
thf(fact_2951_pos__divide__le__eq,axiom,
    ! [C: rat,B2: rat,A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B2 @ C ) @ A )
        = ( ord_less_eq_rat @ B2 @ ( times_times_rat @ A @ C ) ) ) ) ).

% pos_divide_le_eq
thf(fact_2952_neg__le__divide__eq,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B2 @ C ) )
        = ( ord_less_eq_real @ B2 @ ( times_times_real @ A @ C ) ) ) ) ).

% neg_le_divide_eq
thf(fact_2953_neg__le__divide__eq,axiom,
    ! [C: rat,A: rat,B2: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B2 @ C ) )
        = ( ord_less_eq_rat @ B2 @ ( times_times_rat @ A @ C ) ) ) ) ).

% neg_le_divide_eq
thf(fact_2954_neg__divide__le__eq,axiom,
    ! [C: real,B2: real,A: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_eq_real @ ( divide_divide_real @ B2 @ C ) @ A )
        = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B2 ) ) ) ).

% neg_divide_le_eq
thf(fact_2955_neg__divide__le__eq,axiom,
    ! [C: rat,B2: rat,A: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B2 @ C ) @ A )
        = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B2 ) ) ) ).

% neg_divide_le_eq
thf(fact_2956_divide__left__mono,axiom,
    ! [B2: real,A: real,C: real] :
      ( ( ord_less_eq_real @ B2 @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) )
         => ( ord_less_eq_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B2 ) ) ) ) ) ).

% divide_left_mono
thf(fact_2957_divide__left__mono,axiom,
    ! [B2: rat,A: rat,C: rat] :
      ( ( ord_less_eq_rat @ B2 @ A )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B2 ) )
         => ( ord_less_eq_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B2 ) ) ) ) ) ).

% divide_left_mono
thf(fact_2958_le__divide__eq,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B2 @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B2 ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ B2 @ ( times_times_real @ A @ C ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ A @ zero_zero_real ) ) ) ) ) ) ).

% le_divide_eq
thf(fact_2959_le__divide__eq,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B2 @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B2 ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ B2 @ ( times_times_rat @ A @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).

% le_divide_eq
thf(fact_2960_divide__le__eq,axiom,
    ! [B2: real,C: real,A: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ B2 @ C ) @ A )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ B2 @ ( times_times_real @ A @ C ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B2 ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ) ) ).

% divide_le_eq
thf(fact_2961_divide__le__eq,axiom,
    ! [B2: rat,C: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B2 @ C ) @ A )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ B2 @ ( times_times_rat @ A @ C ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B2 ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).

% divide_le_eq
thf(fact_2962_le__divide__eq__1,axiom,
    ! [B2: real,A: real] :
      ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B2 @ A ) )
      = ( ( ( ord_less_real @ zero_zero_real @ A )
          & ( ord_less_eq_real @ A @ B2 ) )
        | ( ( ord_less_real @ A @ zero_zero_real )
          & ( ord_less_eq_real @ B2 @ A ) ) ) ) ).

% le_divide_eq_1
thf(fact_2963_le__divide__eq__1,axiom,
    ! [B2: rat,A: rat] :
      ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B2 @ A ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A )
          & ( ord_less_eq_rat @ A @ B2 ) )
        | ( ( ord_less_rat @ A @ zero_zero_rat )
          & ( ord_less_eq_rat @ B2 @ A ) ) ) ) ).

% le_divide_eq_1
thf(fact_2964_divide__le__eq__1,axiom,
    ! [B2: real,A: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ B2 @ A ) @ one_one_real )
      = ( ( ( ord_less_real @ zero_zero_real @ A )
          & ( ord_less_eq_real @ B2 @ A ) )
        | ( ( ord_less_real @ A @ zero_zero_real )
          & ( ord_less_eq_real @ A @ B2 ) )
        | ( A = zero_zero_real ) ) ) ).

% divide_le_eq_1
thf(fact_2965_divide__le__eq__1,axiom,
    ! [B2: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B2 @ A ) @ one_one_rat )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A )
          & ( ord_less_eq_rat @ B2 @ A ) )
        | ( ( ord_less_rat @ A @ zero_zero_rat )
          & ( ord_less_eq_rat @ A @ B2 ) )
        | ( A = zero_zero_rat ) ) ) ).

% divide_le_eq_1
thf(fact_2966_convex__bound__le,axiom,
    ! [X4: real,A: real,Y3: real,U: real,V: real] :
      ( ( ord_less_eq_real @ X4 @ A )
     => ( ( ord_less_eq_real @ Y3 @ A )
       => ( ( ord_less_eq_real @ zero_zero_real @ U )
         => ( ( ord_less_eq_real @ zero_zero_real @ V )
           => ( ( ( plus_plus_real @ U @ V )
                = one_one_real )
             => ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ U @ X4 ) @ ( times_times_real @ V @ Y3 ) ) @ A ) ) ) ) ) ) ).

% convex_bound_le
thf(fact_2967_convex__bound__le,axiom,
    ! [X4: rat,A: rat,Y3: rat,U: rat,V: rat] :
      ( ( ord_less_eq_rat @ X4 @ A )
     => ( ( ord_less_eq_rat @ Y3 @ A )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ U )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ V )
           => ( ( ( plus_plus_rat @ U @ V )
                = one_one_rat )
             => ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ U @ X4 ) @ ( times_times_rat @ V @ Y3 ) ) @ A ) ) ) ) ) ) ).

% convex_bound_le
thf(fact_2968_convex__bound__le,axiom,
    ! [X4: int,A: int,Y3: int,U: int,V: int] :
      ( ( ord_less_eq_int @ X4 @ A )
     => ( ( ord_less_eq_int @ Y3 @ A )
       => ( ( ord_less_eq_int @ zero_zero_int @ U )
         => ( ( ord_less_eq_int @ zero_zero_int @ V )
           => ( ( ( plus_plus_int @ U @ V )
                = one_one_int )
             => ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ U @ X4 ) @ ( times_times_int @ V @ Y3 ) ) @ A ) ) ) ) ) ) ).

% convex_bound_le
thf(fact_2969_divide__less__eq__numeral_I1_J,axiom,
    ! [B2: real,C: real,W: num] :
      ( ( ord_less_real @ ( divide_divide_real @ B2 @ C ) @ ( numeral_numeral_real @ W ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ B2 @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B2 ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ).

% divide_less_eq_numeral(1)
thf(fact_2970_divide__less__eq__numeral_I1_J,axiom,
    ! [B2: rat,C: rat,W: num] :
      ( ( ord_less_rat @ ( divide_divide_rat @ B2 @ C ) @ ( numeral_numeral_rat @ W ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ B2 @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B2 ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ zero_zero_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ).

% divide_less_eq_numeral(1)
thf(fact_2971_less__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B2: real,C: real] :
      ( ( ord_less_real @ ( numeral_numeral_real @ W ) @ ( divide_divide_real @ B2 @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B2 ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ B2 @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ ( numeral_numeral_real @ W ) @ zero_zero_real ) ) ) ) ) ) ).

% less_divide_eq_numeral(1)
thf(fact_2972_less__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B2: rat,C: rat] :
      ( ( ord_less_rat @ ( numeral_numeral_rat @ W ) @ ( divide_divide_rat @ B2 @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B2 ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ B2 @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ ( numeral_numeral_rat @ W ) @ zero_zero_rat ) ) ) ) ) ) ).

% less_divide_eq_numeral(1)
thf(fact_2973_frac__le__eq,axiom,
    ! [Y3: real,Z: real,X4: real,W: real] :
      ( ( Y3 != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( ( ord_less_eq_real @ ( divide_divide_real @ X4 @ Y3 ) @ ( divide_divide_real @ W @ Z ) )
          = ( ord_less_eq_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X4 @ Z ) @ ( times_times_real @ W @ Y3 ) ) @ ( times_times_real @ Y3 @ Z ) ) @ zero_zero_real ) ) ) ) ).

% frac_le_eq
thf(fact_2974_frac__le__eq,axiom,
    ! [Y3: rat,Z: rat,X4: rat,W: rat] :
      ( ( Y3 != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( ( ord_less_eq_rat @ ( divide_divide_rat @ X4 @ Y3 ) @ ( divide_divide_rat @ W @ Z ) )
          = ( ord_less_eq_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X4 @ Z ) @ ( times_times_rat @ W @ Y3 ) ) @ ( times_times_rat @ Y3 @ Z ) ) @ zero_zero_rat ) ) ) ) ).

% frac_le_eq
thf(fact_2975_frac__less__eq,axiom,
    ! [Y3: real,Z: real,X4: real,W: real] :
      ( ( Y3 != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( ( ord_less_real @ ( divide_divide_real @ X4 @ Y3 ) @ ( divide_divide_real @ W @ Z ) )
          = ( ord_less_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X4 @ Z ) @ ( times_times_real @ W @ Y3 ) ) @ ( times_times_real @ Y3 @ Z ) ) @ zero_zero_real ) ) ) ) ).

% frac_less_eq
thf(fact_2976_frac__less__eq,axiom,
    ! [Y3: rat,Z: rat,X4: rat,W: rat] :
      ( ( Y3 != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( ( ord_less_rat @ ( divide_divide_rat @ X4 @ Y3 ) @ ( divide_divide_rat @ W @ Z ) )
          = ( ord_less_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X4 @ Z ) @ ( times_times_rat @ W @ Y3 ) ) @ ( times_times_rat @ Y3 @ Z ) ) @ zero_zero_rat ) ) ) ) ).

% frac_less_eq
thf(fact_2977_power__Suc__less,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ A @ one_one_real )
       => ( ord_less_real @ ( times_times_real @ A @ ( power_power_real @ A @ N ) ) @ ( power_power_real @ A @ N ) ) ) ) ).

% power_Suc_less
thf(fact_2978_power__Suc__less,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ A @ one_one_rat )
       => ( ord_less_rat @ ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) @ ( power_power_rat @ A @ N ) ) ) ) ).

% power_Suc_less
thf(fact_2979_power__Suc__less,axiom,
    ! [A: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ A @ one_one_nat )
       => ( ord_less_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) @ ( power_power_nat @ A @ N ) ) ) ) ).

% power_Suc_less
thf(fact_2980_power__Suc__less,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ A @ one_one_int )
       => ( ord_less_int @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) @ ( power_power_int @ A @ N ) ) ) ) ).

% power_Suc_less
thf(fact_2981_power__Suc__le__self,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ A @ one_one_real )
       => ( ord_less_eq_real @ ( power_power_real @ A @ ( suc @ N ) ) @ A ) ) ) ).

% power_Suc_le_self
thf(fact_2982_power__Suc__le__self,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ A @ one_one_rat )
       => ( ord_less_eq_rat @ ( power_power_rat @ A @ ( suc @ N ) ) @ A ) ) ) ).

% power_Suc_le_self
thf(fact_2983_power__Suc__le__self,axiom,
    ! [A: nat,N: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ A @ one_one_nat )
       => ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ A ) ) ) ).

% power_Suc_le_self
thf(fact_2984_power__Suc__le__self,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ A @ one_one_int )
       => ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N ) ) @ A ) ) ) ).

% power_Suc_le_self
thf(fact_2985_power__Suc__less__one,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ A @ one_one_real )
       => ( ord_less_real @ ( power_power_real @ A @ ( suc @ N ) ) @ one_one_real ) ) ) ).

% power_Suc_less_one
thf(fact_2986_power__Suc__less__one,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ A @ one_one_rat )
       => ( ord_less_rat @ ( power_power_rat @ A @ ( suc @ N ) ) @ one_one_rat ) ) ) ).

% power_Suc_less_one
thf(fact_2987_power__Suc__less__one,axiom,
    ! [A: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ A @ one_one_nat )
       => ( ord_less_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ one_one_nat ) ) ) ).

% power_Suc_less_one
thf(fact_2988_power__Suc__less__one,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ A @ one_one_int )
       => ( ord_less_int @ ( power_power_int @ A @ ( suc @ N ) ) @ one_one_int ) ) ) ).

% power_Suc_less_one
thf(fact_2989_power__strict__decreasing,axiom,
    ! [N: nat,N3: nat,A: real] :
      ( ( ord_less_nat @ N @ N3 )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ( ( ord_less_real @ A @ one_one_real )
         => ( ord_less_real @ ( power_power_real @ A @ N3 ) @ ( power_power_real @ A @ N ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_2990_power__strict__decreasing,axiom,
    ! [N: nat,N3: nat,A: rat] :
      ( ( ord_less_nat @ N @ N3 )
     => ( ( ord_less_rat @ zero_zero_rat @ A )
       => ( ( ord_less_rat @ A @ one_one_rat )
         => ( ord_less_rat @ ( power_power_rat @ A @ N3 ) @ ( power_power_rat @ A @ N ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_2991_power__strict__decreasing,axiom,
    ! [N: nat,N3: nat,A: nat] :
      ( ( ord_less_nat @ N @ N3 )
     => ( ( ord_less_nat @ zero_zero_nat @ A )
       => ( ( ord_less_nat @ A @ one_one_nat )
         => ( ord_less_nat @ ( power_power_nat @ A @ N3 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_2992_power__strict__decreasing,axiom,
    ! [N: nat,N3: nat,A: int] :
      ( ( ord_less_nat @ N @ N3 )
     => ( ( ord_less_int @ zero_zero_int @ A )
       => ( ( ord_less_int @ A @ one_one_int )
         => ( ord_less_int @ ( power_power_int @ A @ N3 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_2993_power__decreasing,axiom,
    ! [N: nat,N3: nat,A: real] :
      ( ( ord_less_eq_nat @ N @ N3 )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ( ord_less_eq_real @ A @ one_one_real )
         => ( ord_less_eq_real @ ( power_power_real @ A @ N3 ) @ ( power_power_real @ A @ N ) ) ) ) ) ).

% power_decreasing
thf(fact_2994_power__decreasing,axiom,
    ! [N: nat,N3: nat,A: rat] :
      ( ( ord_less_eq_nat @ N @ N3 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
       => ( ( ord_less_eq_rat @ A @ one_one_rat )
         => ( ord_less_eq_rat @ ( power_power_rat @ A @ N3 ) @ ( power_power_rat @ A @ N ) ) ) ) ) ).

% power_decreasing
thf(fact_2995_power__decreasing,axiom,
    ! [N: nat,N3: nat,A: nat] :
      ( ( ord_less_eq_nat @ N @ N3 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ( ord_less_eq_nat @ A @ one_one_nat )
         => ( ord_less_eq_nat @ ( power_power_nat @ A @ N3 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).

% power_decreasing
thf(fact_2996_power__decreasing,axiom,
    ! [N: nat,N3: nat,A: int] :
      ( ( ord_less_eq_nat @ N @ N3 )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ( ord_less_eq_int @ A @ one_one_int )
         => ( ord_less_eq_int @ ( power_power_int @ A @ N3 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).

% power_decreasing
thf(fact_2997_zero__power2,axiom,
    ( ( power_power_rat @ zero_zero_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_rat ) ).

% zero_power2
thf(fact_2998_zero__power2,axiom,
    ( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_nat ) ).

% zero_power2
thf(fact_2999_zero__power2,axiom,
    ( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_real ) ).

% zero_power2
thf(fact_3000_zero__power2,axiom,
    ( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_int ) ).

% zero_power2
thf(fact_3001_zero__power2,axiom,
    ( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_complex ) ).

% zero_power2
thf(fact_3002_self__le__power,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_eq_real @ one_one_real @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_eq_real @ A @ ( power_power_real @ A @ N ) ) ) ) ).

% self_le_power
thf(fact_3003_self__le__power,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_eq_rat @ one_one_rat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_eq_rat @ A @ ( power_power_rat @ A @ N ) ) ) ) ).

% self_le_power
thf(fact_3004_self__le__power,axiom,
    ! [A: nat,N: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_eq_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).

% self_le_power
thf(fact_3005_self__le__power,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_eq_int @ one_one_int @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_eq_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).

% self_le_power
thf(fact_3006_one__less__power,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ) ).

% one_less_power
thf(fact_3007_one__less__power,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A @ N ) ) ) ) ).

% one_less_power
thf(fact_3008_one__less__power,axiom,
    ! [A: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ) ).

% one_less_power
thf(fact_3009_one__less__power,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_int @ one_one_int @ ( power_power_int @ A @ N ) ) ) ) ).

% one_less_power
thf(fact_3010_numeral__2__eq__2,axiom,
    ( ( numeral_numeral_nat @ ( bit0 @ one ) )
    = ( suc @ ( suc @ zero_zero_nat ) ) ) ).

% numeral_2_eq_2
thf(fact_3011_pos2,axiom,
    ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ).

% pos2
thf(fact_3012_power__diff,axiom,
    ! [A: complex,N: nat,M: nat] :
      ( ( A != zero_zero_complex )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( ( power_power_complex @ A @ ( minus_minus_nat @ M @ N ) )
          = ( divide1717551699836669952omplex @ ( power_power_complex @ A @ M ) @ ( power_power_complex @ A @ N ) ) ) ) ) ).

% power_diff
thf(fact_3013_power__diff,axiom,
    ! [A: real,N: nat,M: nat] :
      ( ( A != zero_zero_real )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( ( power_power_real @ A @ ( minus_minus_nat @ M @ N ) )
          = ( divide_divide_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) ) ) ) ) ).

% power_diff
thf(fact_3014_power__diff,axiom,
    ! [A: rat,N: nat,M: nat] :
      ( ( A != zero_zero_rat )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( ( power_power_rat @ A @ ( minus_minus_nat @ M @ N ) )
          = ( divide_divide_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N ) ) ) ) ) ).

% power_diff
thf(fact_3015_power__diff,axiom,
    ! [A: nat,N: nat,M: nat] :
      ( ( A != zero_zero_nat )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( ( power_power_nat @ A @ ( minus_minus_nat @ M @ N ) )
          = ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).

% power_diff
thf(fact_3016_power__diff,axiom,
    ! [A: int,N: nat,M: nat] :
      ( ( A != zero_zero_int )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( ( power_power_int @ A @ ( minus_minus_nat @ M @ N ) )
          = ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) ) ) ) ) ).

% power_diff
thf(fact_3017_even__mult__exp__div__exp__iff,axiom,
    ! [A: code_integer,M: nat,N: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) )
      = ( ( ord_less_nat @ N @ M )
        | ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N )
          = zero_z3403309356797280102nteger )
        | ( ( ord_less_eq_nat @ M @ N )
          & ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).

% even_mult_exp_div_exp_iff
thf(fact_3018_even__mult__exp__div__exp__iff,axiom,
    ! [A: nat,M: nat,N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( times_times_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( ( ord_less_nat @ N @ M )
        | ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
          = zero_zero_nat )
        | ( ( ord_less_eq_nat @ M @ N )
          & ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).

% even_mult_exp_div_exp_iff
thf(fact_3019_even__mult__exp__div__exp__iff,axiom,
    ! [A: int,M: nat,N: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ ( times_times_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
      = ( ( ord_less_nat @ N @ M )
        | ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
          = zero_zero_int )
        | ( ( ord_less_eq_nat @ M @ N )
          & ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).

% even_mult_exp_div_exp_iff
thf(fact_3020_div__if,axiom,
    ( divide_divide_nat
    = ( ^ [M6: nat,N2: nat] :
          ( if_nat
          @ ( ( ord_less_nat @ M6 @ N2 )
            | ( N2 = zero_zero_nat ) )
          @ zero_zero_nat
          @ ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M6 @ N2 ) @ N2 ) ) ) ) ) ).

% div_if
thf(fact_3021_div__geq,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ~ ( ord_less_nat @ M @ N )
       => ( ( divide_divide_nat @ M @ N )
          = ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ) ) ).

% div_geq
thf(fact_3022_Suc__pred_H,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( N
        = ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).

% Suc_pred'
thf(fact_3023_Suc__diff__eq__diff__pred,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( minus_minus_nat @ ( suc @ M ) @ N )
        = ( minus_minus_nat @ M @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).

% Suc_diff_eq_diff_pred
thf(fact_3024_less__eq__div__iff__mult__less__eq,axiom,
    ! [Q2: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ Q2 )
     => ( ( ord_less_eq_nat @ M @ ( divide_divide_nat @ N @ Q2 ) )
        = ( ord_less_eq_nat @ ( times_times_nat @ M @ Q2 ) @ N ) ) ) ).

% less_eq_div_iff_mult_less_eq
thf(fact_3025_add__eq__if,axiom,
    ( plus_plus_nat
    = ( ^ [M6: nat,N2: nat] : ( if_nat @ ( M6 = zero_zero_nat ) @ N2 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M6 @ one_one_nat ) @ N2 ) ) ) ) ) ).

% add_eq_if
thf(fact_3026_dividend__less__times__div,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ord_less_nat @ M @ ( plus_plus_nat @ N @ ( times_times_nat @ N @ ( divide_divide_nat @ M @ N ) ) ) ) ) ).

% dividend_less_times_div
thf(fact_3027_dividend__less__div__times,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ord_less_nat @ M @ ( plus_plus_nat @ N @ ( times_times_nat @ ( divide_divide_nat @ M @ N ) @ N ) ) ) ) ).

% dividend_less_div_times
thf(fact_3028_split__div,axiom,
    ! [P: nat > $o,M: nat,N: nat] :
      ( ( P @ ( divide_divide_nat @ M @ N ) )
      = ( ( ( N = zero_zero_nat )
         => ( P @ zero_zero_nat ) )
        & ( ( N != zero_zero_nat )
         => ! [I2: nat,J3: nat] :
              ( ( ord_less_nat @ J3 @ N )
             => ( ( M
                  = ( plus_plus_nat @ ( times_times_nat @ N @ I2 ) @ J3 ) )
               => ( P @ I2 ) ) ) ) ) ) ).

% split_div
thf(fact_3029_mult__eq__if,axiom,
    ( times_times_nat
    = ( ^ [M6: nat,N2: nat] : ( if_nat @ ( M6 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N2 @ ( times_times_nat @ ( minus_minus_nat @ M6 @ one_one_nat ) @ N2 ) ) ) ) ) ).

% mult_eq_if
thf(fact_3030_split__mod,axiom,
    ! [P: nat > $o,M: nat,N: nat] :
      ( ( P @ ( modulo_modulo_nat @ M @ N ) )
      = ( ( ( N = zero_zero_nat )
         => ( P @ M ) )
        & ( ( N != zero_zero_nat )
         => ! [I2: nat,J3: nat] :
              ( ( ord_less_nat @ J3 @ N )
             => ( ( M
                  = ( plus_plus_nat @ ( times_times_nat @ N @ I2 ) @ J3 ) )
               => ( P @ J3 ) ) ) ) ) ) ).

% split_mod
thf(fact_3031_even__two__times__div__two,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) )
        = A ) ) ).

% even_two_times_div_two
thf(fact_3032_even__two__times__div__two,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = A ) ) ).

% even_two_times_div_two
thf(fact_3033_even__two__times__div__two,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
        = A ) ) ).

% even_two_times_div_two
thf(fact_3034_odd__iff__mod__2__eq__one,axiom,
    ! [A: nat] :
      ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
      = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_nat ) ) ).

% odd_iff_mod_2_eq_one
thf(fact_3035_odd__iff__mod__2__eq__one,axiom,
    ! [A: int] :
      ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
      = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = one_one_int ) ) ).

% odd_iff_mod_2_eq_one
thf(fact_3036_odd__iff__mod__2__eq__one,axiom,
    ! [A: code_integer] :
      ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) )
      = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = one_one_Code_integer ) ) ).

% odd_iff_mod_2_eq_one
thf(fact_3037_power__mono__odd,axiom,
    ! [N: nat,A: real,B2: real] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( ord_less_eq_real @ A @ B2 )
       => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B2 @ N ) ) ) ) ).

% power_mono_odd
thf(fact_3038_power__mono__odd,axiom,
    ! [N: nat,A: rat,B2: rat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( ord_less_eq_rat @ A @ B2 )
       => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B2 @ N ) ) ) ) ).

% power_mono_odd
thf(fact_3039_power__mono__odd,axiom,
    ! [N: nat,A: int,B2: int] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( ord_less_eq_int @ A @ B2 )
       => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B2 @ N ) ) ) ) ).

% power_mono_odd
thf(fact_3040_dvd__power__iff__le,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
     => ( ( dvd_dvd_nat @ ( power_power_nat @ K @ M ) @ ( power_power_nat @ K @ N ) )
        = ( ord_less_eq_nat @ M @ N ) ) ) ).

% dvd_power_iff_le
thf(fact_3041_convex__bound__lt,axiom,
    ! [X4: real,A: real,Y3: real,U: real,V: real] :
      ( ( ord_less_real @ X4 @ A )
     => ( ( ord_less_real @ Y3 @ A )
       => ( ( ord_less_eq_real @ zero_zero_real @ U )
         => ( ( ord_less_eq_real @ zero_zero_real @ V )
           => ( ( ( plus_plus_real @ U @ V )
                = one_one_real )
             => ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ U @ X4 ) @ ( times_times_real @ V @ Y3 ) ) @ A ) ) ) ) ) ) ).

% convex_bound_lt
thf(fact_3042_convex__bound__lt,axiom,
    ! [X4: rat,A: rat,Y3: rat,U: rat,V: rat] :
      ( ( ord_less_rat @ X4 @ A )
     => ( ( ord_less_rat @ Y3 @ A )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ U )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ V )
           => ( ( ( plus_plus_rat @ U @ V )
                = one_one_rat )
             => ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ U @ X4 ) @ ( times_times_rat @ V @ Y3 ) ) @ A ) ) ) ) ) ) ).

% convex_bound_lt
thf(fact_3043_convex__bound__lt,axiom,
    ! [X4: int,A: int,Y3: int,U: int,V: int] :
      ( ( ord_less_int @ X4 @ A )
     => ( ( ord_less_int @ Y3 @ A )
       => ( ( ord_less_eq_int @ zero_zero_int @ U )
         => ( ( ord_less_eq_int @ zero_zero_int @ V )
           => ( ( ( plus_plus_int @ U @ V )
                = one_one_int )
             => ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ U @ X4 ) @ ( times_times_int @ V @ Y3 ) ) @ A ) ) ) ) ) ) ).

% convex_bound_lt
thf(fact_3044_le__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B2: real,C: real] :
      ( ( ord_less_eq_real @ ( numeral_numeral_real @ W ) @ ( divide_divide_real @ B2 @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B2 ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ B2 @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ ( numeral_numeral_real @ W ) @ zero_zero_real ) ) ) ) ) ) ).

% le_divide_eq_numeral(1)
thf(fact_3045_le__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B2: rat,C: rat] :
      ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ W ) @ ( divide_divide_rat @ B2 @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B2 ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ B2 @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( numeral_numeral_rat @ W ) @ zero_zero_rat ) ) ) ) ) ) ).

% le_divide_eq_numeral(1)
thf(fact_3046_divide__le__eq__numeral_I1_J,axiom,
    ! [B2: real,C: real,W: num] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ B2 @ C ) @ ( numeral_numeral_real @ W ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ B2 @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B2 ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ).

% divide_le_eq_numeral(1)
thf(fact_3047_divide__le__eq__numeral_I1_J,axiom,
    ! [B2: rat,C: rat,W: num] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B2 @ C ) @ ( numeral_numeral_rat @ W ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ B2 @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B2 ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ).

% divide_le_eq_numeral(1)
thf(fact_3048_half__gt__zero,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% half_gt_zero
thf(fact_3049_half__gt__zero,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).

% half_gt_zero
thf(fact_3050_half__gt__zero__iff,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
      = ( ord_less_real @ zero_zero_real @ A ) ) ).

% half_gt_zero_iff
thf(fact_3051_half__gt__zero__iff,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
      = ( ord_less_rat @ zero_zero_rat @ A ) ) ).

% half_gt_zero_iff
thf(fact_3052_scaling__mono,axiom,
    ! [U: real,V: real,R2: real,S: real] :
      ( ( ord_less_eq_real @ U @ V )
     => ( ( ord_less_eq_real @ zero_zero_real @ R2 )
       => ( ( ord_less_eq_real @ R2 @ S )
         => ( ord_less_eq_real @ ( plus_plus_real @ U @ ( divide_divide_real @ ( times_times_real @ R2 @ ( minus_minus_real @ V @ U ) ) @ S ) ) @ V ) ) ) ) ).

% scaling_mono
thf(fact_3053_scaling__mono,axiom,
    ! [U: rat,V: rat,R2: rat,S: rat] :
      ( ( ord_less_eq_rat @ U @ V )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ R2 )
       => ( ( ord_less_eq_rat @ R2 @ S )
         => ( ord_less_eq_rat @ ( plus_plus_rat @ U @ ( divide_divide_rat @ ( times_times_rat @ R2 @ ( minus_minus_rat @ V @ U ) ) @ S ) ) @ V ) ) ) ) ).

% scaling_mono
thf(fact_3054_zero__le__power2,axiom,
    ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% zero_le_power2
thf(fact_3055_zero__le__power2,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% zero_le_power2
thf(fact_3056_zero__le__power2,axiom,
    ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% zero_le_power2
thf(fact_3057_power2__eq__imp__eq,axiom,
    ! [X4: real,Y3: real] :
      ( ( ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ X4 )
       => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
         => ( X4 = Y3 ) ) ) ) ).

% power2_eq_imp_eq
thf(fact_3058_power2__eq__imp__eq,axiom,
    ! [X4: rat,Y3: rat] :
      ( ( ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ X4 )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
         => ( X4 = Y3 ) ) ) ) ).

% power2_eq_imp_eq
thf(fact_3059_power2__eq__imp__eq,axiom,
    ! [X4: nat,Y3: nat] :
      ( ( ( power_power_nat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_nat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ X4 )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ Y3 )
         => ( X4 = Y3 ) ) ) ) ).

% power2_eq_imp_eq
thf(fact_3060_power2__eq__imp__eq,axiom,
    ! [X4: int,Y3: int] :
      ( ( ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ X4 )
       => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
         => ( X4 = Y3 ) ) ) ) ).

% power2_eq_imp_eq
thf(fact_3061_power2__le__imp__le,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_eq_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
       => ( ord_less_eq_real @ X4 @ Y3 ) ) ) ).

% power2_le_imp_le
thf(fact_3062_power2__le__imp__le,axiom,
    ! [X4: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
       => ( ord_less_eq_rat @ X4 @ Y3 ) ) ) ).

% power2_le_imp_le
thf(fact_3063_power2__le__imp__le,axiom,
    ! [X4: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ Y3 )
       => ( ord_less_eq_nat @ X4 @ Y3 ) ) ) ).

% power2_le_imp_le
thf(fact_3064_power2__le__imp__le,axiom,
    ! [X4: int,Y3: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
       => ( ord_less_eq_int @ X4 @ Y3 ) ) ) ).

% power2_le_imp_le
thf(fact_3065_power2__less__0,axiom,
    ! [A: real] :
      ~ ( ord_less_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_real ) ).

% power2_less_0
thf(fact_3066_power2__less__0,axiom,
    ! [A: rat] :
      ~ ( ord_less_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_rat ) ).

% power2_less_0
thf(fact_3067_power2__less__0,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_int ) ).

% power2_less_0
thf(fact_3068_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: code_integer,A: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
     => ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ B2 @ C ) )
        = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B2 @ ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ B2 ) @ C ) ) @ ( modulo364778990260209775nteger @ A @ B2 ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
thf(fact_3069_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: nat,A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ C )
     => ( ( modulo_modulo_nat @ A @ ( times_times_nat @ B2 @ C ) )
        = ( plus_plus_nat @ ( times_times_nat @ B2 @ ( modulo_modulo_nat @ ( divide_divide_nat @ A @ B2 ) @ C ) ) @ ( modulo_modulo_nat @ A @ B2 ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
thf(fact_3070_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ C )
     => ( ( modulo_modulo_int @ A @ ( times_times_int @ B2 @ C ) )
        = ( plus_plus_int @ ( times_times_int @ B2 @ ( modulo_modulo_int @ ( divide_divide_int @ A @ B2 ) @ C ) ) @ ( modulo_modulo_int @ A @ B2 ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
thf(fact_3071_exp__add__not__zero__imp__left,axiom,
    ! [M: nat,N: nat] :
      ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
       != zero_zero_nat )
     => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
       != zero_zero_nat ) ) ).

% exp_add_not_zero_imp_left
thf(fact_3072_exp__add__not__zero__imp__left,axiom,
    ! [M: nat,N: nat] :
      ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
       != zero_zero_int )
     => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M )
       != zero_zero_int ) ) ).

% exp_add_not_zero_imp_left
thf(fact_3073_exp__add__not__zero__imp__right,axiom,
    ! [M: nat,N: nat] :
      ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
       != zero_zero_nat )
     => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       != zero_zero_nat ) ) ).

% exp_add_not_zero_imp_right
thf(fact_3074_exp__add__not__zero__imp__right,axiom,
    ! [M: nat,N: nat] :
      ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
       != zero_zero_int )
     => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
       != zero_zero_int ) ) ).

% exp_add_not_zero_imp_right
thf(fact_3075_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [N: nat,M: nat] :
      ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       != zero_zero_nat )
     => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) )
       != zero_zero_nat ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
thf(fact_3076_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [N: nat,M: nat] :
      ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
       != zero_zero_int )
     => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) )
       != zero_zero_int ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
thf(fact_3077_power__diff__power__eq,axiom,
    ! [A: nat,N: nat,M: nat] :
      ( ( A != zero_zero_nat )
     => ( ( ( ord_less_eq_nat @ N @ M )
         => ( ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
            = ( power_power_nat @ A @ ( minus_minus_nat @ M @ N ) ) ) )
        & ( ~ ( ord_less_eq_nat @ N @ M )
         => ( ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
            = ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).

% power_diff_power_eq
thf(fact_3078_power__diff__power__eq,axiom,
    ! [A: int,N: nat,M: nat] :
      ( ( A != zero_zero_int )
     => ( ( ( ord_less_eq_nat @ N @ M )
         => ( ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
            = ( power_power_int @ A @ ( minus_minus_nat @ M @ N ) ) ) )
        & ( ~ ( ord_less_eq_nat @ N @ M )
         => ( ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
            = ( divide_divide_int @ one_one_int @ ( power_power_int @ A @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).

% power_diff_power_eq
thf(fact_3079_less__2__cases,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
     => ( ( N = zero_zero_nat )
        | ( N
          = ( suc @ zero_zero_nat ) ) ) ) ).

% less_2_cases
thf(fact_3080_less__2__cases__iff,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( ( N = zero_zero_nat )
        | ( N
          = ( suc @ zero_zero_nat ) ) ) ) ).

% less_2_cases_iff
thf(fact_3081_nat__induct2,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ zero_zero_nat )
     => ( ( P @ one_one_nat )
       => ( ! [N4: nat] :
              ( ( P @ N4 )
             => ( P @ ( plus_plus_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
         => ( P @ N ) ) ) ) ).

% nat_induct2
thf(fact_3082_power__eq__if,axiom,
    ( power_power_complex
    = ( ^ [P3: complex,M6: nat] : ( if_complex @ ( M6 = zero_zero_nat ) @ one_one_complex @ ( times_times_complex @ P3 @ ( power_power_complex @ P3 @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_3083_power__eq__if,axiom,
    ( power_power_real
    = ( ^ [P3: real,M6: nat] : ( if_real @ ( M6 = zero_zero_nat ) @ one_one_real @ ( times_times_real @ P3 @ ( power_power_real @ P3 @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_3084_power__eq__if,axiom,
    ( power_power_rat
    = ( ^ [P3: rat,M6: nat] : ( if_rat @ ( M6 = zero_zero_nat ) @ one_one_rat @ ( times_times_rat @ P3 @ ( power_power_rat @ P3 @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_3085_power__eq__if,axiom,
    ( power_power_nat
    = ( ^ [P3: nat,M6: nat] : ( if_nat @ ( M6 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ P3 @ ( power_power_nat @ P3 @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_3086_power__eq__if,axiom,
    ( power_power_int
    = ( ^ [P3: int,M6: nat] : ( if_int @ ( M6 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ P3 @ ( power_power_int @ P3 @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_3087_power__minus__mult,axiom,
    ! [N: nat,A: complex] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( times_times_complex @ ( power_power_complex @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
        = ( power_power_complex @ A @ N ) ) ) ).

% power_minus_mult
thf(fact_3088_power__minus__mult,axiom,
    ! [N: nat,A: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( times_times_real @ ( power_power_real @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
        = ( power_power_real @ A @ N ) ) ) ).

% power_minus_mult
thf(fact_3089_power__minus__mult,axiom,
    ! [N: nat,A: rat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( times_times_rat @ ( power_power_rat @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
        = ( power_power_rat @ A @ N ) ) ) ).

% power_minus_mult
thf(fact_3090_power__minus__mult,axiom,
    ! [N: nat,A: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( times_times_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
        = ( power_power_nat @ A @ N ) ) ) ).

% power_minus_mult
thf(fact_3091_power__minus__mult,axiom,
    ! [N: nat,A: int] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( times_times_int @ ( power_power_int @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
        = ( power_power_int @ A @ N ) ) ) ).

% power_minus_mult
thf(fact_3092_le__div__geq,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( ( divide_divide_nat @ M @ N )
          = ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ) ) ).

% le_div_geq
thf(fact_3093_split__div_H,axiom,
    ! [P: nat > $o,M: nat,N: nat] :
      ( ( P @ ( divide_divide_nat @ M @ N ) )
      = ( ( ( N = zero_zero_nat )
          & ( P @ zero_zero_nat ) )
        | ? [Q4: nat] :
            ( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q4 ) @ M )
            & ( ord_less_nat @ M @ ( times_times_nat @ N @ ( suc @ Q4 ) ) )
            & ( P @ Q4 ) ) ) ) ).

% split_div'
thf(fact_3094_Suc__times__mod__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
     => ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ M @ N ) ) @ M )
        = one_one_nat ) ) ).

% Suc_times_mod_eq
thf(fact_3095_oddE,axiom,
    ! [A: code_integer] :
      ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ~ ! [B5: code_integer] :
            ( A
           != ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B5 ) @ one_one_Code_integer ) ) ) ).

% oddE
thf(fact_3096_oddE,axiom,
    ! [A: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ~ ! [B5: nat] :
            ( A
           != ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B5 ) @ one_one_nat ) ) ) ).

% oddE
thf(fact_3097_oddE,axiom,
    ! [A: int] :
      ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ~ ! [B5: int] :
            ( A
           != ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B5 ) @ one_one_int ) ) ) ).

% oddE
thf(fact_3098_power2__less__imp__less,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
       => ( ord_less_real @ X4 @ Y3 ) ) ) ).

% power2_less_imp_less
thf(fact_3099_power2__less__imp__less,axiom,
    ! [X4: rat,Y3: rat] :
      ( ( ord_less_rat @ ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
       => ( ord_less_rat @ X4 @ Y3 ) ) ) ).

% power2_less_imp_less
thf(fact_3100_power2__less__imp__less,axiom,
    ! [X4: nat,Y3: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ Y3 )
       => ( ord_less_nat @ X4 @ Y3 ) ) ) ).

% power2_less_imp_less
thf(fact_3101_power2__less__imp__less,axiom,
    ! [X4: int,Y3: int] :
      ( ( ord_less_int @ ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
       => ( ord_less_int @ X4 @ Y3 ) ) ) ).

% power2_less_imp_less
thf(fact_3102_sum__power2__ge__zero,axiom,
    ! [X4: real,Y3: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sum_power2_ge_zero
thf(fact_3103_sum__power2__ge__zero,axiom,
    ! [X4: rat,Y3: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sum_power2_ge_zero
thf(fact_3104_sum__power2__ge__zero,axiom,
    ! [X4: int,Y3: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sum_power2_ge_zero
thf(fact_3105_sum__power2__le__zero__iff,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_real )
      = ( ( X4 = zero_zero_real )
        & ( Y3 = zero_zero_real ) ) ) ).

% sum_power2_le_zero_iff
thf(fact_3106_sum__power2__le__zero__iff,axiom,
    ! [X4: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_rat )
      = ( ( X4 = zero_zero_rat )
        & ( Y3 = zero_zero_rat ) ) ) ).

% sum_power2_le_zero_iff
thf(fact_3107_sum__power2__le__zero__iff,axiom,
    ! [X4: int,Y3: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_int )
      = ( ( X4 = zero_zero_int )
        & ( Y3 = zero_zero_int ) ) ) ).

% sum_power2_le_zero_iff
thf(fact_3108_not__sum__power2__lt__zero,axiom,
    ! [X4: real,Y3: real] :
      ~ ( ord_less_real @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_real ) ).

% not_sum_power2_lt_zero
thf(fact_3109_not__sum__power2__lt__zero,axiom,
    ! [X4: rat,Y3: rat] :
      ~ ( ord_less_rat @ ( plus_plus_rat @ ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_rat ) ).

% not_sum_power2_lt_zero
thf(fact_3110_not__sum__power2__lt__zero,axiom,
    ! [X4: int,Y3: int] :
      ~ ( ord_less_int @ ( plus_plus_int @ ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_int ) ).

% not_sum_power2_lt_zero
thf(fact_3111_sum__power2__gt__zero__iff,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
      = ( ( X4 != zero_zero_real )
        | ( Y3 != zero_zero_real ) ) ) ).

% sum_power2_gt_zero_iff
thf(fact_3112_sum__power2__gt__zero__iff,axiom,
    ! [X4: rat,Y3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
      = ( ( X4 != zero_zero_rat )
        | ( Y3 != zero_zero_rat ) ) ) ).

% sum_power2_gt_zero_iff
thf(fact_3113_sum__power2__gt__zero__iff,axiom,
    ! [X4: int,Y3: int] :
      ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
      = ( ( X4 != zero_zero_int )
        | ( Y3 != zero_zero_int ) ) ) ).

% sum_power2_gt_zero_iff
thf(fact_3114_divmod__digit__0_I2_J,axiom,
    ! [B2: nat,A: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B2 )
     => ( ( ord_less_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) @ B2 )
       => ( ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) )
          = ( modulo_modulo_nat @ A @ B2 ) ) ) ) ).

% divmod_digit_0(2)
thf(fact_3115_divmod__digit__0_I2_J,axiom,
    ! [B2: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B2 )
     => ( ( ord_less_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) @ B2 )
       => ( ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) )
          = ( modulo_modulo_int @ A @ B2 ) ) ) ) ).

% divmod_digit_0(2)
thf(fact_3116_divmod__digit__0_I2_J,axiom,
    ! [B2: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B2 )
     => ( ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) ) @ B2 )
       => ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) )
          = ( modulo364778990260209775nteger @ A @ B2 ) ) ) ) ).

% divmod_digit_0(2)
thf(fact_3117_bits__stable__imp__add__self,axiom,
    ! [A: nat] :
      ( ( ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = A )
     => ( ( plus_plus_nat @ A @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = zero_zero_nat ) ) ).

% bits_stable_imp_add_self
thf(fact_3118_bits__stable__imp__add__self,axiom,
    ! [A: int] :
      ( ( ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = A )
     => ( ( plus_plus_int @ A @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
        = zero_zero_int ) ) ).

% bits_stable_imp_add_self
thf(fact_3119_bits__stable__imp__add__self,axiom,
    ! [A: code_integer] :
      ( ( ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = A )
     => ( ( plus_p5714425477246183910nteger @ A @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) )
        = zero_z3403309356797280102nteger ) ) ).

% bits_stable_imp_add_self
thf(fact_3120_zero__le__even__power_H,axiom,
    ! [A: real,N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% zero_le_even_power'
thf(fact_3121_zero__le__even__power_H,axiom,
    ! [A: rat,N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% zero_le_even_power'
thf(fact_3122_zero__le__even__power_H,axiom,
    ! [A: int,N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% zero_le_even_power'
thf(fact_3123_nat__bit__induct,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N4: nat] :
            ( ( P @ N4 )
           => ( ( ord_less_nat @ zero_zero_nat @ N4 )
             => ( P @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) )
       => ( ! [N4: nat] :
              ( ( P @ N4 )
             => ( P @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) )
         => ( P @ N ) ) ) ) ).

% nat_bit_induct
thf(fact_3124_div__2__gt__zero,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
     => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% div_2_gt_zero
thf(fact_3125_Suc__n__div__2__gt__zero,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% Suc_n_div_2_gt_zero
thf(fact_3126_verit__le__mono__div,axiom,
    ! [A3: nat,B4: nat,N: nat] :
      ( ( ord_less_nat @ A3 @ B4 )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_eq_nat
          @ ( plus_plus_nat @ ( divide_divide_nat @ A3 @ N )
            @ ( if_nat
              @ ( ( modulo_modulo_nat @ B4 @ N )
                = zero_zero_nat )
              @ one_one_nat
              @ zero_zero_nat ) )
          @ ( divide_divide_nat @ B4 @ N ) ) ) ) ).

% verit_le_mono_div
thf(fact_3127_divmod__digit__0_I1_J,axiom,
    ! [B2: nat,A: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B2 )
     => ( ( ord_less_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) @ B2 )
       => ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) )
          = ( divide_divide_nat @ A @ B2 ) ) ) ) ).

% divmod_digit_0(1)
thf(fact_3128_divmod__digit__0_I1_J,axiom,
    ! [B2: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B2 )
     => ( ( ord_less_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) @ B2 )
       => ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) )
          = ( divide_divide_int @ A @ B2 ) ) ) ) ).

% divmod_digit_0(1)
thf(fact_3129_divmod__digit__0_I1_J,axiom,
    ! [B2: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B2 )
     => ( ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) ) @ B2 )
       => ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) ) )
          = ( divide6298287555418463151nteger @ A @ B2 ) ) ) ) ).

% divmod_digit_0(1)
thf(fact_3130_odd__0__le__power__imp__0__le,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
     => ( ord_less_eq_real @ zero_zero_real @ A ) ) ).

% odd_0_le_power_imp_0_le
thf(fact_3131_odd__0__le__power__imp__0__le,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
     => ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).

% odd_0_le_power_imp_0_le
thf(fact_3132_odd__0__le__power__imp__0__le,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
     => ( ord_less_eq_int @ zero_zero_int @ A ) ) ).

% odd_0_le_power_imp_0_le
thf(fact_3133_odd__power__less__zero,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ord_less_real @ ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ zero_zero_real ) ) ).

% odd_power_less_zero
thf(fact_3134_odd__power__less__zero,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ord_less_rat @ ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ zero_zero_rat ) ) ).

% odd_power_less_zero
thf(fact_3135_odd__power__less__zero,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ord_less_int @ ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ zero_zero_int ) ) ).

% odd_power_less_zero
thf(fact_3136_even__mask__div__iff_H,axiom,
    ! [M: nat,N: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ one_one_Code_integer ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% even_mask_div_iff'
thf(fact_3137_even__mask__div__iff_H,axiom,
    ! [M: nat,N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ one_one_nat ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% even_mask_div_iff'
thf(fact_3138_even__mask__div__iff_H,axiom,
    ! [M: nat,N: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% even_mask_div_iff'
thf(fact_3139_mod__double__modulus,axiom,
    ! [M: code_integer,X4: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ M )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X4 )
       => ( ( ( modulo364778990260209775nteger @ X4 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
            = ( modulo364778990260209775nteger @ X4 @ M ) )
          | ( ( modulo364778990260209775nteger @ X4 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
            = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ X4 @ M ) @ M ) ) ) ) ) ).

% mod_double_modulus
thf(fact_3140_mod__double__modulus,axiom,
    ! [M: nat,X4: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ X4 )
       => ( ( ( modulo_modulo_nat @ X4 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
            = ( modulo_modulo_nat @ X4 @ M ) )
          | ( ( modulo_modulo_nat @ X4 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
            = ( plus_plus_nat @ ( modulo_modulo_nat @ X4 @ M ) @ M ) ) ) ) ) ).

% mod_double_modulus
thf(fact_3141_mod__double__modulus,axiom,
    ! [M: int,X4: int] :
      ( ( ord_less_int @ zero_zero_int @ M )
     => ( ( ord_less_eq_int @ zero_zero_int @ X4 )
       => ( ( ( modulo_modulo_int @ X4 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
            = ( modulo_modulo_int @ X4 @ M ) )
          | ( ( modulo_modulo_int @ X4 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
            = ( plus_plus_int @ ( modulo_modulo_int @ X4 @ M ) @ M ) ) ) ) ) ).

% mod_double_modulus
thf(fact_3142_divmod__digit__1_I2_J,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B2 )
       => ( ( ord_le3102999989581377725nteger @ B2 @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) ) )
         => ( ( minus_8373710615458151222nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) ) @ B2 )
            = ( modulo364778990260209775nteger @ A @ B2 ) ) ) ) ) ).

% divmod_digit_1(2)
thf(fact_3143_divmod__digit__1_I2_J,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ B2 )
       => ( ( ord_less_eq_nat @ B2 @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) )
         => ( ( minus_minus_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) @ B2 )
            = ( modulo_modulo_nat @ A @ B2 ) ) ) ) ) ).

% divmod_digit_1(2)
thf(fact_3144_divmod__digit__1_I2_J,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ zero_zero_int @ B2 )
       => ( ( ord_less_eq_int @ B2 @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) )
         => ( ( minus_minus_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) @ B2 )
            = ( modulo_modulo_int @ A @ B2 ) ) ) ) ) ).

% divmod_digit_1(2)
thf(fact_3145_even__mod__4__div__2,axiom,
    ! [N: nat] :
      ( ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
        = ( suc @ zero_zero_nat ) )
     => ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% even_mod_4_div_2
thf(fact_3146_pow__divides__pow__iff,axiom,
    ! [N: nat,A: nat,B2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B2 @ N ) )
        = ( dvd_dvd_nat @ A @ B2 ) ) ) ).

% pow_divides_pow_iff
thf(fact_3147_pow__divides__pow__iff,axiom,
    ! [N: nat,A: int,B2: int] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B2 @ N ) )
        = ( dvd_dvd_int @ A @ B2 ) ) ) ).

% pow_divides_pow_iff
thf(fact_3148_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
    ! [X4: nat,N: nat,M: nat] :
      ( ( ord_less_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( ord_less_nat @ zero_zero_nat @ M )
         => ( ord_less_nat @ ( vEBT_VEBT_low @ X4 @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).

% VEBT_internal.exp_split_high_low(2)
thf(fact_3149_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
    ! [X4: nat,N: nat,M: nat] :
      ( ( ord_less_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( ord_less_nat @ zero_zero_nat @ M )
         => ( ord_less_nat @ ( vEBT_VEBT_high @ X4 @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) ) ).

% VEBT_internal.exp_split_high_low(1)
thf(fact_3150_even__even__mod__4__iff,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
      = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ).

% even_even_mod_4_iff
thf(fact_3151_double__eq__0__iff,axiom,
    ! [A: real] :
      ( ( ( plus_plus_real @ A @ A )
        = zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% double_eq_0_iff
thf(fact_3152_double__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( ( plus_plus_rat @ A @ A )
        = zero_zero_rat )
      = ( A = zero_zero_rat ) ) ).

% double_eq_0_iff
thf(fact_3153_double__eq__0__iff,axiom,
    ! [A: int] :
      ( ( ( plus_plus_int @ A @ A )
        = zero_zero_int )
      = ( A = zero_zero_int ) ) ).

% double_eq_0_iff
thf(fact_3154_inf__period_I3_J,axiom,
    ! [D: code_integer,D4: code_integer,T: code_integer] :
      ( ( dvd_dvd_Code_integer @ D @ D4 )
     => ! [X5: code_integer,K4: code_integer] :
          ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ T ) )
          = ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ X5 @ ( times_3573771949741848930nteger @ K4 @ D4 ) ) @ T ) ) ) ) ).

% inf_period(3)
thf(fact_3155_inf__period_I3_J,axiom,
    ! [D: real,D4: real,T: real] :
      ( ( dvd_dvd_real @ D @ D4 )
     => ! [X5: real,K4: real] :
          ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ T ) )
          = ( dvd_dvd_real @ D @ ( plus_plus_real @ ( minus_minus_real @ X5 @ ( times_times_real @ K4 @ D4 ) ) @ T ) ) ) ) ).

% inf_period(3)
thf(fact_3156_inf__period_I3_J,axiom,
    ! [D: rat,D4: rat,T: rat] :
      ( ( dvd_dvd_rat @ D @ D4 )
     => ! [X5: rat,K4: rat] :
          ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ T ) )
          = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K4 @ D4 ) ) @ T ) ) ) ) ).

% inf_period(3)
thf(fact_3157_inf__period_I3_J,axiom,
    ! [D: int,D4: int,T: int] :
      ( ( dvd_dvd_int @ D @ D4 )
     => ! [X5: int,K4: int] :
          ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ T ) )
          = ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D4 ) ) @ T ) ) ) ) ).

% inf_period(3)
thf(fact_3158_inf__period_I4_J,axiom,
    ! [D: code_integer,D4: code_integer,T: code_integer] :
      ( ( dvd_dvd_Code_integer @ D @ D4 )
     => ! [X5: code_integer,K4: code_integer] :
          ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ T ) ) )
          = ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ X5 @ ( times_3573771949741848930nteger @ K4 @ D4 ) ) @ T ) ) ) ) ) ).

% inf_period(4)
thf(fact_3159_inf__period_I4_J,axiom,
    ! [D: real,D4: real,T: real] :
      ( ( dvd_dvd_real @ D @ D4 )
     => ! [X5: real,K4: real] :
          ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ T ) ) )
          = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ ( minus_minus_real @ X5 @ ( times_times_real @ K4 @ D4 ) ) @ T ) ) ) ) ) ).

% inf_period(4)
thf(fact_3160_inf__period_I4_J,axiom,
    ! [D: rat,D4: rat,T: rat] :
      ( ( dvd_dvd_rat @ D @ D4 )
     => ! [X5: rat,K4: rat] :
          ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ T ) ) )
          = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K4 @ D4 ) ) @ T ) ) ) ) ) ).

% inf_period(4)
thf(fact_3161_inf__period_I4_J,axiom,
    ! [D: int,D4: int,T: int] :
      ( ( dvd_dvd_int @ D @ D4 )
     => ! [X5: int,K4: int] :
          ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ T ) ) )
          = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D4 ) ) @ T ) ) ) ) ) ).

% inf_period(4)
thf(fact_3162_unity__coeff__ex,axiom,
    ! [P: code_integer > $o,L2: code_integer] :
      ( ( ? [X: code_integer] : ( P @ ( times_3573771949741848930nteger @ L2 @ X ) ) )
      = ( ? [X: code_integer] :
            ( ( dvd_dvd_Code_integer @ L2 @ ( plus_p5714425477246183910nteger @ X @ zero_z3403309356797280102nteger ) )
            & ( P @ X ) ) ) ) ).

% unity_coeff_ex
thf(fact_3163_unity__coeff__ex,axiom,
    ! [P: real > $o,L2: real] :
      ( ( ? [X: real] : ( P @ ( times_times_real @ L2 @ X ) ) )
      = ( ? [X: real] :
            ( ( dvd_dvd_real @ L2 @ ( plus_plus_real @ X @ zero_zero_real ) )
            & ( P @ X ) ) ) ) ).

% unity_coeff_ex
thf(fact_3164_unity__coeff__ex,axiom,
    ! [P: rat > $o,L2: rat] :
      ( ( ? [X: rat] : ( P @ ( times_times_rat @ L2 @ X ) ) )
      = ( ? [X: rat] :
            ( ( dvd_dvd_rat @ L2 @ ( plus_plus_rat @ X @ zero_zero_rat ) )
            & ( P @ X ) ) ) ) ).

% unity_coeff_ex
thf(fact_3165_unity__coeff__ex,axiom,
    ! [P: nat > $o,L2: nat] :
      ( ( ? [X: nat] : ( P @ ( times_times_nat @ L2 @ X ) ) )
      = ( ? [X: nat] :
            ( ( dvd_dvd_nat @ L2 @ ( plus_plus_nat @ X @ zero_zero_nat ) )
            & ( P @ X ) ) ) ) ).

% unity_coeff_ex
thf(fact_3166_unity__coeff__ex,axiom,
    ! [P: int > $o,L2: int] :
      ( ( ? [X: int] : ( P @ ( times_times_int @ L2 @ X ) ) )
      = ( ? [X: int] :
            ( ( dvd_dvd_int @ L2 @ ( plus_plus_int @ X @ zero_zero_int ) )
            & ( P @ X ) ) ) ) ).

% unity_coeff_ex
thf(fact_3167_unset__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se4203085406695923979it_int @ N @ K ) )
      = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).

% unset_bit_nonnegative_int_iff
thf(fact_3168_flip__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se2159334234014336723it_int @ N @ K ) )
      = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).

% flip_bit_nonnegative_int_iff
thf(fact_3169_set__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se7879613467334960850it_int @ N @ K ) )
      = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).

% set_bit_nonnegative_int_iff
thf(fact_3170_set__bit__negative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_int @ ( bit_se7879613467334960850it_int @ N @ K ) @ zero_zero_int )
      = ( ord_less_int @ K @ zero_zero_int ) ) ).

% set_bit_negative_int_iff
thf(fact_3171_flip__bit__negative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_int @ ( bit_se2159334234014336723it_int @ N @ K ) @ zero_zero_int )
      = ( ord_less_int @ K @ zero_zero_int ) ) ).

% flip_bit_negative_int_iff
thf(fact_3172_unset__bit__negative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_int @ ( bit_se4203085406695923979it_int @ N @ K ) @ zero_zero_int )
      = ( ord_less_int @ K @ zero_zero_int ) ) ).

% unset_bit_negative_int_iff
thf(fact_3173_zle__add1__eq__le,axiom,
    ! [W: int,Z: int] :
      ( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
      = ( ord_less_eq_int @ W @ Z ) ) ).

% zle_add1_eq_le
thf(fact_3174_zle__diff1__eq,axiom,
    ! [W: int,Z: int] :
      ( ( ord_less_eq_int @ W @ ( minus_minus_int @ Z @ one_one_int ) )
      = ( ord_less_int @ W @ Z ) ) ).

% zle_diff1_eq
thf(fact_3175_mod__pos__pos__trivial,axiom,
    ! [K: int,L2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ( ( ord_less_int @ K @ L2 )
       => ( ( modulo_modulo_int @ K @ L2 )
          = K ) ) ) ).

% mod_pos_pos_trivial
thf(fact_3176_mod__neg__neg__trivial,axiom,
    ! [K: int,L2: int] :
      ( ( ord_less_eq_int @ K @ zero_zero_int )
     => ( ( ord_less_int @ L2 @ K )
       => ( ( modulo_modulo_int @ K @ L2 )
          = K ) ) ) ).

% mod_neg_neg_trivial
thf(fact_3177_le__imp__0__less,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z ) ) ) ).

% le_imp_0_less
thf(fact_3178_split__zmod,axiom,
    ! [P: int > $o,N: int,K: int] :
      ( ( P @ ( modulo_modulo_int @ N @ K ) )
      = ( ( ( K = zero_zero_int )
         => ( P @ N ) )
        & ( ( ord_less_int @ zero_zero_int @ K )
         => ! [I2: int,J3: int] :
              ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
                & ( ord_less_int @ J3 @ K )
                & ( N
                  = ( plus_plus_int @ ( times_times_int @ K @ I2 ) @ J3 ) ) )
             => ( P @ J3 ) ) )
        & ( ( ord_less_int @ K @ zero_zero_int )
         => ! [I2: int,J3: int] :
              ( ( ( ord_less_int @ K @ J3 )
                & ( ord_less_eq_int @ J3 @ zero_zero_int )
                & ( N
                  = ( plus_plus_int @ ( times_times_int @ K @ I2 ) @ J3 ) ) )
             => ( P @ J3 ) ) ) ) ) ).

% split_zmod
thf(fact_3179_odd__less__0__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z ) @ zero_zero_int )
      = ( ord_less_int @ Z @ zero_zero_int ) ) ).

% odd_less_0_iff
thf(fact_3180_mod__pos__geq,axiom,
    ! [L2: int,K: int] :
      ( ( ord_less_int @ zero_zero_int @ L2 )
     => ( ( ord_less_eq_int @ L2 @ K )
       => ( ( modulo_modulo_int @ K @ L2 )
          = ( modulo_modulo_int @ ( minus_minus_int @ K @ L2 ) @ L2 ) ) ) ) ).

% mod_pos_geq
thf(fact_3181_q__pos__lemma,axiom,
    ! [B: int,Q5: int,R4: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B @ Q5 ) @ R4 ) )
     => ( ( ord_less_int @ R4 @ B )
       => ( ( ord_less_int @ zero_zero_int @ B )
         => ( ord_less_eq_int @ zero_zero_int @ Q5 ) ) ) ) ).

% q_pos_lemma
thf(fact_3182_zmult__zless__mono2,axiom,
    ! [I: int,J: int,K: int] :
      ( ( ord_less_int @ I @ J )
     => ( ( ord_less_int @ zero_zero_int @ K )
       => ( ord_less_int @ ( times_times_int @ K @ I ) @ ( times_times_int @ K @ J ) ) ) ) ).

% zmult_zless_mono2
thf(fact_3183_int__mod__neg__eq,axiom,
    ! [A: int,B2: int,Q2: int,R2: int] :
      ( ( A
        = ( plus_plus_int @ ( times_times_int @ B2 @ Q2 ) @ R2 ) )
     => ( ( ord_less_eq_int @ R2 @ zero_zero_int )
       => ( ( ord_less_int @ B2 @ R2 )
         => ( ( modulo_modulo_int @ A @ B2 )
            = R2 ) ) ) ) ).

% int_mod_neg_eq
thf(fact_3184_int__mod__pos__eq,axiom,
    ! [A: int,B2: int,Q2: int,R2: int] :
      ( ( A
        = ( plus_plus_int @ ( times_times_int @ B2 @ Q2 ) @ R2 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
       => ( ( ord_less_int @ R2 @ B2 )
         => ( ( modulo_modulo_int @ A @ B2 )
            = R2 ) ) ) ) ).

% int_mod_pos_eq
thf(fact_3185_pos__zmult__eq__1__iff,axiom,
    ! [M: int,N: int] :
      ( ( ord_less_int @ zero_zero_int @ M )
     => ( ( ( times_times_int @ M @ N )
          = one_one_int )
        = ( ( M = one_one_int )
          & ( N = one_one_int ) ) ) ) ).

% pos_zmult_eq_1_iff
thf(fact_3186_plusinfinity,axiom,
    ! [D: int,P4: int > $o,P: int > $o] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ! [X3: int,K2: int] :
            ( ( P4 @ X3 )
            = ( P4 @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D ) ) ) )
       => ( ? [Z3: int] :
            ! [X3: int] :
              ( ( ord_less_int @ Z3 @ X3 )
             => ( ( P @ X3 )
                = ( P4 @ X3 ) ) )
         => ( ? [X_12: int] : ( P4 @ X_12 )
           => ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).

% plusinfinity
thf(fact_3187_zdiv__mono2__lemma,axiom,
    ! [B2: int,Q2: int,R2: int,B: int,Q5: int,R4: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ B2 @ Q2 ) @ R2 )
        = ( plus_plus_int @ ( times_times_int @ B @ Q5 ) @ R4 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B @ Q5 ) @ R4 ) )
       => ( ( ord_less_int @ R4 @ B )
         => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
           => ( ( ord_less_int @ zero_zero_int @ B )
             => ( ( ord_less_eq_int @ B @ B2 )
               => ( ord_less_eq_int @ Q2 @ Q5 ) ) ) ) ) ) ) ).

% zdiv_mono2_lemma
thf(fact_3188_minusinfinity,axiom,
    ! [D: int,P1: int > $o,P: int > $o] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ! [X3: int,K2: int] :
            ( ( P1 @ X3 )
            = ( P1 @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D ) ) ) )
       => ( ? [Z3: int] :
            ! [X3: int] :
              ( ( ord_less_int @ X3 @ Z3 )
             => ( ( P @ X3 )
                = ( P1 @ X3 ) ) )
         => ( ? [X_12: int] : ( P1 @ X_12 )
           => ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).

% minusinfinity
thf(fact_3189_decr__mult__lemma,axiom,
    ! [D: int,P: int > $o,K: int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ! [X3: int] :
            ( ( P @ X3 )
           => ( P @ ( minus_minus_int @ X3 @ D ) ) )
       => ( ( ord_less_eq_int @ zero_zero_int @ K )
         => ! [X5: int] :
              ( ( P @ X5 )
             => ( P @ ( minus_minus_int @ X5 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).

% decr_mult_lemma
thf(fact_3190_incr__mult__lemma,axiom,
    ! [D: int,P: int > $o,K: int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ! [X3: int] :
            ( ( P @ X3 )
           => ( P @ ( plus_plus_int @ X3 @ D ) ) )
       => ( ( ord_less_eq_int @ zero_zero_int @ K )
         => ! [X5: int] :
              ( ( P @ X5 )
             => ( P @ ( plus_plus_int @ X5 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).

% incr_mult_lemma
thf(fact_3191_zdiv__mono2__neg__lemma,axiom,
    ! [B2: int,Q2: int,R2: int,B: int,Q5: int,R4: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ B2 @ Q2 ) @ R2 )
        = ( plus_plus_int @ ( times_times_int @ B @ Q5 ) @ R4 ) )
     => ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ B @ Q5 ) @ R4 ) @ zero_zero_int )
       => ( ( ord_less_int @ R2 @ B2 )
         => ( ( ord_less_eq_int @ zero_zero_int @ R4 )
           => ( ( ord_less_int @ zero_zero_int @ B )
             => ( ( ord_less_eq_int @ B @ B2 )
               => ( ord_less_eq_int @ Q5 @ Q2 ) ) ) ) ) ) ) ).

% zdiv_mono2_neg_lemma
thf(fact_3192_int__one__le__iff__zero__less,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ one_one_int @ Z )
      = ( ord_less_int @ zero_zero_int @ Z ) ) ).

% int_one_le_iff_zero_less
thf(fact_3193_unique__quotient__lemma,axiom,
    ! [B2: int,Q5: int,R4: int,Q2: int,R2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B2 @ Q5 ) @ R4 ) @ ( plus_plus_int @ ( times_times_int @ B2 @ Q2 ) @ R2 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ R4 )
       => ( ( ord_less_int @ R4 @ B2 )
         => ( ( ord_less_int @ R2 @ B2 )
           => ( ord_less_eq_int @ Q5 @ Q2 ) ) ) ) ) ).

% unique_quotient_lemma
thf(fact_3194_unique__quotient__lemma__neg,axiom,
    ! [B2: int,Q5: int,R4: int,Q2: int,R2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B2 @ Q5 ) @ R4 ) @ ( plus_plus_int @ ( times_times_int @ B2 @ Q2 ) @ R2 ) )
     => ( ( ord_less_eq_int @ R2 @ zero_zero_int )
       => ( ( ord_less_int @ B2 @ R2 )
         => ( ( ord_less_int @ B2 @ R4 )
           => ( ord_less_eq_int @ Q2 @ Q5 ) ) ) ) ) ).

% unique_quotient_lemma_neg
thf(fact_3195_mod__pos__neg__trivial,axiom,
    ! [K: int,L2: int] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ( ( ord_less_eq_int @ ( plus_plus_int @ K @ L2 ) @ zero_zero_int )
       => ( ( modulo_modulo_int @ K @ L2 )
          = ( plus_plus_int @ K @ L2 ) ) ) ) ).

% mod_pos_neg_trivial
thf(fact_3196_zdvd__reduce,axiom,
    ! [K: int,N: int,M: int] :
      ( ( dvd_dvd_int @ K @ ( plus_plus_int @ N @ ( times_times_int @ K @ M ) ) )
      = ( dvd_dvd_int @ K @ N ) ) ).

% zdvd_reduce
thf(fact_3197_zdvd__period,axiom,
    ! [A: int,D: int,X4: int,T: int,C: int] :
      ( ( dvd_dvd_int @ A @ D )
     => ( ( dvd_dvd_int @ A @ ( plus_plus_int @ X4 @ T ) )
        = ( dvd_dvd_int @ A @ ( plus_plus_int @ ( plus_plus_int @ X4 @ ( times_times_int @ C @ D ) ) @ T ) ) ) ) ).

% zdvd_period
thf(fact_3198_odd__nonzero,axiom,
    ! [Z: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z )
     != zero_zero_int ) ).

% odd_nonzero
thf(fact_3199_times__int__code_I2_J,axiom,
    ! [L2: int] :
      ( ( times_times_int @ zero_zero_int @ L2 )
      = zero_zero_int ) ).

% times_int_code(2)
thf(fact_3200_times__int__code_I1_J,axiom,
    ! [K: int] :
      ( ( times_times_int @ K @ zero_zero_int )
      = zero_zero_int ) ).

% times_int_code(1)
thf(fact_3201_zmod__eq__0D,axiom,
    ! [M: int,D: int] :
      ( ( ( modulo_modulo_int @ M @ D )
        = zero_zero_int )
     => ? [Q3: int] :
          ( M
          = ( times_times_int @ D @ Q3 ) ) ) ).

% zmod_eq_0D
thf(fact_3202_zdvd__mult__cancel,axiom,
    ! [K: int,M: int,N: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ N ) )
     => ( ( K != zero_zero_int )
       => ( dvd_dvd_int @ M @ N ) ) ) ).

% zdvd_mult_cancel
thf(fact_3203_zdvd__mono,axiom,
    ! [K: int,M: int,T: int] :
      ( ( K != zero_zero_int )
     => ( ( dvd_dvd_int @ M @ T )
        = ( dvd_dvd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ T ) ) ) ) ).

% zdvd_mono
thf(fact_3204_zmod__eq__0__iff,axiom,
    ! [M: int,D: int] :
      ( ( ( modulo_modulo_int @ M @ D )
        = zero_zero_int )
      = ( ? [Q4: int] :
            ( M
            = ( times_times_int @ D @ Q4 ) ) ) ) ).

% zmod_eq_0_iff
thf(fact_3205_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
thf(fact_3206_signed__take__bit__mult,axiom,
    ! [N: nat,K: int,L2: int] :
      ( ( bit_ri631733984087533419it_int @ N @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( bit_ri631733984087533419it_int @ N @ L2 ) ) )
      = ( bit_ri631733984087533419it_int @ N @ ( times_times_int @ K @ L2 ) ) ) ).

% signed_take_bit_mult
thf(fact_3207_Euclidean__Division_Opos__mod__bound,axiom,
    ! [L2: int,K: int] :
      ( ( ord_less_int @ zero_zero_int @ L2 )
     => ( ord_less_int @ ( modulo_modulo_int @ K @ L2 ) @ L2 ) ) ).

% Euclidean_Division.pos_mod_bound
thf(fact_3208_neg__mod__bound,axiom,
    ! [L2: int,K: int] :
      ( ( ord_less_int @ L2 @ zero_zero_int )
     => ( ord_less_int @ L2 @ ( modulo_modulo_int @ K @ L2 ) ) ) ).

% neg_mod_bound
thf(fact_3209_Euclidean__Division_Opos__mod__sign,axiom,
    ! [L2: int,K: int] :
      ( ( ord_less_int @ zero_zero_int @ L2 )
     => ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ K @ L2 ) ) ) ).

% Euclidean_Division.pos_mod_sign
thf(fact_3210_neg__mod__sign,axiom,
    ! [L2: int,K: int] :
      ( ( ord_less_int @ L2 @ zero_zero_int )
     => ( ord_less_eq_int @ ( modulo_modulo_int @ K @ L2 ) @ zero_zero_int ) ) ).

% neg_mod_sign
thf(fact_3211_zmod__le__nonneg__dividend,axiom,
    ! [M: int,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ M )
     => ( ord_less_eq_int @ ( modulo_modulo_int @ M @ K ) @ M ) ) ).

% zmod_le_nonneg_dividend
thf(fact_3212_zmod__trivial__iff,axiom,
    ! [I: int,K: int] :
      ( ( ( modulo_modulo_int @ I @ K )
        = I )
      = ( ( K = zero_zero_int )
        | ( ( ord_less_eq_int @ zero_zero_int @ I )
          & ( ord_less_int @ I @ K ) )
        | ( ( ord_less_eq_int @ I @ zero_zero_int )
          & ( ord_less_int @ K @ I ) ) ) ) ).

% zmod_trivial_iff
thf(fact_3213_mod__int__pos__iff,axiom,
    ! [K: int,L2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ K @ L2 ) )
      = ( ( dvd_dvd_int @ L2 @ K )
        | ( ( L2 = zero_zero_int )
          & ( ord_less_eq_int @ zero_zero_int @ K ) )
        | ( ord_less_int @ zero_zero_int @ L2 ) ) ) ).

% mod_int_pos_iff
thf(fact_3214_pos__mod__conj,axiom,
    ! [B2: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ A @ B2 ) )
        & ( ord_less_int @ ( modulo_modulo_int @ A @ B2 ) @ B2 ) ) ) ).

% pos_mod_conj
thf(fact_3215_neg__mod__conj,axiom,
    ! [B2: int,A: int] :
      ( ( ord_less_int @ B2 @ zero_zero_int )
     => ( ( ord_less_eq_int @ ( modulo_modulo_int @ A @ B2 ) @ zero_zero_int )
        & ( ord_less_int @ B2 @ ( modulo_modulo_int @ A @ B2 ) ) ) ) ).

% neg_mod_conj
thf(fact_3216_zdvd__not__zless,axiom,
    ! [M: int,N: int] :
      ( ( ord_less_int @ zero_zero_int @ M )
     => ( ( ord_less_int @ M @ N )
       => ~ ( dvd_dvd_int @ N @ M ) ) ) ).

% zdvd_not_zless
thf(fact_3217_zdvd__imp__le,axiom,
    ! [Z: int,N: int] :
      ( ( dvd_dvd_int @ Z @ N )
     => ( ( ord_less_int @ zero_zero_int @ N )
       => ( ord_less_eq_int @ Z @ N ) ) ) ).

% zdvd_imp_le
thf(fact_3218_less__int__code_I1_J,axiom,
    ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).

% less_int_code(1)
thf(fact_3219_plus__int__code_I2_J,axiom,
    ! [L2: int] :
      ( ( plus_plus_int @ zero_zero_int @ L2 )
      = L2 ) ).

% plus_int_code(2)
thf(fact_3220_plus__int__code_I1_J,axiom,
    ! [K: int] :
      ( ( plus_plus_int @ K @ zero_zero_int )
      = K ) ).

% plus_int_code(1)
thf(fact_3221_minus__int__code_I1_J,axiom,
    ! [K: int] :
      ( ( minus_minus_int @ K @ zero_zero_int )
      = K ) ).

% minus_int_code(1)
thf(fact_3222_zdvd__zdiffD,axiom,
    ! [K: int,M: int,N: int] :
      ( ( dvd_dvd_int @ K @ ( minus_minus_int @ M @ N ) )
     => ( ( dvd_dvd_int @ K @ N )
       => ( dvd_dvd_int @ K @ M ) ) ) ).

% zdvd_zdiffD
thf(fact_3223_signed__take__bit__add,axiom,
    ! [N: nat,K: int,L2: int] :
      ( ( bit_ri631733984087533419it_int @ N @ ( plus_plus_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( bit_ri631733984087533419it_int @ N @ L2 ) ) )
      = ( bit_ri631733984087533419it_int @ N @ ( plus_plus_int @ K @ L2 ) ) ) ).

% signed_take_bit_add
thf(fact_3224_signed__take__bit__diff,axiom,
    ! [N: nat,K: int,L2: int] :
      ( ( bit_ri631733984087533419it_int @ N @ ( minus_minus_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( bit_ri631733984087533419it_int @ N @ L2 ) ) )
      = ( bit_ri631733984087533419it_int @ N @ ( minus_minus_int @ K @ L2 ) ) ) ).

% signed_take_bit_diff
thf(fact_3225_zero__one__enat__neq_I1_J,axiom,
    zero_z5237406670263579293d_enat != one_on7984719198319812577d_enat ).

% zero_one_enat_neq(1)
thf(fact_3226_set__bit__greater__eq,axiom,
    ! [K: int,N: nat] : ( ord_less_eq_int @ K @ ( bit_se7879613467334960850it_int @ N @ K ) ) ).

% set_bit_greater_eq
thf(fact_3227_unset__bit__less__eq,axiom,
    ! [N: nat,K: int] : ( ord_less_eq_int @ ( bit_se4203085406695923979it_int @ N @ K ) @ K ) ).

% unset_bit_less_eq
thf(fact_3228_int__distrib_I4_J,axiom,
    ! [W: int,Z1: int,Z22: int] :
      ( ( times_times_int @ W @ ( minus_minus_int @ Z1 @ Z22 ) )
      = ( minus_minus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).

% int_distrib(4)
thf(fact_3229_int__distrib_I3_J,axiom,
    ! [Z1: int,Z22: int,W: int] :
      ( ( times_times_int @ ( minus_minus_int @ Z1 @ Z22 ) @ W )
      = ( minus_minus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).

% int_distrib(3)
thf(fact_3230_int__distrib_I2_J,axiom,
    ! [W: int,Z1: int,Z22: int] :
      ( ( times_times_int @ W @ ( plus_plus_int @ Z1 @ Z22 ) )
      = ( plus_plus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).

% int_distrib(2)
thf(fact_3231_int__distrib_I1_J,axiom,
    ! [Z1: int,Z22: int,W: int] :
      ( ( times_times_int @ ( plus_plus_int @ Z1 @ Z22 ) @ W )
      = ( plus_plus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).

% int_distrib(1)
thf(fact_3232_zless__imp__add1__zle,axiom,
    ! [W: int,Z: int] :
      ( ( ord_less_int @ W @ Z )
     => ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z ) ) ).

% zless_imp_add1_zle
thf(fact_3233_int__less__induct,axiom,
    ! [I: int,K: int,P: int > $o] :
      ( ( ord_less_int @ I @ K )
     => ( ( P @ ( minus_minus_int @ K @ one_one_int ) )
       => ( ! [I3: int] :
              ( ( ord_less_int @ I3 @ K )
             => ( ( P @ I3 )
               => ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
         => ( P @ I ) ) ) ) ).

% int_less_induct
thf(fact_3234_zless__add1__eq,axiom,
    ! [W: int,Z: int] :
      ( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
      = ( ( ord_less_int @ W @ Z )
        | ( W = Z ) ) ) ).

% zless_add1_eq
thf(fact_3235_int__le__induct,axiom,
    ! [I: int,K: int,P: int > $o] :
      ( ( ord_less_eq_int @ I @ K )
     => ( ( P @ K )
       => ( ! [I3: int] :
              ( ( ord_less_eq_int @ I3 @ K )
             => ( ( P @ I3 )
               => ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
         => ( P @ I ) ) ) ) ).

% int_le_induct
thf(fact_3236_int__gr__induct,axiom,
    ! [K: int,I: int,P: int > $o] :
      ( ( ord_less_int @ K @ I )
     => ( ( P @ ( plus_plus_int @ K @ one_one_int ) )
       => ( ! [I3: int] :
              ( ( ord_less_int @ K @ I3 )
             => ( ( P @ I3 )
               => ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
         => ( P @ I ) ) ) ) ).

% int_gr_induct
thf(fact_3237_int__ge__induct,axiom,
    ! [K: int,I: int,P: int > $o] :
      ( ( ord_less_eq_int @ K @ I )
     => ( ( P @ K )
       => ( ! [I3: int] :
              ( ( ord_less_eq_int @ K @ I3 )
             => ( ( P @ I3 )
               => ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
         => ( P @ I ) ) ) ) ).

% int_ge_induct
thf(fact_3238_add1__zle__eq,axiom,
    ! [W: int,Z: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z )
      = ( ord_less_int @ W @ Z ) ) ).

% add1_zle_eq
thf(fact_3239_int__induct,axiom,
    ! [P: int > $o,K: int,I: int] :
      ( ( P @ K )
     => ( ! [I3: int] :
            ( ( ord_less_eq_int @ K @ I3 )
           => ( ( P @ I3 )
             => ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
       => ( ! [I3: int] :
              ( ( ord_less_eq_int @ I3 @ K )
             => ( ( P @ I3 )
               => ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
         => ( P @ I ) ) ) ) ).

% int_induct
thf(fact_3240_even__diff__iff,axiom,
    ! [K: int,L2: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ K @ L2 ) )
      = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ L2 ) ) ) ).

% even_diff_iff
thf(fact_3241_minf_I7_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X5: real] :
      ( ( ord_less_real @ X5 @ Z2 )
     => ~ ( ord_less_real @ T @ X5 ) ) ).

% minf(7)
thf(fact_3242_minf_I7_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( ( ord_less_rat @ X5 @ Z2 )
     => ~ ( ord_less_rat @ T @ X5 ) ) ).

% minf(7)
thf(fact_3243_minf_I7_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X5: num] :
      ( ( ord_less_num @ X5 @ Z2 )
     => ~ ( ord_less_num @ T @ X5 ) ) ).

% minf(7)
thf(fact_3244_minf_I7_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ X5 @ Z2 )
     => ~ ( ord_less_nat @ T @ X5 ) ) ).

% minf(7)
thf(fact_3245_minf_I7_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( ( ord_less_int @ X5 @ Z2 )
     => ~ ( ord_less_int @ T @ X5 ) ) ).

% minf(7)
thf(fact_3246_minf_I5_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X5: real] :
      ( ( ord_less_real @ X5 @ Z2 )
     => ( ord_less_real @ X5 @ T ) ) ).

% minf(5)
thf(fact_3247_minf_I5_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( ( ord_less_rat @ X5 @ Z2 )
     => ( ord_less_rat @ X5 @ T ) ) ).

% minf(5)
thf(fact_3248_minf_I5_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X5: num] :
      ( ( ord_less_num @ X5 @ Z2 )
     => ( ord_less_num @ X5 @ T ) ) ).

% minf(5)
thf(fact_3249_minf_I5_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ X5 @ Z2 )
     => ( ord_less_nat @ X5 @ T ) ) ).

% minf(5)
thf(fact_3250_minf_I5_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( ( ord_less_int @ X5 @ Z2 )
     => ( ord_less_int @ X5 @ T ) ) ).

% minf(5)
thf(fact_3251_minf_I4_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X5: real] :
      ( ( ord_less_real @ X5 @ Z2 )
     => ( X5 != T ) ) ).

% minf(4)
thf(fact_3252_minf_I4_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( ( ord_less_rat @ X5 @ Z2 )
     => ( X5 != T ) ) ).

% minf(4)
thf(fact_3253_minf_I4_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X5: num] :
      ( ( ord_less_num @ X5 @ Z2 )
     => ( X5 != T ) ) ).

% minf(4)
thf(fact_3254_minf_I4_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ X5 @ Z2 )
     => ( X5 != T ) ) ).

% minf(4)
thf(fact_3255_minf_I4_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( ( ord_less_int @ X5 @ Z2 )
     => ( X5 != T ) ) ).

% minf(4)
thf(fact_3256_minf_I3_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X5: real] :
      ( ( ord_less_real @ X5 @ Z2 )
     => ( X5 != T ) ) ).

% minf(3)
thf(fact_3257_minf_I3_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( ( ord_less_rat @ X5 @ Z2 )
     => ( X5 != T ) ) ).

% minf(3)
thf(fact_3258_minf_I3_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X5: num] :
      ( ( ord_less_num @ X5 @ Z2 )
     => ( X5 != T ) ) ).

% minf(3)
thf(fact_3259_minf_I3_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ X5 @ Z2 )
     => ( X5 != T ) ) ).

% minf(3)
thf(fact_3260_minf_I3_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( ( ord_less_int @ X5 @ Z2 )
     => ( X5 != T ) ) ).

% minf(3)
thf(fact_3261_minf_I2_J,axiom,
    ! [P: real > $o,P4: real > $o,Q: real > $o,Q6: real > $o] :
      ( ? [Z3: real] :
        ! [X3: real] :
          ( ( ord_less_real @ X3 @ Z3 )
         => ( ( P @ X3 )
            = ( P4 @ X3 ) ) )
     => ( ? [Z3: real] :
          ! [X3: real] :
            ( ( ord_less_real @ X3 @ Z3 )
           => ( ( Q @ X3 )
              = ( Q6 @ X3 ) ) )
       => ? [Z2: real] :
          ! [X5: real] :
            ( ( ord_less_real @ X5 @ Z2 )
           => ( ( ( P @ X5 )
                | ( Q @ X5 ) )
              = ( ( P4 @ X5 )
                | ( Q6 @ X5 ) ) ) ) ) ) ).

% minf(2)
thf(fact_3262_minf_I2_J,axiom,
    ! [P: rat > $o,P4: rat > $o,Q: rat > $o,Q6: rat > $o] :
      ( ? [Z3: rat] :
        ! [X3: rat] :
          ( ( ord_less_rat @ X3 @ Z3 )
         => ( ( P @ X3 )
            = ( P4 @ X3 ) ) )
     => ( ? [Z3: rat] :
          ! [X3: rat] :
            ( ( ord_less_rat @ X3 @ Z3 )
           => ( ( Q @ X3 )
              = ( Q6 @ X3 ) ) )
       => ? [Z2: rat] :
          ! [X5: rat] :
            ( ( ord_less_rat @ X5 @ Z2 )
           => ( ( ( P @ X5 )
                | ( Q @ X5 ) )
              = ( ( P4 @ X5 )
                | ( Q6 @ X5 ) ) ) ) ) ) ).

% minf(2)
thf(fact_3263_minf_I2_J,axiom,
    ! [P: num > $o,P4: num > $o,Q: num > $o,Q6: num > $o] :
      ( ? [Z3: num] :
        ! [X3: num] :
          ( ( ord_less_num @ X3 @ Z3 )
         => ( ( P @ X3 )
            = ( P4 @ X3 ) ) )
     => ( ? [Z3: num] :
          ! [X3: num] :
            ( ( ord_less_num @ X3 @ Z3 )
           => ( ( Q @ X3 )
              = ( Q6 @ X3 ) ) )
       => ? [Z2: num] :
          ! [X5: num] :
            ( ( ord_less_num @ X5 @ Z2 )
           => ( ( ( P @ X5 )
                | ( Q @ X5 ) )
              = ( ( P4 @ X5 )
                | ( Q6 @ X5 ) ) ) ) ) ) ).

% minf(2)
thf(fact_3264_minf_I2_J,axiom,
    ! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q6: nat > $o] :
      ( ? [Z3: nat] :
        ! [X3: nat] :
          ( ( ord_less_nat @ X3 @ Z3 )
         => ( ( P @ X3 )
            = ( P4 @ X3 ) ) )
     => ( ? [Z3: nat] :
          ! [X3: nat] :
            ( ( ord_less_nat @ X3 @ Z3 )
           => ( ( Q @ X3 )
              = ( Q6 @ X3 ) ) )
       => ? [Z2: nat] :
          ! [X5: nat] :
            ( ( ord_less_nat @ X5 @ Z2 )
           => ( ( ( P @ X5 )
                | ( Q @ X5 ) )
              = ( ( P4 @ X5 )
                | ( Q6 @ X5 ) ) ) ) ) ) ).

% minf(2)
thf(fact_3265_minf_I2_J,axiom,
    ! [P: int > $o,P4: int > $o,Q: int > $o,Q6: int > $o] :
      ( ? [Z3: int] :
        ! [X3: int] :
          ( ( ord_less_int @ X3 @ Z3 )
         => ( ( P @ X3 )
            = ( P4 @ X3 ) ) )
     => ( ? [Z3: int] :
          ! [X3: int] :
            ( ( ord_less_int @ X3 @ Z3 )
           => ( ( Q @ X3 )
              = ( Q6 @ X3 ) ) )
       => ? [Z2: int] :
          ! [X5: int] :
            ( ( ord_less_int @ X5 @ Z2 )
           => ( ( ( P @ X5 )
                | ( Q @ X5 ) )
              = ( ( P4 @ X5 )
                | ( Q6 @ X5 ) ) ) ) ) ) ).

% minf(2)
thf(fact_3266_minf_I1_J,axiom,
    ! [P: real > $o,P4: real > $o,Q: real > $o,Q6: real > $o] :
      ( ? [Z3: real] :
        ! [X3: real] :
          ( ( ord_less_real @ X3 @ Z3 )
         => ( ( P @ X3 )
            = ( P4 @ X3 ) ) )
     => ( ? [Z3: real] :
          ! [X3: real] :
            ( ( ord_less_real @ X3 @ Z3 )
           => ( ( Q @ X3 )
              = ( Q6 @ X3 ) ) )
       => ? [Z2: real] :
          ! [X5: real] :
            ( ( ord_less_real @ X5 @ Z2 )
           => ( ( ( P @ X5 )
                & ( Q @ X5 ) )
              = ( ( P4 @ X5 )
                & ( Q6 @ X5 ) ) ) ) ) ) ).

% minf(1)
thf(fact_3267_minf_I1_J,axiom,
    ! [P: rat > $o,P4: rat > $o,Q: rat > $o,Q6: rat > $o] :
      ( ? [Z3: rat] :
        ! [X3: rat] :
          ( ( ord_less_rat @ X3 @ Z3 )
         => ( ( P @ X3 )
            = ( P4 @ X3 ) ) )
     => ( ? [Z3: rat] :
          ! [X3: rat] :
            ( ( ord_less_rat @ X3 @ Z3 )
           => ( ( Q @ X3 )
              = ( Q6 @ X3 ) ) )
       => ? [Z2: rat] :
          ! [X5: rat] :
            ( ( ord_less_rat @ X5 @ Z2 )
           => ( ( ( P @ X5 )
                & ( Q @ X5 ) )
              = ( ( P4 @ X5 )
                & ( Q6 @ X5 ) ) ) ) ) ) ).

% minf(1)
thf(fact_3268_minf_I1_J,axiom,
    ! [P: num > $o,P4: num > $o,Q: num > $o,Q6: num > $o] :
      ( ? [Z3: num] :
        ! [X3: num] :
          ( ( ord_less_num @ X3 @ Z3 )
         => ( ( P @ X3 )
            = ( P4 @ X3 ) ) )
     => ( ? [Z3: num] :
          ! [X3: num] :
            ( ( ord_less_num @ X3 @ Z3 )
           => ( ( Q @ X3 )
              = ( Q6 @ X3 ) ) )
       => ? [Z2: num] :
          ! [X5: num] :
            ( ( ord_less_num @ X5 @ Z2 )
           => ( ( ( P @ X5 )
                & ( Q @ X5 ) )
              = ( ( P4 @ X5 )
                & ( Q6 @ X5 ) ) ) ) ) ) ).

% minf(1)
thf(fact_3269_minf_I1_J,axiom,
    ! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q6: nat > $o] :
      ( ? [Z3: nat] :
        ! [X3: nat] :
          ( ( ord_less_nat @ X3 @ Z3 )
         => ( ( P @ X3 )
            = ( P4 @ X3 ) ) )
     => ( ? [Z3: nat] :
          ! [X3: nat] :
            ( ( ord_less_nat @ X3 @ Z3 )
           => ( ( Q @ X3 )
              = ( Q6 @ X3 ) ) )
       => ? [Z2: nat] :
          ! [X5: nat] :
            ( ( ord_less_nat @ X5 @ Z2 )
           => ( ( ( P @ X5 )
                & ( Q @ X5 ) )
              = ( ( P4 @ X5 )
                & ( Q6 @ X5 ) ) ) ) ) ) ).

% minf(1)
thf(fact_3270_minf_I1_J,axiom,
    ! [P: int > $o,P4: int > $o,Q: int > $o,Q6: int > $o] :
      ( ? [Z3: int] :
        ! [X3: int] :
          ( ( ord_less_int @ X3 @ Z3 )
         => ( ( P @ X3 )
            = ( P4 @ X3 ) ) )
     => ( ? [Z3: int] :
          ! [X3: int] :
            ( ( ord_less_int @ X3 @ Z3 )
           => ( ( Q @ X3 )
              = ( Q6 @ X3 ) ) )
       => ? [Z2: int] :
          ! [X5: int] :
            ( ( ord_less_int @ X5 @ Z2 )
           => ( ( ( P @ X5 )
                & ( Q @ X5 ) )
              = ( ( P4 @ X5 )
                & ( Q6 @ X5 ) ) ) ) ) ) ).

% minf(1)
thf(fact_3271_pinf_I7_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X5: real] :
      ( ( ord_less_real @ Z2 @ X5 )
     => ( ord_less_real @ T @ X5 ) ) ).

% pinf(7)
thf(fact_3272_pinf_I7_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( ( ord_less_rat @ Z2 @ X5 )
     => ( ord_less_rat @ T @ X5 ) ) ).

% pinf(7)
thf(fact_3273_pinf_I7_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X5: num] :
      ( ( ord_less_num @ Z2 @ X5 )
     => ( ord_less_num @ T @ X5 ) ) ).

% pinf(7)
thf(fact_3274_pinf_I7_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ Z2 @ X5 )
     => ( ord_less_nat @ T @ X5 ) ) ).

% pinf(7)
thf(fact_3275_pinf_I7_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( ( ord_less_int @ Z2 @ X5 )
     => ( ord_less_int @ T @ X5 ) ) ).

% pinf(7)
thf(fact_3276_pinf_I5_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X5: real] :
      ( ( ord_less_real @ Z2 @ X5 )
     => ~ ( ord_less_real @ X5 @ T ) ) ).

% pinf(5)
thf(fact_3277_pinf_I5_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( ( ord_less_rat @ Z2 @ X5 )
     => ~ ( ord_less_rat @ X5 @ T ) ) ).

% pinf(5)
thf(fact_3278_pinf_I5_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X5: num] :
      ( ( ord_less_num @ Z2 @ X5 )
     => ~ ( ord_less_num @ X5 @ T ) ) ).

% pinf(5)
thf(fact_3279_pinf_I5_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ Z2 @ X5 )
     => ~ ( ord_less_nat @ X5 @ T ) ) ).

% pinf(5)
thf(fact_3280_pinf_I5_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( ( ord_less_int @ Z2 @ X5 )
     => ~ ( ord_less_int @ X5 @ T ) ) ).

% pinf(5)
thf(fact_3281_pinf_I4_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X5: real] :
      ( ( ord_less_real @ Z2 @ X5 )
     => ( X5 != T ) ) ).

% pinf(4)
thf(fact_3282_pinf_I4_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( ( ord_less_rat @ Z2 @ X5 )
     => ( X5 != T ) ) ).

% pinf(4)
thf(fact_3283_pinf_I4_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X5: num] :
      ( ( ord_less_num @ Z2 @ X5 )
     => ( X5 != T ) ) ).

% pinf(4)
thf(fact_3284_pinf_I4_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ Z2 @ X5 )
     => ( X5 != T ) ) ).

% pinf(4)
thf(fact_3285_pinf_I4_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( ( ord_less_int @ Z2 @ X5 )
     => ( X5 != T ) ) ).

% pinf(4)
thf(fact_3286_pinf_I3_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X5: real] :
      ( ( ord_less_real @ Z2 @ X5 )
     => ( X5 != T ) ) ).

% pinf(3)
thf(fact_3287_pinf_I3_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( ( ord_less_rat @ Z2 @ X5 )
     => ( X5 != T ) ) ).

% pinf(3)
thf(fact_3288_pinf_I3_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X5: num] :
      ( ( ord_less_num @ Z2 @ X5 )
     => ( X5 != T ) ) ).

% pinf(3)
thf(fact_3289_pinf_I3_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ Z2 @ X5 )
     => ( X5 != T ) ) ).

% pinf(3)
thf(fact_3290_pinf_I3_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( ( ord_less_int @ Z2 @ X5 )
     => ( X5 != T ) ) ).

% pinf(3)
thf(fact_3291_pinf_I2_J,axiom,
    ! [P: real > $o,P4: real > $o,Q: real > $o,Q6: real > $o] :
      ( ? [Z3: real] :
        ! [X3: real] :
          ( ( ord_less_real @ Z3 @ X3 )
         => ( ( P @ X3 )
            = ( P4 @ X3 ) ) )
     => ( ? [Z3: real] :
          ! [X3: real] :
            ( ( ord_less_real @ Z3 @ X3 )
           => ( ( Q @ X3 )
              = ( Q6 @ X3 ) ) )
       => ? [Z2: real] :
          ! [X5: real] :
            ( ( ord_less_real @ Z2 @ X5 )
           => ( ( ( P @ X5 )
                | ( Q @ X5 ) )
              = ( ( P4 @ X5 )
                | ( Q6 @ X5 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_3292_pinf_I2_J,axiom,
    ! [P: rat > $o,P4: rat > $o,Q: rat > $o,Q6: rat > $o] :
      ( ? [Z3: rat] :
        ! [X3: rat] :
          ( ( ord_less_rat @ Z3 @ X3 )
         => ( ( P @ X3 )
            = ( P4 @ X3 ) ) )
     => ( ? [Z3: rat] :
          ! [X3: rat] :
            ( ( ord_less_rat @ Z3 @ X3 )
           => ( ( Q @ X3 )
              = ( Q6 @ X3 ) ) )
       => ? [Z2: rat] :
          ! [X5: rat] :
            ( ( ord_less_rat @ Z2 @ X5 )
           => ( ( ( P @ X5 )
                | ( Q @ X5 ) )
              = ( ( P4 @ X5 )
                | ( Q6 @ X5 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_3293_pinf_I2_J,axiom,
    ! [P: num > $o,P4: num > $o,Q: num > $o,Q6: num > $o] :
      ( ? [Z3: num] :
        ! [X3: num] :
          ( ( ord_less_num @ Z3 @ X3 )
         => ( ( P @ X3 )
            = ( P4 @ X3 ) ) )
     => ( ? [Z3: num] :
          ! [X3: num] :
            ( ( ord_less_num @ Z3 @ X3 )
           => ( ( Q @ X3 )
              = ( Q6 @ X3 ) ) )
       => ? [Z2: num] :
          ! [X5: num] :
            ( ( ord_less_num @ Z2 @ X5 )
           => ( ( ( P @ X5 )
                | ( Q @ X5 ) )
              = ( ( P4 @ X5 )
                | ( Q6 @ X5 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_3294_pinf_I2_J,axiom,
    ! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q6: nat > $o] :
      ( ? [Z3: nat] :
        ! [X3: nat] :
          ( ( ord_less_nat @ Z3 @ X3 )
         => ( ( P @ X3 )
            = ( P4 @ X3 ) ) )
     => ( ? [Z3: nat] :
          ! [X3: nat] :
            ( ( ord_less_nat @ Z3 @ X3 )
           => ( ( Q @ X3 )
              = ( Q6 @ X3 ) ) )
       => ? [Z2: nat] :
          ! [X5: nat] :
            ( ( ord_less_nat @ Z2 @ X5 )
           => ( ( ( P @ X5 )
                | ( Q @ X5 ) )
              = ( ( P4 @ X5 )
                | ( Q6 @ X5 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_3295_pinf_I2_J,axiom,
    ! [P: int > $o,P4: int > $o,Q: int > $o,Q6: int > $o] :
      ( ? [Z3: int] :
        ! [X3: int] :
          ( ( ord_less_int @ Z3 @ X3 )
         => ( ( P @ X3 )
            = ( P4 @ X3 ) ) )
     => ( ? [Z3: int] :
          ! [X3: int] :
            ( ( ord_less_int @ Z3 @ X3 )
           => ( ( Q @ X3 )
              = ( Q6 @ X3 ) ) )
       => ? [Z2: int] :
          ! [X5: int] :
            ( ( ord_less_int @ Z2 @ X5 )
           => ( ( ( P @ X5 )
                | ( Q @ X5 ) )
              = ( ( P4 @ X5 )
                | ( Q6 @ X5 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_3296_pinf_I1_J,axiom,
    ! [P: real > $o,P4: real > $o,Q: real > $o,Q6: real > $o] :
      ( ? [Z3: real] :
        ! [X3: real] :
          ( ( ord_less_real @ Z3 @ X3 )
         => ( ( P @ X3 )
            = ( P4 @ X3 ) ) )
     => ( ? [Z3: real] :
          ! [X3: real] :
            ( ( ord_less_real @ Z3 @ X3 )
           => ( ( Q @ X3 )
              = ( Q6 @ X3 ) ) )
       => ? [Z2: real] :
          ! [X5: real] :
            ( ( ord_less_real @ Z2 @ X5 )
           => ( ( ( P @ X5 )
                & ( Q @ X5 ) )
              = ( ( P4 @ X5 )
                & ( Q6 @ X5 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_3297_pinf_I1_J,axiom,
    ! [P: rat > $o,P4: rat > $o,Q: rat > $o,Q6: rat > $o] :
      ( ? [Z3: rat] :
        ! [X3: rat] :
          ( ( ord_less_rat @ Z3 @ X3 )
         => ( ( P @ X3 )
            = ( P4 @ X3 ) ) )
     => ( ? [Z3: rat] :
          ! [X3: rat] :
            ( ( ord_less_rat @ Z3 @ X3 )
           => ( ( Q @ X3 )
              = ( Q6 @ X3 ) ) )
       => ? [Z2: rat] :
          ! [X5: rat] :
            ( ( ord_less_rat @ Z2 @ X5 )
           => ( ( ( P @ X5 )
                & ( Q @ X5 ) )
              = ( ( P4 @ X5 )
                & ( Q6 @ X5 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_3298_pinf_I1_J,axiom,
    ! [P: num > $o,P4: num > $o,Q: num > $o,Q6: num > $o] :
      ( ? [Z3: num] :
        ! [X3: num] :
          ( ( ord_less_num @ Z3 @ X3 )
         => ( ( P @ X3 )
            = ( P4 @ X3 ) ) )
     => ( ? [Z3: num] :
          ! [X3: num] :
            ( ( ord_less_num @ Z3 @ X3 )
           => ( ( Q @ X3 )
              = ( Q6 @ X3 ) ) )
       => ? [Z2: num] :
          ! [X5: num] :
            ( ( ord_less_num @ Z2 @ X5 )
           => ( ( ( P @ X5 )
                & ( Q @ X5 ) )
              = ( ( P4 @ X5 )
                & ( Q6 @ X5 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_3299_pinf_I1_J,axiom,
    ! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q6: nat > $o] :
      ( ? [Z3: nat] :
        ! [X3: nat] :
          ( ( ord_less_nat @ Z3 @ X3 )
         => ( ( P @ X3 )
            = ( P4 @ X3 ) ) )
     => ( ? [Z3: nat] :
          ! [X3: nat] :
            ( ( ord_less_nat @ Z3 @ X3 )
           => ( ( Q @ X3 )
              = ( Q6 @ X3 ) ) )
       => ? [Z2: nat] :
          ! [X5: nat] :
            ( ( ord_less_nat @ Z2 @ X5 )
           => ( ( ( P @ X5 )
                & ( Q @ X5 ) )
              = ( ( P4 @ X5 )
                & ( Q6 @ X5 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_3300_pinf_I1_J,axiom,
    ! [P: int > $o,P4: int > $o,Q: int > $o,Q6: int > $o] :
      ( ? [Z3: int] :
        ! [X3: int] :
          ( ( ord_less_int @ Z3 @ X3 )
         => ( ( P @ X3 )
            = ( P4 @ X3 ) ) )
     => ( ? [Z3: int] :
          ! [X3: int] :
            ( ( ord_less_int @ Z3 @ X3 )
           => ( ( Q @ X3 )
              = ( Q6 @ X3 ) ) )
       => ? [Z2: int] :
          ! [X5: int] :
            ( ( ord_less_int @ Z2 @ X5 )
           => ( ( ( P @ X5 )
                & ( Q @ X5 ) )
              = ( ( P4 @ X5 )
                & ( Q6 @ X5 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_3301_vebt__buildup_Ocases,axiom,
    ! [X4: nat] :
      ( ( X4 != zero_zero_nat )
     => ( ( X4
         != ( suc @ zero_zero_nat ) )
       => ~ ! [Va: nat] :
              ( X4
             != ( suc @ ( suc @ Va ) ) ) ) ) ).

% vebt_buildup.cases
thf(fact_3302_Euclid__induct,axiom,
    ! [P: nat > nat > $o,A: nat,B2: nat] :
      ( ! [A5: nat,B5: nat] :
          ( ( P @ A5 @ B5 )
          = ( P @ B5 @ A5 ) )
     => ( ! [A5: nat] : ( P @ A5 @ zero_zero_nat )
       => ( ! [A5: nat,B5: nat] :
              ( ( P @ A5 @ B5 )
             => ( P @ A5 @ ( plus_plus_nat @ A5 @ B5 ) ) )
         => ( P @ A @ B2 ) ) ) ) ).

% Euclid_induct
thf(fact_3303_minf_I8_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X5: real] :
      ( ( ord_less_real @ X5 @ Z2 )
     => ~ ( ord_less_eq_real @ T @ X5 ) ) ).

% minf(8)
thf(fact_3304_minf_I8_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( ( ord_less_rat @ X5 @ Z2 )
     => ~ ( ord_less_eq_rat @ T @ X5 ) ) ).

% minf(8)
thf(fact_3305_minf_I8_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X5: num] :
      ( ( ord_less_num @ X5 @ Z2 )
     => ~ ( ord_less_eq_num @ T @ X5 ) ) ).

% minf(8)
thf(fact_3306_minf_I8_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ X5 @ Z2 )
     => ~ ( ord_less_eq_nat @ T @ X5 ) ) ).

% minf(8)
thf(fact_3307_minf_I8_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( ( ord_less_int @ X5 @ Z2 )
     => ~ ( ord_less_eq_int @ T @ X5 ) ) ).

% minf(8)
thf(fact_3308_minf_I6_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X5: real] :
      ( ( ord_less_real @ X5 @ Z2 )
     => ( ord_less_eq_real @ X5 @ T ) ) ).

% minf(6)
thf(fact_3309_minf_I6_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( ( ord_less_rat @ X5 @ Z2 )
     => ( ord_less_eq_rat @ X5 @ T ) ) ).

% minf(6)
thf(fact_3310_minf_I6_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X5: num] :
      ( ( ord_less_num @ X5 @ Z2 )
     => ( ord_less_eq_num @ X5 @ T ) ) ).

% minf(6)
thf(fact_3311_minf_I6_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ X5 @ Z2 )
     => ( ord_less_eq_nat @ X5 @ T ) ) ).

% minf(6)
thf(fact_3312_minf_I6_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( ( ord_less_int @ X5 @ Z2 )
     => ( ord_less_eq_int @ X5 @ T ) ) ).

% minf(6)
thf(fact_3313_pinf_I8_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X5: real] :
      ( ( ord_less_real @ Z2 @ X5 )
     => ( ord_less_eq_real @ T @ X5 ) ) ).

% pinf(8)
thf(fact_3314_pinf_I8_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( ( ord_less_rat @ Z2 @ X5 )
     => ( ord_less_eq_rat @ T @ X5 ) ) ).

% pinf(8)
thf(fact_3315_pinf_I8_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X5: num] :
      ( ( ord_less_num @ Z2 @ X5 )
     => ( ord_less_eq_num @ T @ X5 ) ) ).

% pinf(8)
thf(fact_3316_pinf_I8_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ Z2 @ X5 )
     => ( ord_less_eq_nat @ T @ X5 ) ) ).

% pinf(8)
thf(fact_3317_pinf_I8_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( ( ord_less_int @ Z2 @ X5 )
     => ( ord_less_eq_int @ T @ X5 ) ) ).

% pinf(8)
thf(fact_3318_pinf_I6_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X5: real] :
      ( ( ord_less_real @ Z2 @ X5 )
     => ~ ( ord_less_eq_real @ X5 @ T ) ) ).

% pinf(6)
thf(fact_3319_pinf_I6_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( ( ord_less_rat @ Z2 @ X5 )
     => ~ ( ord_less_eq_rat @ X5 @ T ) ) ).

% pinf(6)
thf(fact_3320_pinf_I6_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X5: num] :
      ( ( ord_less_num @ Z2 @ X5 )
     => ~ ( ord_less_eq_num @ X5 @ T ) ) ).

% pinf(6)
thf(fact_3321_pinf_I6_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ Z2 @ X5 )
     => ~ ( ord_less_eq_nat @ X5 @ T ) ) ).

% pinf(6)
thf(fact_3322_pinf_I6_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( ( ord_less_int @ Z2 @ X5 )
     => ~ ( ord_less_eq_int @ X5 @ T ) ) ).

% pinf(6)
thf(fact_3323_inf__period_I2_J,axiom,
    ! [P: real > $o,D4: real,Q: real > $o] :
      ( ! [X3: real,K2: real] :
          ( ( P @ X3 )
          = ( P @ ( minus_minus_real @ X3 @ ( times_times_real @ K2 @ D4 ) ) ) )
     => ( ! [X3: real,K2: real] :
            ( ( Q @ X3 )
            = ( Q @ ( minus_minus_real @ X3 @ ( times_times_real @ K2 @ D4 ) ) ) )
       => ! [X5: real,K4: real] :
            ( ( ( P @ X5 )
              | ( Q @ X5 ) )
            = ( ( P @ ( minus_minus_real @ X5 @ ( times_times_real @ K4 @ D4 ) ) )
              | ( Q @ ( minus_minus_real @ X5 @ ( times_times_real @ K4 @ D4 ) ) ) ) ) ) ) ).

% inf_period(2)
thf(fact_3324_inf__period_I2_J,axiom,
    ! [P: rat > $o,D4: rat,Q: rat > $o] :
      ( ! [X3: rat,K2: rat] :
          ( ( P @ X3 )
          = ( P @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K2 @ D4 ) ) ) )
     => ( ! [X3: rat,K2: rat] :
            ( ( Q @ X3 )
            = ( Q @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K2 @ D4 ) ) ) )
       => ! [X5: rat,K4: rat] :
            ( ( ( P @ X5 )
              | ( Q @ X5 ) )
            = ( ( P @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K4 @ D4 ) ) )
              | ( Q @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K4 @ D4 ) ) ) ) ) ) ) ).

% inf_period(2)
thf(fact_3325_inf__period_I2_J,axiom,
    ! [P: int > $o,D4: int,Q: int > $o] :
      ( ! [X3: int,K2: int] :
          ( ( P @ X3 )
          = ( P @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D4 ) ) ) )
     => ( ! [X3: int,K2: int] :
            ( ( Q @ X3 )
            = ( Q @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D4 ) ) ) )
       => ! [X5: int,K4: int] :
            ( ( ( P @ X5 )
              | ( Q @ X5 ) )
            = ( ( P @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D4 ) ) )
              | ( Q @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D4 ) ) ) ) ) ) ) ).

% inf_period(2)
thf(fact_3326_inf__period_I1_J,axiom,
    ! [P: real > $o,D4: real,Q: real > $o] :
      ( ! [X3: real,K2: real] :
          ( ( P @ X3 )
          = ( P @ ( minus_minus_real @ X3 @ ( times_times_real @ K2 @ D4 ) ) ) )
     => ( ! [X3: real,K2: real] :
            ( ( Q @ X3 )
            = ( Q @ ( minus_minus_real @ X3 @ ( times_times_real @ K2 @ D4 ) ) ) )
       => ! [X5: real,K4: real] :
            ( ( ( P @ X5 )
              & ( Q @ X5 ) )
            = ( ( P @ ( minus_minus_real @ X5 @ ( times_times_real @ K4 @ D4 ) ) )
              & ( Q @ ( minus_minus_real @ X5 @ ( times_times_real @ K4 @ D4 ) ) ) ) ) ) ) ).

% inf_period(1)
thf(fact_3327_inf__period_I1_J,axiom,
    ! [P: rat > $o,D4: rat,Q: rat > $o] :
      ( ! [X3: rat,K2: rat] :
          ( ( P @ X3 )
          = ( P @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K2 @ D4 ) ) ) )
     => ( ! [X3: rat,K2: rat] :
            ( ( Q @ X3 )
            = ( Q @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K2 @ D4 ) ) ) )
       => ! [X5: rat,K4: rat] :
            ( ( ( P @ X5 )
              & ( Q @ X5 ) )
            = ( ( P @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K4 @ D4 ) ) )
              & ( Q @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K4 @ D4 ) ) ) ) ) ) ) ).

% inf_period(1)
thf(fact_3328_inf__period_I1_J,axiom,
    ! [P: int > $o,D4: int,Q: int > $o] :
      ( ! [X3: int,K2: int] :
          ( ( P @ X3 )
          = ( P @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D4 ) ) ) )
     => ( ! [X3: int,K2: int] :
            ( ( Q @ X3 )
            = ( Q @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D4 ) ) ) )
       => ! [X5: int,K4: int] :
            ( ( ( P @ X5 )
              & ( Q @ X5 ) )
            = ( ( P @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D4 ) ) )
              & ( Q @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D4 ) ) ) ) ) ) ) ).

% inf_period(1)
thf(fact_3329_dvd__productE,axiom,
    ! [P2: nat,A: nat,B2: nat] :
      ( ( dvd_dvd_nat @ P2 @ ( times_times_nat @ A @ B2 ) )
     => ~ ! [X3: nat,Y4: nat] :
            ( ( P2
              = ( times_times_nat @ X3 @ Y4 ) )
           => ( ( dvd_dvd_nat @ X3 @ A )
             => ~ ( dvd_dvd_nat @ Y4 @ B2 ) ) ) ) ).

% dvd_productE
thf(fact_3330_dvd__productE,axiom,
    ! [P2: int,A: int,B2: int] :
      ( ( dvd_dvd_int @ P2 @ ( times_times_int @ A @ B2 ) )
     => ~ ! [X3: int,Y4: int] :
            ( ( P2
              = ( times_times_int @ X3 @ Y4 ) )
           => ( ( dvd_dvd_int @ X3 @ A )
             => ~ ( dvd_dvd_int @ Y4 @ B2 ) ) ) ) ).

% dvd_productE
thf(fact_3331_division__decomp,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ ( times_times_nat @ B2 @ C ) )
     => ? [B6: nat,C4: nat] :
          ( ( A
            = ( times_times_nat @ B6 @ C4 ) )
          & ( dvd_dvd_nat @ B6 @ B2 )
          & ( dvd_dvd_nat @ C4 @ C ) ) ) ).

% division_decomp
thf(fact_3332_division__decomp,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( dvd_dvd_int @ A @ ( times_times_int @ B2 @ C ) )
     => ? [B6: int,C4: int] :
          ( ( A
            = ( times_times_int @ B6 @ C4 ) )
          & ( dvd_dvd_int @ B6 @ B2 )
          & ( dvd_dvd_int @ C4 @ C ) ) ) ).

% division_decomp
thf(fact_3333_dvd__pos__nat,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( dvd_dvd_nat @ M @ N )
       => ( ord_less_nat @ zero_zero_nat @ M ) ) ) ).

% dvd_pos_nat
thf(fact_3334_gcd__nat__induct,axiom,
    ! [P: nat > nat > $o,M: nat,N: nat] :
      ( ! [M4: nat] : ( P @ M4 @ zero_zero_nat )
     => ( ! [M4: nat,N4: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N4 )
           => ( ( P @ N4 @ ( modulo_modulo_nat @ M4 @ N4 ) )
             => ( P @ M4 @ N4 ) ) )
       => ( P @ M @ N ) ) ) ).

% gcd_nat_induct
thf(fact_3335_pinf_I9_J,axiom,
    ! [D: code_integer,S: code_integer] :
    ? [Z2: code_integer] :
    ! [X5: code_integer] :
      ( ( ord_le6747313008572928689nteger @ Z2 @ X5 )
     => ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S ) )
        = ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S ) ) ) ) ).

% pinf(9)
thf(fact_3336_pinf_I9_J,axiom,
    ! [D: real,S: real] :
    ? [Z2: real] :
    ! [X5: real] :
      ( ( ord_less_real @ Z2 @ X5 )
     => ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S ) )
        = ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S ) ) ) ) ).

% pinf(9)
thf(fact_3337_pinf_I9_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( ( ord_less_rat @ Z2 @ X5 )
     => ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S ) )
        = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S ) ) ) ) ).

% pinf(9)
thf(fact_3338_pinf_I9_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ Z2 @ X5 )
     => ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) )
        = ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) ) ) ) ).

% pinf(9)
thf(fact_3339_pinf_I9_J,axiom,
    ! [D: int,S: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( ( ord_less_int @ Z2 @ X5 )
     => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) )
        = ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) ) ) ) ).

% pinf(9)
thf(fact_3340_pinf_I10_J,axiom,
    ! [D: code_integer,S: code_integer] :
    ? [Z2: code_integer] :
    ! [X5: code_integer] :
      ( ( ord_le6747313008572928689nteger @ Z2 @ X5 )
     => ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S ) ) )
        = ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S ) ) ) ) ) ).

% pinf(10)
thf(fact_3341_pinf_I10_J,axiom,
    ! [D: real,S: real] :
    ? [Z2: real] :
    ! [X5: real] :
      ( ( ord_less_real @ Z2 @ X5 )
     => ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S ) ) )
        = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S ) ) ) ) ) ).

% pinf(10)
thf(fact_3342_pinf_I10_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( ( ord_less_rat @ Z2 @ X5 )
     => ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S ) ) )
        = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S ) ) ) ) ) ).

% pinf(10)
thf(fact_3343_pinf_I10_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ Z2 @ X5 )
     => ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) ) )
        = ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) ) ) ) ) ).

% pinf(10)
thf(fact_3344_pinf_I10_J,axiom,
    ! [D: int,S: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( ( ord_less_int @ Z2 @ X5 )
     => ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) ) )
        = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) ) ) ) ) ).

% pinf(10)
thf(fact_3345_minf_I9_J,axiom,
    ! [D: code_integer,S: code_integer] :
    ? [Z2: code_integer] :
    ! [X5: code_integer] :
      ( ( ord_le6747313008572928689nteger @ X5 @ Z2 )
     => ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S ) )
        = ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S ) ) ) ) ).

% minf(9)
thf(fact_3346_minf_I9_J,axiom,
    ! [D: real,S: real] :
    ? [Z2: real] :
    ! [X5: real] :
      ( ( ord_less_real @ X5 @ Z2 )
     => ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S ) )
        = ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S ) ) ) ) ).

% minf(9)
thf(fact_3347_minf_I9_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( ( ord_less_rat @ X5 @ Z2 )
     => ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S ) )
        = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S ) ) ) ) ).

% minf(9)
thf(fact_3348_minf_I9_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ X5 @ Z2 )
     => ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) )
        = ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) ) ) ) ).

% minf(9)
thf(fact_3349_minf_I9_J,axiom,
    ! [D: int,S: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( ( ord_less_int @ X5 @ Z2 )
     => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) )
        = ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) ) ) ) ).

% minf(9)
thf(fact_3350_minf_I10_J,axiom,
    ! [D: code_integer,S: code_integer] :
    ? [Z2: code_integer] :
    ! [X5: code_integer] :
      ( ( ord_le6747313008572928689nteger @ X5 @ Z2 )
     => ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S ) ) )
        = ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S ) ) ) ) ) ).

% minf(10)
thf(fact_3351_minf_I10_J,axiom,
    ! [D: real,S: real] :
    ? [Z2: real] :
    ! [X5: real] :
      ( ( ord_less_real @ X5 @ Z2 )
     => ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S ) ) )
        = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S ) ) ) ) ) ).

% minf(10)
thf(fact_3352_minf_I10_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( ( ord_less_rat @ X5 @ Z2 )
     => ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S ) ) )
        = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S ) ) ) ) ) ).

% minf(10)
thf(fact_3353_minf_I10_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ X5 @ Z2 )
     => ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) ) )
        = ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) ) ) ) ) ).

% minf(10)
thf(fact_3354_minf_I10_J,axiom,
    ! [D: int,S: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( ( ord_less_int @ X5 @ Z2 )
     => ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) ) )
        = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) ) ) ) ) ).

% minf(10)
thf(fact_3355_bezout__lemma__nat,axiom,
    ! [D: nat,A: nat,B2: nat,X4: nat,Y3: nat] :
      ( ( dvd_dvd_nat @ D @ A )
     => ( ( dvd_dvd_nat @ D @ B2 )
       => ( ( ( ( times_times_nat @ A @ X4 )
              = ( plus_plus_nat @ ( times_times_nat @ B2 @ Y3 ) @ D ) )
            | ( ( times_times_nat @ B2 @ X4 )
              = ( plus_plus_nat @ ( times_times_nat @ A @ Y3 ) @ D ) ) )
         => ? [X3: nat,Y4: nat] :
              ( ( dvd_dvd_nat @ D @ A )
              & ( dvd_dvd_nat @ D @ ( plus_plus_nat @ A @ B2 ) )
              & ( ( ( times_times_nat @ A @ X3 )
                  = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B2 ) @ Y4 ) @ D ) )
                | ( ( times_times_nat @ ( plus_plus_nat @ A @ B2 ) @ X3 )
                  = ( plus_plus_nat @ ( times_times_nat @ A @ Y4 ) @ D ) ) ) ) ) ) ) ).

% bezout_lemma_nat
thf(fact_3356_bezout__add__nat,axiom,
    ! [A: nat,B2: nat] :
    ? [D3: nat,X3: nat,Y4: nat] :
      ( ( dvd_dvd_nat @ D3 @ A )
      & ( dvd_dvd_nat @ D3 @ B2 )
      & ( ( ( times_times_nat @ A @ X3 )
          = ( plus_plus_nat @ ( times_times_nat @ B2 @ Y4 ) @ D3 ) )
        | ( ( times_times_nat @ B2 @ X3 )
          = ( plus_plus_nat @ ( times_times_nat @ A @ Y4 ) @ D3 ) ) ) ) ).

% bezout_add_nat
thf(fact_3357_bezout__add__strong__nat,axiom,
    ! [A: nat,B2: nat] :
      ( ( A != zero_zero_nat )
     => ? [D3: nat,X3: nat,Y4: nat] :
          ( ( dvd_dvd_nat @ D3 @ A )
          & ( dvd_dvd_nat @ D3 @ B2 )
          & ( ( times_times_nat @ A @ X3 )
            = ( plus_plus_nat @ ( times_times_nat @ B2 @ Y4 ) @ D3 ) ) ) ) ).

% bezout_add_strong_nat
thf(fact_3358_bezout1__nat,axiom,
    ! [A: nat,B2: nat] :
    ? [D3: nat,X3: nat,Y4: nat] :
      ( ( dvd_dvd_nat @ D3 @ A )
      & ( dvd_dvd_nat @ D3 @ B2 )
      & ( ( ( minus_minus_nat @ ( times_times_nat @ A @ X3 ) @ ( times_times_nat @ B2 @ Y4 ) )
          = D3 )
        | ( ( minus_minus_nat @ ( times_times_nat @ B2 @ X3 ) @ ( times_times_nat @ A @ Y4 ) )
          = D3 ) ) ) ).

% bezout1_nat
thf(fact_3359_Bolzano,axiom,
    ! [A: real,B2: real,P: real > real > $o] :
      ( ( ord_less_eq_real @ A @ B2 )
     => ( ! [A5: real,B5: real,C3: real] :
            ( ( P @ A5 @ B5 )
           => ( ( P @ B5 @ C3 )
             => ( ( ord_less_eq_real @ A5 @ B5 )
               => ( ( ord_less_eq_real @ B5 @ C3 )
                 => ( P @ A5 @ C3 ) ) ) ) )
       => ( ! [X3: real] :
              ( ( ord_less_eq_real @ A @ X3 )
             => ( ( ord_less_eq_real @ X3 @ B2 )
               => ? [D5: real] :
                    ( ( ord_less_real @ zero_zero_real @ D5 )
                    & ! [A5: real,B5: real] :
                        ( ( ( ord_less_eq_real @ A5 @ X3 )
                          & ( ord_less_eq_real @ X3 @ B5 )
                          & ( ord_less_real @ ( minus_minus_real @ B5 @ A5 ) @ D5 ) )
                       => ( P @ A5 @ B5 ) ) ) ) )
         => ( P @ A @ B2 ) ) ) ) ).

% Bolzano
thf(fact_3360_mult__le__cancel__iff1,axiom,
    ! [Z: real,X4: real,Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ Z )
     => ( ( ord_less_eq_real @ ( times_times_real @ X4 @ Z ) @ ( times_times_real @ Y3 @ Z ) )
        = ( ord_less_eq_real @ X4 @ Y3 ) ) ) ).

% mult_le_cancel_iff1
thf(fact_3361_mult__le__cancel__iff1,axiom,
    ! [Z: rat,X4: rat,Y3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Z )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ X4 @ Z ) @ ( times_times_rat @ Y3 @ Z ) )
        = ( ord_less_eq_rat @ X4 @ Y3 ) ) ) ).

% mult_le_cancel_iff1
thf(fact_3362_mult__le__cancel__iff1,axiom,
    ! [Z: int,X4: int,Y3: int] :
      ( ( ord_less_int @ zero_zero_int @ Z )
     => ( ( ord_less_eq_int @ ( times_times_int @ X4 @ Z ) @ ( times_times_int @ Y3 @ Z ) )
        = ( ord_less_eq_int @ X4 @ Y3 ) ) ) ).

% mult_le_cancel_iff1
thf(fact_3363_mult__le__cancel__iff2,axiom,
    ! [Z: real,X4: real,Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ Z )
     => ( ( ord_less_eq_real @ ( times_times_real @ Z @ X4 ) @ ( times_times_real @ Z @ Y3 ) )
        = ( ord_less_eq_real @ X4 @ Y3 ) ) ) ).

% mult_le_cancel_iff2
thf(fact_3364_mult__le__cancel__iff2,axiom,
    ! [Z: rat,X4: rat,Y3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Z )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ Z @ X4 ) @ ( times_times_rat @ Z @ Y3 ) )
        = ( ord_less_eq_rat @ X4 @ Y3 ) ) ) ).

% mult_le_cancel_iff2
thf(fact_3365_mult__le__cancel__iff2,axiom,
    ! [Z: int,X4: int,Y3: int] :
      ( ( ord_less_int @ zero_zero_int @ Z )
     => ( ( ord_less_eq_int @ ( times_times_int @ Z @ X4 ) @ ( times_times_int @ Z @ Y3 ) )
        = ( ord_less_eq_int @ X4 @ Y3 ) ) ) ).

% mult_le_cancel_iff2
thf(fact_3366_triangle__def,axiom,
    ( nat_triangle
    = ( ^ [N2: nat] : ( divide_divide_nat @ ( times_times_nat @ N2 @ ( suc @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% triangle_def
thf(fact_3367_signed__take__bit__rec,axiom,
    ( bit_ri6519982836138164636nteger
    = ( ^ [N2: nat,A4: code_integer] : ( if_Code_integer @ ( N2 = zero_zero_nat ) @ ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( divide6298287555418463151nteger @ A4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% signed_take_bit_rec
thf(fact_3368_signed__take__bit__rec,axiom,
    ( bit_ri631733984087533419it_int
    = ( ^ [N2: nat,A4: int] : ( if_int @ ( N2 = zero_zero_nat ) @ ( uminus_uminus_int @ ( modulo_modulo_int @ A4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( plus_plus_int @ ( modulo_modulo_int @ A4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( divide_divide_int @ A4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% signed_take_bit_rec
thf(fact_3369_verit__minus__simplify_I4_J,axiom,
    ! [B2: real] :
      ( ( uminus_uminus_real @ ( uminus_uminus_real @ B2 ) )
      = B2 ) ).

% verit_minus_simplify(4)
thf(fact_3370_verit__minus__simplify_I4_J,axiom,
    ! [B2: int] :
      ( ( uminus_uminus_int @ ( uminus_uminus_int @ B2 ) )
      = B2 ) ).

% verit_minus_simplify(4)
thf(fact_3371_verit__minus__simplify_I4_J,axiom,
    ! [B2: complex] :
      ( ( uminus1482373934393186551omplex @ ( uminus1482373934393186551omplex @ B2 ) )
      = B2 ) ).

% verit_minus_simplify(4)
thf(fact_3372_verit__minus__simplify_I4_J,axiom,
    ! [B2: code_integer] :
      ( ( uminus1351360451143612070nteger @ ( uminus1351360451143612070nteger @ B2 ) )
      = B2 ) ).

% verit_minus_simplify(4)
thf(fact_3373_verit__minus__simplify_I4_J,axiom,
    ! [B2: rat] :
      ( ( uminus_uminus_rat @ ( uminus_uminus_rat @ B2 ) )
      = B2 ) ).

% verit_minus_simplify(4)
thf(fact_3374_add_Oinverse__inverse,axiom,
    ! [A: real] :
      ( ( uminus_uminus_real @ ( uminus_uminus_real @ A ) )
      = A ) ).

% add.inverse_inverse
thf(fact_3375_add_Oinverse__inverse,axiom,
    ! [A: int] :
      ( ( uminus_uminus_int @ ( uminus_uminus_int @ A ) )
      = A ) ).

% add.inverse_inverse
thf(fact_3376_add_Oinverse__inverse,axiom,
    ! [A: complex] :
      ( ( uminus1482373934393186551omplex @ ( uminus1482373934393186551omplex @ A ) )
      = A ) ).

% add.inverse_inverse
thf(fact_3377_add_Oinverse__inverse,axiom,
    ! [A: code_integer] :
      ( ( uminus1351360451143612070nteger @ ( uminus1351360451143612070nteger @ A ) )
      = A ) ).

% add.inverse_inverse
thf(fact_3378_add_Oinverse__inverse,axiom,
    ! [A: rat] :
      ( ( uminus_uminus_rat @ ( uminus_uminus_rat @ A ) )
      = A ) ).

% add.inverse_inverse
thf(fact_3379_neg__equal__iff__equal,axiom,
    ! [A: real,B2: real] :
      ( ( ( uminus_uminus_real @ A )
        = ( uminus_uminus_real @ B2 ) )
      = ( A = B2 ) ) ).

% neg_equal_iff_equal
thf(fact_3380_neg__equal__iff__equal,axiom,
    ! [A: int,B2: int] :
      ( ( ( uminus_uminus_int @ A )
        = ( uminus_uminus_int @ B2 ) )
      = ( A = B2 ) ) ).

% neg_equal_iff_equal
thf(fact_3381_neg__equal__iff__equal,axiom,
    ! [A: complex,B2: complex] :
      ( ( ( uminus1482373934393186551omplex @ A )
        = ( uminus1482373934393186551omplex @ B2 ) )
      = ( A = B2 ) ) ).

% neg_equal_iff_equal
thf(fact_3382_neg__equal__iff__equal,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( ( uminus1351360451143612070nteger @ A )
        = ( uminus1351360451143612070nteger @ B2 ) )
      = ( A = B2 ) ) ).

% neg_equal_iff_equal
thf(fact_3383_neg__equal__iff__equal,axiom,
    ! [A: rat,B2: rat] :
      ( ( ( uminus_uminus_rat @ A )
        = ( uminus_uminus_rat @ B2 ) )
      = ( A = B2 ) ) ).

% neg_equal_iff_equal
thf(fact_3384_neg__le__iff__le,axiom,
    ! [B2: real,A: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ B2 ) @ ( uminus_uminus_real @ A ) )
      = ( ord_less_eq_real @ A @ B2 ) ) ).

% neg_le_iff_le
thf(fact_3385_neg__le__iff__le,axiom,
    ! [B2: code_integer,A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B2 ) @ ( uminus1351360451143612070nteger @ A ) )
      = ( ord_le3102999989581377725nteger @ A @ B2 ) ) ).

% neg_le_iff_le
thf(fact_3386_neg__le__iff__le,axiom,
    ! [B2: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ B2 ) @ ( uminus_uminus_rat @ A ) )
      = ( ord_less_eq_rat @ A @ B2 ) ) ).

% neg_le_iff_le
thf(fact_3387_neg__le__iff__le,axiom,
    ! [B2: int,A: int] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ B2 ) @ ( uminus_uminus_int @ A ) )
      = ( ord_less_eq_int @ A @ B2 ) ) ).

% neg_le_iff_le
thf(fact_3388_add_Oinverse__neutral,axiom,
    ( ( uminus_uminus_real @ zero_zero_real )
    = zero_zero_real ) ).

% add.inverse_neutral
thf(fact_3389_add_Oinverse__neutral,axiom,
    ( ( uminus_uminus_int @ zero_zero_int )
    = zero_zero_int ) ).

% add.inverse_neutral
thf(fact_3390_add_Oinverse__neutral,axiom,
    ( ( uminus1482373934393186551omplex @ zero_zero_complex )
    = zero_zero_complex ) ).

% add.inverse_neutral
thf(fact_3391_add_Oinverse__neutral,axiom,
    ( ( uminus1351360451143612070nteger @ zero_z3403309356797280102nteger )
    = zero_z3403309356797280102nteger ) ).

% add.inverse_neutral
thf(fact_3392_add_Oinverse__neutral,axiom,
    ( ( uminus_uminus_rat @ zero_zero_rat )
    = zero_zero_rat ) ).

% add.inverse_neutral
thf(fact_3393_neg__0__equal__iff__equal,axiom,
    ! [A: real] :
      ( ( zero_zero_real
        = ( uminus_uminus_real @ A ) )
      = ( zero_zero_real = A ) ) ).

% neg_0_equal_iff_equal
thf(fact_3394_neg__0__equal__iff__equal,axiom,
    ! [A: int] :
      ( ( zero_zero_int
        = ( uminus_uminus_int @ A ) )
      = ( zero_zero_int = A ) ) ).

% neg_0_equal_iff_equal
thf(fact_3395_neg__0__equal__iff__equal,axiom,
    ! [A: complex] :
      ( ( zero_zero_complex
        = ( uminus1482373934393186551omplex @ A ) )
      = ( zero_zero_complex = A ) ) ).

% neg_0_equal_iff_equal
thf(fact_3396_neg__0__equal__iff__equal,axiom,
    ! [A: code_integer] :
      ( ( zero_z3403309356797280102nteger
        = ( uminus1351360451143612070nteger @ A ) )
      = ( zero_z3403309356797280102nteger = A ) ) ).

% neg_0_equal_iff_equal
thf(fact_3397_neg__0__equal__iff__equal,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat
        = ( uminus_uminus_rat @ A ) )
      = ( zero_zero_rat = A ) ) ).

% neg_0_equal_iff_equal
thf(fact_3398_neg__equal__0__iff__equal,axiom,
    ! [A: real] :
      ( ( ( uminus_uminus_real @ A )
        = zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% neg_equal_0_iff_equal
thf(fact_3399_neg__equal__0__iff__equal,axiom,
    ! [A: int] :
      ( ( ( uminus_uminus_int @ A )
        = zero_zero_int )
      = ( A = zero_zero_int ) ) ).

% neg_equal_0_iff_equal
thf(fact_3400_neg__equal__0__iff__equal,axiom,
    ! [A: complex] :
      ( ( ( uminus1482373934393186551omplex @ A )
        = zero_zero_complex )
      = ( A = zero_zero_complex ) ) ).

% neg_equal_0_iff_equal
thf(fact_3401_neg__equal__0__iff__equal,axiom,
    ! [A: code_integer] :
      ( ( ( uminus1351360451143612070nteger @ A )
        = zero_z3403309356797280102nteger )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% neg_equal_0_iff_equal
thf(fact_3402_neg__equal__0__iff__equal,axiom,
    ! [A: rat] :
      ( ( ( uminus_uminus_rat @ A )
        = zero_zero_rat )
      = ( A = zero_zero_rat ) ) ).

% neg_equal_0_iff_equal
thf(fact_3403_equal__neg__zero,axiom,
    ! [A: real] :
      ( ( A
        = ( uminus_uminus_real @ A ) )
      = ( A = zero_zero_real ) ) ).

% equal_neg_zero
thf(fact_3404_equal__neg__zero,axiom,
    ! [A: int] :
      ( ( A
        = ( uminus_uminus_int @ A ) )
      = ( A = zero_zero_int ) ) ).

% equal_neg_zero
thf(fact_3405_equal__neg__zero,axiom,
    ! [A: code_integer] :
      ( ( A
        = ( uminus1351360451143612070nteger @ A ) )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% equal_neg_zero
thf(fact_3406_equal__neg__zero,axiom,
    ! [A: rat] :
      ( ( A
        = ( uminus_uminus_rat @ A ) )
      = ( A = zero_zero_rat ) ) ).

% equal_neg_zero
thf(fact_3407_neg__equal__zero,axiom,
    ! [A: real] :
      ( ( ( uminus_uminus_real @ A )
        = A )
      = ( A = zero_zero_real ) ) ).

% neg_equal_zero
thf(fact_3408_neg__equal__zero,axiom,
    ! [A: int] :
      ( ( ( uminus_uminus_int @ A )
        = A )
      = ( A = zero_zero_int ) ) ).

% neg_equal_zero
thf(fact_3409_neg__equal__zero,axiom,
    ! [A: code_integer] :
      ( ( ( uminus1351360451143612070nteger @ A )
        = A )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% neg_equal_zero
thf(fact_3410_neg__equal__zero,axiom,
    ! [A: rat] :
      ( ( ( uminus_uminus_rat @ A )
        = A )
      = ( A = zero_zero_rat ) ) ).

% neg_equal_zero
thf(fact_3411_neg__less__iff__less,axiom,
    ! [B2: real,A: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ B2 ) @ ( uminus_uminus_real @ A ) )
      = ( ord_less_real @ A @ B2 ) ) ).

% neg_less_iff_less
thf(fact_3412_neg__less__iff__less,axiom,
    ! [B2: int,A: int] :
      ( ( ord_less_int @ ( uminus_uminus_int @ B2 ) @ ( uminus_uminus_int @ A ) )
      = ( ord_less_int @ A @ B2 ) ) ).

% neg_less_iff_less
thf(fact_3413_neg__less__iff__less,axiom,
    ! [B2: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B2 ) @ ( uminus1351360451143612070nteger @ A ) )
      = ( ord_le6747313008572928689nteger @ A @ B2 ) ) ).

% neg_less_iff_less
thf(fact_3414_neg__less__iff__less,axiom,
    ! [B2: rat,A: rat] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ B2 ) @ ( uminus_uminus_rat @ A ) )
      = ( ord_less_rat @ A @ B2 ) ) ).

% neg_less_iff_less
thf(fact_3415_neg__numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ M ) )
        = ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
      = ( M = N ) ) ).

% neg_numeral_eq_iff
thf(fact_3416_neg__numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( ( uminus_uminus_int @ ( numeral_numeral_int @ M ) )
        = ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( M = N ) ) ).

% neg_numeral_eq_iff
thf(fact_3417_neg__numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) )
        = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
      = ( M = N ) ) ).

% neg_numeral_eq_iff
thf(fact_3418_neg__numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) )
        = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
      = ( M = N ) ) ).

% neg_numeral_eq_iff
thf(fact_3419_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
thf(fact_3420_mult__minus__right,axiom,
    ! [A: real,B2: real] :
      ( ( times_times_real @ A @ ( uminus_uminus_real @ B2 ) )
      = ( uminus_uminus_real @ ( times_times_real @ A @ B2 ) ) ) ).

% mult_minus_right
thf(fact_3421_mult__minus__right,axiom,
    ! [A: int,B2: int] :
      ( ( times_times_int @ A @ ( uminus_uminus_int @ B2 ) )
      = ( uminus_uminus_int @ ( times_times_int @ A @ B2 ) ) ) ).

% mult_minus_right
thf(fact_3422_mult__minus__right,axiom,
    ! [A: complex,B2: complex] :
      ( ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ B2 ) )
      = ( uminus1482373934393186551omplex @ ( times_times_complex @ A @ B2 ) ) ) ).

% mult_minus_right
thf(fact_3423_mult__minus__right,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( times_3573771949741848930nteger @ A @ ( uminus1351360451143612070nteger @ B2 ) )
      = ( uminus1351360451143612070nteger @ ( times_3573771949741848930nteger @ A @ B2 ) ) ) ).

% mult_minus_right
thf(fact_3424_mult__minus__right,axiom,
    ! [A: rat,B2: rat] :
      ( ( times_times_rat @ A @ ( uminus_uminus_rat @ B2 ) )
      = ( uminus_uminus_rat @ ( times_times_rat @ A @ B2 ) ) ) ).

% mult_minus_right
thf(fact_3425_minus__mult__minus,axiom,
    ! [A: real,B2: real] :
      ( ( times_times_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B2 ) )
      = ( times_times_real @ A @ B2 ) ) ).

% minus_mult_minus
thf(fact_3426_minus__mult__minus,axiom,
    ! [A: int,B2: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B2 ) )
      = ( times_times_int @ A @ B2 ) ) ).

% minus_mult_minus
thf(fact_3427_minus__mult__minus,axiom,
    ! [A: complex,B2: complex] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B2 ) )
      = ( times_times_complex @ A @ B2 ) ) ).

% minus_mult_minus
thf(fact_3428_minus__mult__minus,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B2 ) )
      = ( times_3573771949741848930nteger @ A @ B2 ) ) ).

% minus_mult_minus
thf(fact_3429_minus__mult__minus,axiom,
    ! [A: rat,B2: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B2 ) )
      = ( times_times_rat @ A @ B2 ) ) ).

% minus_mult_minus
thf(fact_3430_mult__minus__left,axiom,
    ! [A: real,B2: real] :
      ( ( times_times_real @ ( uminus_uminus_real @ A ) @ B2 )
      = ( uminus_uminus_real @ ( times_times_real @ A @ B2 ) ) ) ).

% mult_minus_left
thf(fact_3431_mult__minus__left,axiom,
    ! [A: int,B2: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ A ) @ B2 )
      = ( uminus_uminus_int @ ( times_times_int @ A @ B2 ) ) ) ).

% mult_minus_left
thf(fact_3432_mult__minus__left,axiom,
    ! [A: complex,B2: complex] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ A ) @ B2 )
      = ( uminus1482373934393186551omplex @ ( times_times_complex @ A @ B2 ) ) ) ).

% mult_minus_left
thf(fact_3433_mult__minus__left,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A ) @ B2 )
      = ( uminus1351360451143612070nteger @ ( times_3573771949741848930nteger @ A @ B2 ) ) ) ).

% mult_minus_left
thf(fact_3434_mult__minus__left,axiom,
    ! [A: rat,B2: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ A ) @ B2 )
      = ( uminus_uminus_rat @ ( times_times_rat @ A @ B2 ) ) ) ).

% mult_minus_left
thf(fact_3435_minus__add__distrib,axiom,
    ! [A: real,B2: real] :
      ( ( uminus_uminus_real @ ( plus_plus_real @ A @ B2 ) )
      = ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B2 ) ) ) ).

% minus_add_distrib
thf(fact_3436_minus__add__distrib,axiom,
    ! [A: int,B2: int] :
      ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B2 ) )
      = ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B2 ) ) ) ).

% minus_add_distrib
thf(fact_3437_minus__add__distrib,axiom,
    ! [A: complex,B2: complex] :
      ( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A @ B2 ) )
      = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B2 ) ) ) ).

% minus_add_distrib
thf(fact_3438_minus__add__distrib,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A @ B2 ) )
      = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B2 ) ) ) ).

% minus_add_distrib
thf(fact_3439_minus__add__distrib,axiom,
    ! [A: rat,B2: rat] :
      ( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B2 ) )
      = ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B2 ) ) ) ).

% minus_add_distrib
thf(fact_3440_minus__add__cancel,axiom,
    ! [A: real,B2: real] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( plus_plus_real @ A @ B2 ) )
      = B2 ) ).

% minus_add_cancel
thf(fact_3441_minus__add__cancel,axiom,
    ! [A: int,B2: int] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( plus_plus_int @ A @ B2 ) )
      = B2 ) ).

% minus_add_cancel
thf(fact_3442_minus__add__cancel,axiom,
    ! [A: complex,B2: complex] :
      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( plus_plus_complex @ A @ B2 ) )
      = B2 ) ).

% minus_add_cancel
thf(fact_3443_minus__add__cancel,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ ( plus_p5714425477246183910nteger @ A @ B2 ) )
      = B2 ) ).

% minus_add_cancel
thf(fact_3444_minus__add__cancel,axiom,
    ! [A: rat,B2: rat] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( plus_plus_rat @ A @ B2 ) )
      = B2 ) ).

% minus_add_cancel
thf(fact_3445_add__minus__cancel,axiom,
    ! [A: real,B2: real] :
      ( ( plus_plus_real @ A @ ( plus_plus_real @ ( uminus_uminus_real @ A ) @ B2 ) )
      = B2 ) ).

% add_minus_cancel
thf(fact_3446_add__minus__cancel,axiom,
    ! [A: int,B2: int] :
      ( ( plus_plus_int @ A @ ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B2 ) )
      = B2 ) ).

% add_minus_cancel
thf(fact_3447_add__minus__cancel,axiom,
    ! [A: complex,B2: complex] :
      ( ( plus_plus_complex @ A @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ B2 ) )
      = B2 ) ).

% add_minus_cancel
thf(fact_3448_add__minus__cancel,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( plus_p5714425477246183910nteger @ A @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ B2 ) )
      = B2 ) ).

% add_minus_cancel
thf(fact_3449_add__minus__cancel,axiom,
    ! [A: rat,B2: rat] :
      ( ( plus_plus_rat @ A @ ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ B2 ) )
      = B2 ) ).

% add_minus_cancel
thf(fact_3450_minus__diff__eq,axiom,
    ! [A: real,B2: real] :
      ( ( uminus_uminus_real @ ( minus_minus_real @ A @ B2 ) )
      = ( minus_minus_real @ B2 @ A ) ) ).

% minus_diff_eq
thf(fact_3451_minus__diff__eq,axiom,
    ! [A: int,B2: int] :
      ( ( uminus_uminus_int @ ( minus_minus_int @ A @ B2 ) )
      = ( minus_minus_int @ B2 @ A ) ) ).

% minus_diff_eq
thf(fact_3452_minus__diff__eq,axiom,
    ! [A: complex,B2: complex] :
      ( ( uminus1482373934393186551omplex @ ( minus_minus_complex @ A @ B2 ) )
      = ( minus_minus_complex @ B2 @ A ) ) ).

% minus_diff_eq
thf(fact_3453_minus__diff__eq,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( uminus1351360451143612070nteger @ ( minus_8373710615458151222nteger @ A @ B2 ) )
      = ( minus_8373710615458151222nteger @ B2 @ A ) ) ).

% minus_diff_eq
thf(fact_3454_minus__diff__eq,axiom,
    ! [A: rat,B2: rat] :
      ( ( uminus_uminus_rat @ ( minus_minus_rat @ A @ B2 ) )
      = ( minus_minus_rat @ B2 @ A ) ) ).

% minus_diff_eq
thf(fact_3455_div__minus__minus,axiom,
    ! [A: int,B2: int] :
      ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B2 ) )
      = ( divide_divide_int @ A @ B2 ) ) ).

% div_minus_minus
thf(fact_3456_div__minus__minus,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B2 ) )
      = ( divide6298287555418463151nteger @ A @ B2 ) ) ).

% div_minus_minus
thf(fact_3457_dvd__minus__iff,axiom,
    ! [X4: real,Y3: real] :
      ( ( dvd_dvd_real @ X4 @ ( uminus_uminus_real @ Y3 ) )
      = ( dvd_dvd_real @ X4 @ Y3 ) ) ).

% dvd_minus_iff
thf(fact_3458_dvd__minus__iff,axiom,
    ! [X4: int,Y3: int] :
      ( ( dvd_dvd_int @ X4 @ ( uminus_uminus_int @ Y3 ) )
      = ( dvd_dvd_int @ X4 @ Y3 ) ) ).

% dvd_minus_iff
thf(fact_3459_dvd__minus__iff,axiom,
    ! [X4: complex,Y3: complex] :
      ( ( dvd_dvd_complex @ X4 @ ( uminus1482373934393186551omplex @ Y3 ) )
      = ( dvd_dvd_complex @ X4 @ Y3 ) ) ).

% dvd_minus_iff
thf(fact_3460_dvd__minus__iff,axiom,
    ! [X4: code_integer,Y3: code_integer] :
      ( ( dvd_dvd_Code_integer @ X4 @ ( uminus1351360451143612070nteger @ Y3 ) )
      = ( dvd_dvd_Code_integer @ X4 @ Y3 ) ) ).

% dvd_minus_iff
thf(fact_3461_dvd__minus__iff,axiom,
    ! [X4: rat,Y3: rat] :
      ( ( dvd_dvd_rat @ X4 @ ( uminus_uminus_rat @ Y3 ) )
      = ( dvd_dvd_rat @ X4 @ Y3 ) ) ).

% dvd_minus_iff
thf(fact_3462_minus__dvd__iff,axiom,
    ! [X4: real,Y3: real] :
      ( ( dvd_dvd_real @ ( uminus_uminus_real @ X4 ) @ Y3 )
      = ( dvd_dvd_real @ X4 @ Y3 ) ) ).

% minus_dvd_iff
thf(fact_3463_minus__dvd__iff,axiom,
    ! [X4: int,Y3: int] :
      ( ( dvd_dvd_int @ ( uminus_uminus_int @ X4 ) @ Y3 )
      = ( dvd_dvd_int @ X4 @ Y3 ) ) ).

% minus_dvd_iff
thf(fact_3464_minus__dvd__iff,axiom,
    ! [X4: complex,Y3: complex] :
      ( ( dvd_dvd_complex @ ( uminus1482373934393186551omplex @ X4 ) @ Y3 )
      = ( dvd_dvd_complex @ X4 @ Y3 ) ) ).

% minus_dvd_iff
thf(fact_3465_minus__dvd__iff,axiom,
    ! [X4: code_integer,Y3: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( uminus1351360451143612070nteger @ X4 ) @ Y3 )
      = ( dvd_dvd_Code_integer @ X4 @ Y3 ) ) ).

% minus_dvd_iff
thf(fact_3466_minus__dvd__iff,axiom,
    ! [X4: rat,Y3: rat] :
      ( ( dvd_dvd_rat @ ( uminus_uminus_rat @ X4 ) @ Y3 )
      = ( dvd_dvd_rat @ X4 @ Y3 ) ) ).

% minus_dvd_iff
thf(fact_3467_mod__minus__minus,axiom,
    ! [A: int,B2: int] :
      ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B2 ) )
      = ( uminus_uminus_int @ ( modulo_modulo_int @ A @ B2 ) ) ) ).

% mod_minus_minus
thf(fact_3468_mod__minus__minus,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B2 ) )
      = ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A @ B2 ) ) ) ).

% mod_minus_minus
thf(fact_3469_real__add__minus__iff,axiom,
    ! [X4: real,A: real] :
      ( ( ( plus_plus_real @ X4 @ ( uminus_uminus_real @ A ) )
        = zero_zero_real )
      = ( X4 = A ) ) ).

% real_add_minus_iff
thf(fact_3470_neg__less__eq__nonneg,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ A )
      = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).

% neg_less_eq_nonneg
thf(fact_3471_neg__less__eq__nonneg,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
      = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).

% neg_less_eq_nonneg
thf(fact_3472_neg__less__eq__nonneg,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ A )
      = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).

% neg_less_eq_nonneg
thf(fact_3473_neg__less__eq__nonneg,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ A )
      = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).

% neg_less_eq_nonneg
thf(fact_3474_less__eq__neg__nonpos,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ A ) )
      = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).

% less_eq_neg_nonpos
thf(fact_3475_less__eq__neg__nonpos,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
      = ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) ) ).

% less_eq_neg_nonpos
thf(fact_3476_less__eq__neg__nonpos,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ A ) )
      = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).

% less_eq_neg_nonpos
thf(fact_3477_less__eq__neg__nonpos,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ A ) )
      = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).

% less_eq_neg_nonpos
thf(fact_3478_neg__le__0__iff__le,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
      = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).

% neg_le_0_iff_le
thf(fact_3479_neg__le__0__iff__le,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ zero_z3403309356797280102nteger )
      = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).

% neg_le_0_iff_le
thf(fact_3480_neg__le__0__iff__le,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ zero_zero_rat )
      = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).

% neg_le_0_iff_le
thf(fact_3481_neg__le__0__iff__le,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
      = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).

% neg_le_0_iff_le
thf(fact_3482_neg__0__le__iff__le,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
      = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).

% neg_0_le_iff_le
thf(fact_3483_neg__0__le__iff__le,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ A ) )
      = ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) ) ).

% neg_0_le_iff_le
thf(fact_3484_neg__0__le__iff__le,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ A ) )
      = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).

% neg_0_le_iff_le
thf(fact_3485_neg__0__le__iff__le,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
      = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).

% neg_0_le_iff_le
thf(fact_3486_neg__less__0__iff__less,axiom,
    ! [A: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
      = ( ord_less_real @ zero_zero_real @ A ) ) ).

% neg_less_0_iff_less
thf(fact_3487_neg__less__0__iff__less,axiom,
    ! [A: int] :
      ( ( ord_less_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
      = ( ord_less_int @ zero_zero_int @ A ) ) ).

% neg_less_0_iff_less
thf(fact_3488_neg__less__0__iff__less,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ zero_z3403309356797280102nteger )
      = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).

% neg_less_0_iff_less
thf(fact_3489_neg__less__0__iff__less,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ zero_zero_rat )
      = ( ord_less_rat @ zero_zero_rat @ A ) ) ).

% neg_less_0_iff_less
thf(fact_3490_neg__0__less__iff__less,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
      = ( ord_less_real @ A @ zero_zero_real ) ) ).

% neg_0_less_iff_less
thf(fact_3491_neg__0__less__iff__less,axiom,
    ! [A: int] :
      ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
      = ( ord_less_int @ A @ zero_zero_int ) ) ).

% neg_0_less_iff_less
thf(fact_3492_neg__0__less__iff__less,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ A ) )
      = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).

% neg_0_less_iff_less
thf(fact_3493_neg__0__less__iff__less,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ A ) )
      = ( ord_less_rat @ A @ zero_zero_rat ) ) ).

% neg_0_less_iff_less
thf(fact_3494_neg__less__pos,axiom,
    ! [A: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ A ) @ A )
      = ( ord_less_real @ zero_zero_real @ A ) ) ).

% neg_less_pos
thf(fact_3495_neg__less__pos,axiom,
    ! [A: int] :
      ( ( ord_less_int @ ( uminus_uminus_int @ A ) @ A )
      = ( ord_less_int @ zero_zero_int @ A ) ) ).

% neg_less_pos
thf(fact_3496_neg__less__pos,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
      = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).

% neg_less_pos
thf(fact_3497_neg__less__pos,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ A )
      = ( ord_less_rat @ zero_zero_rat @ A ) ) ).

% neg_less_pos
thf(fact_3498_less__neg__neg,axiom,
    ! [A: real] :
      ( ( ord_less_real @ A @ ( uminus_uminus_real @ A ) )
      = ( ord_less_real @ A @ zero_zero_real ) ) ).

% less_neg_neg
thf(fact_3499_less__neg__neg,axiom,
    ! [A: int] :
      ( ( ord_less_int @ A @ ( uminus_uminus_int @ A ) )
      = ( ord_less_int @ A @ zero_zero_int ) ) ).

% less_neg_neg
thf(fact_3500_less__neg__neg,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
      = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).

% less_neg_neg
thf(fact_3501_less__neg__neg,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ A ) )
      = ( ord_less_rat @ A @ zero_zero_rat ) ) ).

% less_neg_neg
thf(fact_3502_ab__left__minus,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ A )
      = zero_zero_real ) ).

% ab_left_minus
thf(fact_3503_ab__left__minus,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
      = zero_zero_int ) ).

% ab_left_minus
thf(fact_3504_ab__left__minus,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ A )
      = zero_zero_complex ) ).

% ab_left_minus
thf(fact_3505_ab__left__minus,axiom,
    ! [A: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
      = zero_z3403309356797280102nteger ) ).

% ab_left_minus
thf(fact_3506_ab__left__minus,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ A )
      = zero_zero_rat ) ).

% ab_left_minus
thf(fact_3507_add_Oright__inverse,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ A @ ( uminus_uminus_real @ A ) )
      = zero_zero_real ) ).

% add.right_inverse
thf(fact_3508_add_Oright__inverse,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ ( uminus_uminus_int @ A ) )
      = zero_zero_int ) ).

% add.right_inverse
thf(fact_3509_add_Oright__inverse,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ A @ ( uminus1482373934393186551omplex @ A ) )
      = zero_zero_complex ) ).

% add.right_inverse
thf(fact_3510_add_Oright__inverse,axiom,
    ! [A: code_integer] :
      ( ( plus_p5714425477246183910nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
      = zero_z3403309356797280102nteger ) ).

% add.right_inverse
thf(fact_3511_add_Oright__inverse,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ A @ ( uminus_uminus_rat @ A ) )
      = zero_zero_rat ) ).

% add.right_inverse
thf(fact_3512_diff__0,axiom,
    ! [A: real] :
      ( ( minus_minus_real @ zero_zero_real @ A )
      = ( uminus_uminus_real @ A ) ) ).

% diff_0
thf(fact_3513_diff__0,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ zero_zero_int @ A )
      = ( uminus_uminus_int @ A ) ) ).

% diff_0
thf(fact_3514_diff__0,axiom,
    ! [A: complex] :
      ( ( minus_minus_complex @ zero_zero_complex @ A )
      = ( uminus1482373934393186551omplex @ A ) ) ).

% diff_0
thf(fact_3515_diff__0,axiom,
    ! [A: code_integer] :
      ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ A )
      = ( uminus1351360451143612070nteger @ A ) ) ).

% diff_0
thf(fact_3516_diff__0,axiom,
    ! [A: rat] :
      ( ( minus_minus_rat @ zero_zero_rat @ A )
      = ( uminus_uminus_rat @ A ) ) ).

% diff_0
thf(fact_3517_verit__minus__simplify_I3_J,axiom,
    ! [B2: real] :
      ( ( minus_minus_real @ zero_zero_real @ B2 )
      = ( uminus_uminus_real @ B2 ) ) ).

% verit_minus_simplify(3)
thf(fact_3518_verit__minus__simplify_I3_J,axiom,
    ! [B2: int] :
      ( ( minus_minus_int @ zero_zero_int @ B2 )
      = ( uminus_uminus_int @ B2 ) ) ).

% verit_minus_simplify(3)
thf(fact_3519_verit__minus__simplify_I3_J,axiom,
    ! [B2: complex] :
      ( ( minus_minus_complex @ zero_zero_complex @ B2 )
      = ( uminus1482373934393186551omplex @ B2 ) ) ).

% verit_minus_simplify(3)
thf(fact_3520_verit__minus__simplify_I3_J,axiom,
    ! [B2: code_integer] :
      ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ B2 )
      = ( uminus1351360451143612070nteger @ B2 ) ) ).

% verit_minus_simplify(3)
thf(fact_3521_verit__minus__simplify_I3_J,axiom,
    ! [B2: rat] :
      ( ( minus_minus_rat @ zero_zero_rat @ B2 )
      = ( uminus_uminus_rat @ B2 ) ) ).

% verit_minus_simplify(3)
thf(fact_3522_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
      = ( uminus_uminus_real @ ( plus_plus_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) ) ) ) ).

% add_neg_numeral_simps(3)
thf(fact_3523_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( uminus_uminus_int @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) ) ) ) ).

% add_neg_numeral_simps(3)
thf(fact_3524_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
      = ( uminus1482373934393186551omplex @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ M ) @ ( numera6690914467698888265omplex @ N ) ) ) ) ).

% add_neg_numeral_simps(3)
thf(fact_3525_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
      = ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ N ) ) ) ) ).

% add_neg_numeral_simps(3)
thf(fact_3526_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)
thf(fact_3527_mult__minus1,axiom,
    ! [Z: real] :
      ( ( times_times_real @ ( uminus_uminus_real @ one_one_real ) @ Z )
      = ( uminus_uminus_real @ Z ) ) ).

% mult_minus1
thf(fact_3528_mult__minus1,axiom,
    ! [Z: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ one_one_int ) @ Z )
      = ( uminus_uminus_int @ Z ) ) ).

% mult_minus1
thf(fact_3529_mult__minus1,axiom,
    ! [Z: complex] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ Z )
      = ( uminus1482373934393186551omplex @ Z ) ) ).

% mult_minus1
thf(fact_3530_mult__minus1,axiom,
    ! [Z: code_integer] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ Z )
      = ( uminus1351360451143612070nteger @ Z ) ) ).

% mult_minus1
thf(fact_3531_mult__minus1,axiom,
    ! [Z: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ one_one_rat ) @ Z )
      = ( uminus_uminus_rat @ Z ) ) ).

% mult_minus1
thf(fact_3532_mult__minus1__right,axiom,
    ! [Z: real] :
      ( ( times_times_real @ Z @ ( uminus_uminus_real @ one_one_real ) )
      = ( uminus_uminus_real @ Z ) ) ).

% mult_minus1_right
thf(fact_3533_mult__minus1__right,axiom,
    ! [Z: int] :
      ( ( times_times_int @ Z @ ( uminus_uminus_int @ one_one_int ) )
      = ( uminus_uminus_int @ Z ) ) ).

% mult_minus1_right
thf(fact_3534_mult__minus1__right,axiom,
    ! [Z: complex] :
      ( ( times_times_complex @ Z @ ( uminus1482373934393186551omplex @ one_one_complex ) )
      = ( uminus1482373934393186551omplex @ Z ) ) ).

% mult_minus1_right
thf(fact_3535_mult__minus1__right,axiom,
    ! [Z: code_integer] :
      ( ( times_3573771949741848930nteger @ Z @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( uminus1351360451143612070nteger @ Z ) ) ).

% mult_minus1_right
thf(fact_3536_mult__minus1__right,axiom,
    ! [Z: rat] :
      ( ( times_times_rat @ Z @ ( uminus_uminus_rat @ one_one_rat ) )
      = ( uminus_uminus_rat @ Z ) ) ).

% mult_minus1_right
thf(fact_3537_uminus__add__conv__diff,axiom,
    ! [A: real,B2: real] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ B2 )
      = ( minus_minus_real @ B2 @ A ) ) ).

% uminus_add_conv_diff
thf(fact_3538_uminus__add__conv__diff,axiom,
    ! [A: int,B2: int] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B2 )
      = ( minus_minus_int @ B2 @ A ) ) ).

% uminus_add_conv_diff
thf(fact_3539_uminus__add__conv__diff,axiom,
    ! [A: complex,B2: complex] :
      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ B2 )
      = ( minus_minus_complex @ B2 @ A ) ) ).

% uminus_add_conv_diff
thf(fact_3540_uminus__add__conv__diff,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ B2 )
      = ( minus_8373710615458151222nteger @ B2 @ A ) ) ).

% uminus_add_conv_diff
thf(fact_3541_uminus__add__conv__diff,axiom,
    ! [A: rat,B2: rat] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ B2 )
      = ( minus_minus_rat @ B2 @ A ) ) ).

% uminus_add_conv_diff
thf(fact_3542_diff__minus__eq__add,axiom,
    ! [A: real,B2: real] :
      ( ( minus_minus_real @ A @ ( uminus_uminus_real @ B2 ) )
      = ( plus_plus_real @ A @ B2 ) ) ).

% diff_minus_eq_add
thf(fact_3543_diff__minus__eq__add,axiom,
    ! [A: int,B2: int] :
      ( ( minus_minus_int @ A @ ( uminus_uminus_int @ B2 ) )
      = ( plus_plus_int @ A @ B2 ) ) ).

% diff_minus_eq_add
thf(fact_3544_diff__minus__eq__add,axiom,
    ! [A: complex,B2: complex] :
      ( ( minus_minus_complex @ A @ ( uminus1482373934393186551omplex @ B2 ) )
      = ( plus_plus_complex @ A @ B2 ) ) ).

% diff_minus_eq_add
thf(fact_3545_diff__minus__eq__add,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( minus_8373710615458151222nteger @ A @ ( uminus1351360451143612070nteger @ B2 ) )
      = ( plus_p5714425477246183910nteger @ A @ B2 ) ) ).

% diff_minus_eq_add
thf(fact_3546_diff__minus__eq__add,axiom,
    ! [A: rat,B2: rat] :
      ( ( minus_minus_rat @ A @ ( uminus_uminus_rat @ B2 ) )
      = ( plus_plus_rat @ A @ B2 ) ) ).

% diff_minus_eq_add
thf(fact_3547_div__minus1__right,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ A @ ( uminus_uminus_int @ one_one_int ) )
      = ( uminus_uminus_int @ A ) ) ).

% div_minus1_right
thf(fact_3548_div__minus1__right,axiom,
    ! [A: code_integer] :
      ( ( divide6298287555418463151nteger @ A @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( uminus1351360451143612070nteger @ A ) ) ).

% div_minus1_right
thf(fact_3549_divide__minus1,axiom,
    ! [X4: real] :
      ( ( divide_divide_real @ X4 @ ( uminus_uminus_real @ one_one_real ) )
      = ( uminus_uminus_real @ X4 ) ) ).

% divide_minus1
thf(fact_3550_divide__minus1,axiom,
    ! [X4: complex] :
      ( ( divide1717551699836669952omplex @ X4 @ ( uminus1482373934393186551omplex @ one_one_complex ) )
      = ( uminus1482373934393186551omplex @ X4 ) ) ).

% divide_minus1
thf(fact_3551_divide__minus1,axiom,
    ! [X4: rat] :
      ( ( divide_divide_rat @ X4 @ ( uminus_uminus_rat @ one_one_rat ) )
      = ( uminus_uminus_rat @ X4 ) ) ).

% divide_minus1
thf(fact_3552_minus__mod__self1,axiom,
    ! [B2: int,A: int] :
      ( ( modulo_modulo_int @ ( minus_minus_int @ B2 @ A ) @ B2 )
      = ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B2 ) ) ).

% minus_mod_self1
thf(fact_3553_minus__mod__self1,axiom,
    ! [B2: code_integer,A: code_integer] :
      ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ B2 @ A ) @ B2 )
      = ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B2 ) ) ).

% minus_mod_self1
thf(fact_3554_signed__take__bit__of__minus__1,axiom,
    ! [N: nat] :
      ( ( bit_ri6519982836138164636nteger @ N @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% signed_take_bit_of_minus_1
thf(fact_3555_signed__take__bit__of__minus__1,axiom,
    ! [N: nat] :
      ( ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
      = ( uminus_uminus_int @ one_one_int ) ) ).

% signed_take_bit_of_minus_1
thf(fact_3556_dbl__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_numeral_dbl_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ K ) ) )
      = ( uminus_uminus_real @ ( neg_numeral_dbl_real @ ( numeral_numeral_real @ K ) ) ) ) ).

% dbl_simps(1)
thf(fact_3557_dbl__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_numeral_dbl_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
      = ( uminus_uminus_int @ ( neg_numeral_dbl_int @ ( numeral_numeral_int @ K ) ) ) ) ).

% dbl_simps(1)
thf(fact_3558_dbl__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu7009210354673126013omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) )
      = ( uminus1482373934393186551omplex @ ( neg_nu7009210354673126013omplex @ ( numera6690914467698888265omplex @ K ) ) ) ) ).

% dbl_simps(1)
thf(fact_3559_dbl__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu8804712462038260780nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
      = ( uminus1351360451143612070nteger @ ( neg_nu8804712462038260780nteger @ ( numera6620942414471956472nteger @ K ) ) ) ) ).

% dbl_simps(1)
thf(fact_3560_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)
thf(fact_3561_triangle__Suc,axiom,
    ! [N: nat] :
      ( ( nat_triangle @ ( suc @ N ) )
      = ( plus_plus_nat @ ( nat_triangle @ N ) @ ( suc @ N ) ) ) ).

% triangle_Suc
thf(fact_3562_add__neg__numeral__special_I8_J,axiom,
    ( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real )
    = zero_zero_real ) ).

% add_neg_numeral_special(8)
thf(fact_3563_add__neg__numeral__special_I8_J,axiom,
    ( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int )
    = zero_zero_int ) ).

% add_neg_numeral_special(8)
thf(fact_3564_add__neg__numeral__special_I8_J,axiom,
    ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ one_one_complex )
    = zero_zero_complex ) ).

% add_neg_numeral_special(8)
thf(fact_3565_add__neg__numeral__special_I8_J,axiom,
    ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer )
    = zero_z3403309356797280102nteger ) ).

% add_neg_numeral_special(8)
thf(fact_3566_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)
thf(fact_3567_add__neg__numeral__special_I7_J,axiom,
    ( ( plus_plus_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) )
    = zero_zero_real ) ).

% add_neg_numeral_special(7)
thf(fact_3568_add__neg__numeral__special_I7_J,axiom,
    ( ( plus_plus_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) )
    = zero_zero_int ) ).

% add_neg_numeral_special(7)
thf(fact_3569_add__neg__numeral__special_I7_J,axiom,
    ( ( plus_plus_complex @ one_one_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
    = zero_zero_complex ) ).

% add_neg_numeral_special(7)
thf(fact_3570_add__neg__numeral__special_I7_J,axiom,
    ( ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = zero_z3403309356797280102nteger ) ).

% add_neg_numeral_special(7)
thf(fact_3571_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)
thf(fact_3572_diff__numeral__special_I12_J,axiom,
    ( ( minus_minus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ one_one_real ) )
    = zero_zero_real ) ).

% diff_numeral_special(12)
thf(fact_3573_diff__numeral__special_I12_J,axiom,
    ( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ one_one_int ) )
    = zero_zero_int ) ).

% diff_numeral_special(12)
thf(fact_3574_diff__numeral__special_I12_J,axiom,
    ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
    = zero_zero_complex ) ).

% diff_numeral_special(12)
thf(fact_3575_diff__numeral__special_I12_J,axiom,
    ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = zero_z3403309356797280102nteger ) ).

% diff_numeral_special(12)
thf(fact_3576_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)
thf(fact_3577_neg__one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( ( uminus_uminus_real @ one_one_real )
        = ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
      = ( N = one ) ) ).

% neg_one_eq_numeral_iff
thf(fact_3578_neg__one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( ( uminus_uminus_int @ one_one_int )
        = ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( N = one ) ) ).

% neg_one_eq_numeral_iff
thf(fact_3579_neg__one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( ( uminus1482373934393186551omplex @ one_one_complex )
        = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
      = ( N = one ) ) ).

% neg_one_eq_numeral_iff
thf(fact_3580_neg__one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( ( uminus1351360451143612070nteger @ one_one_Code_integer )
        = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
      = ( N = one ) ) ).

% neg_one_eq_numeral_iff
thf(fact_3581_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
thf(fact_3582_numeral__eq__neg__one__iff,axiom,
    ! [N: num] :
      ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ N ) )
        = ( uminus_uminus_real @ one_one_real ) )
      = ( N = one ) ) ).

% numeral_eq_neg_one_iff
thf(fact_3583_numeral__eq__neg__one__iff,axiom,
    ! [N: num] :
      ( ( ( uminus_uminus_int @ ( numeral_numeral_int @ N ) )
        = ( uminus_uminus_int @ one_one_int ) )
      = ( N = one ) ) ).

% numeral_eq_neg_one_iff
thf(fact_3584_numeral__eq__neg__one__iff,axiom,
    ! [N: num] :
      ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) )
        = ( uminus1482373934393186551omplex @ one_one_complex ) )
      = ( N = one ) ) ).

% numeral_eq_neg_one_iff
thf(fact_3585_numeral__eq__neg__one__iff,axiom,
    ! [N: num] :
      ( ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) )
        = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( N = one ) ) ).

% numeral_eq_neg_one_iff
thf(fact_3586_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
thf(fact_3587_minus__one__mult__self,axiom,
    ! [N: nat] :
      ( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) )
      = one_one_real ) ).

% minus_one_mult_self
thf(fact_3588_minus__one__mult__self,axiom,
    ! [N: nat] :
      ( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) )
      = one_one_int ) ).

% minus_one_mult_self
thf(fact_3589_minus__one__mult__self,axiom,
    ! [N: nat] :
      ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) )
      = one_one_complex ) ).

% minus_one_mult_self
thf(fact_3590_minus__one__mult__self,axiom,
    ! [N: nat] :
      ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) )
      = one_one_Code_integer ) ).

% minus_one_mult_self
thf(fact_3591_minus__one__mult__self,axiom,
    ! [N: nat] :
      ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) )
      = one_one_rat ) ).

% minus_one_mult_self
thf(fact_3592_left__minus__one__mult__self,axiom,
    ! [N: nat,A: real] :
      ( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ A ) )
      = A ) ).

% left_minus_one_mult_self
thf(fact_3593_left__minus__one__mult__self,axiom,
    ! [N: nat,A: int] :
      ( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ A ) )
      = A ) ).

% left_minus_one_mult_self
thf(fact_3594_left__minus__one__mult__self,axiom,
    ! [N: nat,A: complex] :
      ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ A ) )
      = A ) ).

% left_minus_one_mult_self
thf(fact_3595_left__minus__one__mult__self,axiom,
    ! [N: nat,A: code_integer] :
      ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ A ) )
      = A ) ).

% left_minus_one_mult_self
thf(fact_3596_left__minus__one__mult__self,axiom,
    ! [N: nat,A: rat] :
      ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ A ) )
      = A ) ).

% left_minus_one_mult_self
thf(fact_3597_mod__minus1__right,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ one_one_int ) )
      = zero_zero_int ) ).

% mod_minus1_right
thf(fact_3598_mod__minus1__right,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger @ A @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = zero_z3403309356797280102nteger ) ).

% mod_minus1_right
thf(fact_3599_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y3: real] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ Y3 ) )
      = ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ V @ W ) ) ) @ Y3 ) ) ).

% semiring_norm(168)
thf(fact_3600_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y3: int] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y3 ) )
      = ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W ) ) ) @ Y3 ) ) ).

% semiring_norm(168)
thf(fact_3601_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y3: complex] :
      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ V ) ) @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ Y3 ) )
      = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ V @ W ) ) ) @ Y3 ) ) ).

% semiring_norm(168)
thf(fact_3602_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y3: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) @ Y3 ) )
      = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ V @ W ) ) ) @ Y3 ) ) ).

% semiring_norm(168)
thf(fact_3603_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y3: rat] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ Y3 ) )
      = ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ V @ W ) ) ) @ Y3 ) ) ).

% semiring_norm(168)
thf(fact_3604_diff__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) )
      = ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ M @ N ) ) ) ) ).

% diff_numeral_simps(3)
thf(fact_3605_diff__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ M @ N ) ) ) ) ).

% diff_numeral_simps(3)
thf(fact_3606_diff__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( numera6690914467698888265omplex @ N ) )
      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ N ) ) ) ) ).

% diff_numeral_simps(3)
thf(fact_3607_diff__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) )
      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ M @ N ) ) ) ) ).

% diff_numeral_simps(3)
thf(fact_3608_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 @ ( plus_plus_num @ M @ N ) ) ) ) ).

% diff_numeral_simps(3)
thf(fact_3609_diff__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] :
      ( ( minus_minus_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
      = ( numeral_numeral_real @ ( plus_plus_num @ M @ N ) ) ) ).

% diff_numeral_simps(2)
thf(fact_3610_diff__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] :
      ( ( minus_minus_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( numeral_numeral_int @ ( plus_plus_num @ M @ N ) ) ) ).

% diff_numeral_simps(2)
thf(fact_3611_diff__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] :
      ( ( minus_minus_complex @ ( numera6690914467698888265omplex @ M ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
      = ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ N ) ) ) ).

% diff_numeral_simps(2)
thf(fact_3612_diff__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] :
      ( ( minus_8373710615458151222nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
      = ( numera6620942414471956472nteger @ ( plus_plus_num @ M @ N ) ) ) ).

% diff_numeral_simps(2)
thf(fact_3613_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 @ ( plus_plus_num @ M @ N ) ) ) ).

% diff_numeral_simps(2)
thf(fact_3614_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y3: real] :
      ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ Y3 ) )
      = ( times_times_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) @ Y3 ) ) ).

% semiring_norm(172)
thf(fact_3615_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y3: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y3 ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) @ Y3 ) ) ).

% semiring_norm(172)
thf(fact_3616_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y3: complex] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ Y3 ) )
      = ( times_times_complex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) @ Y3 ) ) ).

% semiring_norm(172)
thf(fact_3617_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y3: code_integer] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) @ ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) @ Y3 ) )
      = ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V @ W ) ) @ Y3 ) ) ).

% semiring_norm(172)
thf(fact_3618_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y3: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ Y3 ) )
      = ( times_times_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) @ Y3 ) ) ).

% semiring_norm(172)
thf(fact_3619_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y3: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ Y3 ) )
      = ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).

% semiring_norm(171)
thf(fact_3620_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y3: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y3 ) )
      = ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).

% semiring_norm(171)
thf(fact_3621_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y3: complex] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ Y3 ) )
      = ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).

% semiring_norm(171)
thf(fact_3622_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y3: code_integer] :
      ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ V ) @ ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) @ Y3 ) )
      = ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).

% semiring_norm(171)
thf(fact_3623_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y3: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ Y3 ) )
      = ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).

% semiring_norm(171)
thf(fact_3624_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y3: real] :
      ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ ( numeral_numeral_real @ W ) @ Y3 ) )
      = ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).

% semiring_norm(170)
thf(fact_3625_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y3: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ ( numeral_numeral_int @ W ) @ Y3 ) )
      = ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).

% semiring_norm(170)
thf(fact_3626_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y3: complex] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ Y3 ) )
      = ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).

% semiring_norm(170)
thf(fact_3627_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y3: code_integer] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ W ) @ Y3 ) )
      = ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).

% semiring_norm(170)
thf(fact_3628_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y3: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ Y3 ) )
      = ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).

% semiring_norm(170)
thf(fact_3629_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( times_times_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
      = ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ M @ N ) ) ) ) ).

% mult_neg_numeral_simps(3)
thf(fact_3630_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( times_times_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ) ).

% mult_neg_numeral_simps(3)
thf(fact_3631_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ M ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ M @ N ) ) ) ) ).

% mult_neg_numeral_simps(3)
thf(fact_3632_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ M @ N ) ) ) ) ).

% mult_neg_numeral_simps(3)
thf(fact_3633_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ M @ N ) ) ) ) ).

% mult_neg_numeral_simps(3)
thf(fact_3634_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] :
      ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) )
      = ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ M @ N ) ) ) ) ).

% mult_neg_numeral_simps(2)
thf(fact_3635_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] :
      ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ) ).

% mult_neg_numeral_simps(2)
thf(fact_3636_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( numera6690914467698888265omplex @ N ) )
      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ M @ N ) ) ) ) ).

% mult_neg_numeral_simps(2)
thf(fact_3637_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) )
      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ M @ N ) ) ) ) ).

% mult_neg_numeral_simps(2)
thf(fact_3638_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ M @ N ) ) ) ) ).

% mult_neg_numeral_simps(2)
thf(fact_3639_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N: num] :
      ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
      = ( numeral_numeral_real @ ( times_times_num @ M @ N ) ) ) ).

% mult_neg_numeral_simps(1)
thf(fact_3640_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N: num] :
      ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ).

% mult_neg_numeral_simps(1)
thf(fact_3641_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N: num] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
      = ( numera6690914467698888265omplex @ ( times_times_num @ M @ N ) ) ) ).

% mult_neg_numeral_simps(1)
thf(fact_3642_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N: num] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
      = ( numera6620942414471956472nteger @ ( times_times_num @ M @ N ) ) ) ).

% mult_neg_numeral_simps(1)
thf(fact_3643_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N: num] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
      = ( numeral_numeral_rat @ ( times_times_num @ M @ N ) ) ) ).

% mult_neg_numeral_simps(1)
thf(fact_3644_neg__numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
      = ( ord_less_eq_num @ N @ M ) ) ).

% neg_numeral_le_iff
thf(fact_3645_neg__numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
      = ( ord_less_eq_num @ N @ M ) ) ).

% neg_numeral_le_iff
thf(fact_3646_neg__numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
      = ( ord_less_eq_num @ N @ M ) ) ).

% neg_numeral_le_iff
thf(fact_3647_neg__numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( ord_less_eq_num @ N @ M ) ) ).

% neg_numeral_le_iff
thf(fact_3648_neg__numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
      = ( ord_less_num @ N @ M ) ) ).

% neg_numeral_less_iff
thf(fact_3649_neg__numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( ord_less_num @ N @ M ) ) ).

% neg_numeral_less_iff
thf(fact_3650_neg__numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
      = ( ord_less_num @ N @ M ) ) ).

% neg_numeral_less_iff
thf(fact_3651_neg__numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
      = ( ord_less_num @ N @ M ) ) ).

% neg_numeral_less_iff
thf(fact_3652_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M: num] :
      ( ( ~ ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) )
      = ( M != one ) ) ).

% not_neg_one_le_neg_numeral_iff
thf(fact_3653_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M: num] :
      ( ( ~ ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) )
      = ( M != one ) ) ).

% not_neg_one_le_neg_numeral_iff
thf(fact_3654_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M: num] :
      ( ( ~ ( 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
thf(fact_3655_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M: num] :
      ( ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) )
      = ( M != one ) ) ).

% not_neg_one_le_neg_numeral_iff
thf(fact_3656_divide__le__eq__numeral1_I2_J,axiom,
    ! [B2: real,W: num,A: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ B2 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ A )
      = ( ord_less_eq_real @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ B2 ) ) ).

% divide_le_eq_numeral1(2)
thf(fact_3657_divide__le__eq__numeral1_I2_J,axiom,
    ! [B2: rat,W: num,A: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B2 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ A )
      = ( ord_less_eq_rat @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ B2 ) ) ).

% divide_le_eq_numeral1(2)
thf(fact_3658_le__divide__eq__numeral1_I2_J,axiom,
    ! [A: real,B2: real,W: num] :
      ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B2 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) )
      = ( ord_less_eq_real @ B2 @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ).

% le_divide_eq_numeral1(2)
thf(fact_3659_le__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B2: rat,W: num] :
      ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B2 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) )
      = ( ord_less_eq_rat @ B2 @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ).

% le_divide_eq_numeral1(2)
thf(fact_3660_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B2: real,W: num,A: real] :
      ( ( ( divide_divide_real @ B2 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
        = A )
      = ( ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
           != zero_zero_real )
         => ( B2
            = ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) )
        & ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
            = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% divide_eq_eq_numeral1(2)
thf(fact_3661_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B2: complex,W: num,A: complex] :
      ( ( ( divide1717551699836669952omplex @ B2 @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
        = A )
      = ( ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
           != zero_zero_complex )
         => ( B2
            = ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ) )
        & ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
            = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% divide_eq_eq_numeral1(2)
thf(fact_3662_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B2: rat,W: num,A: rat] :
      ( ( ( divide_divide_rat @ B2 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
        = A )
      = ( ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
           != zero_zero_rat )
         => ( B2
            = ( 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)
thf(fact_3663_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: real,B2: real,W: num] :
      ( ( A
        = ( divide_divide_real @ B2 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) )
      = ( ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
           != zero_zero_real )
         => ( ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
            = B2 ) )
        & ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
            = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral1(2)
thf(fact_3664_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: complex,B2: complex,W: num] :
      ( ( A
        = ( divide1717551699836669952omplex @ B2 @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) )
      = ( ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
           != zero_zero_complex )
         => ( ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
            = B2 ) )
        & ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
            = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% eq_divide_eq_numeral1(2)
thf(fact_3665_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B2: rat,W: num] :
      ( ( A
        = ( divide_divide_rat @ B2 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) )
      = ( ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
           != zero_zero_rat )
         => ( ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
            = B2 ) )
        & ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
            = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral1(2)
thf(fact_3666_neg__numeral__less__neg__one__iff,axiom,
    ! [M: num] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ one_one_real ) )
      = ( M != one ) ) ).

% neg_numeral_less_neg_one_iff
thf(fact_3667_neg__numeral__less__neg__one__iff,axiom,
    ! [M: num] :
      ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) )
      = ( M != one ) ) ).

% neg_numeral_less_neg_one_iff
thf(fact_3668_neg__numeral__less__neg__one__iff,axiom,
    ! [M: num] :
      ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( M != one ) ) ).

% neg_numeral_less_neg_one_iff
thf(fact_3669_neg__numeral__less__neg__one__iff,axiom,
    ! [M: num] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ one_one_rat ) )
      = ( M != one ) ) ).

% neg_numeral_less_neg_one_iff
thf(fact_3670_divide__less__eq__numeral1_I2_J,axiom,
    ! [B2: real,W: num,A: real] :
      ( ( ord_less_real @ ( divide_divide_real @ B2 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ A )
      = ( ord_less_real @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ B2 ) ) ).

% divide_less_eq_numeral1(2)
thf(fact_3671_divide__less__eq__numeral1_I2_J,axiom,
    ! [B2: rat,W: num,A: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ B2 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ A )
      = ( ord_less_rat @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ B2 ) ) ).

% divide_less_eq_numeral1(2)
thf(fact_3672_less__divide__eq__numeral1_I2_J,axiom,
    ! [A: real,B2: real,W: num] :
      ( ( ord_less_real @ A @ ( divide_divide_real @ B2 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) )
      = ( ord_less_real @ B2 @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ).

% less_divide_eq_numeral1(2)
thf(fact_3673_less__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B2: rat,W: num] :
      ( ( ord_less_rat @ A @ ( divide_divide_rat @ B2 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) )
      = ( ord_less_rat @ B2 @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ).

% less_divide_eq_numeral1(2)
thf(fact_3674_power2__minus,axiom,
    ! [A: real] :
      ( ( power_power_real @ ( uminus_uminus_real @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_minus
thf(fact_3675_power2__minus,axiom,
    ! [A: int] :
      ( ( power_power_int @ ( uminus_uminus_int @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_minus
thf(fact_3676_power2__minus,axiom,
    ! [A: complex] :
      ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_minus
thf(fact_3677_power2__minus,axiom,
    ! [A: code_integer] :
      ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_minus
thf(fact_3678_power2__minus,axiom,
    ! [A: rat] :
      ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_minus
thf(fact_3679_add__neg__numeral__special_I9_J,axiom,
    ( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ one_one_real ) )
    = ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% add_neg_numeral_special(9)
thf(fact_3680_add__neg__numeral__special_I9_J,axiom,
    ( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ one_one_int ) )
    = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% add_neg_numeral_special(9)
thf(fact_3681_add__neg__numeral__special_I9_J,axiom,
    ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
    = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).

% add_neg_numeral_special(9)
thf(fact_3682_add__neg__numeral__special_I9_J,axiom,
    ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% add_neg_numeral_special(9)
thf(fact_3683_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 @ ( bit0 @ one ) ) ) ) ).

% add_neg_numeral_special(9)
thf(fact_3684_diff__numeral__special_I11_J,axiom,
    ( ( minus_minus_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) )
    = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).

% diff_numeral_special(11)
thf(fact_3685_diff__numeral__special_I11_J,axiom,
    ( ( minus_minus_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) )
    = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).

% diff_numeral_special(11)
thf(fact_3686_diff__numeral__special_I11_J,axiom,
    ( ( minus_minus_complex @ one_one_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
    = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).

% diff_numeral_special(11)
thf(fact_3687_diff__numeral__special_I11_J,axiom,
    ( ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ).

% diff_numeral_special(11)
thf(fact_3688_diff__numeral__special_I11_J,axiom,
    ( ( minus_minus_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) )
    = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).

% diff_numeral_special(11)
thf(fact_3689_diff__numeral__special_I10_J,axiom,
    ( ( minus_minus_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real )
    = ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% diff_numeral_special(10)
thf(fact_3690_diff__numeral__special_I10_J,axiom,
    ( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int )
    = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% diff_numeral_special(10)
thf(fact_3691_diff__numeral__special_I10_J,axiom,
    ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ one_one_complex )
    = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).

% diff_numeral_special(10)
thf(fact_3692_diff__numeral__special_I10_J,axiom,
    ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer )
    = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% diff_numeral_special(10)
thf(fact_3693_diff__numeral__special_I10_J,axiom,
    ( ( minus_minus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat )
    = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).

% diff_numeral_special(10)
thf(fact_3694_minus__1__div__2__eq,axiom,
    ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
    = ( uminus_uminus_int @ one_one_int ) ) ).

% minus_1_div_2_eq
thf(fact_3695_minus__1__div__2__eq,axiom,
    ( ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
    = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% minus_1_div_2_eq
thf(fact_3696_minus__1__mod__2__eq,axiom,
    ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
    = one_one_int ) ).

% minus_1_mod_2_eq
thf(fact_3697_minus__1__mod__2__eq,axiom,
    ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
    = one_one_Code_integer ) ).

% minus_1_mod_2_eq
thf(fact_3698_bits__minus__1__mod__2__eq,axiom,
    ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
    = one_one_int ) ).

% bits_minus_1_mod_2_eq
thf(fact_3699_bits__minus__1__mod__2__eq,axiom,
    ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
    = one_one_Code_integer ) ).

% bits_minus_1_mod_2_eq
thf(fact_3700_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: real,N: nat] :
      ( ( power_power_real @ ( uminus_uminus_real @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% Power.ring_1_class.power_minus_even
thf(fact_3701_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: int,N: nat] :
      ( ( power_power_int @ ( uminus_uminus_int @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% Power.ring_1_class.power_minus_even
thf(fact_3702_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: complex,N: nat] :
      ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( power_power_complex @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% Power.ring_1_class.power_minus_even
thf(fact_3703_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: code_integer,N: nat] :
      ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( power_8256067586552552935nteger @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% Power.ring_1_class.power_minus_even
thf(fact_3704_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: rat,N: nat] :
      ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( power_power_rat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% Power.ring_1_class.power_minus_even
thf(fact_3705_power__minus__odd,axiom,
    ! [N: nat,A: real] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
        = ( uminus_uminus_real @ ( power_power_real @ A @ N ) ) ) ) ).

% power_minus_odd
thf(fact_3706_power__minus__odd,axiom,
    ! [N: nat,A: int] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
        = ( uminus_uminus_int @ ( power_power_int @ A @ N ) ) ) ) ).

% power_minus_odd
thf(fact_3707_power__minus__odd,axiom,
    ! [N: nat,A: complex] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
        = ( uminus1482373934393186551omplex @ ( power_power_complex @ A @ N ) ) ) ) ).

% power_minus_odd
thf(fact_3708_power__minus__odd,axiom,
    ! [N: nat,A: code_integer] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
        = ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ A @ N ) ) ) ) ).

% power_minus_odd
thf(fact_3709_power__minus__odd,axiom,
    ! [N: nat,A: rat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
        = ( uminus_uminus_rat @ ( power_power_rat @ A @ N ) ) ) ) ).

% power_minus_odd
thf(fact_3710_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [N: nat,A: real] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
        = ( power_power_real @ A @ N ) ) ) ).

% Parity.ring_1_class.power_minus_even
thf(fact_3711_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [N: nat,A: int] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
        = ( power_power_int @ A @ N ) ) ) ).

% Parity.ring_1_class.power_minus_even
thf(fact_3712_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [N: nat,A: complex] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
        = ( power_power_complex @ A @ N ) ) ) ).

% Parity.ring_1_class.power_minus_even
thf(fact_3713_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [N: nat,A: code_integer] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
        = ( power_8256067586552552935nteger @ A @ N ) ) ) ).

% Parity.ring_1_class.power_minus_even
thf(fact_3714_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [N: nat,A: rat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
        = ( power_power_rat @ A @ N ) ) ) ).

% Parity.ring_1_class.power_minus_even
thf(fact_3715_diff__numeral__special_I3_J,axiom,
    ! [N: num] :
      ( ( minus_minus_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
      = ( numeral_numeral_real @ ( plus_plus_num @ one @ N ) ) ) ).

% diff_numeral_special(3)
thf(fact_3716_diff__numeral__special_I3_J,axiom,
    ! [N: num] :
      ( ( minus_minus_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( numeral_numeral_int @ ( plus_plus_num @ one @ N ) ) ) ).

% diff_numeral_special(3)
thf(fact_3717_diff__numeral__special_I3_J,axiom,
    ! [N: num] :
      ( ( minus_minus_complex @ one_one_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
      = ( numera6690914467698888265omplex @ ( plus_plus_num @ one @ N ) ) ) ).

% diff_numeral_special(3)
thf(fact_3718_diff__numeral__special_I3_J,axiom,
    ! [N: num] :
      ( ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
      = ( numera6620942414471956472nteger @ ( plus_plus_num @ one @ N ) ) ) ).

% diff_numeral_special(3)
thf(fact_3719_diff__numeral__special_I3_J,axiom,
    ! [N: num] :
      ( ( minus_minus_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
      = ( numeral_numeral_rat @ ( plus_plus_num @ one @ N ) ) ) ).

% diff_numeral_special(3)
thf(fact_3720_diff__numeral__special_I4_J,axiom,
    ! [M: num] :
      ( ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ one_one_real )
      = ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ M @ one ) ) ) ) ).

% diff_numeral_special(4)
thf(fact_3721_diff__numeral__special_I4_J,axiom,
    ! [M: num] :
      ( ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ M @ one ) ) ) ) ).

% diff_numeral_special(4)
thf(fact_3722_diff__numeral__special_I4_J,axiom,
    ! [M: num] :
      ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ one_one_complex )
      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ one ) ) ) ) ).

% diff_numeral_special(4)
thf(fact_3723_diff__numeral__special_I4_J,axiom,
    ! [M: num] :
      ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer )
      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ M @ one ) ) ) ) ).

% diff_numeral_special(4)
thf(fact_3724_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 @ ( plus_plus_num @ M @ one ) ) ) ) ).

% diff_numeral_special(4)
thf(fact_3725_signed__take__bit__Suc__minus__bit0,axiom,
    ! [N: nat,K: num] :
      ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
      = ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% signed_take_bit_Suc_minus_bit0
thf(fact_3726_dbl__simps_I4_J,axiom,
    ( ( neg_numeral_dbl_real @ ( uminus_uminus_real @ one_one_real ) )
    = ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% dbl_simps(4)
thf(fact_3727_dbl__simps_I4_J,axiom,
    ( ( neg_numeral_dbl_int @ ( uminus_uminus_int @ one_one_int ) )
    = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% dbl_simps(4)
thf(fact_3728_dbl__simps_I4_J,axiom,
    ( ( neg_nu7009210354673126013omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
    = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).

% dbl_simps(4)
thf(fact_3729_dbl__simps_I4_J,axiom,
    ( ( neg_nu8804712462038260780nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% dbl_simps(4)
thf(fact_3730_dbl__simps_I4_J,axiom,
    ( ( neg_numeral_dbl_rat @ ( uminus_uminus_rat @ one_one_rat ) )
    = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).

% dbl_simps(4)
thf(fact_3731_power__minus1__even,axiom,
    ! [N: nat] :
      ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = one_one_real ) ).

% power_minus1_even
thf(fact_3732_power__minus1__even,axiom,
    ! [N: nat] :
      ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = one_one_int ) ).

% power_minus1_even
thf(fact_3733_power__minus1__even,axiom,
    ! [N: nat] :
      ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = one_one_complex ) ).

% power_minus1_even
thf(fact_3734_power__minus1__even,axiom,
    ! [N: nat] :
      ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = one_one_Code_integer ) ).

% power_minus1_even
thf(fact_3735_power__minus1__even,axiom,
    ! [N: nat] :
      ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = one_one_rat ) ).

% power_minus1_even
thf(fact_3736_neg__one__odd__power,axiom,
    ! [N: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N )
        = ( uminus_uminus_real @ one_one_real ) ) ) ).

% neg_one_odd_power
thf(fact_3737_neg__one__odd__power,axiom,
    ! [N: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
        = ( uminus_uminus_int @ one_one_int ) ) ) ).

% neg_one_odd_power
thf(fact_3738_neg__one__odd__power,axiom,
    ! [N: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N )
        = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ).

% neg_one_odd_power
thf(fact_3739_neg__one__odd__power,axiom,
    ! [N: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N )
        = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ).

% neg_one_odd_power
thf(fact_3740_neg__one__odd__power,axiom,
    ! [N: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N )
        = ( uminus_uminus_rat @ one_one_rat ) ) ) ).

% neg_one_odd_power
thf(fact_3741_neg__one__even__power,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N )
        = one_one_real ) ) ).

% neg_one_even_power
thf(fact_3742_neg__one__even__power,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
        = one_one_int ) ) ).

% neg_one_even_power
thf(fact_3743_neg__one__even__power,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N )
        = one_one_complex ) ) ).

% neg_one_even_power
thf(fact_3744_neg__one__even__power,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N )
        = one_one_Code_integer ) ) ).

% neg_one_even_power
thf(fact_3745_neg__one__even__power,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N )
        = one_one_rat ) ) ).

% neg_one_even_power
thf(fact_3746_signed__take__bit__0,axiom,
    ! [A: code_integer] :
      ( ( bit_ri6519982836138164636nteger @ zero_zero_nat @ A )
      = ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).

% signed_take_bit_0
thf(fact_3747_signed__take__bit__0,axiom,
    ! [A: int] :
      ( ( bit_ri631733984087533419it_int @ zero_zero_nat @ A )
      = ( uminus_uminus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).

% signed_take_bit_0
thf(fact_3748_verit__negate__coefficient_I3_J,axiom,
    ! [A: real,B2: real] :
      ( ( A = B2 )
     => ( ( uminus_uminus_real @ A )
        = ( uminus_uminus_real @ B2 ) ) ) ).

% verit_negate_coefficient(3)
thf(fact_3749_verit__negate__coefficient_I3_J,axiom,
    ! [A: int,B2: int] :
      ( ( A = B2 )
     => ( ( uminus_uminus_int @ A )
        = ( uminus_uminus_int @ B2 ) ) ) ).

% verit_negate_coefficient(3)
thf(fact_3750_verit__negate__coefficient_I3_J,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( A = B2 )
     => ( ( uminus1351360451143612070nteger @ A )
        = ( uminus1351360451143612070nteger @ B2 ) ) ) ).

% verit_negate_coefficient(3)
thf(fact_3751_verit__negate__coefficient_I3_J,axiom,
    ! [A: rat,B2: rat] :
      ( ( A = B2 )
     => ( ( uminus_uminus_rat @ A )
        = ( uminus_uminus_rat @ B2 ) ) ) ).

% verit_negate_coefficient(3)
thf(fact_3752_equation__minus__iff,axiom,
    ! [A: real,B2: real] :
      ( ( A
        = ( uminus_uminus_real @ B2 ) )
      = ( B2
        = ( uminus_uminus_real @ A ) ) ) ).

% equation_minus_iff
thf(fact_3753_equation__minus__iff,axiom,
    ! [A: int,B2: int] :
      ( ( A
        = ( uminus_uminus_int @ B2 ) )
      = ( B2
        = ( uminus_uminus_int @ A ) ) ) ).

% equation_minus_iff
thf(fact_3754_equation__minus__iff,axiom,
    ! [A: complex,B2: complex] :
      ( ( A
        = ( uminus1482373934393186551omplex @ B2 ) )
      = ( B2
        = ( uminus1482373934393186551omplex @ A ) ) ) ).

% equation_minus_iff
thf(fact_3755_equation__minus__iff,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( A
        = ( uminus1351360451143612070nteger @ B2 ) )
      = ( B2
        = ( uminus1351360451143612070nteger @ A ) ) ) ).

% equation_minus_iff
thf(fact_3756_equation__minus__iff,axiom,
    ! [A: rat,B2: rat] :
      ( ( A
        = ( uminus_uminus_rat @ B2 ) )
      = ( B2
        = ( uminus_uminus_rat @ A ) ) ) ).

% equation_minus_iff
thf(fact_3757_minus__equation__iff,axiom,
    ! [A: real,B2: real] :
      ( ( ( uminus_uminus_real @ A )
        = B2 )
      = ( ( uminus_uminus_real @ B2 )
        = A ) ) ).

% minus_equation_iff
thf(fact_3758_minus__equation__iff,axiom,
    ! [A: int,B2: int] :
      ( ( ( uminus_uminus_int @ A )
        = B2 )
      = ( ( uminus_uminus_int @ B2 )
        = A ) ) ).

% minus_equation_iff
thf(fact_3759_minus__equation__iff,axiom,
    ! [A: complex,B2: complex] :
      ( ( ( uminus1482373934393186551omplex @ A )
        = B2 )
      = ( ( uminus1482373934393186551omplex @ B2 )
        = A ) ) ).

% minus_equation_iff
thf(fact_3760_minus__equation__iff,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( ( uminus1351360451143612070nteger @ A )
        = B2 )
      = ( ( uminus1351360451143612070nteger @ B2 )
        = A ) ) ).

% minus_equation_iff
thf(fact_3761_minus__equation__iff,axiom,
    ! [A: rat,B2: rat] :
      ( ( ( uminus_uminus_rat @ A )
        = B2 )
      = ( ( uminus_uminus_rat @ B2 )
        = A ) ) ).

% minus_equation_iff
thf(fact_3762_le__imp__neg__le,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ A @ B2 )
     => ( ord_less_eq_real @ ( uminus_uminus_real @ B2 ) @ ( uminus_uminus_real @ A ) ) ) ).

% le_imp_neg_le
thf(fact_3763_le__imp__neg__le,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ B2 )
     => ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B2 ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).

% le_imp_neg_le
thf(fact_3764_le__imp__neg__le,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_eq_rat @ A @ B2 )
     => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B2 ) @ ( uminus_uminus_rat @ A ) ) ) ).

% le_imp_neg_le
thf(fact_3765_le__imp__neg__le,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ A @ B2 )
     => ( ord_less_eq_int @ ( uminus_uminus_int @ B2 ) @ ( uminus_uminus_int @ A ) ) ) ).

% le_imp_neg_le
thf(fact_3766_minus__le__iff,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B2 )
      = ( ord_less_eq_real @ ( uminus_uminus_real @ B2 ) @ A ) ) ).

% minus_le_iff
thf(fact_3767_minus__le__iff,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B2 )
      = ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B2 ) @ A ) ) ).

% minus_le_iff
thf(fact_3768_minus__le__iff,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B2 )
      = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B2 ) @ A ) ) ).

% minus_le_iff
thf(fact_3769_minus__le__iff,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B2 )
      = ( ord_less_eq_int @ ( uminus_uminus_int @ B2 ) @ A ) ) ).

% minus_le_iff
thf(fact_3770_le__minus__iff,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ B2 ) )
      = ( ord_less_eq_real @ B2 @ ( uminus_uminus_real @ A ) ) ) ).

% le_minus_iff
thf(fact_3771_le__minus__iff,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ ( uminus1351360451143612070nteger @ B2 ) )
      = ( ord_le3102999989581377725nteger @ B2 @ ( uminus1351360451143612070nteger @ A ) ) ) ).

% le_minus_iff
thf(fact_3772_le__minus__iff,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ B2 ) )
      = ( ord_less_eq_rat @ B2 @ ( uminus_uminus_rat @ A ) ) ) ).

% le_minus_iff
thf(fact_3773_le__minus__iff,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ B2 ) )
      = ( ord_less_eq_int @ B2 @ ( uminus_uminus_int @ A ) ) ) ).

% le_minus_iff
thf(fact_3774_minus__less__iff,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ A ) @ B2 )
      = ( ord_less_real @ ( uminus_uminus_real @ B2 ) @ A ) ) ).

% minus_less_iff
thf(fact_3775_minus__less__iff,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_int @ ( uminus_uminus_int @ A ) @ B2 )
      = ( ord_less_int @ ( uminus_uminus_int @ B2 ) @ A ) ) ).

% minus_less_iff
thf(fact_3776_minus__less__iff,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ B2 )
      = ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B2 ) @ A ) ) ).

% minus_less_iff
thf(fact_3777_minus__less__iff,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ B2 )
      = ( ord_less_rat @ ( uminus_uminus_rat @ B2 ) @ A ) ) ).

% minus_less_iff
thf(fact_3778_less__minus__iff,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ A @ ( uminus_uminus_real @ B2 ) )
      = ( ord_less_real @ B2 @ ( uminus_uminus_real @ A ) ) ) ).

% less_minus_iff
thf(fact_3779_less__minus__iff,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_int @ A @ ( uminus_uminus_int @ B2 ) )
      = ( ord_less_int @ B2 @ ( uminus_uminus_int @ A ) ) ) ).

% less_minus_iff
thf(fact_3780_less__minus__iff,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ ( uminus1351360451143612070nteger @ B2 ) )
      = ( ord_le6747313008572928689nteger @ B2 @ ( uminus1351360451143612070nteger @ A ) ) ) ).

% less_minus_iff
thf(fact_3781_less__minus__iff,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ B2 ) )
      = ( ord_less_rat @ B2 @ ( uminus_uminus_rat @ A ) ) ) ).

% less_minus_iff
thf(fact_3782_verit__negate__coefficient_I2_J,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ord_less_real @ ( uminus_uminus_real @ B2 ) @ ( uminus_uminus_real @ A ) ) ) ).

% verit_negate_coefficient(2)
thf(fact_3783_verit__negate__coefficient_I2_J,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_int @ A @ B2 )
     => ( ord_less_int @ ( uminus_uminus_int @ B2 ) @ ( uminus_uminus_int @ A ) ) ) ).

% verit_negate_coefficient(2)
thf(fact_3784_verit__negate__coefficient_I2_J,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ B2 )
     => ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B2 ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).

% verit_negate_coefficient(2)
thf(fact_3785_verit__negate__coefficient_I2_J,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_rat @ A @ B2 )
     => ( ord_less_rat @ ( uminus_uminus_rat @ B2 ) @ ( uminus_uminus_rat @ A ) ) ) ).

% verit_negate_coefficient(2)
thf(fact_3786_neg__numeral__neq__numeral,axiom,
    ! [M: num,N: num] :
      ( ( uminus_uminus_real @ ( numeral_numeral_real @ M ) )
     != ( numeral_numeral_real @ N ) ) ).

% neg_numeral_neq_numeral
thf(fact_3787_neg__numeral__neq__numeral,axiom,
    ! [M: num,N: num] :
      ( ( uminus_uminus_int @ ( numeral_numeral_int @ M ) )
     != ( numeral_numeral_int @ N ) ) ).

% neg_numeral_neq_numeral
thf(fact_3788_neg__numeral__neq__numeral,axiom,
    ! [M: num,N: num] :
      ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) )
     != ( numera6690914467698888265omplex @ N ) ) ).

% neg_numeral_neq_numeral
thf(fact_3789_neg__numeral__neq__numeral,axiom,
    ! [M: num,N: num] :
      ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) )
     != ( numera6620942414471956472nteger @ N ) ) ).

% neg_numeral_neq_numeral
thf(fact_3790_neg__numeral__neq__numeral,axiom,
    ! [M: num,N: num] :
      ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) )
     != ( numeral_numeral_rat @ N ) ) ).

% neg_numeral_neq_numeral
thf(fact_3791_numeral__neq__neg__numeral,axiom,
    ! [M: num,N: num] :
      ( ( numeral_numeral_real @ M )
     != ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).

% numeral_neq_neg_numeral
thf(fact_3792_numeral__neq__neg__numeral,axiom,
    ! [M: num,N: num] :
      ( ( numeral_numeral_int @ M )
     != ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).

% numeral_neq_neg_numeral
thf(fact_3793_numeral__neq__neg__numeral,axiom,
    ! [M: num,N: num] :
      ( ( numera6690914467698888265omplex @ M )
     != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) ) ).

% numeral_neq_neg_numeral
thf(fact_3794_numeral__neq__neg__numeral,axiom,
    ! [M: num,N: num] :
      ( ( numera6620942414471956472nteger @ M )
     != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).

% numeral_neq_neg_numeral
thf(fact_3795_numeral__neq__neg__numeral,axiom,
    ! [M: num,N: num] :
      ( ( numeral_numeral_rat @ M )
     != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).

% numeral_neq_neg_numeral
thf(fact_3796_minus__mult__commute,axiom,
    ! [A: real,B2: real] :
      ( ( times_times_real @ ( uminus_uminus_real @ A ) @ B2 )
      = ( times_times_real @ A @ ( uminus_uminus_real @ B2 ) ) ) ).

% minus_mult_commute
thf(fact_3797_minus__mult__commute,axiom,
    ! [A: int,B2: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ A ) @ B2 )
      = ( times_times_int @ A @ ( uminus_uminus_int @ B2 ) ) ) ).

% minus_mult_commute
thf(fact_3798_minus__mult__commute,axiom,
    ! [A: complex,B2: complex] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ A ) @ B2 )
      = ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ B2 ) ) ) ).

% minus_mult_commute
thf(fact_3799_minus__mult__commute,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A ) @ B2 )
      = ( times_3573771949741848930nteger @ A @ ( uminus1351360451143612070nteger @ B2 ) ) ) ).

% minus_mult_commute
thf(fact_3800_minus__mult__commute,axiom,
    ! [A: rat,B2: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ A ) @ B2 )
      = ( times_times_rat @ A @ ( uminus_uminus_rat @ B2 ) ) ) ).

% minus_mult_commute
thf(fact_3801_square__eq__iff,axiom,
    ! [A: real,B2: real] :
      ( ( ( times_times_real @ A @ A )
        = ( times_times_real @ B2 @ B2 ) )
      = ( ( A = B2 )
        | ( A
          = ( uminus_uminus_real @ B2 ) ) ) ) ).

% square_eq_iff
thf(fact_3802_square__eq__iff,axiom,
    ! [A: int,B2: int] :
      ( ( ( times_times_int @ A @ A )
        = ( times_times_int @ B2 @ B2 ) )
      = ( ( A = B2 )
        | ( A
          = ( uminus_uminus_int @ B2 ) ) ) ) ).

% square_eq_iff
thf(fact_3803_square__eq__iff,axiom,
    ! [A: complex,B2: complex] :
      ( ( ( times_times_complex @ A @ A )
        = ( times_times_complex @ B2 @ B2 ) )
      = ( ( A = B2 )
        | ( A
          = ( uminus1482373934393186551omplex @ B2 ) ) ) ) ).

% square_eq_iff
thf(fact_3804_square__eq__iff,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( ( times_3573771949741848930nteger @ A @ A )
        = ( times_3573771949741848930nteger @ B2 @ B2 ) )
      = ( ( A = B2 )
        | ( A
          = ( uminus1351360451143612070nteger @ B2 ) ) ) ) ).

% square_eq_iff
thf(fact_3805_square__eq__iff,axiom,
    ! [A: rat,B2: rat] :
      ( ( ( times_times_rat @ A @ A )
        = ( times_times_rat @ B2 @ B2 ) )
      = ( ( A = B2 )
        | ( A
          = ( uminus_uminus_rat @ B2 ) ) ) ) ).

% square_eq_iff
thf(fact_3806_one__neq__neg__one,axiom,
    ( one_one_real
   != ( uminus_uminus_real @ one_one_real ) ) ).

% one_neq_neg_one
thf(fact_3807_one__neq__neg__one,axiom,
    ( one_one_int
   != ( uminus_uminus_int @ one_one_int ) ) ).

% one_neq_neg_one
thf(fact_3808_one__neq__neg__one,axiom,
    ( one_one_complex
   != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% one_neq_neg_one
thf(fact_3809_one__neq__neg__one,axiom,
    ( one_one_Code_integer
   != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% one_neq_neg_one
thf(fact_3810_one__neq__neg__one,axiom,
    ( one_one_rat
   != ( uminus_uminus_rat @ one_one_rat ) ) ).

% one_neq_neg_one
thf(fact_3811_add_Oinverse__distrib__swap,axiom,
    ! [A: real,B2: real] :
      ( ( uminus_uminus_real @ ( plus_plus_real @ A @ B2 ) )
      = ( plus_plus_real @ ( uminus_uminus_real @ B2 ) @ ( uminus_uminus_real @ A ) ) ) ).

% add.inverse_distrib_swap
thf(fact_3812_add_Oinverse__distrib__swap,axiom,
    ! [A: int,B2: int] :
      ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B2 ) )
      = ( plus_plus_int @ ( uminus_uminus_int @ B2 ) @ ( uminus_uminus_int @ A ) ) ) ).

% add.inverse_distrib_swap
thf(fact_3813_add_Oinverse__distrib__swap,axiom,
    ! [A: complex,B2: complex] :
      ( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A @ B2 ) )
      = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ B2 ) @ ( uminus1482373934393186551omplex @ A ) ) ) ).

% add.inverse_distrib_swap
thf(fact_3814_add_Oinverse__distrib__swap,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A @ B2 ) )
      = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ B2 ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).

% add.inverse_distrib_swap
thf(fact_3815_add_Oinverse__distrib__swap,axiom,
    ! [A: rat,B2: rat] :
      ( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B2 ) )
      = ( plus_plus_rat @ ( uminus_uminus_rat @ B2 ) @ ( uminus_uminus_rat @ A ) ) ) ).

% add.inverse_distrib_swap
thf(fact_3816_group__cancel_Oneg1,axiom,
    ! [A3: real,K: real,A: real] :
      ( ( A3
        = ( plus_plus_real @ K @ A ) )
     => ( ( uminus_uminus_real @ A3 )
        = ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( uminus_uminus_real @ A ) ) ) ) ).

% group_cancel.neg1
thf(fact_3817_group__cancel_Oneg1,axiom,
    ! [A3: int,K: int,A: int] :
      ( ( A3
        = ( plus_plus_int @ K @ A ) )
     => ( ( uminus_uminus_int @ A3 )
        = ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( uminus_uminus_int @ A ) ) ) ) ).

% group_cancel.neg1
thf(fact_3818_group__cancel_Oneg1,axiom,
    ! [A3: complex,K: complex,A: complex] :
      ( ( A3
        = ( plus_plus_complex @ K @ A ) )
     => ( ( uminus1482373934393186551omplex @ A3 )
        = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ K ) @ ( uminus1482373934393186551omplex @ A ) ) ) ) ).

% group_cancel.neg1
thf(fact_3819_group__cancel_Oneg1,axiom,
    ! [A3: code_integer,K: code_integer,A: code_integer] :
      ( ( A3
        = ( plus_p5714425477246183910nteger @ K @ A ) )
     => ( ( uminus1351360451143612070nteger @ A3 )
        = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ K ) @ ( uminus1351360451143612070nteger @ A ) ) ) ) ).

% group_cancel.neg1
thf(fact_3820_group__cancel_Oneg1,axiom,
    ! [A3: rat,K: rat,A: rat] :
      ( ( A3
        = ( plus_plus_rat @ K @ A ) )
     => ( ( uminus_uminus_rat @ A3 )
        = ( plus_plus_rat @ ( uminus_uminus_rat @ K ) @ ( uminus_uminus_rat @ A ) ) ) ) ).

% group_cancel.neg1
thf(fact_3821_is__num__normalize_I8_J,axiom,
    ! [A: real,B2: real] :
      ( ( uminus_uminus_real @ ( plus_plus_real @ A @ B2 ) )
      = ( plus_plus_real @ ( uminus_uminus_real @ B2 ) @ ( uminus_uminus_real @ A ) ) ) ).

% is_num_normalize(8)
thf(fact_3822_is__num__normalize_I8_J,axiom,
    ! [A: int,B2: int] :
      ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B2 ) )
      = ( plus_plus_int @ ( uminus_uminus_int @ B2 ) @ ( uminus_uminus_int @ A ) ) ) ).

% is_num_normalize(8)
thf(fact_3823_is__num__normalize_I8_J,axiom,
    ! [A: complex,B2: complex] :
      ( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A @ B2 ) )
      = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ B2 ) @ ( uminus1482373934393186551omplex @ A ) ) ) ).

% is_num_normalize(8)
thf(fact_3824_is__num__normalize_I8_J,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A @ B2 ) )
      = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ B2 ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).

% is_num_normalize(8)
thf(fact_3825_is__num__normalize_I8_J,axiom,
    ! [A: rat,B2: rat] :
      ( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B2 ) )
      = ( plus_plus_rat @ ( uminus_uminus_rat @ B2 ) @ ( uminus_uminus_rat @ A ) ) ) ).

% is_num_normalize(8)
thf(fact_3826_minus__diff__commute,axiom,
    ! [B2: real,A: real] :
      ( ( minus_minus_real @ ( uminus_uminus_real @ B2 ) @ A )
      = ( minus_minus_real @ ( uminus_uminus_real @ A ) @ B2 ) ) ).

% minus_diff_commute
thf(fact_3827_minus__diff__commute,axiom,
    ! [B2: int,A: int] :
      ( ( minus_minus_int @ ( uminus_uminus_int @ B2 ) @ A )
      = ( minus_minus_int @ ( uminus_uminus_int @ A ) @ B2 ) ) ).

% minus_diff_commute
thf(fact_3828_minus__diff__commute,axiom,
    ! [B2: complex,A: complex] :
      ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ B2 ) @ A )
      = ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A ) @ B2 ) ) ).

% minus_diff_commute
thf(fact_3829_minus__diff__commute,axiom,
    ! [B2: code_integer,A: code_integer] :
      ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ B2 ) @ A )
      = ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ A ) @ B2 ) ) ).

% minus_diff_commute
thf(fact_3830_minus__diff__commute,axiom,
    ! [B2: rat,A: rat] :
      ( ( minus_minus_rat @ ( uminus_uminus_rat @ B2 ) @ A )
      = ( minus_minus_rat @ ( uminus_uminus_rat @ A ) @ B2 ) ) ).

% minus_diff_commute
thf(fact_3831_minus__diff__minus,axiom,
    ! [A: real,B2: real] :
      ( ( minus_minus_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B2 ) )
      = ( uminus_uminus_real @ ( minus_minus_real @ A @ B2 ) ) ) ).

% minus_diff_minus
thf(fact_3832_minus__diff__minus,axiom,
    ! [A: int,B2: int] :
      ( ( minus_minus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B2 ) )
      = ( uminus_uminus_int @ ( minus_minus_int @ A @ B2 ) ) ) ).

% minus_diff_minus
thf(fact_3833_minus__diff__minus,axiom,
    ! [A: complex,B2: complex] :
      ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B2 ) )
      = ( uminus1482373934393186551omplex @ ( minus_minus_complex @ A @ B2 ) ) ) ).

% minus_diff_minus
thf(fact_3834_minus__diff__minus,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B2 ) )
      = ( uminus1351360451143612070nteger @ ( minus_8373710615458151222nteger @ A @ B2 ) ) ) ).

% minus_diff_minus
thf(fact_3835_minus__diff__minus,axiom,
    ! [A: rat,B2: rat] :
      ( ( minus_minus_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B2 ) )
      = ( uminus_uminus_rat @ ( minus_minus_rat @ A @ B2 ) ) ) ).

% minus_diff_minus
thf(fact_3836_div__minus__right,axiom,
    ! [A: int,B2: int] :
      ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B2 ) )
      = ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B2 ) ) ).

% div_minus_right
thf(fact_3837_div__minus__right,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( divide6298287555418463151nteger @ A @ ( uminus1351360451143612070nteger @ B2 ) )
      = ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ A ) @ B2 ) ) ).

% div_minus_right
thf(fact_3838_minus__divide__right,axiom,
    ! [A: real,B2: real] :
      ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B2 ) )
      = ( divide_divide_real @ A @ ( uminus_uminus_real @ B2 ) ) ) ).

% minus_divide_right
thf(fact_3839_minus__divide__right,axiom,
    ! [A: complex,B2: complex] :
      ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B2 ) )
      = ( divide1717551699836669952omplex @ A @ ( uminus1482373934393186551omplex @ B2 ) ) ) ).

% minus_divide_right
thf(fact_3840_minus__divide__right,axiom,
    ! [A: rat,B2: rat] :
      ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B2 ) )
      = ( divide_divide_rat @ A @ ( uminus_uminus_rat @ B2 ) ) ) ).

% minus_divide_right
thf(fact_3841_minus__divide__divide,axiom,
    ! [A: real,B2: real] :
      ( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B2 ) )
      = ( divide_divide_real @ A @ B2 ) ) ).

% minus_divide_divide
thf(fact_3842_minus__divide__divide,axiom,
    ! [A: complex,B2: complex] :
      ( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B2 ) )
      = ( divide1717551699836669952omplex @ A @ B2 ) ) ).

% minus_divide_divide
thf(fact_3843_minus__divide__divide,axiom,
    ! [A: rat,B2: rat] :
      ( ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B2 ) )
      = ( divide_divide_rat @ A @ B2 ) ) ).

% minus_divide_divide
thf(fact_3844_minus__divide__left,axiom,
    ! [A: real,B2: real] :
      ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B2 ) )
      = ( divide_divide_real @ ( uminus_uminus_real @ A ) @ B2 ) ) ).

% minus_divide_left
thf(fact_3845_minus__divide__left,axiom,
    ! [A: complex,B2: complex] :
      ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B2 ) )
      = ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ B2 ) ) ).

% minus_divide_left
thf(fact_3846_minus__divide__left,axiom,
    ! [A: rat,B2: rat] :
      ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B2 ) )
      = ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ B2 ) ) ).

% minus_divide_left
thf(fact_3847_mod__minus__eq,axiom,
    ! [A: int,B2: int] :
      ( ( modulo_modulo_int @ ( uminus_uminus_int @ ( modulo_modulo_int @ A @ B2 ) ) @ B2 )
      = ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B2 ) ) ).

% mod_minus_eq
thf(fact_3848_mod__minus__eq,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A @ B2 ) ) @ B2 )
      = ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B2 ) ) ).

% mod_minus_eq
thf(fact_3849_mod__minus__cong,axiom,
    ! [A: int,B2: int,A2: int] :
      ( ( ( modulo_modulo_int @ A @ B2 )
        = ( modulo_modulo_int @ A2 @ B2 ) )
     => ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B2 )
        = ( modulo_modulo_int @ ( uminus_uminus_int @ A2 ) @ B2 ) ) ) ).

% mod_minus_cong
thf(fact_3850_mod__minus__cong,axiom,
    ! [A: code_integer,B2: code_integer,A2: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ B2 )
        = ( modulo364778990260209775nteger @ A2 @ B2 ) )
     => ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B2 )
        = ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A2 ) @ B2 ) ) ) ).

% mod_minus_cong
thf(fact_3851_mod__minus__right,axiom,
    ! [A: int,B2: int] :
      ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ B2 ) )
      = ( uminus_uminus_int @ ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B2 ) ) ) ).

% mod_minus_right
thf(fact_3852_mod__minus__right,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( modulo364778990260209775nteger @ A @ ( uminus1351360451143612070nteger @ B2 ) )
      = ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B2 ) ) ) ).

% mod_minus_right
thf(fact_3853_signed__take__bit__minus,axiom,
    ! [N: nat,K: int] :
      ( ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ ( bit_ri631733984087533419it_int @ N @ K ) ) )
      = ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ K ) ) ) ).

% signed_take_bit_minus
thf(fact_3854_neg__numeral__le__numeral,axiom,
    ! [M: num,N: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) ) ).

% neg_numeral_le_numeral
thf(fact_3855_neg__numeral__le__numeral,axiom,
    ! [M: num,N: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) ) ).

% neg_numeral_le_numeral
thf(fact_3856_neg__numeral__le__numeral,axiom,
    ! [M: num,N: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) ) ).

% neg_numeral_le_numeral
thf(fact_3857_neg__numeral__le__numeral,axiom,
    ! [M: num,N: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) ) ).

% neg_numeral_le_numeral
thf(fact_3858_not__numeral__le__neg__numeral,axiom,
    ! [M: num,N: num] :
      ~ ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).

% not_numeral_le_neg_numeral
thf(fact_3859_not__numeral__le__neg__numeral,axiom,
    ! [M: num,N: num] :
      ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).

% not_numeral_le_neg_numeral
thf(fact_3860_not__numeral__le__neg__numeral,axiom,
    ! [M: num,N: num] :
      ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).

% not_numeral_le_neg_numeral
thf(fact_3861_not__numeral__le__neg__numeral,axiom,
    ! [M: num,N: num] :
      ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).

% not_numeral_le_neg_numeral
thf(fact_3862_zero__neq__neg__numeral,axiom,
    ! [N: num] :
      ( zero_zero_real
     != ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).

% zero_neq_neg_numeral
thf(fact_3863_zero__neq__neg__numeral,axiom,
    ! [N: num] :
      ( zero_zero_int
     != ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).

% zero_neq_neg_numeral
thf(fact_3864_zero__neq__neg__numeral,axiom,
    ! [N: num] :
      ( zero_zero_complex
     != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) ) ).

% zero_neq_neg_numeral
thf(fact_3865_zero__neq__neg__numeral,axiom,
    ! [N: num] :
      ( zero_z3403309356797280102nteger
     != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).

% zero_neq_neg_numeral
thf(fact_3866_zero__neq__neg__numeral,axiom,
    ! [N: num] :
      ( zero_zero_rat
     != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).

% zero_neq_neg_numeral
thf(fact_3867_not__numeral__less__neg__numeral,axiom,
    ! [M: num,N: num] :
      ~ ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).

% not_numeral_less_neg_numeral
thf(fact_3868_not__numeral__less__neg__numeral,axiom,
    ! [M: num,N: num] :
      ~ ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).

% not_numeral_less_neg_numeral
thf(fact_3869_not__numeral__less__neg__numeral,axiom,
    ! [M: num,N: num] :
      ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).

% not_numeral_less_neg_numeral
thf(fact_3870_not__numeral__less__neg__numeral,axiom,
    ! [M: num,N: num] :
      ~ ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).

% not_numeral_less_neg_numeral
thf(fact_3871_neg__numeral__less__numeral,axiom,
    ! [M: num,N: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) ) ).

% neg_numeral_less_numeral
thf(fact_3872_neg__numeral__less__numeral,axiom,
    ! [M: num,N: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) ) ).

% neg_numeral_less_numeral
thf(fact_3873_neg__numeral__less__numeral,axiom,
    ! [M: num,N: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) ) ).

% neg_numeral_less_numeral
thf(fact_3874_neg__numeral__less__numeral,axiom,
    ! [M: num,N: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) ) ).

% neg_numeral_less_numeral
thf(fact_3875_le__minus__one__simps_I2_J,axiom,
    ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ).

% le_minus_one_simps(2)
thf(fact_3876_le__minus__one__simps_I2_J,axiom,
    ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ).

% le_minus_one_simps(2)
thf(fact_3877_le__minus__one__simps_I2_J,axiom,
    ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat ).

% le_minus_one_simps(2)
thf(fact_3878_le__minus__one__simps_I2_J,axiom,
    ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).

% le_minus_one_simps(2)
thf(fact_3879_le__minus__one__simps_I4_J,axiom,
    ~ ( ord_less_eq_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ).

% le_minus_one_simps(4)
thf(fact_3880_le__minus__one__simps_I4_J,axiom,
    ~ ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% le_minus_one_simps(4)
thf(fact_3881_le__minus__one__simps_I4_J,axiom,
    ~ ( ord_less_eq_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).

% le_minus_one_simps(4)
thf(fact_3882_le__minus__one__simps_I4_J,axiom,
    ~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).

% le_minus_one_simps(4)
thf(fact_3883_zero__neq__neg__one,axiom,
    ( zero_zero_real
   != ( uminus_uminus_real @ one_one_real ) ) ).

% zero_neq_neg_one
thf(fact_3884_zero__neq__neg__one,axiom,
    ( zero_zero_int
   != ( uminus_uminus_int @ one_one_int ) ) ).

% zero_neq_neg_one
thf(fact_3885_zero__neq__neg__one,axiom,
    ( zero_zero_complex
   != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% zero_neq_neg_one
thf(fact_3886_zero__neq__neg__one,axiom,
    ( zero_z3403309356797280102nteger
   != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% zero_neq_neg_one
thf(fact_3887_zero__neq__neg__one,axiom,
    ( zero_zero_rat
   != ( uminus_uminus_rat @ one_one_rat ) ) ).

% zero_neq_neg_one
thf(fact_3888_neg__eq__iff__add__eq__0,axiom,
    ! [A: real,B2: real] :
      ( ( ( uminus_uminus_real @ A )
        = B2 )
      = ( ( plus_plus_real @ A @ B2 )
        = zero_zero_real ) ) ).

% neg_eq_iff_add_eq_0
thf(fact_3889_neg__eq__iff__add__eq__0,axiom,
    ! [A: int,B2: int] :
      ( ( ( uminus_uminus_int @ A )
        = B2 )
      = ( ( plus_plus_int @ A @ B2 )
        = zero_zero_int ) ) ).

% neg_eq_iff_add_eq_0
thf(fact_3890_neg__eq__iff__add__eq__0,axiom,
    ! [A: complex,B2: complex] :
      ( ( ( uminus1482373934393186551omplex @ A )
        = B2 )
      = ( ( plus_plus_complex @ A @ B2 )
        = zero_zero_complex ) ) ).

% neg_eq_iff_add_eq_0
thf(fact_3891_neg__eq__iff__add__eq__0,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( ( uminus1351360451143612070nteger @ A )
        = B2 )
      = ( ( plus_p5714425477246183910nteger @ A @ B2 )
        = zero_z3403309356797280102nteger ) ) ).

% neg_eq_iff_add_eq_0
thf(fact_3892_neg__eq__iff__add__eq__0,axiom,
    ! [A: rat,B2: rat] :
      ( ( ( uminus_uminus_rat @ A )
        = B2 )
      = ( ( plus_plus_rat @ A @ B2 )
        = zero_zero_rat ) ) ).

% neg_eq_iff_add_eq_0
thf(fact_3893_eq__neg__iff__add__eq__0,axiom,
    ! [A: real,B2: real] :
      ( ( A
        = ( uminus_uminus_real @ B2 ) )
      = ( ( plus_plus_real @ A @ B2 )
        = zero_zero_real ) ) ).

% eq_neg_iff_add_eq_0
thf(fact_3894_eq__neg__iff__add__eq__0,axiom,
    ! [A: int,B2: int] :
      ( ( A
        = ( uminus_uminus_int @ B2 ) )
      = ( ( plus_plus_int @ A @ B2 )
        = zero_zero_int ) ) ).

% eq_neg_iff_add_eq_0
thf(fact_3895_eq__neg__iff__add__eq__0,axiom,
    ! [A: complex,B2: complex] :
      ( ( A
        = ( uminus1482373934393186551omplex @ B2 ) )
      = ( ( plus_plus_complex @ A @ B2 )
        = zero_zero_complex ) ) ).

% eq_neg_iff_add_eq_0
thf(fact_3896_eq__neg__iff__add__eq__0,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( A
        = ( uminus1351360451143612070nteger @ B2 ) )
      = ( ( plus_p5714425477246183910nteger @ A @ B2 )
        = zero_z3403309356797280102nteger ) ) ).

% eq_neg_iff_add_eq_0
thf(fact_3897_eq__neg__iff__add__eq__0,axiom,
    ! [A: rat,B2: rat] :
      ( ( A
        = ( uminus_uminus_rat @ B2 ) )
      = ( ( plus_plus_rat @ A @ B2 )
        = zero_zero_rat ) ) ).

% eq_neg_iff_add_eq_0
thf(fact_3898_add_Oinverse__unique,axiom,
    ! [A: real,B2: real] :
      ( ( ( plus_plus_real @ A @ B2 )
        = zero_zero_real )
     => ( ( uminus_uminus_real @ A )
        = B2 ) ) ).

% add.inverse_unique
thf(fact_3899_add_Oinverse__unique,axiom,
    ! [A: int,B2: int] :
      ( ( ( plus_plus_int @ A @ B2 )
        = zero_zero_int )
     => ( ( uminus_uminus_int @ A )
        = B2 ) ) ).

% add.inverse_unique
thf(fact_3900_add_Oinverse__unique,axiom,
    ! [A: complex,B2: complex] :
      ( ( ( plus_plus_complex @ A @ B2 )
        = zero_zero_complex )
     => ( ( uminus1482373934393186551omplex @ A )
        = B2 ) ) ).

% add.inverse_unique
thf(fact_3901_add_Oinverse__unique,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( ( plus_p5714425477246183910nteger @ A @ B2 )
        = zero_z3403309356797280102nteger )
     => ( ( uminus1351360451143612070nteger @ A )
        = B2 ) ) ).

% add.inverse_unique
thf(fact_3902_add_Oinverse__unique,axiom,
    ! [A: rat,B2: rat] :
      ( ( ( plus_plus_rat @ A @ B2 )
        = zero_zero_rat )
     => ( ( uminus_uminus_rat @ A )
        = B2 ) ) ).

% add.inverse_unique
thf(fact_3903_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ A )
      = zero_zero_real ) ).

% ab_group_add_class.ab_left_minus
thf(fact_3904_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
      = zero_zero_int ) ).

% ab_group_add_class.ab_left_minus
thf(fact_3905_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ A )
      = zero_zero_complex ) ).

% ab_group_add_class.ab_left_minus
thf(fact_3906_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
      = zero_z3403309356797280102nteger ) ).

% ab_group_add_class.ab_left_minus
thf(fact_3907_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
thf(fact_3908_add__eq__0__iff,axiom,
    ! [A: real,B2: real] :
      ( ( ( plus_plus_real @ A @ B2 )
        = zero_zero_real )
      = ( B2
        = ( uminus_uminus_real @ A ) ) ) ).

% add_eq_0_iff
thf(fact_3909_add__eq__0__iff,axiom,
    ! [A: int,B2: int] :
      ( ( ( plus_plus_int @ A @ B2 )
        = zero_zero_int )
      = ( B2
        = ( uminus_uminus_int @ A ) ) ) ).

% add_eq_0_iff
thf(fact_3910_add__eq__0__iff,axiom,
    ! [A: complex,B2: complex] :
      ( ( ( plus_plus_complex @ A @ B2 )
        = zero_zero_complex )
      = ( B2
        = ( uminus1482373934393186551omplex @ A ) ) ) ).

% add_eq_0_iff
thf(fact_3911_add__eq__0__iff,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( ( plus_p5714425477246183910nteger @ A @ B2 )
        = zero_z3403309356797280102nteger )
      = ( B2
        = ( uminus1351360451143612070nteger @ A ) ) ) ).

% add_eq_0_iff
thf(fact_3912_add__eq__0__iff,axiom,
    ! [A: rat,B2: rat] :
      ( ( ( plus_plus_rat @ A @ B2 )
        = zero_zero_rat )
      = ( B2
        = ( uminus_uminus_rat @ A ) ) ) ).

% add_eq_0_iff
thf(fact_3913_less__minus__one__simps_I2_J,axiom,
    ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ).

% less_minus_one_simps(2)
thf(fact_3914_less__minus__one__simps_I2_J,axiom,
    ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).

% less_minus_one_simps(2)
thf(fact_3915_less__minus__one__simps_I2_J,axiom,
    ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ).

% less_minus_one_simps(2)
thf(fact_3916_less__minus__one__simps_I2_J,axiom,
    ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat ).

% less_minus_one_simps(2)
thf(fact_3917_less__minus__one__simps_I4_J,axiom,
    ~ ( ord_less_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ).

% less_minus_one_simps(4)
thf(fact_3918_less__minus__one__simps_I4_J,axiom,
    ~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).

% less_minus_one_simps(4)
thf(fact_3919_less__minus__one__simps_I4_J,axiom,
    ~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% less_minus_one_simps(4)
thf(fact_3920_less__minus__one__simps_I4_J,axiom,
    ~ ( ord_less_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).

% less_minus_one_simps(4)
thf(fact_3921_numeral__times__minus__swap,axiom,
    ! [W: num,X4: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ W ) @ ( uminus_uminus_real @ X4 ) )
      = ( times_times_real @ X4 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ).

% numeral_times_minus_swap
thf(fact_3922_numeral__times__minus__swap,axiom,
    ! [W: num,X4: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ W ) @ ( uminus_uminus_int @ X4 ) )
      = ( times_times_int @ X4 @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) ) ) ).

% numeral_times_minus_swap
thf(fact_3923_numeral__times__minus__swap,axiom,
    ! [W: num,X4: complex] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ ( uminus1482373934393186551omplex @ X4 ) )
      = ( times_times_complex @ X4 @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ) ).

% numeral_times_minus_swap
thf(fact_3924_numeral__times__minus__swap,axiom,
    ! [W: num,X4: code_integer] :
      ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ W ) @ ( uminus1351360451143612070nteger @ X4 ) )
      = ( times_3573771949741848930nteger @ X4 @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) ) ) ).

% numeral_times_minus_swap
thf(fact_3925_numeral__times__minus__swap,axiom,
    ! [W: num,X4: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ W ) @ ( uminus_uminus_rat @ X4 ) )
      = ( times_times_rat @ X4 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ).

% numeral_times_minus_swap
thf(fact_3926_nonzero__minus__divide__right,axiom,
    ! [B2: real,A: real] :
      ( ( B2 != zero_zero_real )
     => ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B2 ) )
        = ( divide_divide_real @ A @ ( uminus_uminus_real @ B2 ) ) ) ) ).

% nonzero_minus_divide_right
thf(fact_3927_nonzero__minus__divide__right,axiom,
    ! [B2: complex,A: complex] :
      ( ( B2 != zero_zero_complex )
     => ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B2 ) )
        = ( divide1717551699836669952omplex @ A @ ( uminus1482373934393186551omplex @ B2 ) ) ) ) ).

% nonzero_minus_divide_right
thf(fact_3928_nonzero__minus__divide__right,axiom,
    ! [B2: rat,A: rat] :
      ( ( B2 != zero_zero_rat )
     => ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B2 ) )
        = ( divide_divide_rat @ A @ ( uminus_uminus_rat @ B2 ) ) ) ) ).

% nonzero_minus_divide_right
thf(fact_3929_nonzero__minus__divide__divide,axiom,
    ! [B2: real,A: real] :
      ( ( B2 != zero_zero_real )
     => ( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B2 ) )
        = ( divide_divide_real @ A @ B2 ) ) ) ).

% nonzero_minus_divide_divide
thf(fact_3930_nonzero__minus__divide__divide,axiom,
    ! [B2: complex,A: complex] :
      ( ( B2 != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B2 ) )
        = ( divide1717551699836669952omplex @ A @ B2 ) ) ) ).

% nonzero_minus_divide_divide
thf(fact_3931_nonzero__minus__divide__divide,axiom,
    ! [B2: rat,A: rat] :
      ( ( B2 != zero_zero_rat )
     => ( ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B2 ) )
        = ( divide_divide_rat @ A @ B2 ) ) ) ).

% nonzero_minus_divide_divide
thf(fact_3932_one__neq__neg__numeral,axiom,
    ! [N: num] :
      ( one_one_real
     != ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).

% one_neq_neg_numeral
thf(fact_3933_one__neq__neg__numeral,axiom,
    ! [N: num] :
      ( one_one_int
     != ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).

% one_neq_neg_numeral
thf(fact_3934_one__neq__neg__numeral,axiom,
    ! [N: num] :
      ( one_one_complex
     != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) ) ).

% one_neq_neg_numeral
thf(fact_3935_one__neq__neg__numeral,axiom,
    ! [N: num] :
      ( one_one_Code_integer
     != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).

% one_neq_neg_numeral
thf(fact_3936_one__neq__neg__numeral,axiom,
    ! [N: num] :
      ( one_one_rat
     != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).

% one_neq_neg_numeral
thf(fact_3937_numeral__neq__neg__one,axiom,
    ! [N: num] :
      ( ( numeral_numeral_real @ N )
     != ( uminus_uminus_real @ one_one_real ) ) ).

% numeral_neq_neg_one
thf(fact_3938_numeral__neq__neg__one,axiom,
    ! [N: num] :
      ( ( numeral_numeral_int @ N )
     != ( uminus_uminus_int @ one_one_int ) ) ).

% numeral_neq_neg_one
thf(fact_3939_numeral__neq__neg__one,axiom,
    ! [N: num] :
      ( ( numera6690914467698888265omplex @ N )
     != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% numeral_neq_neg_one
thf(fact_3940_numeral__neq__neg__one,axiom,
    ! [N: num] :
      ( ( numera6620942414471956472nteger @ N )
     != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% numeral_neq_neg_one
thf(fact_3941_numeral__neq__neg__one,axiom,
    ! [N: num] :
      ( ( numeral_numeral_rat @ N )
     != ( uminus_uminus_rat @ one_one_rat ) ) ).

% numeral_neq_neg_one
thf(fact_3942_square__eq__1__iff,axiom,
    ! [X4: real] :
      ( ( ( times_times_real @ X4 @ X4 )
        = one_one_real )
      = ( ( X4 = one_one_real )
        | ( X4
          = ( uminus_uminus_real @ one_one_real ) ) ) ) ).

% square_eq_1_iff
thf(fact_3943_square__eq__1__iff,axiom,
    ! [X4: int] :
      ( ( ( times_times_int @ X4 @ X4 )
        = one_one_int )
      = ( ( X4 = one_one_int )
        | ( X4
          = ( uminus_uminus_int @ one_one_int ) ) ) ) ).

% square_eq_1_iff
thf(fact_3944_square__eq__1__iff,axiom,
    ! [X4: complex] :
      ( ( ( times_times_complex @ X4 @ X4 )
        = one_one_complex )
      = ( ( X4 = one_one_complex )
        | ( X4
          = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ) ).

% square_eq_1_iff
thf(fact_3945_square__eq__1__iff,axiom,
    ! [X4: code_integer] :
      ( ( ( times_3573771949741848930nteger @ X4 @ X4 )
        = one_one_Code_integer )
      = ( ( X4 = one_one_Code_integer )
        | ( X4
          = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).

% square_eq_1_iff
thf(fact_3946_square__eq__1__iff,axiom,
    ! [X4: rat] :
      ( ( ( times_times_rat @ X4 @ X4 )
        = one_one_rat )
      = ( ( X4 = one_one_rat )
        | ( X4
          = ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).

% square_eq_1_iff
thf(fact_3947_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_minus_real
    = ( ^ [A4: real,B3: real] : ( plus_plus_real @ A4 @ ( uminus_uminus_real @ B3 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_3948_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_minus_int
    = ( ^ [A4: int,B3: int] : ( plus_plus_int @ A4 @ ( uminus_uminus_int @ B3 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_3949_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_minus_complex
    = ( ^ [A4: complex,B3: complex] : ( plus_plus_complex @ A4 @ ( uminus1482373934393186551omplex @ B3 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_3950_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_8373710615458151222nteger
    = ( ^ [A4: code_integer,B3: code_integer] : ( plus_p5714425477246183910nteger @ A4 @ ( uminus1351360451143612070nteger @ B3 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_3951_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_minus_rat
    = ( ^ [A4: rat,B3: rat] : ( plus_plus_rat @ A4 @ ( uminus_uminus_rat @ B3 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_3952_diff__conv__add__uminus,axiom,
    ( minus_minus_real
    = ( ^ [A4: real,B3: real] : ( plus_plus_real @ A4 @ ( uminus_uminus_real @ B3 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_3953_diff__conv__add__uminus,axiom,
    ( minus_minus_int
    = ( ^ [A4: int,B3: int] : ( plus_plus_int @ A4 @ ( uminus_uminus_int @ B3 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_3954_diff__conv__add__uminus,axiom,
    ( minus_minus_complex
    = ( ^ [A4: complex,B3: complex] : ( plus_plus_complex @ A4 @ ( uminus1482373934393186551omplex @ B3 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_3955_diff__conv__add__uminus,axiom,
    ( minus_8373710615458151222nteger
    = ( ^ [A4: code_integer,B3: code_integer] : ( plus_p5714425477246183910nteger @ A4 @ ( uminus1351360451143612070nteger @ B3 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_3956_diff__conv__add__uminus,axiom,
    ( minus_minus_rat
    = ( ^ [A4: rat,B3: rat] : ( plus_plus_rat @ A4 @ ( uminus_uminus_rat @ B3 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_3957_group__cancel_Osub2,axiom,
    ! [B4: real,K: real,B2: real,A: real] :
      ( ( B4
        = ( plus_plus_real @ K @ B2 ) )
     => ( ( minus_minus_real @ A @ B4 )
        = ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( minus_minus_real @ A @ B2 ) ) ) ) ).

% group_cancel.sub2
thf(fact_3958_group__cancel_Osub2,axiom,
    ! [B4: int,K: int,B2: int,A: int] :
      ( ( B4
        = ( plus_plus_int @ K @ B2 ) )
     => ( ( minus_minus_int @ A @ B4 )
        = ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( minus_minus_int @ A @ B2 ) ) ) ) ).

% group_cancel.sub2
thf(fact_3959_group__cancel_Osub2,axiom,
    ! [B4: complex,K: complex,B2: complex,A: complex] :
      ( ( B4
        = ( plus_plus_complex @ K @ B2 ) )
     => ( ( minus_minus_complex @ A @ B4 )
        = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ K ) @ ( minus_minus_complex @ A @ B2 ) ) ) ) ).

% group_cancel.sub2
thf(fact_3960_group__cancel_Osub2,axiom,
    ! [B4: code_integer,K: code_integer,B2: code_integer,A: code_integer] :
      ( ( B4
        = ( plus_p5714425477246183910nteger @ K @ B2 ) )
     => ( ( minus_8373710615458151222nteger @ A @ B4 )
        = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ K ) @ ( minus_8373710615458151222nteger @ A @ B2 ) ) ) ) ).

% group_cancel.sub2
thf(fact_3961_group__cancel_Osub2,axiom,
    ! [B4: rat,K: rat,B2: rat,A: rat] :
      ( ( B4
        = ( plus_plus_rat @ K @ B2 ) )
     => ( ( minus_minus_rat @ A @ B4 )
        = ( plus_plus_rat @ ( uminus_uminus_rat @ K ) @ ( minus_minus_rat @ A @ B2 ) ) ) ) ).

% group_cancel.sub2
thf(fact_3962_dvd__div__neg,axiom,
    ! [B2: real,A: real] :
      ( ( dvd_dvd_real @ B2 @ A )
     => ( ( divide_divide_real @ A @ ( uminus_uminus_real @ B2 ) )
        = ( uminus_uminus_real @ ( divide_divide_real @ A @ B2 ) ) ) ) ).

% dvd_div_neg
thf(fact_3963_dvd__div__neg,axiom,
    ! [B2: int,A: int] :
      ( ( dvd_dvd_int @ B2 @ A )
     => ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B2 ) )
        = ( uminus_uminus_int @ ( divide_divide_int @ A @ B2 ) ) ) ) ).

% dvd_div_neg
thf(fact_3964_dvd__div__neg,axiom,
    ! [B2: complex,A: complex] :
      ( ( dvd_dvd_complex @ B2 @ A )
     => ( ( divide1717551699836669952omplex @ A @ ( uminus1482373934393186551omplex @ B2 ) )
        = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B2 ) ) ) ) ).

% dvd_div_neg
thf(fact_3965_dvd__div__neg,axiom,
    ! [B2: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ B2 @ A )
     => ( ( divide6298287555418463151nteger @ A @ ( uminus1351360451143612070nteger @ B2 ) )
        = ( uminus1351360451143612070nteger @ ( divide6298287555418463151nteger @ A @ B2 ) ) ) ) ).

% dvd_div_neg
thf(fact_3966_dvd__div__neg,axiom,
    ! [B2: rat,A: rat] :
      ( ( dvd_dvd_rat @ B2 @ A )
     => ( ( divide_divide_rat @ A @ ( uminus_uminus_rat @ B2 ) )
        = ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B2 ) ) ) ) ).

% dvd_div_neg
thf(fact_3967_dvd__neg__div,axiom,
    ! [B2: real,A: real] :
      ( ( dvd_dvd_real @ B2 @ A )
     => ( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ B2 )
        = ( uminus_uminus_real @ ( divide_divide_real @ A @ B2 ) ) ) ) ).

% dvd_neg_div
thf(fact_3968_dvd__neg__div,axiom,
    ! [B2: int,A: int] :
      ( ( dvd_dvd_int @ B2 @ A )
     => ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B2 )
        = ( uminus_uminus_int @ ( divide_divide_int @ A @ B2 ) ) ) ) ).

% dvd_neg_div
thf(fact_3969_dvd__neg__div,axiom,
    ! [B2: complex,A: complex] :
      ( ( dvd_dvd_complex @ B2 @ A )
     => ( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ B2 )
        = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B2 ) ) ) ) ).

% dvd_neg_div
thf(fact_3970_dvd__neg__div,axiom,
    ! [B2: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ B2 @ A )
     => ( ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ A ) @ B2 )
        = ( uminus1351360451143612070nteger @ ( divide6298287555418463151nteger @ A @ B2 ) ) ) ) ).

% dvd_neg_div
thf(fact_3971_dvd__neg__div,axiom,
    ! [B2: rat,A: rat] :
      ( ( dvd_dvd_rat @ B2 @ A )
     => ( ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ B2 )
        = ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B2 ) ) ) ) ).

% dvd_neg_div
thf(fact_3972_real__minus__mult__self__le,axiom,
    ! [U: real,X4: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( times_times_real @ U @ U ) ) @ ( times_times_real @ X4 @ X4 ) ) ).

% real_minus_mult_self_le
thf(fact_3973_pos__zmult__eq__1__iff__lemma,axiom,
    ! [M: int,N: int] :
      ( ( ( times_times_int @ M @ N )
        = one_one_int )
     => ( ( M = one_one_int )
        | ( M
          = ( uminus_uminus_int @ one_one_int ) ) ) ) ).

% pos_zmult_eq_1_iff_lemma
thf(fact_3974_zmult__eq__1__iff,axiom,
    ! [M: int,N: int] :
      ( ( ( times_times_int @ M @ N )
        = one_one_int )
      = ( ( ( M = one_one_int )
          & ( N = one_one_int ) )
        | ( ( M
            = ( uminus_uminus_int @ one_one_int ) )
          & ( N
            = ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).

% zmult_eq_1_iff
thf(fact_3975_minus__int__code_I2_J,axiom,
    ! [L2: int] :
      ( ( minus_minus_int @ zero_zero_int @ L2 )
      = ( uminus_uminus_int @ L2 ) ) ).

% minus_int_code(2)
thf(fact_3976_zmod__zminus2__not__zero,axiom,
    ! [K: int,L2: int] :
      ( ( ( modulo_modulo_int @ K @ ( uminus_uminus_int @ L2 ) )
       != zero_zero_int )
     => ( ( modulo_modulo_int @ K @ L2 )
       != zero_zero_int ) ) ).

% zmod_zminus2_not_zero
thf(fact_3977_zmod__zminus1__not__zero,axiom,
    ! [K: int,L2: int] :
      ( ( ( modulo_modulo_int @ ( uminus_uminus_int @ K ) @ L2 )
       != zero_zero_int )
     => ( ( modulo_modulo_int @ K @ L2 )
       != zero_zero_int ) ) ).

% zmod_zminus1_not_zero
thf(fact_3978_minus__real__def,axiom,
    ( minus_minus_real
    = ( ^ [X: real,Y: real] : ( plus_plus_real @ X @ ( uminus_uminus_real @ Y ) ) ) ) ).

% minus_real_def
thf(fact_3979_not__zero__le__neg__numeral,axiom,
    ! [N: num] :
      ~ ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).

% not_zero_le_neg_numeral
thf(fact_3980_not__zero__le__neg__numeral,axiom,
    ! [N: num] :
      ~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).

% not_zero_le_neg_numeral
thf(fact_3981_not__zero__le__neg__numeral,axiom,
    ! [N: num] :
      ~ ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).

% not_zero_le_neg_numeral
thf(fact_3982_not__zero__le__neg__numeral,axiom,
    ! [N: num] :
      ~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).

% not_zero_le_neg_numeral
thf(fact_3983_neg__numeral__le__zero,axiom,
    ! [N: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) @ zero_zero_real ) ).

% neg_numeral_le_zero
thf(fact_3984_neg__numeral__le__zero,axiom,
    ! [N: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) @ zero_z3403309356797280102nteger ) ).

% neg_numeral_le_zero
thf(fact_3985_neg__numeral__le__zero,axiom,
    ! [N: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) @ zero_zero_rat ) ).

% neg_numeral_le_zero
thf(fact_3986_neg__numeral__le__zero,axiom,
    ! [N: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ zero_zero_int ) ).

% neg_numeral_le_zero
thf(fact_3987_not__zero__less__neg__numeral,axiom,
    ! [N: num] :
      ~ ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).

% not_zero_less_neg_numeral
thf(fact_3988_not__zero__less__neg__numeral,axiom,
    ! [N: num] :
      ~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).

% not_zero_less_neg_numeral
thf(fact_3989_not__zero__less__neg__numeral,axiom,
    ! [N: num] :
      ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).

% not_zero_less_neg_numeral
thf(fact_3990_not__zero__less__neg__numeral,axiom,
    ! [N: num] :
      ~ ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).

% not_zero_less_neg_numeral
thf(fact_3991_neg__numeral__less__zero,axiom,
    ! [N: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) @ zero_zero_real ) ).

% neg_numeral_less_zero
thf(fact_3992_neg__numeral__less__zero,axiom,
    ! [N: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ zero_zero_int ) ).

% neg_numeral_less_zero
thf(fact_3993_neg__numeral__less__zero,axiom,
    ! [N: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) @ zero_z3403309356797280102nteger ) ).

% neg_numeral_less_zero
thf(fact_3994_neg__numeral__less__zero,axiom,
    ! [N: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) @ zero_zero_rat ) ).

% neg_numeral_less_zero
thf(fact_3995_le__minus__one__simps_I3_J,axiom,
    ~ ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ one_one_real ) ) ).

% le_minus_one_simps(3)
thf(fact_3996_le__minus__one__simps_I3_J,axiom,
    ~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% le_minus_one_simps(3)
thf(fact_3997_le__minus__one__simps_I3_J,axiom,
    ~ ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).

% le_minus_one_simps(3)
thf(fact_3998_le__minus__one__simps_I3_J,axiom,
    ~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).

% le_minus_one_simps(3)
thf(fact_3999_le__minus__one__simps_I1_J,axiom,
    ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ zero_zero_real ).

% le_minus_one_simps(1)
thf(fact_4000_le__minus__one__simps_I1_J,axiom,
    ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).

% le_minus_one_simps(1)
thf(fact_4001_le__minus__one__simps_I1_J,axiom,
    ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ zero_zero_rat ).

% le_minus_one_simps(1)
thf(fact_4002_le__minus__one__simps_I1_J,axiom,
    ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).

% le_minus_one_simps(1)
thf(fact_4003_less__minus__one__simps_I3_J,axiom,
    ~ ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ one_one_real ) ) ).

% less_minus_one_simps(3)
thf(fact_4004_less__minus__one__simps_I3_J,axiom,
    ~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).

% less_minus_one_simps(3)
thf(fact_4005_less__minus__one__simps_I3_J,axiom,
    ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% less_minus_one_simps(3)
thf(fact_4006_less__minus__one__simps_I3_J,axiom,
    ~ ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).

% less_minus_one_simps(3)
thf(fact_4007_less__minus__one__simps_I1_J,axiom,
    ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ zero_zero_real ).

% less_minus_one_simps(1)
thf(fact_4008_less__minus__one__simps_I1_J,axiom,
    ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).

% less_minus_one_simps(1)
thf(fact_4009_less__minus__one__simps_I1_J,axiom,
    ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).

% less_minus_one_simps(1)
thf(fact_4010_less__minus__one__simps_I1_J,axiom,
    ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ zero_zero_rat ).

% less_minus_one_simps(1)
thf(fact_4011_not__one__le__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_less_eq_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).

% not_one_le_neg_numeral
thf(fact_4012_not__one__le__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).

% not_one_le_neg_numeral
thf(fact_4013_not__one__le__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_less_eq_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).

% not_one_le_neg_numeral
thf(fact_4014_not__one__le__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).

% not_one_le_neg_numeral
thf(fact_4015_not__numeral__le__neg__one,axiom,
    ! [M: num] :
      ~ ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ one_one_real ) ) ).

% not_numeral_le_neg_one
thf(fact_4016_not__numeral__le__neg__one,axiom,
    ! [M: num] :
      ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% not_numeral_le_neg_one
thf(fact_4017_not__numeral__le__neg__one,axiom,
    ! [M: num] :
      ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).

% not_numeral_le_neg_one
thf(fact_4018_not__numeral__le__neg__one,axiom,
    ! [M: num] :
      ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) ) ).

% not_numeral_le_neg_one
thf(fact_4019_neg__numeral__le__neg__one,axiom,
    ! [M: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ one_one_real ) ) ).

% neg_numeral_le_neg_one
thf(fact_4020_neg__numeral__le__neg__one,axiom,
    ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% neg_numeral_le_neg_one
thf(fact_4021_neg__numeral__le__neg__one,axiom,
    ! [M: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).

% neg_numeral_le_neg_one
thf(fact_4022_neg__numeral__le__neg__one,axiom,
    ! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) ) ).

% neg_numeral_le_neg_one
thf(fact_4023_neg__one__le__numeral,axiom,
    ! [M: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( numeral_numeral_real @ M ) ) ).

% neg_one_le_numeral
thf(fact_4024_neg__one__le__numeral,axiom,
    ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ M ) ) ).

% neg_one_le_numeral
thf(fact_4025_neg__one__le__numeral,axiom,
    ! [M: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ M ) ) ).

% neg_one_le_numeral
thf(fact_4026_neg__one__le__numeral,axiom,
    ! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ M ) ) ).

% neg_one_le_numeral
thf(fact_4027_neg__numeral__le__one,axiom,
    ! [M: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ one_one_real ) ).

% neg_numeral_le_one
thf(fact_4028_neg__numeral__le__one,axiom,
    ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer ) ).

% neg_numeral_le_one
thf(fact_4029_neg__numeral__le__one,axiom,
    ! [M: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ one_one_rat ) ).

% neg_numeral_le_one
thf(fact_4030_neg__numeral__le__one,axiom,
    ! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) ).

% neg_numeral_le_one
thf(fact_4031_not__neg__one__less__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).

% not_neg_one_less_neg_numeral
thf(fact_4032_not__neg__one__less__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).

% not_neg_one_less_neg_numeral
thf(fact_4033_not__neg__one__less__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).

% not_neg_one_less_neg_numeral
thf(fact_4034_not__neg__one__less__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).

% not_neg_one_less_neg_numeral
thf(fact_4035_not__one__less__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_less_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).

% not_one_less_neg_numeral
thf(fact_4036_not__one__less__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).

% not_one_less_neg_numeral
thf(fact_4037_not__one__less__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).

% not_one_less_neg_numeral
thf(fact_4038_not__one__less__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_less_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).

% not_one_less_neg_numeral
thf(fact_4039_not__numeral__less__neg__one,axiom,
    ! [M: num] :
      ~ ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ one_one_real ) ) ).

% not_numeral_less_neg_one
thf(fact_4040_not__numeral__less__neg__one,axiom,
    ! [M: num] :
      ~ ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) ) ).

% not_numeral_less_neg_one
thf(fact_4041_not__numeral__less__neg__one,axiom,
    ! [M: num] :
      ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% not_numeral_less_neg_one
thf(fact_4042_not__numeral__less__neg__one,axiom,
    ! [M: num] :
      ~ ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).

% not_numeral_less_neg_one
thf(fact_4043_neg__one__less__numeral,axiom,
    ! [M: num] : ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( numeral_numeral_real @ M ) ) ).

% neg_one_less_numeral
thf(fact_4044_neg__one__less__numeral,axiom,
    ! [M: num] : ( ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ M ) ) ).

% neg_one_less_numeral
thf(fact_4045_neg__one__less__numeral,axiom,
    ! [M: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ M ) ) ).

% neg_one_less_numeral
thf(fact_4046_neg__one__less__numeral,axiom,
    ! [M: num] : ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ M ) ) ).

% neg_one_less_numeral
thf(fact_4047_neg__numeral__less__one,axiom,
    ! [M: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ one_one_real ) ).

% neg_numeral_less_one
thf(fact_4048_neg__numeral__less__one,axiom,
    ! [M: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) ).

% neg_numeral_less_one
thf(fact_4049_neg__numeral__less__one,axiom,
    ! [M: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer ) ).

% neg_numeral_less_one
thf(fact_4050_neg__numeral__less__one,axiom,
    ! [M: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ one_one_rat ) ).

% neg_numeral_less_one
thf(fact_4051_eq__minus__divide__eq,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( A
        = ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) )
      = ( ( ( C != zero_zero_real )
         => ( ( times_times_real @ A @ C )
            = ( uminus_uminus_real @ B2 ) ) )
        & ( ( C = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% eq_minus_divide_eq
thf(fact_4052_eq__minus__divide__eq,axiom,
    ! [A: complex,B2: complex,C: complex] :
      ( ( A
        = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ B2 @ C ) ) )
      = ( ( ( C != zero_zero_complex )
         => ( ( times_times_complex @ A @ C )
            = ( uminus1482373934393186551omplex @ B2 ) ) )
        & ( ( C = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% eq_minus_divide_eq
thf(fact_4053_eq__minus__divide__eq,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( A
        = ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C ) ) )
      = ( ( ( C != zero_zero_rat )
         => ( ( times_times_rat @ A @ C )
            = ( uminus_uminus_rat @ B2 ) ) )
        & ( ( C = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% eq_minus_divide_eq
thf(fact_4054_minus__divide__eq__eq,axiom,
    ! [B2: real,C: real,A: real] :
      ( ( ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) )
        = A )
      = ( ( ( C != zero_zero_real )
         => ( ( uminus_uminus_real @ B2 )
            = ( times_times_real @ A @ C ) ) )
        & ( ( C = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% minus_divide_eq_eq
thf(fact_4055_minus__divide__eq__eq,axiom,
    ! [B2: complex,C: complex,A: complex] :
      ( ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ B2 @ C ) )
        = A )
      = ( ( ( C != zero_zero_complex )
         => ( ( uminus1482373934393186551omplex @ B2 )
            = ( times_times_complex @ A @ C ) ) )
        & ( ( C = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% minus_divide_eq_eq
thf(fact_4056_minus__divide__eq__eq,axiom,
    ! [B2: rat,C: rat,A: rat] :
      ( ( ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C ) )
        = A )
      = ( ( ( C != zero_zero_rat )
         => ( ( uminus_uminus_rat @ B2 )
            = ( times_times_rat @ A @ C ) ) )
        & ( ( C = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% minus_divide_eq_eq
thf(fact_4057_nonzero__neg__divide__eq__eq,axiom,
    ! [B2: real,A: real,C: real] :
      ( ( B2 != zero_zero_real )
     => ( ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B2 ) )
          = C )
        = ( ( uminus_uminus_real @ A )
          = ( times_times_real @ C @ B2 ) ) ) ) ).

% nonzero_neg_divide_eq_eq
thf(fact_4058_nonzero__neg__divide__eq__eq,axiom,
    ! [B2: complex,A: complex,C: complex] :
      ( ( B2 != zero_zero_complex )
     => ( ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B2 ) )
          = C )
        = ( ( uminus1482373934393186551omplex @ A )
          = ( times_times_complex @ C @ B2 ) ) ) ) ).

% nonzero_neg_divide_eq_eq
thf(fact_4059_nonzero__neg__divide__eq__eq,axiom,
    ! [B2: rat,A: rat,C: rat] :
      ( ( B2 != zero_zero_rat )
     => ( ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B2 ) )
          = C )
        = ( ( uminus_uminus_rat @ A )
          = ( times_times_rat @ C @ B2 ) ) ) ) ).

% nonzero_neg_divide_eq_eq
thf(fact_4060_nonzero__neg__divide__eq__eq2,axiom,
    ! [B2: real,C: real,A: real] :
      ( ( B2 != zero_zero_real )
     => ( ( C
          = ( uminus_uminus_real @ ( divide_divide_real @ A @ B2 ) ) )
        = ( ( times_times_real @ C @ B2 )
          = ( uminus_uminus_real @ A ) ) ) ) ).

% nonzero_neg_divide_eq_eq2
thf(fact_4061_nonzero__neg__divide__eq__eq2,axiom,
    ! [B2: complex,C: complex,A: complex] :
      ( ( B2 != zero_zero_complex )
     => ( ( C
          = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B2 ) ) )
        = ( ( times_times_complex @ C @ B2 )
          = ( uminus1482373934393186551omplex @ A ) ) ) ) ).

% nonzero_neg_divide_eq_eq2
thf(fact_4062_nonzero__neg__divide__eq__eq2,axiom,
    ! [B2: rat,C: rat,A: rat] :
      ( ( B2 != zero_zero_rat )
     => ( ( C
          = ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B2 ) ) )
        = ( ( times_times_rat @ C @ B2 )
          = ( uminus_uminus_rat @ A ) ) ) ) ).

% nonzero_neg_divide_eq_eq2
thf(fact_4063_mult__1s__ring__1_I2_J,axiom,
    ! [B2: real] :
      ( ( times_times_real @ B2 @ ( uminus_uminus_real @ ( numeral_numeral_real @ one ) ) )
      = ( uminus_uminus_real @ B2 ) ) ).

% mult_1s_ring_1(2)
thf(fact_4064_mult__1s__ring__1_I2_J,axiom,
    ! [B2: int] :
      ( ( times_times_int @ B2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ one ) ) )
      = ( uminus_uminus_int @ B2 ) ) ).

% mult_1s_ring_1(2)
thf(fact_4065_mult__1s__ring__1_I2_J,axiom,
    ! [B2: complex] :
      ( ( times_times_complex @ B2 @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ one ) ) )
      = ( uminus1482373934393186551omplex @ B2 ) ) ).

% mult_1s_ring_1(2)
thf(fact_4066_mult__1s__ring__1_I2_J,axiom,
    ! [B2: code_integer] :
      ( ( times_3573771949741848930nteger @ B2 @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) ) )
      = ( uminus1351360451143612070nteger @ B2 ) ) ).

% mult_1s_ring_1(2)
thf(fact_4067_mult__1s__ring__1_I2_J,axiom,
    ! [B2: rat] :
      ( ( times_times_rat @ B2 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) ) )
      = ( uminus_uminus_rat @ B2 ) ) ).

% mult_1s_ring_1(2)
thf(fact_4068_mult__1s__ring__1_I1_J,axiom,
    ! [B2: real] :
      ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ one ) ) @ B2 )
      = ( uminus_uminus_real @ B2 ) ) ).

% mult_1s_ring_1(1)
thf(fact_4069_mult__1s__ring__1_I1_J,axiom,
    ! [B2: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ one ) ) @ B2 )
      = ( uminus_uminus_int @ B2 ) ) ).

% mult_1s_ring_1(1)
thf(fact_4070_mult__1s__ring__1_I1_J,axiom,
    ! [B2: complex] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ one ) ) @ B2 )
      = ( uminus1482373934393186551omplex @ B2 ) ) ).

% mult_1s_ring_1(1)
thf(fact_4071_mult__1s__ring__1_I1_J,axiom,
    ! [B2: code_integer] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) ) @ B2 )
      = ( uminus1351360451143612070nteger @ B2 ) ) ).

% mult_1s_ring_1(1)
thf(fact_4072_mult__1s__ring__1_I1_J,axiom,
    ! [B2: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) ) @ B2 )
      = ( uminus_uminus_rat @ B2 ) ) ).

% mult_1s_ring_1(1)
thf(fact_4073_divide__eq__minus__1__iff,axiom,
    ! [A: real,B2: real] :
      ( ( ( divide_divide_real @ A @ B2 )
        = ( uminus_uminus_real @ one_one_real ) )
      = ( ( B2 != zero_zero_real )
        & ( A
          = ( uminus_uminus_real @ B2 ) ) ) ) ).

% divide_eq_minus_1_iff
thf(fact_4074_divide__eq__minus__1__iff,axiom,
    ! [A: complex,B2: complex] :
      ( ( ( divide1717551699836669952omplex @ A @ B2 )
        = ( uminus1482373934393186551omplex @ one_one_complex ) )
      = ( ( B2 != zero_zero_complex )
        & ( A
          = ( uminus1482373934393186551omplex @ B2 ) ) ) ) ).

% divide_eq_minus_1_iff
thf(fact_4075_divide__eq__minus__1__iff,axiom,
    ! [A: rat,B2: rat] :
      ( ( ( divide_divide_rat @ A @ B2 )
        = ( uminus_uminus_rat @ one_one_rat ) )
      = ( ( B2 != zero_zero_rat )
        & ( A
          = ( uminus_uminus_rat @ B2 ) ) ) ) ).

% divide_eq_minus_1_iff
thf(fact_4076_uminus__numeral__One,axiom,
    ( ( uminus_uminus_real @ ( numeral_numeral_real @ one ) )
    = ( uminus_uminus_real @ one_one_real ) ) ).

% uminus_numeral_One
thf(fact_4077_uminus__numeral__One,axiom,
    ( ( uminus_uminus_int @ ( numeral_numeral_int @ one ) )
    = ( uminus_uminus_int @ one_one_int ) ) ).

% uminus_numeral_One
thf(fact_4078_uminus__numeral__One,axiom,
    ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ one ) )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% uminus_numeral_One
thf(fact_4079_uminus__numeral__One,axiom,
    ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) )
    = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% uminus_numeral_One
thf(fact_4080_uminus__numeral__One,axiom,
    ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) )
    = ( uminus_uminus_rat @ one_one_rat ) ) ).

% uminus_numeral_One
thf(fact_4081_power__minus,axiom,
    ! [A: real,N: nat] :
      ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
      = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( power_power_real @ A @ N ) ) ) ).

% power_minus
thf(fact_4082_power__minus,axiom,
    ! [A: int,N: nat] :
      ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
      = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( power_power_int @ A @ N ) ) ) ).

% power_minus
thf(fact_4083_power__minus,axiom,
    ! [A: complex,N: nat] :
      ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
      = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( power_power_complex @ A @ N ) ) ) ).

% power_minus
thf(fact_4084_power__minus,axiom,
    ! [A: code_integer,N: nat] :
      ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
      = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ ( power_8256067586552552935nteger @ A @ N ) ) ) ).

% power_minus
thf(fact_4085_power__minus,axiom,
    ! [A: rat,N: nat] :
      ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
      = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( power_power_rat @ A @ N ) ) ) ).

% power_minus
thf(fact_4086_power__minus__Bit0,axiom,
    ! [X4: real,K: num] :
      ( ( power_power_real @ ( uminus_uminus_real @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
      = ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).

% power_minus_Bit0
thf(fact_4087_power__minus__Bit0,axiom,
    ! [X4: int,K: num] :
      ( ( power_power_int @ ( uminus_uminus_int @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
      = ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).

% power_minus_Bit0
thf(fact_4088_power__minus__Bit0,axiom,
    ! [X4: complex,K: num] :
      ( ( power_power_complex @ ( uminus1482373934393186551omplex @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
      = ( power_power_complex @ X4 @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).

% power_minus_Bit0
thf(fact_4089_power__minus__Bit0,axiom,
    ! [X4: code_integer,K: num] :
      ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
      = ( power_8256067586552552935nteger @ X4 @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).

% power_minus_Bit0
thf(fact_4090_power__minus__Bit0,axiom,
    ! [X4: rat,K: num] :
      ( ( power_power_rat @ ( uminus_uminus_rat @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
      = ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).

% power_minus_Bit0
thf(fact_4091_real__add__less__0__iff,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_real @ ( plus_plus_real @ X4 @ Y3 ) @ zero_zero_real )
      = ( ord_less_real @ Y3 @ ( uminus_uminus_real @ X4 ) ) ) ).

% real_add_less_0_iff
thf(fact_4092_real__0__less__add__iff,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X4 @ Y3 ) )
      = ( ord_less_real @ ( uminus_uminus_real @ X4 ) @ Y3 ) ) ).

% real_0_less_add_iff
thf(fact_4093_real__0__le__add__iff,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ X4 @ Y3 ) )
      = ( ord_less_eq_real @ ( uminus_uminus_real @ X4 ) @ Y3 ) ) ).

% real_0_le_add_iff
thf(fact_4094_real__add__le__0__iff,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ X4 @ Y3 ) @ zero_zero_real )
      = ( ord_less_eq_real @ Y3 @ ( uminus_uminus_real @ X4 ) ) ) ).

% real_add_le_0_iff
thf(fact_4095_zmod__zminus2__eq__if,axiom,
    ! [A: int,B2: int] :
      ( ( ( ( modulo_modulo_int @ A @ B2 )
          = zero_zero_int )
       => ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ B2 ) )
          = zero_zero_int ) )
      & ( ( ( modulo_modulo_int @ A @ B2 )
         != zero_zero_int )
       => ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ B2 ) )
          = ( minus_minus_int @ ( modulo_modulo_int @ A @ B2 ) @ B2 ) ) ) ) ).

% zmod_zminus2_eq_if
thf(fact_4096_zmod__zminus1__eq__if,axiom,
    ! [A: int,B2: int] :
      ( ( ( ( modulo_modulo_int @ A @ B2 )
          = zero_zero_int )
       => ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B2 )
          = zero_zero_int ) )
      & ( ( ( modulo_modulo_int @ A @ B2 )
         != zero_zero_int )
       => ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B2 )
          = ( minus_minus_int @ B2 @ ( modulo_modulo_int @ A @ B2 ) ) ) ) ) ).

% zmod_zminus1_eq_if
thf(fact_4097_pos__minus__divide__less__eq,axiom,
    ! [C: real,B2: real,A: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) @ A )
        = ( ord_less_real @ ( uminus_uminus_real @ B2 ) @ ( times_times_real @ A @ C ) ) ) ) ).

% pos_minus_divide_less_eq
thf(fact_4098_pos__minus__divide__less__eq,axiom,
    ! [C: rat,B2: rat,A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C ) ) @ A )
        = ( ord_less_rat @ ( uminus_uminus_rat @ B2 ) @ ( times_times_rat @ A @ C ) ) ) ) ).

% pos_minus_divide_less_eq
thf(fact_4099_pos__less__minus__divide__eq,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) )
        = ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B2 ) ) ) ) ).

% pos_less_minus_divide_eq
thf(fact_4100_pos__less__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C ) ) )
        = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B2 ) ) ) ) ).

% pos_less_minus_divide_eq
thf(fact_4101_neg__minus__divide__less__eq,axiom,
    ! [C: real,B2: real,A: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) @ A )
        = ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B2 ) ) ) ) ).

% neg_minus_divide_less_eq
thf(fact_4102_neg__minus__divide__less__eq,axiom,
    ! [C: rat,B2: rat,A: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C ) ) @ A )
        = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B2 ) ) ) ) ).

% neg_minus_divide_less_eq
thf(fact_4103_neg__less__minus__divide__eq,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) )
        = ( ord_less_real @ ( uminus_uminus_real @ B2 ) @ ( times_times_real @ A @ C ) ) ) ) ).

% neg_less_minus_divide_eq
thf(fact_4104_neg__less__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B2: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C ) ) )
        = ( ord_less_rat @ ( uminus_uminus_rat @ B2 ) @ ( times_times_rat @ A @ C ) ) ) ) ).

% neg_less_minus_divide_eq
thf(fact_4105_minus__divide__less__eq,axiom,
    ! [B2: real,C: real,A: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) @ A )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ ( uminus_uminus_real @ B2 ) @ ( times_times_real @ A @ C ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B2 ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ zero_zero_real @ A ) ) ) ) ) ) ).

% minus_divide_less_eq
thf(fact_4106_minus__divide__less__eq,axiom,
    ! [B2: rat,C: rat,A: rat] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C ) ) @ A )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ ( uminus_uminus_rat @ B2 ) @ ( times_times_rat @ A @ C ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B2 ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).

% minus_divide_less_eq
thf(fact_4107_less__minus__divide__eq,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B2 ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ ( uminus_uminus_real @ B2 ) @ ( times_times_real @ A @ C ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ A @ zero_zero_real ) ) ) ) ) ) ).

% less_minus_divide_eq
thf(fact_4108_less__minus__divide__eq,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C ) ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B2 ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ ( uminus_uminus_rat @ B2 ) @ ( times_times_rat @ A @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).

% less_minus_divide_eq
thf(fact_4109_divide__eq__eq__numeral_I2_J,axiom,
    ! [B2: real,C: real,W: num] :
      ( ( ( divide_divide_real @ B2 @ C )
        = ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
      = ( ( ( C != zero_zero_real )
         => ( B2
            = ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
        & ( ( C = zero_zero_real )
         => ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
            = zero_zero_real ) ) ) ) ).

% divide_eq_eq_numeral(2)
thf(fact_4110_divide__eq__eq__numeral_I2_J,axiom,
    ! [B2: complex,C: complex,W: num] :
      ( ( ( divide1717551699836669952omplex @ B2 @ C )
        = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
      = ( ( ( C != zero_zero_complex )
         => ( B2
            = ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ C ) ) )
        & ( ( C = zero_zero_complex )
         => ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
            = zero_zero_complex ) ) ) ) ).

% divide_eq_eq_numeral(2)
thf(fact_4111_divide__eq__eq__numeral_I2_J,axiom,
    ! [B2: rat,C: rat,W: num] :
      ( ( ( divide_divide_rat @ B2 @ C )
        = ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
      = ( ( ( C != zero_zero_rat )
         => ( B2
            = ( 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)
thf(fact_4112_eq__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B2: real,C: real] :
      ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
        = ( divide_divide_real @ B2 @ C ) )
      = ( ( ( C != zero_zero_real )
         => ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C )
            = B2 ) )
        & ( ( C = zero_zero_real )
         => ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
            = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral(2)
thf(fact_4113_eq__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B2: complex,C: complex] :
      ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
        = ( divide1717551699836669952omplex @ B2 @ C ) )
      = ( ( ( C != zero_zero_complex )
         => ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ C )
            = B2 ) )
        & ( ( C = zero_zero_complex )
         => ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
            = zero_zero_complex ) ) ) ) ).

% eq_divide_eq_numeral(2)
thf(fact_4114_eq__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B2: rat,C: rat] :
      ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
        = ( divide_divide_rat @ B2 @ C ) )
      = ( ( ( C != zero_zero_rat )
         => ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C )
            = B2 ) )
        & ( ( C = zero_zero_rat )
         => ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
            = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral(2)
thf(fact_4115_add__divide__eq__if__simps_I3_J,axiom,
    ! [Z: real,A: real,B2: real] :
      ( ( ( Z = zero_zero_real )
       => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B2 )
          = B2 ) )
      & ( ( Z != zero_zero_real )
       => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B2 )
          = ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( times_times_real @ B2 @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(3)
thf(fact_4116_add__divide__eq__if__simps_I3_J,axiom,
    ! [Z: complex,A: complex,B2: complex] :
      ( ( ( Z = zero_zero_complex )
       => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z ) ) @ B2 )
          = B2 ) )
      & ( ( Z != zero_zero_complex )
       => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z ) ) @ B2 )
          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_complex @ B2 @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(3)
thf(fact_4117_add__divide__eq__if__simps_I3_J,axiom,
    ! [Z: rat,A: rat,B2: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B2 )
          = B2 ) )
      & ( ( Z != zero_zero_rat )
       => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B2 )
          = ( divide_divide_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_rat @ B2 @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(3)
thf(fact_4118_minus__divide__add__eq__iff,axiom,
    ! [Z: real,X4: real,Y3: real] :
      ( ( Z != zero_zero_real )
     => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X4 @ Z ) ) @ Y3 )
        = ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ X4 ) @ ( times_times_real @ Y3 @ Z ) ) @ Z ) ) ) ).

% minus_divide_add_eq_iff
thf(fact_4119_minus__divide__add__eq__iff,axiom,
    ! [Z: complex,X4: complex,Y3: complex] :
      ( ( Z != zero_zero_complex )
     => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ X4 @ Z ) ) @ Y3 )
        = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ X4 ) @ ( times_times_complex @ Y3 @ Z ) ) @ Z ) ) ) ).

% minus_divide_add_eq_iff
thf(fact_4120_minus__divide__add__eq__iff,axiom,
    ! [Z: rat,X4: rat,Y3: rat] :
      ( ( Z != zero_zero_rat )
     => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ X4 @ Z ) ) @ Y3 )
        = ( divide_divide_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ X4 ) @ ( times_times_rat @ Y3 @ Z ) ) @ Z ) ) ) ).

% minus_divide_add_eq_iff
thf(fact_4121_add__divide__eq__if__simps_I6_J,axiom,
    ! [Z: real,A: real,B2: real] :
      ( ( ( Z = zero_zero_real )
       => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B2 )
          = ( uminus_uminus_real @ B2 ) ) )
      & ( ( Z != zero_zero_real )
       => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B2 )
          = ( divide_divide_real @ ( minus_minus_real @ ( uminus_uminus_real @ A ) @ ( times_times_real @ B2 @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(6)
thf(fact_4122_add__divide__eq__if__simps_I6_J,axiom,
    ! [Z: complex,A: complex,B2: complex] :
      ( ( ( Z = zero_zero_complex )
       => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z ) ) @ B2 )
          = ( uminus1482373934393186551omplex @ B2 ) ) )
      & ( ( Z != zero_zero_complex )
       => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z ) ) @ B2 )
          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_complex @ B2 @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(6)
thf(fact_4123_add__divide__eq__if__simps_I6_J,axiom,
    ! [Z: rat,A: rat,B2: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B2 )
          = ( uminus_uminus_rat @ B2 ) ) )
      & ( ( Z != zero_zero_rat )
       => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B2 )
          = ( divide_divide_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_rat @ B2 @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(6)
thf(fact_4124_add__divide__eq__if__simps_I5_J,axiom,
    ! [Z: real,A: real,B2: real] :
      ( ( ( Z = zero_zero_real )
       => ( ( minus_minus_real @ ( divide_divide_real @ A @ Z ) @ B2 )
          = ( uminus_uminus_real @ B2 ) ) )
      & ( ( Z != zero_zero_real )
       => ( ( minus_minus_real @ ( divide_divide_real @ A @ Z ) @ B2 )
          = ( divide_divide_real @ ( minus_minus_real @ A @ ( times_times_real @ B2 @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(5)
thf(fact_4125_add__divide__eq__if__simps_I5_J,axiom,
    ! [Z: complex,A: complex,B2: complex] :
      ( ( ( Z = zero_zero_complex )
       => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ Z ) @ B2 )
          = ( uminus1482373934393186551omplex @ B2 ) ) )
      & ( ( Z != zero_zero_complex )
       => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ Z ) @ B2 )
          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ A @ ( times_times_complex @ B2 @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(5)
thf(fact_4126_add__divide__eq__if__simps_I5_J,axiom,
    ! [Z: rat,A: rat,B2: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( ( minus_minus_rat @ ( divide_divide_rat @ A @ Z ) @ B2 )
          = ( uminus_uminus_rat @ B2 ) ) )
      & ( ( Z != zero_zero_rat )
       => ( ( minus_minus_rat @ ( divide_divide_rat @ A @ Z ) @ B2 )
          = ( divide_divide_rat @ ( minus_minus_rat @ A @ ( times_times_rat @ B2 @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(5)
thf(fact_4127_minus__divide__diff__eq__iff,axiom,
    ! [Z: real,X4: real,Y3: real] :
      ( ( Z != zero_zero_real )
     => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X4 @ Z ) ) @ Y3 )
        = ( divide_divide_real @ ( minus_minus_real @ ( uminus_uminus_real @ X4 ) @ ( times_times_real @ Y3 @ Z ) ) @ Z ) ) ) ).

% minus_divide_diff_eq_iff
thf(fact_4128_minus__divide__diff__eq__iff,axiom,
    ! [Z: complex,X4: complex,Y3: complex] :
      ( ( Z != zero_zero_complex )
     => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ X4 @ Z ) ) @ Y3 )
        = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ X4 ) @ ( times_times_complex @ Y3 @ Z ) ) @ Z ) ) ) ).

% minus_divide_diff_eq_iff
thf(fact_4129_minus__divide__diff__eq__iff,axiom,
    ! [Z: rat,X4: rat,Y3: rat] :
      ( ( Z != zero_zero_rat )
     => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ X4 @ Z ) ) @ Y3 )
        = ( divide_divide_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ X4 ) @ ( times_times_rat @ Y3 @ Z ) ) @ Z ) ) ) ).

% minus_divide_diff_eq_iff
thf(fact_4130_even__minus,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( uminus_uminus_int @ A ) )
      = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ).

% even_minus
thf(fact_4131_even__minus,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( uminus1351360451143612070nteger @ A ) )
      = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ).

% even_minus
thf(fact_4132_power2__eq__iff,axiom,
    ! [X4: real,Y3: real] :
      ( ( ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( ( X4 = Y3 )
        | ( X4
          = ( uminus_uminus_real @ Y3 ) ) ) ) ).

% power2_eq_iff
thf(fact_4133_power2__eq__iff,axiom,
    ! [X4: int,Y3: int] :
      ( ( ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( ( X4 = Y3 )
        | ( X4
          = ( uminus_uminus_int @ Y3 ) ) ) ) ).

% power2_eq_iff
thf(fact_4134_power2__eq__iff,axiom,
    ! [X4: complex,Y3: complex] :
      ( ( ( power_power_complex @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_complex @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( ( X4 = Y3 )
        | ( X4
          = ( uminus1482373934393186551omplex @ Y3 ) ) ) ) ).

% power2_eq_iff
thf(fact_4135_power2__eq__iff,axiom,
    ! [X4: code_integer,Y3: code_integer] :
      ( ( ( power_8256067586552552935nteger @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_8256067586552552935nteger @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( ( X4 = Y3 )
        | ( X4
          = ( uminus1351360451143612070nteger @ Y3 ) ) ) ) ).

% power2_eq_iff
thf(fact_4136_power2__eq__iff,axiom,
    ! [X4: rat,Y3: rat] :
      ( ( ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( ( X4 = Y3 )
        | ( X4
          = ( uminus_uminus_rat @ Y3 ) ) ) ) ).

% power2_eq_iff
thf(fact_4137_verit__less__mono__div__int2,axiom,
    ! [A3: int,B4: int,N: int] :
      ( ( ord_less_eq_int @ A3 @ B4 )
     => ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ N ) )
       => ( ord_less_eq_int @ ( divide_divide_int @ B4 @ N ) @ ( divide_divide_int @ A3 @ N ) ) ) ) ).

% verit_less_mono_div_int2
thf(fact_4138_div__eq__minus1,axiom,
    ! [B2: int] :
      ( ( ord_less_int @ zero_zero_int @ B2 )
     => ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ B2 )
        = ( uminus_uminus_int @ one_one_int ) ) ) ).

% div_eq_minus1
thf(fact_4139_pos__minus__divide__le__eq,axiom,
    ! [C: real,B2: real,A: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) @ A )
        = ( ord_less_eq_real @ ( uminus_uminus_real @ B2 ) @ ( times_times_real @ A @ C ) ) ) ) ).

% pos_minus_divide_le_eq
thf(fact_4140_pos__minus__divide__le__eq,axiom,
    ! [C: rat,B2: rat,A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C ) ) @ A )
        = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B2 ) @ ( times_times_rat @ A @ C ) ) ) ) ).

% pos_minus_divide_le_eq
thf(fact_4141_pos__le__minus__divide__eq,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) )
        = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B2 ) ) ) ) ).

% pos_le_minus_divide_eq
thf(fact_4142_pos__le__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C ) ) )
        = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B2 ) ) ) ) ).

% pos_le_minus_divide_eq
thf(fact_4143_neg__minus__divide__le__eq,axiom,
    ! [C: real,B2: real,A: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) @ A )
        = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B2 ) ) ) ) ).

% neg_minus_divide_le_eq
thf(fact_4144_neg__minus__divide__le__eq,axiom,
    ! [C: rat,B2: rat,A: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C ) ) @ A )
        = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B2 ) ) ) ) ).

% neg_minus_divide_le_eq
thf(fact_4145_neg__le__minus__divide__eq,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) )
        = ( ord_less_eq_real @ ( uminus_uminus_real @ B2 ) @ ( times_times_real @ A @ C ) ) ) ) ).

% neg_le_minus_divide_eq
thf(fact_4146_neg__le__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B2: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C ) ) )
        = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B2 ) @ ( times_times_rat @ A @ C ) ) ) ) ).

% neg_le_minus_divide_eq
thf(fact_4147_minus__divide__le__eq,axiom,
    ! [B2: real,C: real,A: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) @ A )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ ( uminus_uminus_real @ B2 ) @ ( times_times_real @ A @ C ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B2 ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ) ) ).

% minus_divide_le_eq
thf(fact_4148_minus__divide__le__eq,axiom,
    ! [B2: rat,C: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C ) ) @ A )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B2 ) @ ( times_times_rat @ A @ C ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B2 ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).

% minus_divide_le_eq
thf(fact_4149_le__minus__divide__eq,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B2 ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ ( uminus_uminus_real @ B2 ) @ ( times_times_real @ A @ C ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ A @ zero_zero_real ) ) ) ) ) ) ).

% le_minus_divide_eq
thf(fact_4150_le__minus__divide__eq,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C ) ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B2 ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B2 ) @ ( times_times_rat @ A @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).

% le_minus_divide_eq
thf(fact_4151_less__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B2: real,C: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ ( divide_divide_real @ B2 @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B2 ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ B2 @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ zero_zero_real ) ) ) ) ) ) ).

% less_divide_eq_numeral(2)
thf(fact_4152_less__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B2: rat,C: rat] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ ( divide_divide_rat @ B2 @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B2 ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ B2 @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ zero_zero_rat ) ) ) ) ) ) ).

% less_divide_eq_numeral(2)
thf(fact_4153_divide__less__eq__numeral_I2_J,axiom,
    ! [B2: real,C: real,W: num] :
      ( ( ord_less_real @ ( divide_divide_real @ B2 @ C ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ B2 @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B2 ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ) ).

% divide_less_eq_numeral(2)
thf(fact_4154_divide__less__eq__numeral_I2_J,axiom,
    ! [B2: rat,C: rat,W: num] :
      ( ( ord_less_rat @ ( divide_divide_rat @ B2 @ C ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ B2 @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B2 ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ) ).

% divide_less_eq_numeral(2)
thf(fact_4155_power2__eq__1__iff,axiom,
    ! [A: real] :
      ( ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_real )
      = ( ( A = one_one_real )
        | ( A
          = ( uminus_uminus_real @ one_one_real ) ) ) ) ).

% power2_eq_1_iff
thf(fact_4156_power2__eq__1__iff,axiom,
    ! [A: int] :
      ( ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_int )
      = ( ( A = one_one_int )
        | ( A
          = ( uminus_uminus_int @ one_one_int ) ) ) ) ).

% power2_eq_1_iff
thf(fact_4157_power2__eq__1__iff,axiom,
    ! [A: complex] :
      ( ( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_complex )
      = ( ( A = one_one_complex )
        | ( A
          = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ) ).

% power2_eq_1_iff
thf(fact_4158_power2__eq__1__iff,axiom,
    ! [A: code_integer] :
      ( ( ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_Code_integer )
      = ( ( A = one_one_Code_integer )
        | ( A
          = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).

% power2_eq_1_iff
thf(fact_4159_power2__eq__1__iff,axiom,
    ! [A: rat] :
      ( ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_rat )
      = ( ( A = one_one_rat )
        | ( A
          = ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).

% power2_eq_1_iff
thf(fact_4160_uminus__power__if,axiom,
    ! [N: nat,A: real] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
          = ( power_power_real @ A @ N ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
          = ( uminus_uminus_real @ ( power_power_real @ A @ N ) ) ) ) ) ).

% uminus_power_if
thf(fact_4161_uminus__power__if,axiom,
    ! [N: nat,A: int] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
          = ( power_power_int @ A @ N ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
          = ( uminus_uminus_int @ ( power_power_int @ A @ N ) ) ) ) ) ).

% uminus_power_if
thf(fact_4162_uminus__power__if,axiom,
    ! [N: nat,A: complex] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
          = ( power_power_complex @ A @ N ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
          = ( uminus1482373934393186551omplex @ ( power_power_complex @ A @ N ) ) ) ) ) ).

% uminus_power_if
thf(fact_4163_uminus__power__if,axiom,
    ! [N: nat,A: code_integer] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
          = ( power_8256067586552552935nteger @ A @ N ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
          = ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ A @ N ) ) ) ) ) ).

% uminus_power_if
thf(fact_4164_uminus__power__if,axiom,
    ! [N: nat,A: rat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
          = ( power_power_rat @ A @ N ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
          = ( uminus_uminus_rat @ ( power_power_rat @ A @ N ) ) ) ) ) ).

% uminus_power_if
thf(fact_4165_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( plus_plus_nat @ N @ K ) )
        = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
thf(fact_4166_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( plus_plus_nat @ N @ K ) )
        = ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
thf(fact_4167_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( plus_plus_nat @ N @ K ) )
        = ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
thf(fact_4168_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( plus_plus_nat @ N @ K ) )
        = ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
thf(fact_4169_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( plus_plus_nat @ N @ K ) )
        = ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
thf(fact_4170_realpow__square__minus__le,axiom,
    ! [U: real,X4: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( power_power_real @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% realpow_square_minus_le
thf(fact_4171_minus__mod__int__eq,axiom,
    ! [L2: int,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ L2 )
     => ( ( modulo_modulo_int @ ( uminus_uminus_int @ K ) @ L2 )
        = ( minus_minus_int @ ( minus_minus_int @ L2 @ one_one_int ) @ ( modulo_modulo_int @ ( minus_minus_int @ K @ one_one_int ) @ L2 ) ) ) ) ).

% minus_mod_int_eq
thf(fact_4172_zmod__minus1,axiom,
    ! [B2: int] :
      ( ( ord_less_int @ zero_zero_int @ B2 )
     => ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ B2 )
        = ( minus_minus_int @ B2 @ one_one_int ) ) ) ).

% zmod_minus1
thf(fact_4173_zdiv__zminus1__eq__if,axiom,
    ! [B2: int,A: int] :
      ( ( B2 != zero_zero_int )
     => ( ( ( ( modulo_modulo_int @ A @ B2 )
            = zero_zero_int )
         => ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B2 )
            = ( uminus_uminus_int @ ( divide_divide_int @ A @ B2 ) ) ) )
        & ( ( ( modulo_modulo_int @ A @ B2 )
           != zero_zero_int )
         => ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B2 )
            = ( minus_minus_int @ ( uminus_uminus_int @ ( divide_divide_int @ A @ B2 ) ) @ one_one_int ) ) ) ) ) ).

% zdiv_zminus1_eq_if
thf(fact_4174_zdiv__zminus2__eq__if,axiom,
    ! [B2: int,A: int] :
      ( ( B2 != zero_zero_int )
     => ( ( ( ( modulo_modulo_int @ A @ B2 )
            = zero_zero_int )
         => ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B2 ) )
            = ( uminus_uminus_int @ ( divide_divide_int @ A @ B2 ) ) ) )
        & ( ( ( modulo_modulo_int @ A @ B2 )
           != zero_zero_int )
         => ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B2 ) )
            = ( minus_minus_int @ ( uminus_uminus_int @ ( divide_divide_int @ A @ B2 ) ) @ one_one_int ) ) ) ) ) ).

% zdiv_zminus2_eq_if
thf(fact_4175_le__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B2: real,C: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ ( divide_divide_real @ B2 @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B2 ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ B2 @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ zero_zero_real ) ) ) ) ) ) ).

% le_divide_eq_numeral(2)
thf(fact_4176_le__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B2: rat,C: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ ( divide_divide_rat @ B2 @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B2 ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ B2 @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ zero_zero_rat ) ) ) ) ) ) ).

% le_divide_eq_numeral(2)
thf(fact_4177_divide__le__eq__numeral_I2_J,axiom,
    ! [B2: real,C: real,W: num] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ B2 @ C ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ B2 @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B2 ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ) ).

% divide_le_eq_numeral(2)
thf(fact_4178_divide__le__eq__numeral_I2_J,axiom,
    ! [B2: rat,C: rat,W: num] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B2 @ C ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ B2 @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B2 ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ) ).

% divide_le_eq_numeral(2)
thf(fact_4179_square__le__1,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X4 )
     => ( ( ord_less_eq_real @ X4 @ one_one_real )
       => ( ord_less_eq_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ).

% square_le_1
thf(fact_4180_square__le__1,axiom,
    ! [X4: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ X4 )
     => ( ( ord_le3102999989581377725nteger @ X4 @ one_one_Code_integer )
       => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer ) ) ) ).

% square_le_1
thf(fact_4181_square__le__1,axiom,
    ! [X4: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ X4 )
     => ( ( ord_less_eq_rat @ X4 @ one_one_rat )
       => ( ord_less_eq_rat @ ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat ) ) ) ).

% square_le_1
thf(fact_4182_square__le__1,axiom,
    ! [X4: int] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ X4 )
     => ( ( ord_less_eq_int @ X4 @ one_one_int )
       => ( ord_less_eq_int @ ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int ) ) ) ).

% square_le_1
thf(fact_4183_minus__power__mult__self,axiom,
    ! [A: real,N: nat] :
      ( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ A ) @ N ) @ ( power_power_real @ ( uminus_uminus_real @ A ) @ N ) )
      = ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% minus_power_mult_self
thf(fact_4184_minus__power__mult__self,axiom,
    ! [A: int,N: nat] :
      ( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ A ) @ N ) @ ( power_power_int @ ( uminus_uminus_int @ A ) @ N ) )
      = ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% minus_power_mult_self
thf(fact_4185_minus__power__mult__self,axiom,
    ! [A: complex,N: nat] :
      ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N ) @ ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N ) )
      = ( power_power_complex @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% minus_power_mult_self
thf(fact_4186_minus__power__mult__self,axiom,
    ! [A: code_integer,N: nat] :
      ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N ) )
      = ( power_8256067586552552935nteger @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% minus_power_mult_self
thf(fact_4187_minus__power__mult__self,axiom,
    ! [A: rat,N: nat] :
      ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N ) @ ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N ) )
      = ( power_power_rat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% minus_power_mult_self
thf(fact_4188_minus__one__power__iff,axiom,
    ! [N: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N )
          = one_one_real ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N )
          = ( uminus_uminus_real @ one_one_real ) ) ) ) ).

% minus_one_power_iff
thf(fact_4189_minus__one__power__iff,axiom,
    ! [N: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
          = one_one_int ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
          = ( uminus_uminus_int @ one_one_int ) ) ) ) ).

% minus_one_power_iff
thf(fact_4190_minus__one__power__iff,axiom,
    ! [N: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N )
          = one_one_complex ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N )
          = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ) ).

% minus_one_power_iff
thf(fact_4191_minus__one__power__iff,axiom,
    ! [N: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N )
          = one_one_Code_integer ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N )
          = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).

% minus_one_power_iff
thf(fact_4192_minus__one__power__iff,axiom,
    ! [N: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N )
          = one_one_rat ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N )
          = ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).

% minus_one_power_iff
thf(fact_4193_minus__1__div__exp__eq__int,axiom,
    ! [N: nat] :
      ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
      = ( uminus_uminus_int @ one_one_int ) ) ).

% minus_1_div_exp_eq_int
thf(fact_4194_div__pos__neg__trivial,axiom,
    ! [K: int,L2: int] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ( ( ord_less_eq_int @ ( plus_plus_int @ K @ L2 ) @ zero_zero_int )
       => ( ( divide_divide_int @ K @ L2 )
          = ( uminus_uminus_int @ one_one_int ) ) ) ) ).

% div_pos_neg_trivial
thf(fact_4195_signed__take__bit__int__less__eq__self__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_eq_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ K )
      = ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ K ) ) ).

% signed_take_bit_int_less_eq_self_iff
thf(fact_4196_signed__take__bit__int__greater__eq__minus__exp,axiom,
    ! [N: nat,K: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ ( bit_ri631733984087533419it_int @ N @ K ) ) ).

% signed_take_bit_int_greater_eq_minus_exp
thf(fact_4197_signed__take__bit__int__greater__self__iff,axiom,
    ! [K: int,N: nat] :
      ( ( ord_less_int @ K @ ( bit_ri631733984087533419it_int @ N @ K ) )
      = ( ord_less_int @ K @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% signed_take_bit_int_greater_self_iff
thf(fact_4198_power__minus1__odd,axiom,
    ! [N: nat] :
      ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( uminus_uminus_real @ one_one_real ) ) ).

% power_minus1_odd
thf(fact_4199_power__minus1__odd,axiom,
    ! [N: nat] :
      ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( uminus_uminus_int @ one_one_int ) ) ).

% power_minus1_odd
thf(fact_4200_power__minus1__odd,axiom,
    ! [N: nat] :
      ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% power_minus1_odd
thf(fact_4201_power__minus1__odd,axiom,
    ! [N: nat] :
      ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% power_minus1_odd
thf(fact_4202_power__minus1__odd,axiom,
    ! [N: nat] :
      ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( uminus_uminus_rat @ one_one_rat ) ) ).

% power_minus1_odd
thf(fact_4203_int__bit__induct,axiom,
    ! [P: int > $o,K: int] :
      ( ( P @ zero_zero_int )
     => ( ( P @ ( uminus_uminus_int @ one_one_int ) )
       => ( ! [K2: int] :
              ( ( P @ K2 )
             => ( ( K2 != zero_zero_int )
               => ( P @ ( times_times_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) )
         => ( ! [K2: int] :
                ( ( P @ K2 )
               => ( ( K2
                   != ( uminus_uminus_int @ one_one_int ) )
                 => ( P @ ( plus_plus_int @ one_one_int @ ( times_times_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) )
           => ( P @ K ) ) ) ) ) ).

% int_bit_induct
thf(fact_4204_signed__take__bit__int__eq__self,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ K )
     => ( ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
       => ( ( bit_ri631733984087533419it_int @ N @ K )
          = K ) ) ) ).

% signed_take_bit_int_eq_self
thf(fact_4205_signed__take__bit__int__eq__self__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ( bit_ri631733984087533419it_int @ N @ K )
        = K )
      = ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ K )
        & ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% signed_take_bit_int_eq_self_iff
thf(fact_4206_signed__take__bit__int__greater__eq,axiom,
    ! [K: int,N: nat] :
      ( ( ord_less_int @ K @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N ) ) ) @ ( bit_ri631733984087533419it_int @ N @ K ) ) ) ).

% signed_take_bit_int_greater_eq
thf(fact_4207_mult__less__iff1,axiom,
    ! [Z: real,X4: real,Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ Z )
     => ( ( ord_less_real @ ( times_times_real @ X4 @ Z ) @ ( times_times_real @ Y3 @ Z ) )
        = ( ord_less_real @ X4 @ Y3 ) ) ) ).

% mult_less_iff1
thf(fact_4208_mult__less__iff1,axiom,
    ! [Z: rat,X4: rat,Y3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Z )
     => ( ( ord_less_rat @ ( times_times_rat @ X4 @ Z ) @ ( times_times_rat @ Y3 @ Z ) )
        = ( ord_less_rat @ X4 @ Y3 ) ) ) ).

% mult_less_iff1
thf(fact_4209_mult__less__iff1,axiom,
    ! [Z: int,X4: int,Y3: int] :
      ( ( ord_less_int @ zero_zero_int @ Z )
     => ( ( ord_less_int @ ( times_times_int @ X4 @ Z ) @ ( times_times_int @ Y3 @ Z ) )
        = ( ord_less_int @ X4 @ Y3 ) ) ) ).

% mult_less_iff1
thf(fact_4210_compl__le__compl__iff,axiom,
    ! [X4: set_nat,Y3: set_nat] :
      ( ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ X4 ) @ ( uminus5710092332889474511et_nat @ Y3 ) )
      = ( ord_less_eq_set_nat @ Y3 @ X4 ) ) ).

% compl_le_compl_iff
thf(fact_4211_signed__take__bit__Suc__minus__bit1,axiom,
    ! [N: nat,K: num] :
      ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% signed_take_bit_Suc_minus_bit1
thf(fact_4212_concat__bit__Suc,axiom,
    ! [N: nat,K: int,L2: int] :
      ( ( bit_concat_bit @ ( suc @ N ) @ K @ L2 )
      = ( plus_plus_int @ ( modulo_modulo_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_concat_bit @ N @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ L2 ) ) ) ) ).

% concat_bit_Suc
thf(fact_4213_flip__bit__0,axiom,
    ! [A: code_integer] :
      ( ( bit_se1345352211410354436nteger @ zero_zero_nat @ A )
      = ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ).

% flip_bit_0
thf(fact_4214_flip__bit__0,axiom,
    ! [A: int] :
      ( ( bit_se2159334234014336723it_int @ zero_zero_nat @ A )
      = ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ).

% flip_bit_0
thf(fact_4215_flip__bit__0,axiom,
    ! [A: nat] :
      ( ( bit_se2161824704523386999it_nat @ zero_zero_nat @ A )
      = ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% flip_bit_0
thf(fact_4216_set__decode__0,axiom,
    ! [X4: nat] :
      ( ( member_nat @ zero_zero_nat @ ( nat_set_decode @ X4 ) )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X4 ) ) ) ).

% set_decode_0
thf(fact_4217_semiring__norm_I90_J,axiom,
    ! [M: num,N: num] :
      ( ( ( bit1 @ M )
        = ( bit1 @ N ) )
      = ( M = N ) ) ).

% semiring_norm(90)
thf(fact_4218_verit__eq__simplify_I9_J,axiom,
    ! [X32: num,Y32: num] :
      ( ( ( bit1 @ X32 )
        = ( bit1 @ Y32 ) )
      = ( X32 = Y32 ) ) ).

% verit_eq_simplify(9)
thf(fact_4219_semiring__norm_I88_J,axiom,
    ! [M: num,N: num] :
      ( ( bit0 @ M )
     != ( bit1 @ N ) ) ).

% semiring_norm(88)
thf(fact_4220_semiring__norm_I89_J,axiom,
    ! [M: num,N: num] :
      ( ( bit1 @ M )
     != ( bit0 @ N ) ) ).

% semiring_norm(89)
thf(fact_4221_semiring__norm_I84_J,axiom,
    ! [N: num] :
      ( one
     != ( bit1 @ N ) ) ).

% semiring_norm(84)
thf(fact_4222_semiring__norm_I86_J,axiom,
    ! [M: num] :
      ( ( bit1 @ M )
     != one ) ).

% semiring_norm(86)
thf(fact_4223_of__bool__less__eq__iff,axiom,
    ! [P: $o,Q: $o] :
      ( ( ord_less_eq_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ ( zero_n2052037380579107095ol_rat @ Q ) )
      = ( P
       => Q ) ) ).

% of_bool_less_eq_iff
thf(fact_4224_of__bool__less__eq__iff,axiom,
    ! [P: $o,Q: $o] :
      ( ( ord_less_eq_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q ) )
      = ( P
       => Q ) ) ).

% of_bool_less_eq_iff
thf(fact_4225_of__bool__less__eq__iff,axiom,
    ! [P: $o,Q: $o] :
      ( ( ord_less_eq_int @ ( zero_n2684676970156552555ol_int @ P ) @ ( zero_n2684676970156552555ol_int @ Q ) )
      = ( P
       => Q ) ) ).

% of_bool_less_eq_iff
thf(fact_4226_of__bool__less__eq__iff,axiom,
    ! [P: $o,Q: $o] :
      ( ( ord_le3102999989581377725nteger @ ( zero_n356916108424825756nteger @ P ) @ ( zero_n356916108424825756nteger @ Q ) )
      = ( P
       => Q ) ) ).

% of_bool_less_eq_iff
thf(fact_4227_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n3304061248610475627l_real @ P )
        = zero_zero_real )
      = ~ P ) ).

% of_bool_eq_0_iff
thf(fact_4228_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n2052037380579107095ol_rat @ P )
        = zero_zero_rat )
      = ~ P ) ).

% of_bool_eq_0_iff
thf(fact_4229_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n2687167440665602831ol_nat @ P )
        = zero_zero_nat )
      = ~ P ) ).

% of_bool_eq_0_iff
thf(fact_4230_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n2684676970156552555ol_int @ P )
        = zero_zero_int )
      = ~ P ) ).

% of_bool_eq_0_iff
thf(fact_4231_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n356916108424825756nteger @ P )
        = zero_z3403309356797280102nteger )
      = ~ P ) ).

% of_bool_eq_0_iff
thf(fact_4232_of__bool__eq_I1_J,axiom,
    ( ( zero_n3304061248610475627l_real @ $false )
    = zero_zero_real ) ).

% of_bool_eq(1)
thf(fact_4233_of__bool__eq_I1_J,axiom,
    ( ( zero_n2052037380579107095ol_rat @ $false )
    = zero_zero_rat ) ).

% of_bool_eq(1)
thf(fact_4234_of__bool__eq_I1_J,axiom,
    ( ( zero_n2687167440665602831ol_nat @ $false )
    = zero_zero_nat ) ).

% of_bool_eq(1)
thf(fact_4235_of__bool__eq_I1_J,axiom,
    ( ( zero_n2684676970156552555ol_int @ $false )
    = zero_zero_int ) ).

% of_bool_eq(1)
thf(fact_4236_of__bool__eq_I1_J,axiom,
    ( ( zero_n356916108424825756nteger @ $false )
    = zero_z3403309356797280102nteger ) ).

% of_bool_eq(1)
thf(fact_4237_of__bool__less__iff,axiom,
    ! [P: $o,Q: $o] :
      ( ( ord_less_real @ ( zero_n3304061248610475627l_real @ P ) @ ( zero_n3304061248610475627l_real @ Q ) )
      = ( ~ P
        & Q ) ) ).

% of_bool_less_iff
thf(fact_4238_of__bool__less__iff,axiom,
    ! [P: $o,Q: $o] :
      ( ( ord_less_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ ( zero_n2052037380579107095ol_rat @ Q ) )
      = ( ~ P
        & Q ) ) ).

% of_bool_less_iff
thf(fact_4239_of__bool__less__iff,axiom,
    ! [P: $o,Q: $o] :
      ( ( ord_less_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q ) )
      = ( ~ P
        & Q ) ) ).

% of_bool_less_iff
thf(fact_4240_of__bool__less__iff,axiom,
    ! [P: $o,Q: $o] :
      ( ( ord_less_int @ ( zero_n2684676970156552555ol_int @ P ) @ ( zero_n2684676970156552555ol_int @ Q ) )
      = ( ~ P
        & Q ) ) ).

% of_bool_less_iff
thf(fact_4241_of__bool__less__iff,axiom,
    ! [P: $o,Q: $o] :
      ( ( ord_le6747313008572928689nteger @ ( zero_n356916108424825756nteger @ P ) @ ( zero_n356916108424825756nteger @ Q ) )
      = ( ~ P
        & Q ) ) ).

% of_bool_less_iff
thf(fact_4242_of__bool__eq_I2_J,axiom,
    ( ( zero_n1201886186963655149omplex @ $true )
    = one_one_complex ) ).

% of_bool_eq(2)
thf(fact_4243_of__bool__eq_I2_J,axiom,
    ( ( zero_n3304061248610475627l_real @ $true )
    = one_one_real ) ).

% of_bool_eq(2)
thf(fact_4244_of__bool__eq_I2_J,axiom,
    ( ( zero_n2052037380579107095ol_rat @ $true )
    = one_one_rat ) ).

% of_bool_eq(2)
thf(fact_4245_of__bool__eq_I2_J,axiom,
    ( ( zero_n2687167440665602831ol_nat @ $true )
    = one_one_nat ) ).

% of_bool_eq(2)
thf(fact_4246_of__bool__eq_I2_J,axiom,
    ( ( zero_n2684676970156552555ol_int @ $true )
    = one_one_int ) ).

% of_bool_eq(2)
thf(fact_4247_of__bool__eq_I2_J,axiom,
    ( ( zero_n356916108424825756nteger @ $true )
    = one_one_Code_integer ) ).

% of_bool_eq(2)
thf(fact_4248_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n1201886186963655149omplex @ P )
        = one_one_complex )
      = P ) ).

% of_bool_eq_1_iff
thf(fact_4249_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n3304061248610475627l_real @ P )
        = one_one_real )
      = P ) ).

% of_bool_eq_1_iff
thf(fact_4250_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n2052037380579107095ol_rat @ P )
        = one_one_rat )
      = P ) ).

% of_bool_eq_1_iff
thf(fact_4251_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n2687167440665602831ol_nat @ P )
        = one_one_nat )
      = P ) ).

% of_bool_eq_1_iff
thf(fact_4252_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n2684676970156552555ol_int @ P )
        = one_one_int )
      = P ) ).

% of_bool_eq_1_iff
thf(fact_4253_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n356916108424825756nteger @ P )
        = one_one_Code_integer )
      = P ) ).

% of_bool_eq_1_iff
thf(fact_4254_concat__bit__0,axiom,
    ! [K: int,L2: int] :
      ( ( bit_concat_bit @ zero_zero_nat @ K @ L2 )
      = L2 ) ).

% concat_bit_0
thf(fact_4255_semiring__norm_I73_J,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
      = ( ord_less_eq_num @ M @ N ) ) ).

% semiring_norm(73)
thf(fact_4256_semiring__norm_I80_J,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
      = ( ord_less_num @ M @ N ) ) ).

% semiring_norm(80)
thf(fact_4257_zero__less__of__bool__iff,axiom,
    ! [P: $o] :
      ( ( ord_less_real @ zero_zero_real @ ( zero_n3304061248610475627l_real @ P ) )
      = P ) ).

% zero_less_of_bool_iff
thf(fact_4258_zero__less__of__bool__iff,axiom,
    ! [P: $o] :
      ( ( ord_less_rat @ zero_zero_rat @ ( zero_n2052037380579107095ol_rat @ P ) )
      = P ) ).

% zero_less_of_bool_iff
thf(fact_4259_zero__less__of__bool__iff,axiom,
    ! [P: $o] :
      ( ( ord_less_nat @ zero_zero_nat @ ( zero_n2687167440665602831ol_nat @ P ) )
      = P ) ).

% zero_less_of_bool_iff
thf(fact_4260_zero__less__of__bool__iff,axiom,
    ! [P: $o] :
      ( ( ord_less_int @ zero_zero_int @ ( zero_n2684676970156552555ol_int @ P ) )
      = P ) ).

% zero_less_of_bool_iff
thf(fact_4261_zero__less__of__bool__iff,axiom,
    ! [P: $o] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( zero_n356916108424825756nteger @ P ) )
      = P ) ).

% zero_less_of_bool_iff
thf(fact_4262_of__bool__less__one__iff,axiom,
    ! [P: $o] :
      ( ( ord_less_real @ ( zero_n3304061248610475627l_real @ P ) @ one_one_real )
      = ~ P ) ).

% of_bool_less_one_iff
thf(fact_4263_of__bool__less__one__iff,axiom,
    ! [P: $o] :
      ( ( ord_less_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ one_one_rat )
      = ~ P ) ).

% of_bool_less_one_iff
thf(fact_4264_of__bool__less__one__iff,axiom,
    ! [P: $o] :
      ( ( ord_less_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ one_one_nat )
      = ~ P ) ).

% of_bool_less_one_iff
thf(fact_4265_of__bool__less__one__iff,axiom,
    ! [P: $o] :
      ( ( ord_less_int @ ( zero_n2684676970156552555ol_int @ P ) @ one_one_int )
      = ~ P ) ).

% of_bool_less_one_iff
thf(fact_4266_of__bool__less__one__iff,axiom,
    ! [P: $o] :
      ( ( ord_le6747313008572928689nteger @ ( zero_n356916108424825756nteger @ P ) @ one_one_Code_integer )
      = ~ P ) ).

% of_bool_less_one_iff
thf(fact_4267_of__bool__not__iff,axiom,
    ! [P: $o] :
      ( ( zero_n1201886186963655149omplex @ ~ P )
      = ( minus_minus_complex @ one_one_complex @ ( zero_n1201886186963655149omplex @ P ) ) ) ).

% of_bool_not_iff
thf(fact_4268_of__bool__not__iff,axiom,
    ! [P: $o] :
      ( ( zero_n3304061248610475627l_real @ ~ P )
      = ( minus_minus_real @ one_one_real @ ( zero_n3304061248610475627l_real @ P ) ) ) ).

% of_bool_not_iff
thf(fact_4269_of__bool__not__iff,axiom,
    ! [P: $o] :
      ( ( zero_n2052037380579107095ol_rat @ ~ P )
      = ( minus_minus_rat @ one_one_rat @ ( zero_n2052037380579107095ol_rat @ P ) ) ) ).

% of_bool_not_iff
thf(fact_4270_of__bool__not__iff,axiom,
    ! [P: $o] :
      ( ( zero_n2684676970156552555ol_int @ ~ P )
      = ( minus_minus_int @ one_one_int @ ( zero_n2684676970156552555ol_int @ P ) ) ) ).

% of_bool_not_iff
thf(fact_4271_of__bool__not__iff,axiom,
    ! [P: $o] :
      ( ( zero_n356916108424825756nteger @ ~ P )
      = ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( zero_n356916108424825756nteger @ P ) ) ) ).

% of_bool_not_iff
thf(fact_4272_Suc__0__mod__eq,axiom,
    ! [N: nat] :
      ( ( modulo_modulo_nat @ ( suc @ zero_zero_nat ) @ N )
      = ( zero_n2687167440665602831ol_nat
        @ ( N
         != ( suc @ zero_zero_nat ) ) ) ) ).

% Suc_0_mod_eq
thf(fact_4273_semiring__norm_I9_J,axiom,
    ! [M: num,N: num] :
      ( ( plus_plus_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
      = ( bit1 @ ( plus_plus_num @ M @ N ) ) ) ).

% semiring_norm(9)
thf(fact_4274_semiring__norm_I7_J,axiom,
    ! [M: num,N: num] :
      ( ( plus_plus_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
      = ( bit1 @ ( plus_plus_num @ M @ N ) ) ) ).

% semiring_norm(7)
thf(fact_4275_semiring__norm_I15_J,axiom,
    ! [M: num,N: num] :
      ( ( times_times_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
      = ( bit0 @ ( times_times_num @ ( bit1 @ M ) @ N ) ) ) ).

% semiring_norm(15)
thf(fact_4276_semiring__norm_I14_J,axiom,
    ! [M: num,N: num] :
      ( ( times_times_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
      = ( bit0 @ ( times_times_num @ M @ ( bit1 @ N ) ) ) ) ).

% semiring_norm(14)
thf(fact_4277_semiring__norm_I72_J,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
      = ( ord_less_eq_num @ M @ N ) ) ).

% semiring_norm(72)
thf(fact_4278_semiring__norm_I81_J,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
      = ( ord_less_num @ M @ N ) ) ).

% semiring_norm(81)
thf(fact_4279_semiring__norm_I70_J,axiom,
    ! [M: num] :
      ~ ( ord_less_eq_num @ ( bit1 @ M ) @ one ) ).

% semiring_norm(70)
thf(fact_4280_semiring__norm_I77_J,axiom,
    ! [N: num] : ( ord_less_num @ one @ ( bit1 @ N ) ) ).

% semiring_norm(77)
thf(fact_4281_concat__bit__nonnegative__iff,axiom,
    ! [N: nat,K: int,L2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_concat_bit @ N @ K @ L2 ) )
      = ( ord_less_eq_int @ zero_zero_int @ L2 ) ) ).

% concat_bit_nonnegative_iff
thf(fact_4282_concat__bit__negative__iff,axiom,
    ! [N: nat,K: int,L2: int] :
      ( ( ord_less_int @ ( bit_concat_bit @ N @ K @ L2 ) @ zero_zero_int )
      = ( ord_less_int @ L2 @ zero_zero_int ) ) ).

% concat_bit_negative_iff
thf(fact_4283_zdiv__numeral__Bit1,axiom,
    ! [V: num,W: num] :
      ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit1 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
      = ( divide_divide_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) ).

% zdiv_numeral_Bit1
thf(fact_4284_semiring__norm_I3_J,axiom,
    ! [N: num] :
      ( ( plus_plus_num @ one @ ( bit0 @ N ) )
      = ( bit1 @ N ) ) ).

% semiring_norm(3)
thf(fact_4285_semiring__norm_I4_J,axiom,
    ! [N: num] :
      ( ( plus_plus_num @ one @ ( bit1 @ N ) )
      = ( bit0 @ ( plus_plus_num @ N @ one ) ) ) ).

% semiring_norm(4)
thf(fact_4286_semiring__norm_I5_J,axiom,
    ! [M: num] :
      ( ( plus_plus_num @ ( bit0 @ M ) @ one )
      = ( bit1 @ M ) ) ).

% semiring_norm(5)
thf(fact_4287_semiring__norm_I8_J,axiom,
    ! [M: num] :
      ( ( plus_plus_num @ ( bit1 @ M ) @ one )
      = ( bit0 @ ( plus_plus_num @ M @ one ) ) ) ).

% semiring_norm(8)
thf(fact_4288_semiring__norm_I10_J,axiom,
    ! [M: num,N: num] :
      ( ( plus_plus_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
      = ( bit0 @ ( plus_plus_num @ ( plus_plus_num @ M @ N ) @ one ) ) ) ).

% semiring_norm(10)
thf(fact_4289_semiring__norm_I16_J,axiom,
    ! [M: num,N: num] :
      ( ( times_times_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
      = ( bit1 @ ( plus_plus_num @ ( plus_plus_num @ M @ N ) @ ( bit0 @ ( times_times_num @ M @ N ) ) ) ) ) ).

% semiring_norm(16)
thf(fact_4290_semiring__norm_I74_J,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
      = ( ord_less_num @ M @ N ) ) ).

% semiring_norm(74)
thf(fact_4291_semiring__norm_I79_J,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
      = ( ord_less_eq_num @ M @ N ) ) ).

% semiring_norm(79)
thf(fact_4292_odd__of__bool__self,axiom,
    ! [P2: $o] :
      ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( zero_n2687167440665602831ol_nat @ P2 ) ) )
      = P2 ) ).

% odd_of_bool_self
thf(fact_4293_odd__of__bool__self,axiom,
    ! [P2: $o] :
      ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( zero_n2684676970156552555ol_int @ P2 ) ) )
      = P2 ) ).

% odd_of_bool_self
thf(fact_4294_odd__of__bool__self,axiom,
    ! [P2: $o] :
      ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( zero_n356916108424825756nteger @ P2 ) ) )
      = P2 ) ).

% odd_of_bool_self
thf(fact_4295_of__bool__half__eq__0,axiom,
    ! [B2: $o] :
      ( ( divide_divide_nat @ ( zero_n2687167440665602831ol_nat @ B2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = zero_zero_nat ) ).

% of_bool_half_eq_0
thf(fact_4296_of__bool__half__eq__0,axiom,
    ! [B2: $o] :
      ( ( divide_divide_int @ ( zero_n2684676970156552555ol_int @ B2 ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
      = zero_zero_int ) ).

% of_bool_half_eq_0
thf(fact_4297_of__bool__half__eq__0,axiom,
    ! [B2: $o] :
      ( ( divide6298287555418463151nteger @ ( zero_n356916108424825756nteger @ B2 ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
      = zero_z3403309356797280102nteger ) ).

% of_bool_half_eq_0
thf(fact_4298_Suc__div__eq__add3__div__numeral,axiom,
    ! [M: nat,V: num] :
      ( ( divide_divide_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ ( numeral_numeral_nat @ V ) )
      = ( divide_divide_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ ( numeral_numeral_nat @ V ) ) ) ).

% Suc_div_eq_add3_div_numeral
thf(fact_4299_div__Suc__eq__div__add3,axiom,
    ! [M: nat,N: nat] :
      ( ( divide_divide_nat @ M @ ( suc @ ( suc @ ( suc @ N ) ) ) )
      = ( divide_divide_nat @ M @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N ) ) ) ).

% div_Suc_eq_div_add3
thf(fact_4300_mod__Suc__eq__mod__add3,axiom,
    ! [M: nat,N: nat] :
      ( ( modulo_modulo_nat @ M @ ( suc @ ( suc @ ( suc @ N ) ) ) )
      = ( modulo_modulo_nat @ M @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N ) ) ) ).

% mod_Suc_eq_mod_add3
thf(fact_4301_Suc__mod__eq__add3__mod__numeral,axiom,
    ! [M: nat,V: num] :
      ( ( modulo_modulo_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ ( numeral_numeral_nat @ V ) )
      = ( modulo_modulo_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ ( numeral_numeral_nat @ V ) ) ) ).

% Suc_mod_eq_add3_mod_numeral
thf(fact_4302_set__decode__Suc,axiom,
    ! [N: nat,X4: nat] :
      ( ( member_nat @ ( suc @ N ) @ ( nat_set_decode @ X4 ) )
      = ( member_nat @ N @ ( nat_set_decode @ ( divide_divide_nat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% set_decode_Suc
thf(fact_4303_bits__1__div__exp,axiom,
    ! [N: nat] :
      ( ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( zero_n2687167440665602831ol_nat @ ( N = zero_zero_nat ) ) ) ).

% bits_1_div_exp
thf(fact_4304_bits__1__div__exp,axiom,
    ! [N: nat] :
      ( ( divide_divide_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
      = ( zero_n2684676970156552555ol_int @ ( N = zero_zero_nat ) ) ) ).

% bits_1_div_exp
thf(fact_4305_bits__1__div__exp,axiom,
    ! [N: nat] :
      ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
      = ( zero_n356916108424825756nteger @ ( N = zero_zero_nat ) ) ) ).

% bits_1_div_exp
thf(fact_4306_one__div__2__pow__eq,axiom,
    ! [N: nat] :
      ( ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( zero_n2687167440665602831ol_nat @ ( N = zero_zero_nat ) ) ) ).

% one_div_2_pow_eq
thf(fact_4307_one__div__2__pow__eq,axiom,
    ! [N: nat] :
      ( ( divide_divide_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
      = ( zero_n2684676970156552555ol_int @ ( N = zero_zero_nat ) ) ) ).

% one_div_2_pow_eq
thf(fact_4308_one__div__2__pow__eq,axiom,
    ! [N: nat] :
      ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
      = ( zero_n356916108424825756nteger @ ( N = zero_zero_nat ) ) ) ).

% one_div_2_pow_eq
thf(fact_4309_zmod__numeral__Bit1,axiom,
    ! [V: num,W: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) @ one_one_int ) ) ).

% zmod_numeral_Bit1
thf(fact_4310_one__mod__2__pow__eq,axiom,
    ! [N: nat] :
      ( ( modulo_modulo_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% one_mod_2_pow_eq
thf(fact_4311_one__mod__2__pow__eq,axiom,
    ! [N: nat] :
      ( ( modulo_modulo_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
      = ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% one_mod_2_pow_eq
thf(fact_4312_one__mod__2__pow__eq,axiom,
    ! [N: nat] :
      ( ( modulo364778990260209775nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
      = ( zero_n356916108424825756nteger @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% one_mod_2_pow_eq
thf(fact_4313_signed__take__bit__Suc__bit1,axiom,
    ! [N: nat,K: num] :
      ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% signed_take_bit_Suc_bit1
thf(fact_4314_of__bool__eq__iff,axiom,
    ! [P2: $o,Q2: $o] :
      ( ( ( zero_n2687167440665602831ol_nat @ P2 )
        = ( zero_n2687167440665602831ol_nat @ Q2 ) )
      = ( P2 = Q2 ) ) ).

% of_bool_eq_iff
thf(fact_4315_of__bool__eq__iff,axiom,
    ! [P2: $o,Q2: $o] :
      ( ( ( zero_n2684676970156552555ol_int @ P2 )
        = ( zero_n2684676970156552555ol_int @ Q2 ) )
      = ( P2 = Q2 ) ) ).

% of_bool_eq_iff
thf(fact_4316_of__bool__eq__iff,axiom,
    ! [P2: $o,Q2: $o] :
      ( ( ( zero_n356916108424825756nteger @ P2 )
        = ( zero_n356916108424825756nteger @ Q2 ) )
      = ( P2 = Q2 ) ) ).

% of_bool_eq_iff
thf(fact_4317_of__bool__conj,axiom,
    ! [P: $o,Q: $o] :
      ( ( zero_n3304061248610475627l_real
        @ ( P
          & Q ) )
      = ( times_times_real @ ( zero_n3304061248610475627l_real @ P ) @ ( zero_n3304061248610475627l_real @ Q ) ) ) ).

% of_bool_conj
thf(fact_4318_of__bool__conj,axiom,
    ! [P: $o,Q: $o] :
      ( ( zero_n2052037380579107095ol_rat
        @ ( P
          & Q ) )
      = ( times_times_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ ( zero_n2052037380579107095ol_rat @ Q ) ) ) ).

% of_bool_conj
thf(fact_4319_of__bool__conj,axiom,
    ! [P: $o,Q: $o] :
      ( ( zero_n2687167440665602831ol_nat
        @ ( P
          & Q ) )
      = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q ) ) ) ).

% of_bool_conj
thf(fact_4320_of__bool__conj,axiom,
    ! [P: $o,Q: $o] :
      ( ( zero_n2684676970156552555ol_int
        @ ( P
          & Q ) )
      = ( times_times_int @ ( zero_n2684676970156552555ol_int @ P ) @ ( zero_n2684676970156552555ol_int @ Q ) ) ) ).

% of_bool_conj
thf(fact_4321_of__bool__conj,axiom,
    ! [P: $o,Q: $o] :
      ( ( zero_n356916108424825756nteger
        @ ( P
          & Q ) )
      = ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ P ) @ ( zero_n356916108424825756nteger @ Q ) ) ) ).

% of_bool_conj
thf(fact_4322_verit__eq__simplify_I14_J,axiom,
    ! [X2: num,X32: num] :
      ( ( bit0 @ X2 )
     != ( bit1 @ X32 ) ) ).

% verit_eq_simplify(14)
thf(fact_4323_verit__eq__simplify_I12_J,axiom,
    ! [X32: num] :
      ( one
     != ( bit1 @ X32 ) ) ).

% verit_eq_simplify(12)
thf(fact_4324_concat__bit__assoc,axiom,
    ! [N: nat,K: int,M: nat,L2: int,R2: int] :
      ( ( bit_concat_bit @ N @ K @ ( bit_concat_bit @ M @ L2 @ R2 ) )
      = ( bit_concat_bit @ ( plus_plus_nat @ M @ N ) @ ( bit_concat_bit @ N @ K @ L2 ) @ R2 ) ) ).

% concat_bit_assoc
thf(fact_4325_subset__decode__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_set_nat @ ( nat_set_decode @ M ) @ ( nat_set_decode @ N ) )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% subset_decode_imp_le
thf(fact_4326_zero__less__eq__of__bool,axiom,
    ! [P: $o] : ( ord_less_eq_real @ zero_zero_real @ ( zero_n3304061248610475627l_real @ P ) ) ).

% zero_less_eq_of_bool
thf(fact_4327_zero__less__eq__of__bool,axiom,
    ! [P: $o] : ( ord_less_eq_rat @ zero_zero_rat @ ( zero_n2052037380579107095ol_rat @ P ) ) ).

% zero_less_eq_of_bool
thf(fact_4328_zero__less__eq__of__bool,axiom,
    ! [P: $o] : ( ord_less_eq_nat @ zero_zero_nat @ ( zero_n2687167440665602831ol_nat @ P ) ) ).

% zero_less_eq_of_bool
thf(fact_4329_zero__less__eq__of__bool,axiom,
    ! [P: $o] : ( ord_less_eq_int @ zero_zero_int @ ( zero_n2684676970156552555ol_int @ P ) ) ).

% zero_less_eq_of_bool
thf(fact_4330_zero__less__eq__of__bool,axiom,
    ! [P: $o] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( zero_n356916108424825756nteger @ P ) ) ).

% zero_less_eq_of_bool
thf(fact_4331_of__bool__less__eq__one,axiom,
    ! [P: $o] : ( ord_less_eq_real @ ( zero_n3304061248610475627l_real @ P ) @ one_one_real ) ).

% of_bool_less_eq_one
thf(fact_4332_of__bool__less__eq__one,axiom,
    ! [P: $o] : ( ord_less_eq_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ one_one_rat ) ).

% of_bool_less_eq_one
thf(fact_4333_of__bool__less__eq__one,axiom,
    ! [P: $o] : ( ord_less_eq_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ one_one_nat ) ).

% of_bool_less_eq_one
thf(fact_4334_of__bool__less__eq__one,axiom,
    ! [P: $o] : ( ord_less_eq_int @ ( zero_n2684676970156552555ol_int @ P ) @ one_one_int ) ).

% of_bool_less_eq_one
thf(fact_4335_of__bool__less__eq__one,axiom,
    ! [P: $o] : ( ord_le3102999989581377725nteger @ ( zero_n356916108424825756nteger @ P ) @ one_one_Code_integer ) ).

% of_bool_less_eq_one
thf(fact_4336_split__of__bool__asm,axiom,
    ! [P: complex > $o,P2: $o] :
      ( ( P @ ( zero_n1201886186963655149omplex @ P2 ) )
      = ( ~ ( ( P2
              & ~ ( P @ one_one_complex ) )
            | ( ~ P2
              & ~ ( P @ zero_zero_complex ) ) ) ) ) ).

% split_of_bool_asm
thf(fact_4337_split__of__bool__asm,axiom,
    ! [P: real > $o,P2: $o] :
      ( ( P @ ( zero_n3304061248610475627l_real @ P2 ) )
      = ( ~ ( ( P2
              & ~ ( P @ one_one_real ) )
            | ( ~ P2
              & ~ ( P @ zero_zero_real ) ) ) ) ) ).

% split_of_bool_asm
thf(fact_4338_split__of__bool__asm,axiom,
    ! [P: rat > $o,P2: $o] :
      ( ( P @ ( zero_n2052037380579107095ol_rat @ P2 ) )
      = ( ~ ( ( P2
              & ~ ( P @ one_one_rat ) )
            | ( ~ P2
              & ~ ( P @ zero_zero_rat ) ) ) ) ) ).

% split_of_bool_asm
thf(fact_4339_split__of__bool__asm,axiom,
    ! [P: nat > $o,P2: $o] :
      ( ( P @ ( zero_n2687167440665602831ol_nat @ P2 ) )
      = ( ~ ( ( P2
              & ~ ( P @ one_one_nat ) )
            | ( ~ P2
              & ~ ( P @ zero_zero_nat ) ) ) ) ) ).

% split_of_bool_asm
thf(fact_4340_split__of__bool__asm,axiom,
    ! [P: int > $o,P2: $o] :
      ( ( P @ ( zero_n2684676970156552555ol_int @ P2 ) )
      = ( ~ ( ( P2
              & ~ ( P @ one_one_int ) )
            | ( ~ P2
              & ~ ( P @ zero_zero_int ) ) ) ) ) ).

% split_of_bool_asm
thf(fact_4341_split__of__bool__asm,axiom,
    ! [P: code_integer > $o,P2: $o] :
      ( ( P @ ( zero_n356916108424825756nteger @ P2 ) )
      = ( ~ ( ( P2
              & ~ ( P @ one_one_Code_integer ) )
            | ( ~ P2
              & ~ ( P @ zero_z3403309356797280102nteger ) ) ) ) ) ).

% split_of_bool_asm
thf(fact_4342_split__of__bool,axiom,
    ! [P: complex > $o,P2: $o] :
      ( ( P @ ( zero_n1201886186963655149omplex @ P2 ) )
      = ( ( P2
         => ( P @ one_one_complex ) )
        & ( ~ P2
         => ( P @ zero_zero_complex ) ) ) ) ).

% split_of_bool
thf(fact_4343_split__of__bool,axiom,
    ! [P: real > $o,P2: $o] :
      ( ( P @ ( zero_n3304061248610475627l_real @ P2 ) )
      = ( ( P2
         => ( P @ one_one_real ) )
        & ( ~ P2
         => ( P @ zero_zero_real ) ) ) ) ).

% split_of_bool
thf(fact_4344_split__of__bool,axiom,
    ! [P: rat > $o,P2: $o] :
      ( ( P @ ( zero_n2052037380579107095ol_rat @ P2 ) )
      = ( ( P2
         => ( P @ one_one_rat ) )
        & ( ~ P2
         => ( P @ zero_zero_rat ) ) ) ) ).

% split_of_bool
thf(fact_4345_split__of__bool,axiom,
    ! [P: nat > $o,P2: $o] :
      ( ( P @ ( zero_n2687167440665602831ol_nat @ P2 ) )
      = ( ( P2
         => ( P @ one_one_nat ) )
        & ( ~ P2
         => ( P @ zero_zero_nat ) ) ) ) ).

% split_of_bool
thf(fact_4346_split__of__bool,axiom,
    ! [P: int > $o,P2: $o] :
      ( ( P @ ( zero_n2684676970156552555ol_int @ P2 ) )
      = ( ( P2
         => ( P @ one_one_int ) )
        & ( ~ P2
         => ( P @ zero_zero_int ) ) ) ) ).

% split_of_bool
thf(fact_4347_split__of__bool,axiom,
    ! [P: code_integer > $o,P2: $o] :
      ( ( P @ ( zero_n356916108424825756nteger @ P2 ) )
      = ( ( P2
         => ( P @ one_one_Code_integer ) )
        & ( ~ P2
         => ( P @ zero_z3403309356797280102nteger ) ) ) ) ).

% split_of_bool
thf(fact_4348_of__bool__def,axiom,
    ( zero_n1201886186963655149omplex
    = ( ^ [P3: $o] : ( if_complex @ P3 @ one_one_complex @ zero_zero_complex ) ) ) ).

% of_bool_def
thf(fact_4349_of__bool__def,axiom,
    ( zero_n3304061248610475627l_real
    = ( ^ [P3: $o] : ( if_real @ P3 @ one_one_real @ zero_zero_real ) ) ) ).

% of_bool_def
thf(fact_4350_of__bool__def,axiom,
    ( zero_n2052037380579107095ol_rat
    = ( ^ [P3: $o] : ( if_rat @ P3 @ one_one_rat @ zero_zero_rat ) ) ) ).

% of_bool_def
thf(fact_4351_of__bool__def,axiom,
    ( zero_n2687167440665602831ol_nat
    = ( ^ [P3: $o] : ( if_nat @ P3 @ one_one_nat @ zero_zero_nat ) ) ) ).

% of_bool_def
thf(fact_4352_of__bool__def,axiom,
    ( zero_n2684676970156552555ol_int
    = ( ^ [P3: $o] : ( if_int @ P3 @ one_one_int @ zero_zero_int ) ) ) ).

% of_bool_def
thf(fact_4353_of__bool__def,axiom,
    ( zero_n356916108424825756nteger
    = ( ^ [P3: $o] : ( if_Code_integer @ P3 @ one_one_Code_integer @ zero_z3403309356797280102nteger ) ) ) ).

% of_bool_def
thf(fact_4354_num_Oexhaust,axiom,
    ! [Y3: num] :
      ( ( Y3 != one )
     => ( ! [X22: num] :
            ( Y3
           != ( bit0 @ X22 ) )
       => ~ ! [X33: num] :
              ( Y3
             != ( bit1 @ X33 ) ) ) ) ).

% num.exhaust
thf(fact_4355_numeral__Bit1,axiom,
    ! [N: num] :
      ( ( numera6690914467698888265omplex @ ( bit1 @ N ) )
      = ( plus_plus_complex @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ ( numera6690914467698888265omplex @ N ) ) @ one_one_complex ) ) ).

% numeral_Bit1
thf(fact_4356_numeral__Bit1,axiom,
    ! [N: num] :
      ( ( numeral_numeral_real @ ( bit1 @ N ) )
      = ( plus_plus_real @ ( plus_plus_real @ ( numeral_numeral_real @ N ) @ ( numeral_numeral_real @ N ) ) @ one_one_real ) ) ).

% numeral_Bit1
thf(fact_4357_numeral__Bit1,axiom,
    ! [N: num] :
      ( ( numeral_numeral_rat @ ( bit1 @ N ) )
      = ( plus_plus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ ( numeral_numeral_rat @ N ) ) @ one_one_rat ) ) ).

% numeral_Bit1
thf(fact_4358_numeral__Bit1,axiom,
    ! [N: num] :
      ( ( numeral_numeral_nat @ ( bit1 @ N ) )
      = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) @ one_one_nat ) ) ).

% numeral_Bit1
thf(fact_4359_numeral__Bit1,axiom,
    ! [N: num] :
      ( ( numeral_numeral_int @ ( bit1 @ N ) )
      = ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) @ one_one_int ) ) ).

% numeral_Bit1
thf(fact_4360_eval__nat__numeral_I3_J,axiom,
    ! [N: num] :
      ( ( numeral_numeral_nat @ ( bit1 @ N ) )
      = ( suc @ ( numeral_numeral_nat @ ( bit0 @ N ) ) ) ) ).

% eval_nat_numeral(3)
thf(fact_4361_cong__exp__iff__simps_I10_J,axiom,
    ! [M: num,Q2: num,N: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
     != ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(10)
thf(fact_4362_cong__exp__iff__simps_I10_J,axiom,
    ! [M: num,Q2: num,N: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
     != ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(10)
thf(fact_4363_cong__exp__iff__simps_I10_J,axiom,
    ! [M: num,Q2: num,N: num] :
      ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
     != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(10)
thf(fact_4364_cong__exp__iff__simps_I12_J,axiom,
    ! [M: num,Q2: num,N: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
     != ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(12)
thf(fact_4365_cong__exp__iff__simps_I12_J,axiom,
    ! [M: num,Q2: num,N: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
     != ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(12)
thf(fact_4366_cong__exp__iff__simps_I12_J,axiom,
    ! [M: num,Q2: num,N: num] :
      ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
     != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(12)
thf(fact_4367_cong__exp__iff__simps_I13_J,axiom,
    ! [M: num,Q2: num,N: num] :
      ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) )
      = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q2 ) )
        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(13)
thf(fact_4368_cong__exp__iff__simps_I13_J,axiom,
    ! [M: num,Q2: num,N: num] :
      ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
        = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) )
      = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q2 ) )
        = ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(13)
thf(fact_4369_cong__exp__iff__simps_I13_J,axiom,
    ! [M: num,Q2: num,N: num] :
      ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
        = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) )
      = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ Q2 ) )
        = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(13)
thf(fact_4370_power__minus__Bit1,axiom,
    ! [X4: real,K: num] :
      ( ( power_power_real @ ( uminus_uminus_real @ X4 ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
      = ( uminus_uminus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).

% power_minus_Bit1
thf(fact_4371_power__minus__Bit1,axiom,
    ! [X4: int,K: num] :
      ( ( power_power_int @ ( uminus_uminus_int @ X4 ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
      = ( uminus_uminus_int @ ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).

% power_minus_Bit1
thf(fact_4372_power__minus__Bit1,axiom,
    ! [X4: complex,K: num] :
      ( ( power_power_complex @ ( uminus1482373934393186551omplex @ X4 ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
      = ( uminus1482373934393186551omplex @ ( power_power_complex @ X4 @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).

% power_minus_Bit1
thf(fact_4373_power__minus__Bit1,axiom,
    ! [X4: code_integer,K: num] :
      ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ X4 ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
      = ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ X4 @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).

% power_minus_Bit1
thf(fact_4374_power__minus__Bit1,axiom,
    ! [X4: rat,K: num] :
      ( ( power_power_rat @ ( uminus_uminus_rat @ X4 ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
      = ( uminus_uminus_rat @ ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).

% power_minus_Bit1
thf(fact_4375_numeral__Bit1__div__2,axiom,
    ! [N: num] :
      ( ( divide_divide_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( numeral_numeral_nat @ N ) ) ).

% numeral_Bit1_div_2
thf(fact_4376_numeral__Bit1__div__2,axiom,
    ! [N: num] :
      ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
      = ( numeral_numeral_int @ N ) ) ).

% numeral_Bit1_div_2
thf(fact_4377_odd__numeral,axiom,
    ! [N: num] :
      ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) ) ).

% odd_numeral
thf(fact_4378_odd__numeral,axiom,
    ! [N: num] :
      ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bit1 @ N ) ) ) ).

% odd_numeral
thf(fact_4379_odd__numeral,axiom,
    ! [N: num] :
      ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) ).

% odd_numeral
thf(fact_4380_cong__exp__iff__simps_I3_J,axiom,
    ! [N: num,Q2: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
     != zero_zero_nat ) ).

% cong_exp_iff_simps(3)
thf(fact_4381_cong__exp__iff__simps_I3_J,axiom,
    ! [N: num,Q2: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
     != zero_zero_int ) ).

% cong_exp_iff_simps(3)
thf(fact_4382_cong__exp__iff__simps_I3_J,axiom,
    ! [N: num,Q2: num] :
      ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
     != zero_z3403309356797280102nteger ) ).

% cong_exp_iff_simps(3)
thf(fact_4383_power3__eq__cube,axiom,
    ! [A: complex] :
      ( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
      = ( times_times_complex @ ( times_times_complex @ A @ A ) @ A ) ) ).

% power3_eq_cube
thf(fact_4384_power3__eq__cube,axiom,
    ! [A: real] :
      ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
      = ( times_times_real @ ( times_times_real @ A @ A ) @ A ) ) ).

% power3_eq_cube
thf(fact_4385_power3__eq__cube,axiom,
    ! [A: rat] :
      ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
      = ( times_times_rat @ ( times_times_rat @ A @ A ) @ A ) ) ).

% power3_eq_cube
thf(fact_4386_power3__eq__cube,axiom,
    ! [A: nat] :
      ( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
      = ( times_times_nat @ ( times_times_nat @ A @ A ) @ A ) ) ).

% power3_eq_cube
thf(fact_4387_power3__eq__cube,axiom,
    ! [A: int] :
      ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
      = ( times_times_int @ ( times_times_int @ A @ A ) @ A ) ) ).

% power3_eq_cube
thf(fact_4388_numeral__3__eq__3,axiom,
    ( ( numeral_numeral_nat @ ( bit1 @ one ) )
    = ( suc @ ( suc @ ( suc @ zero_zero_nat ) ) ) ) ).

% numeral_3_eq_3
thf(fact_4389_Suc3__eq__add__3,axiom,
    ! [N: nat] :
      ( ( suc @ ( suc @ ( suc @ N ) ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N ) ) ).

% Suc3_eq_add_3
thf(fact_4390_of__bool__odd__eq__mod__2,axiom,
    ! [A: nat] :
      ( ( zero_n2687167440665602831ol_nat
        @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
      = ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% of_bool_odd_eq_mod_2
thf(fact_4391_of__bool__odd__eq__mod__2,axiom,
    ! [A: int] :
      ( ( zero_n2684676970156552555ol_int
        @ ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
      = ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% of_bool_odd_eq_mod_2
thf(fact_4392_of__bool__odd__eq__mod__2,axiom,
    ! [A: code_integer] :
      ( ( zero_n356916108424825756nteger
        @ ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) )
      = ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% of_bool_odd_eq_mod_2
thf(fact_4393_cong__exp__iff__simps_I7_J,axiom,
    ! [Q2: num,N: num] :
      ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) )
      = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q2 ) )
        = zero_zero_nat ) ) ).

% cong_exp_iff_simps(7)
thf(fact_4394_cong__exp__iff__simps_I7_J,axiom,
    ! [Q2: num,N: num] :
      ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
        = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) )
      = ( ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q2 ) )
        = zero_zero_int ) ) ).

% cong_exp_iff_simps(7)
thf(fact_4395_cong__exp__iff__simps_I7_J,axiom,
    ! [Q2: num,N: num] :
      ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
        = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) )
      = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q2 ) )
        = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(7)
thf(fact_4396_cong__exp__iff__simps_I11_J,axiom,
    ! [M: num,Q2: num] :
      ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) )
      = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q2 ) )
        = zero_zero_nat ) ) ).

% cong_exp_iff_simps(11)
thf(fact_4397_cong__exp__iff__simps_I11_J,axiom,
    ! [M: num,Q2: num] :
      ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
        = ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) )
      = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q2 ) )
        = zero_zero_int ) ) ).

% cong_exp_iff_simps(11)
thf(fact_4398_cong__exp__iff__simps_I11_J,axiom,
    ! [M: num,Q2: num] :
      ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
        = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) )
      = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ Q2 ) )
        = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(11)
thf(fact_4399_Suc__div__eq__add3__div,axiom,
    ! [M: nat,N: nat] :
      ( ( divide_divide_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ N )
      = ( divide_divide_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ N ) ) ).

% Suc_div_eq_add3_div
thf(fact_4400_Suc__mod__eq__add3__mod,axiom,
    ! [M: nat,N: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ N )
      = ( modulo_modulo_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ N ) ) ).

% Suc_mod_eq_add3_mod
thf(fact_4401_bits__induct,axiom,
    ! [P: nat > $o,A: nat] :
      ( ! [A5: nat] :
          ( ( ( divide_divide_nat @ A5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = A5 )
         => ( P @ A5 ) )
     => ( ! [A5: nat,B5: $o] :
            ( ( P @ A5 )
           => ( ( ( divide_divide_nat @ ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ B5 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A5 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
                = A5 )
             => ( P @ ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ B5 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A5 ) ) ) ) )
       => ( P @ A ) ) ) ).

% bits_induct
thf(fact_4402_bits__induct,axiom,
    ! [P: int > $o,A: int] :
      ( ! [A5: int] :
          ( ( ( divide_divide_int @ A5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
            = A5 )
         => ( P @ A5 ) )
     => ( ! [A5: int,B5: $o] :
            ( ( P @ A5 )
           => ( ( ( divide_divide_int @ ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ B5 ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A5 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
                = A5 )
             => ( P @ ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ B5 ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A5 ) ) ) ) )
       => ( P @ A ) ) ) ).

% bits_induct
thf(fact_4403_bits__induct,axiom,
    ! [P: code_integer > $o,A: code_integer] :
      ( ! [A5: code_integer] :
          ( ( ( divide6298287555418463151nteger @ A5 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
            = A5 )
         => ( P @ A5 ) )
     => ( ! [A5: code_integer,B5: $o] :
            ( ( P @ A5 )
           => ( ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ B5 ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A5 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
                = A5 )
             => ( P @ ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ B5 ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A5 ) ) ) ) )
       => ( P @ A ) ) ) ).

% bits_induct
thf(fact_4404_compl__le__swap2,axiom,
    ! [Y3: set_nat,X4: set_nat] :
      ( ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ Y3 ) @ X4 )
     => ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ X4 ) @ Y3 ) ) ).

% compl_le_swap2
thf(fact_4405_compl__le__swap1,axiom,
    ! [Y3: set_nat,X4: set_nat] :
      ( ( ord_less_eq_set_nat @ Y3 @ ( uminus5710092332889474511et_nat @ X4 ) )
     => ( ord_less_eq_set_nat @ X4 @ ( uminus5710092332889474511et_nat @ Y3 ) ) ) ).

% compl_le_swap1
thf(fact_4406_compl__mono,axiom,
    ! [X4: set_nat,Y3: set_nat] :
      ( ( ord_less_eq_set_nat @ X4 @ Y3 )
     => ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ Y3 ) @ ( uminus5710092332889474511et_nat @ X4 ) ) ) ).

% compl_mono
thf(fact_4407_exp__mod__exp,axiom,
    ! [M: nat,N: nat] :
      ( ( modulo_modulo_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ M @ N ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).

% exp_mod_exp
thf(fact_4408_exp__mod__exp,axiom,
    ! [M: nat,N: nat] :
      ( ( modulo_modulo_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
      = ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ M @ N ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) ) ).

% exp_mod_exp
thf(fact_4409_exp__mod__exp,axiom,
    ! [M: nat,N: nat] :
      ( ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
      = ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ ( ord_less_nat @ M @ N ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) ) ).

% exp_mod_exp
thf(fact_4410_exp__div__exp__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( times_times_nat
        @ ( zero_n2687167440665602831ol_nat
          @ ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
             != zero_zero_nat )
            & ( ord_less_eq_nat @ N @ M ) ) )
        @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) ) ) ) ).

% exp_div_exp_eq
thf(fact_4411_exp__div__exp__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( divide_divide_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
      = ( times_times_int
        @ ( zero_n2684676970156552555ol_int
          @ ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M )
             != zero_zero_int )
            & ( ord_less_eq_nat @ N @ M ) ) )
        @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) ) ) ) ).

% exp_div_exp_eq
thf(fact_4412_exp__div__exp__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
      = ( times_3573771949741848930nteger
        @ ( zero_n356916108424825756nteger
          @ ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M )
             != zero_z3403309356797280102nteger )
            & ( ord_less_eq_nat @ N @ M ) ) )
        @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) ) ) ) ).

% exp_div_exp_eq
thf(fact_4413_odd__mod__4__div__2,axiom,
    ! [N: nat] :
      ( ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
        = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
     => ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% odd_mod_4_div_2
thf(fact_4414_mod__exhaust__less__4,axiom,
    ! [M: nat] :
      ( ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
        = zero_zero_nat )
      | ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
        = one_one_nat )
      | ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
        = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      | ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
        = ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) ).

% mod_exhaust_less_4
thf(fact_4415_signed__take__bit__numeral__minus__bit1,axiom,
    ! [L2: num,K: num] :
      ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L2 ) @ ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% signed_take_bit_numeral_minus_bit1
thf(fact_4416_dbl__dec__simps_I4_J,axiom,
    ( ( neg_nu6075765906172075777c_real @ ( uminus_uminus_real @ one_one_real ) )
    = ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) ) ).

% dbl_dec_simps(4)
thf(fact_4417_dbl__dec__simps_I4_J,axiom,
    ( ( neg_nu3811975205180677377ec_int @ ( uminus_uminus_int @ one_one_int ) )
    = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ one ) ) ) ) ).

% dbl_dec_simps(4)
thf(fact_4418_dbl__dec__simps_I4_J,axiom,
    ( ( neg_nu6511756317524482435omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
    = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit1 @ one ) ) ) ) ).

% dbl_dec_simps(4)
thf(fact_4419_dbl__dec__simps_I4_J,axiom,
    ( ( neg_nu7757733837767384882nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit1 @ one ) ) ) ) ).

% dbl_dec_simps(4)
thf(fact_4420_dbl__dec__simps_I4_J,axiom,
    ( ( neg_nu3179335615603231917ec_rat @ ( uminus_uminus_rat @ one_one_rat ) )
    = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit1 @ one ) ) ) ) ).

% dbl_dec_simps(4)
thf(fact_4421_signed__take__bit__numeral__bit1,axiom,
    ! [L2: num,K: num] :
      ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L2 ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L2 ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% signed_take_bit_numeral_bit1
thf(fact_4422_dbl__inc__simps_I3_J,axiom,
    ( ( neg_nu8557863876264182079omplex @ one_one_complex )
    = ( numera6690914467698888265omplex @ ( bit1 @ one ) ) ) ).

% dbl_inc_simps(3)
thf(fact_4423_dbl__inc__simps_I3_J,axiom,
    ( ( neg_nu8295874005876285629c_real @ one_one_real )
    = ( numeral_numeral_real @ ( bit1 @ one ) ) ) ).

% dbl_inc_simps(3)
thf(fact_4424_dbl__inc__simps_I3_J,axiom,
    ( ( neg_nu5219082963157363817nc_rat @ one_one_rat )
    = ( numeral_numeral_rat @ ( bit1 @ one ) ) ) ).

% dbl_inc_simps(3)
thf(fact_4425_dbl__inc__simps_I3_J,axiom,
    ( ( neg_nu5851722552734809277nc_int @ one_one_int )
    = ( numeral_numeral_int @ ( bit1 @ one ) ) ) ).

% dbl_inc_simps(3)
thf(fact_4426_add__scale__eq__noteq,axiom,
    ! [R2: real,A: real,B2: real,C: real,D: real] :
      ( ( R2 != zero_zero_real )
     => ( ( ( A = B2 )
          & ( C != D ) )
       => ( ( plus_plus_real @ A @ ( times_times_real @ R2 @ C ) )
         != ( plus_plus_real @ B2 @ ( times_times_real @ R2 @ D ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_4427_add__scale__eq__noteq,axiom,
    ! [R2: rat,A: rat,B2: rat,C: rat,D: rat] :
      ( ( R2 != zero_zero_rat )
     => ( ( ( A = B2 )
          & ( C != D ) )
       => ( ( plus_plus_rat @ A @ ( times_times_rat @ R2 @ C ) )
         != ( plus_plus_rat @ B2 @ ( times_times_rat @ R2 @ D ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_4428_add__scale__eq__noteq,axiom,
    ! [R2: nat,A: nat,B2: nat,C: nat,D: nat] :
      ( ( R2 != zero_zero_nat )
     => ( ( ( A = B2 )
          & ( C != D ) )
       => ( ( plus_plus_nat @ A @ ( times_times_nat @ R2 @ C ) )
         != ( plus_plus_nat @ B2 @ ( times_times_nat @ R2 @ D ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_4429_add__scale__eq__noteq,axiom,
    ! [R2: int,A: int,B2: int,C: int,D: int] :
      ( ( R2 != zero_zero_int )
     => ( ( ( A = B2 )
          & ( C != D ) )
       => ( ( plus_plus_int @ A @ ( times_times_int @ R2 @ C ) )
         != ( plus_plus_int @ B2 @ ( times_times_int @ R2 @ D ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_4430_num_Osize__gen_I3_J,axiom,
    ! [X32: num] :
      ( ( size_num @ ( bit1 @ X32 ) )
      = ( plus_plus_nat @ ( size_num @ X32 ) @ ( suc @ zero_zero_nat ) ) ) ).

% num.size_gen(3)
thf(fact_4431_dbl__dec__simps_I3_J,axiom,
    ( ( neg_nu6511756317524482435omplex @ one_one_complex )
    = one_one_complex ) ).

% dbl_dec_simps(3)
thf(fact_4432_dbl__dec__simps_I3_J,axiom,
    ( ( neg_nu6075765906172075777c_real @ one_one_real )
    = one_one_real ) ).

% dbl_dec_simps(3)
thf(fact_4433_dbl__dec__simps_I3_J,axiom,
    ( ( neg_nu3179335615603231917ec_rat @ one_one_rat )
    = one_one_rat ) ).

% dbl_dec_simps(3)
thf(fact_4434_dbl__dec__simps_I3_J,axiom,
    ( ( neg_nu3811975205180677377ec_int @ one_one_int )
    = one_one_int ) ).

% dbl_dec_simps(3)
thf(fact_4435_pred__numeral__simps_I1_J,axiom,
    ( ( pred_numeral @ one )
    = zero_zero_nat ) ).

% pred_numeral_simps(1)
thf(fact_4436_eq__numeral__Suc,axiom,
    ! [K: num,N: nat] :
      ( ( ( numeral_numeral_nat @ K )
        = ( suc @ N ) )
      = ( ( pred_numeral @ K )
        = N ) ) ).

% eq_numeral_Suc
thf(fact_4437_Suc__eq__numeral,axiom,
    ! [N: nat,K: num] :
      ( ( ( suc @ N )
        = ( numeral_numeral_nat @ K ) )
      = ( N
        = ( pred_numeral @ K ) ) ) ).

% Suc_eq_numeral
thf(fact_4438_dbl__inc__simps_I2_J,axiom,
    ( ( neg_nu8557863876264182079omplex @ zero_zero_complex )
    = one_one_complex ) ).

% dbl_inc_simps(2)
thf(fact_4439_dbl__inc__simps_I2_J,axiom,
    ( ( neg_nu8295874005876285629c_real @ zero_zero_real )
    = one_one_real ) ).

% dbl_inc_simps(2)
thf(fact_4440_dbl__inc__simps_I2_J,axiom,
    ( ( neg_nu5219082963157363817nc_rat @ zero_zero_rat )
    = one_one_rat ) ).

% dbl_inc_simps(2)
thf(fact_4441_dbl__inc__simps_I2_J,axiom,
    ( ( neg_nu5851722552734809277nc_int @ zero_zero_int )
    = one_one_int ) ).

% dbl_inc_simps(2)
thf(fact_4442_dbl__inc__simps_I4_J,axiom,
    ( ( neg_nu8295874005876285629c_real @ ( uminus_uminus_real @ one_one_real ) )
    = ( uminus_uminus_real @ one_one_real ) ) ).

% dbl_inc_simps(4)
thf(fact_4443_dbl__inc__simps_I4_J,axiom,
    ( ( neg_nu5851722552734809277nc_int @ ( uminus_uminus_int @ one_one_int ) )
    = ( uminus_uminus_int @ one_one_int ) ) ).

% dbl_inc_simps(4)
thf(fact_4444_dbl__inc__simps_I4_J,axiom,
    ( ( neg_nu8557863876264182079omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% dbl_inc_simps(4)
thf(fact_4445_dbl__inc__simps_I4_J,axiom,
    ( ( neg_nu5831290666863070958nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% dbl_inc_simps(4)
thf(fact_4446_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)
thf(fact_4447_dbl__inc__simps_I5_J,axiom,
    ! [K: num] :
      ( ( neg_nu8557863876264182079omplex @ ( numera6690914467698888265omplex @ K ) )
      = ( numera6690914467698888265omplex @ ( bit1 @ K ) ) ) ).

% dbl_inc_simps(5)
thf(fact_4448_dbl__inc__simps_I5_J,axiom,
    ! [K: num] :
      ( ( neg_nu8295874005876285629c_real @ ( numeral_numeral_real @ K ) )
      = ( numeral_numeral_real @ ( bit1 @ K ) ) ) ).

% dbl_inc_simps(5)
thf(fact_4449_dbl__inc__simps_I5_J,axiom,
    ! [K: num] :
      ( ( neg_nu5219082963157363817nc_rat @ ( numeral_numeral_rat @ K ) )
      = ( numeral_numeral_rat @ ( bit1 @ K ) ) ) ).

% dbl_inc_simps(5)
thf(fact_4450_dbl__inc__simps_I5_J,axiom,
    ! [K: num] :
      ( ( neg_nu5851722552734809277nc_int @ ( numeral_numeral_int @ K ) )
      = ( numeral_numeral_int @ ( bit1 @ K ) ) ) ).

% dbl_inc_simps(5)
thf(fact_4451_less__numeral__Suc,axiom,
    ! [K: num,N: nat] :
      ( ( ord_less_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
      = ( ord_less_nat @ ( pred_numeral @ K ) @ N ) ) ).

% less_numeral_Suc
thf(fact_4452_less__Suc__numeral,axiom,
    ! [N: nat,K: num] :
      ( ( ord_less_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
      = ( ord_less_nat @ N @ ( pred_numeral @ K ) ) ) ).

% less_Suc_numeral
thf(fact_4453_pred__numeral__simps_I3_J,axiom,
    ! [K: num] :
      ( ( pred_numeral @ ( bit1 @ K ) )
      = ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ).

% pred_numeral_simps(3)
thf(fact_4454_le__numeral__Suc,axiom,
    ! [K: num,N: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
      = ( ord_less_eq_nat @ ( pred_numeral @ K ) @ N ) ) ).

% le_numeral_Suc
thf(fact_4455_le__Suc__numeral,axiom,
    ! [N: nat,K: num] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
      = ( ord_less_eq_nat @ N @ ( pred_numeral @ K ) ) ) ).

% le_Suc_numeral
thf(fact_4456_diff__Suc__numeral,axiom,
    ! [N: nat,K: num] :
      ( ( minus_minus_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
      = ( minus_minus_nat @ N @ ( pred_numeral @ K ) ) ) ).

% diff_Suc_numeral
thf(fact_4457_diff__numeral__Suc,axiom,
    ! [K: num,N: nat] :
      ( ( minus_minus_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
      = ( minus_minus_nat @ ( pred_numeral @ K ) @ N ) ) ).

% diff_numeral_Suc
thf(fact_4458_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu6075765906172075777c_real @ zero_zero_real )
    = ( uminus_uminus_real @ one_one_real ) ) ).

% dbl_dec_simps(2)
thf(fact_4459_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu3811975205180677377ec_int @ zero_zero_int )
    = ( uminus_uminus_int @ one_one_int ) ) ).

% dbl_dec_simps(2)
thf(fact_4460_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu6511756317524482435omplex @ zero_zero_complex )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% dbl_dec_simps(2)
thf(fact_4461_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu7757733837767384882nteger @ zero_z3403309356797280102nteger )
    = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% dbl_dec_simps(2)
thf(fact_4462_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu3179335615603231917ec_rat @ zero_zero_rat )
    = ( uminus_uminus_rat @ one_one_rat ) ) ).

% dbl_dec_simps(2)
thf(fact_4463_dbl__dec__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu6075765906172075777c_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ K ) ) )
      = ( uminus_uminus_real @ ( neg_nu8295874005876285629c_real @ ( numeral_numeral_real @ K ) ) ) ) ).

% dbl_dec_simps(1)
thf(fact_4464_dbl__dec__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu3811975205180677377ec_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
      = ( uminus_uminus_int @ ( neg_nu5851722552734809277nc_int @ ( numeral_numeral_int @ K ) ) ) ) ).

% dbl_dec_simps(1)
thf(fact_4465_dbl__dec__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu6511756317524482435omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) )
      = ( uminus1482373934393186551omplex @ ( neg_nu8557863876264182079omplex @ ( numera6690914467698888265omplex @ K ) ) ) ) ).

% dbl_dec_simps(1)
thf(fact_4466_dbl__dec__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu7757733837767384882nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
      = ( uminus1351360451143612070nteger @ ( neg_nu5831290666863070958nteger @ ( numera6620942414471956472nteger @ K ) ) ) ) ).

% dbl_dec_simps(1)
thf(fact_4467_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)
thf(fact_4468_dbl__inc__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu8295874005876285629c_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ K ) ) )
      = ( uminus_uminus_real @ ( neg_nu6075765906172075777c_real @ ( numeral_numeral_real @ K ) ) ) ) ).

% dbl_inc_simps(1)
thf(fact_4469_dbl__inc__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu5851722552734809277nc_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
      = ( uminus_uminus_int @ ( neg_nu3811975205180677377ec_int @ ( numeral_numeral_int @ K ) ) ) ) ).

% dbl_inc_simps(1)
thf(fact_4470_dbl__inc__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu8557863876264182079omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) )
      = ( uminus1482373934393186551omplex @ ( neg_nu6511756317524482435omplex @ ( numera6690914467698888265omplex @ K ) ) ) ) ).

% dbl_inc_simps(1)
thf(fact_4471_dbl__inc__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu5831290666863070958nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
      = ( uminus1351360451143612070nteger @ ( neg_nu7757733837767384882nteger @ ( numera6620942414471956472nteger @ K ) ) ) ) ).

% dbl_inc_simps(1)
thf(fact_4472_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)
thf(fact_4473_signed__take__bit__numeral__bit0,axiom,
    ! [L2: num,K: num] :
      ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L2 ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
      = ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L2 ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% signed_take_bit_numeral_bit0
thf(fact_4474_signed__take__bit__numeral__minus__bit0,axiom,
    ! [L2: num,K: num] :
      ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
      = ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% signed_take_bit_numeral_minus_bit0
thf(fact_4475_numeral__eq__Suc,axiom,
    ( numeral_numeral_nat
    = ( ^ [K3: num] : ( suc @ ( pred_numeral @ K3 ) ) ) ) ).

% numeral_eq_Suc
thf(fact_4476_pred__numeral__def,axiom,
    ( pred_numeral
    = ( ^ [K3: num] : ( minus_minus_nat @ ( numeral_numeral_nat @ K3 ) @ one_one_nat ) ) ) ).

% pred_numeral_def
thf(fact_4477_dbl__inc__def,axiom,
    ( neg_nu8557863876264182079omplex
    = ( ^ [X: complex] : ( plus_plus_complex @ ( plus_plus_complex @ X @ X ) @ one_one_complex ) ) ) ).

% dbl_inc_def
thf(fact_4478_dbl__inc__def,axiom,
    ( neg_nu8295874005876285629c_real
    = ( ^ [X: real] : ( plus_plus_real @ ( plus_plus_real @ X @ X ) @ one_one_real ) ) ) ).

% dbl_inc_def
thf(fact_4479_dbl__inc__def,axiom,
    ( neg_nu5219082963157363817nc_rat
    = ( ^ [X: rat] : ( plus_plus_rat @ ( plus_plus_rat @ X @ X ) @ one_one_rat ) ) ) ).

% dbl_inc_def
thf(fact_4480_dbl__inc__def,axiom,
    ( neg_nu5851722552734809277nc_int
    = ( ^ [X: int] : ( plus_plus_int @ ( plus_plus_int @ X @ X ) @ one_one_int ) ) ) ).

% dbl_inc_def
thf(fact_4481_num_Osize__gen_I1_J,axiom,
    ( ( size_num @ one )
    = zero_zero_nat ) ).

% num.size_gen(1)
thf(fact_4482_dbl__dec__def,axiom,
    ( neg_nu6511756317524482435omplex
    = ( ^ [X: complex] : ( minus_minus_complex @ ( plus_plus_complex @ X @ X ) @ one_one_complex ) ) ) ).

% dbl_dec_def
thf(fact_4483_dbl__dec__def,axiom,
    ( neg_nu6075765906172075777c_real
    = ( ^ [X: real] : ( minus_minus_real @ ( plus_plus_real @ X @ X ) @ one_one_real ) ) ) ).

% dbl_dec_def
thf(fact_4484_dbl__dec__def,axiom,
    ( neg_nu3179335615603231917ec_rat
    = ( ^ [X: rat] : ( minus_minus_rat @ ( plus_plus_rat @ X @ X ) @ one_one_rat ) ) ) ).

% dbl_dec_def
thf(fact_4485_dbl__dec__def,axiom,
    ( neg_nu3811975205180677377ec_int
    = ( ^ [X: int] : ( minus_minus_int @ ( plus_plus_int @ X @ X ) @ one_one_int ) ) ) ).

% dbl_dec_def
thf(fact_4486_add__0__iff,axiom,
    ! [B2: real,A: real] :
      ( ( B2
        = ( plus_plus_real @ B2 @ A ) )
      = ( A = zero_zero_real ) ) ).

% add_0_iff
thf(fact_4487_add__0__iff,axiom,
    ! [B2: rat,A: rat] :
      ( ( B2
        = ( plus_plus_rat @ B2 @ A ) )
      = ( A = zero_zero_rat ) ) ).

% add_0_iff
thf(fact_4488_add__0__iff,axiom,
    ! [B2: nat,A: nat] :
      ( ( B2
        = ( plus_plus_nat @ B2 @ A ) )
      = ( A = zero_zero_nat ) ) ).

% add_0_iff
thf(fact_4489_add__0__iff,axiom,
    ! [B2: int,A: int] :
      ( ( B2
        = ( plus_plus_int @ B2 @ A ) )
      = ( A = zero_zero_int ) ) ).

% add_0_iff
thf(fact_4490_crossproduct__noteq,axiom,
    ! [A: real,B2: real,C: real,D: real] :
      ( ( ( A != B2 )
        & ( C != D ) )
      = ( ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ D ) )
       != ( plus_plus_real @ ( times_times_real @ A @ D ) @ ( times_times_real @ B2 @ C ) ) ) ) ).

% crossproduct_noteq
thf(fact_4491_crossproduct__noteq,axiom,
    ! [A: rat,B2: rat,C: rat,D: rat] :
      ( ( ( A != B2 )
        & ( C != D ) )
      = ( ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ D ) )
       != ( plus_plus_rat @ ( times_times_rat @ A @ D ) @ ( times_times_rat @ B2 @ C ) ) ) ) ).

% crossproduct_noteq
thf(fact_4492_crossproduct__noteq,axiom,
    ! [A: nat,B2: nat,C: nat,D: nat] :
      ( ( ( A != B2 )
        & ( C != D ) )
      = ( ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ D ) )
       != ( plus_plus_nat @ ( times_times_nat @ A @ D ) @ ( times_times_nat @ B2 @ C ) ) ) ) ).

% crossproduct_noteq
thf(fact_4493_crossproduct__noteq,axiom,
    ! [A: int,B2: int,C: int,D: int] :
      ( ( ( A != B2 )
        & ( C != D ) )
      = ( ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ D ) )
       != ( plus_plus_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ B2 @ C ) ) ) ) ).

% crossproduct_noteq
thf(fact_4494_crossproduct__eq,axiom,
    ! [W: real,Y3: real,X4: real,Z: real] :
      ( ( ( plus_plus_real @ ( times_times_real @ W @ Y3 ) @ ( times_times_real @ X4 @ Z ) )
        = ( plus_plus_real @ ( times_times_real @ W @ Z ) @ ( times_times_real @ X4 @ Y3 ) ) )
      = ( ( W = X4 )
        | ( Y3 = Z ) ) ) ).

% crossproduct_eq
thf(fact_4495_crossproduct__eq,axiom,
    ! [W: rat,Y3: rat,X4: rat,Z: rat] :
      ( ( ( plus_plus_rat @ ( times_times_rat @ W @ Y3 ) @ ( times_times_rat @ X4 @ Z ) )
        = ( plus_plus_rat @ ( times_times_rat @ W @ Z ) @ ( times_times_rat @ X4 @ Y3 ) ) )
      = ( ( W = X4 )
        | ( Y3 = Z ) ) ) ).

% crossproduct_eq
thf(fact_4496_crossproduct__eq,axiom,
    ! [W: nat,Y3: nat,X4: nat,Z: nat] :
      ( ( ( plus_plus_nat @ ( times_times_nat @ W @ Y3 ) @ ( times_times_nat @ X4 @ Z ) )
        = ( plus_plus_nat @ ( times_times_nat @ W @ Z ) @ ( times_times_nat @ X4 @ Y3 ) ) )
      = ( ( W = X4 )
        | ( Y3 = Z ) ) ) ).

% crossproduct_eq
thf(fact_4497_crossproduct__eq,axiom,
    ! [W: int,Y3: int,X4: int,Z: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ W @ Y3 ) @ ( times_times_int @ X4 @ Z ) )
        = ( plus_plus_int @ ( times_times_int @ W @ Z ) @ ( times_times_int @ X4 @ Y3 ) ) )
      = ( ( W = X4 )
        | ( Y3 = Z ) ) ) ).

% crossproduct_eq
thf(fact_4498_eq__diff__eq_H,axiom,
    ! [X4: real,Y3: real,Z: real] :
      ( ( X4
        = ( minus_minus_real @ Y3 @ Z ) )
      = ( Y3
        = ( plus_plus_real @ X4 @ Z ) ) ) ).

% eq_diff_eq'
thf(fact_4499_num_Osize__gen_I2_J,axiom,
    ! [X2: num] :
      ( ( size_num @ ( bit0 @ X2 ) )
      = ( plus_plus_nat @ ( size_num @ X2 ) @ ( suc @ zero_zero_nat ) ) ) ).

% num.size_gen(2)
thf(fact_4500_mask__numeral,axiom,
    ! [N: num] :
      ( ( bit_se2002935070580805687sk_nat @ ( numeral_numeral_nat @ N ) )
      = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2002935070580805687sk_nat @ ( pred_numeral @ N ) ) ) ) ) ).

% mask_numeral
thf(fact_4501_mask__numeral,axiom,
    ! [N: num] :
      ( ( bit_se2000444600071755411sk_int @ ( numeral_numeral_nat @ N ) )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2000444600071755411sk_int @ ( pred_numeral @ N ) ) ) ) ) ).

% mask_numeral
thf(fact_4502_take__bit__rec,axiom,
    ( bit_se1745604003318907178nteger
    = ( ^ [N2: nat,A4: code_integer] : ( if_Code_integer @ ( N2 = zero_zero_nat ) @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( bit_se1745604003318907178nteger @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( divide6298287555418463151nteger @ A4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( modulo364778990260209775nteger @ A4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% take_bit_rec
thf(fact_4503_take__bit__rec,axiom,
    ( bit_se2923211474154528505it_int
    = ( ^ [N2: nat,A4: int] : ( if_int @ ( N2 = zero_zero_nat ) @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( divide_divide_int @ A4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ A4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).

% take_bit_rec
thf(fact_4504_take__bit__rec,axiom,
    ( bit_se2925701944663578781it_nat
    = ( ^ [N2: nat,A4: nat] : ( if_nat @ ( N2 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( divide_divide_nat @ A4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ A4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% take_bit_rec
thf(fact_4505_and__int__unfold,axiom,
    ( bit_se725231765392027082nd_int
    = ( ^ [K3: int,L: int] :
          ( if_int
          @ ( ( K3 = zero_zero_int )
            | ( L = zero_zero_int ) )
          @ zero_zero_int
          @ ( if_int
            @ ( K3
              = ( uminus_uminus_int @ one_one_int ) )
            @ L
            @ ( if_int
              @ ( L
                = ( uminus_uminus_int @ one_one_int ) )
              @ K3
              @ ( plus_plus_int @ ( times_times_int @ ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).

% and_int_unfold
thf(fact_4506_exp__lower__Taylor__quadratic,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ord_less_eq_real @ ( plus_plus_real @ ( plus_plus_real @ one_one_real @ X4 ) @ ( divide_divide_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( exp_real @ X4 ) ) ) ).

% exp_lower_Taylor_quadratic
thf(fact_4507_sqrt__sum__squares__half__less,axiom,
    ! [X4: real,U: real,Y3: real] :
      ( ( ord_less_real @ X4 @ ( divide_divide_real @ U @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_real @ Y3 @ ( divide_divide_real @ U @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( ord_less_eq_real @ zero_zero_real @ X4 )
         => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
           => ( ord_less_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ U ) ) ) ) ) ).

% sqrt_sum_squares_half_less
thf(fact_4508_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X4: num,N: nat] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X4 ) ) @ N ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N ) ) ) ).

% of_int_less_neg_numeral_power_cancel_iff
thf(fact_4509_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X4: num,N: nat] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N ) ) ) ).

% of_int_less_neg_numeral_power_cancel_iff
thf(fact_4510_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X4: num,N: nat] :
      ( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ A ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X4 ) ) @ N ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N ) ) ) ).

% of_int_less_neg_numeral_power_cancel_iff
thf(fact_4511_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X4: num,N: nat] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X4 ) ) @ N ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N ) ) ) ).

% of_int_less_neg_numeral_power_cancel_iff
thf(fact_4512_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X4: num,N: nat,A: int] :
      ( ( ord_less_real @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X4 ) ) @ N ) @ ( ring_1_of_int_real @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N ) @ A ) ) ).

% neg_numeral_power_less_of_int_cancel_iff
thf(fact_4513_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X4: num,N: nat,A: int] :
      ( ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N ) @ ( ring_1_of_int_int @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N ) @ A ) ) ).

% neg_numeral_power_less_of_int_cancel_iff
thf(fact_4514_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X4: num,N: nat,A: int] :
      ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X4 ) ) @ N ) @ ( ring_18347121197199848620nteger @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N ) @ A ) ) ).

% neg_numeral_power_less_of_int_cancel_iff
thf(fact_4515_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X4: num,N: nat,A: int] :
      ( ( ord_less_rat @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X4 ) ) @ N ) @ ( ring_1_of_int_rat @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N ) @ A ) ) ).

% neg_numeral_power_less_of_int_cancel_iff
thf(fact_4516_and_Oright__idem,axiom,
    ! [A: int,B2: int] :
      ( ( bit_se725231765392027082nd_int @ ( bit_se725231765392027082nd_int @ A @ B2 ) @ B2 )
      = ( bit_se725231765392027082nd_int @ A @ B2 ) ) ).

% and.right_idem
thf(fact_4517_and_Oright__idem,axiom,
    ! [A: nat,B2: nat] :
      ( ( bit_se727722235901077358nd_nat @ ( bit_se727722235901077358nd_nat @ A @ B2 ) @ B2 )
      = ( bit_se727722235901077358nd_nat @ A @ B2 ) ) ).

% and.right_idem
thf(fact_4518_and_Oleft__idem,axiom,
    ! [A: int,B2: int] :
      ( ( bit_se725231765392027082nd_int @ A @ ( bit_se725231765392027082nd_int @ A @ B2 ) )
      = ( bit_se725231765392027082nd_int @ A @ B2 ) ) ).

% and.left_idem
thf(fact_4519_and_Oleft__idem,axiom,
    ! [A: nat,B2: nat] :
      ( ( bit_se727722235901077358nd_nat @ A @ ( bit_se727722235901077358nd_nat @ A @ B2 ) )
      = ( bit_se727722235901077358nd_nat @ A @ B2 ) ) ).

% and.left_idem
thf(fact_4520_and_Oidem,axiom,
    ! [A: int] :
      ( ( bit_se725231765392027082nd_int @ A @ A )
      = A ) ).

% and.idem
thf(fact_4521_and_Oidem,axiom,
    ! [A: nat] :
      ( ( bit_se727722235901077358nd_nat @ A @ A )
      = A ) ).

% and.idem
thf(fact_4522_real__sqrt__eq__iff,axiom,
    ! [X4: real,Y3: real] :
      ( ( ( sqrt @ X4 )
        = ( sqrt @ Y3 ) )
      = ( X4 = Y3 ) ) ).

% real_sqrt_eq_iff
thf(fact_4523_mask__nat__positive__iff,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( bit_se2002935070580805687sk_nat @ N ) )
      = ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% mask_nat_positive_iff
thf(fact_4524_take__bit__of__0,axiom,
    ! [N: nat] :
      ( ( bit_se2923211474154528505it_int @ N @ zero_zero_int )
      = zero_zero_int ) ).

% take_bit_of_0
thf(fact_4525_take__bit__of__0,axiom,
    ! [N: nat] :
      ( ( bit_se2925701944663578781it_nat @ N @ zero_zero_nat )
      = zero_zero_nat ) ).

% take_bit_of_0
thf(fact_4526_bit_Oconj__zero__right,axiom,
    ! [X4: int] :
      ( ( bit_se725231765392027082nd_int @ X4 @ zero_zero_int )
      = zero_zero_int ) ).

% bit.conj_zero_right
thf(fact_4527_bit_Oconj__zero__left,axiom,
    ! [X4: int] :
      ( ( bit_se725231765392027082nd_int @ zero_zero_int @ X4 )
      = zero_zero_int ) ).

% bit.conj_zero_left
thf(fact_4528_zero__and__eq,axiom,
    ! [A: int] :
      ( ( bit_se725231765392027082nd_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% zero_and_eq
thf(fact_4529_zero__and__eq,axiom,
    ! [A: nat] :
      ( ( bit_se727722235901077358nd_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% zero_and_eq
thf(fact_4530_and__zero__eq,axiom,
    ! [A: int] :
      ( ( bit_se725231765392027082nd_int @ A @ zero_zero_int )
      = zero_zero_int ) ).

% and_zero_eq
thf(fact_4531_and__zero__eq,axiom,
    ! [A: nat] :
      ( ( bit_se727722235901077358nd_nat @ A @ zero_zero_nat )
      = zero_zero_nat ) ).

% and_zero_eq
thf(fact_4532_real__sqrt__zero,axiom,
    ( ( sqrt @ zero_zero_real )
    = zero_zero_real ) ).

% real_sqrt_zero
thf(fact_4533_real__sqrt__eq__zero__cancel__iff,axiom,
    ! [X4: real] :
      ( ( ( sqrt @ X4 )
        = zero_zero_real )
      = ( X4 = zero_zero_real ) ) ).

% real_sqrt_eq_zero_cancel_iff
thf(fact_4534_real__sqrt__less__iff,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_real @ ( sqrt @ X4 ) @ ( sqrt @ Y3 ) )
      = ( ord_less_real @ X4 @ Y3 ) ) ).

% real_sqrt_less_iff
thf(fact_4535_real__sqrt__le__iff,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_eq_real @ ( sqrt @ X4 ) @ ( sqrt @ Y3 ) )
      = ( ord_less_eq_real @ X4 @ Y3 ) ) ).

% real_sqrt_le_iff
thf(fact_4536_real__sqrt__one,axiom,
    ( ( sqrt @ one_one_real )
    = one_one_real ) ).

% real_sqrt_one
thf(fact_4537_real__sqrt__eq__1__iff,axiom,
    ! [X4: real] :
      ( ( ( sqrt @ X4 )
        = one_one_real )
      = ( X4 = one_one_real ) ) ).

% real_sqrt_eq_1_iff
thf(fact_4538_take__bit__and,axiom,
    ! [N: nat,A: int,B2: int] :
      ( ( bit_se2923211474154528505it_int @ N @ ( bit_se725231765392027082nd_int @ A @ B2 ) )
      = ( bit_se725231765392027082nd_int @ ( bit_se2923211474154528505it_int @ N @ A ) @ ( bit_se2923211474154528505it_int @ N @ B2 ) ) ) ).

% take_bit_and
thf(fact_4539_take__bit__and,axiom,
    ! [N: nat,A: nat,B2: nat] :
      ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se727722235901077358nd_nat @ A @ B2 ) )
      = ( bit_se727722235901077358nd_nat @ ( bit_se2925701944663578781it_nat @ N @ A ) @ ( bit_se2925701944663578781it_nat @ N @ B2 ) ) ) ).

% take_bit_and
thf(fact_4540_concat__bit__of__zero__2,axiom,
    ! [N: nat,K: int] :
      ( ( bit_concat_bit @ N @ K @ zero_zero_int )
      = ( bit_se2923211474154528505it_int @ N @ K ) ) ).

% concat_bit_of_zero_2
thf(fact_4541_take__bit__0,axiom,
    ! [A: int] :
      ( ( bit_se2923211474154528505it_int @ zero_zero_nat @ A )
      = zero_zero_int ) ).

% take_bit_0
thf(fact_4542_take__bit__0,axiom,
    ! [A: nat] :
      ( ( bit_se2925701944663578781it_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% take_bit_0
thf(fact_4543_take__bit__Suc__1,axiom,
    ! [N: nat] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ one_one_int )
      = one_one_int ) ).

% take_bit_Suc_1
thf(fact_4544_take__bit__Suc__1,axiom,
    ! [N: nat] :
      ( ( bit_se2925701944663578781it_nat @ ( suc @ N ) @ one_one_nat )
      = one_one_nat ) ).

% take_bit_Suc_1
thf(fact_4545_and_Oleft__neutral,axiom,
    ! [A: code_integer] :
      ( ( bit_se3949692690581998587nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ A )
      = A ) ).

% and.left_neutral
thf(fact_4546_and_Oleft__neutral,axiom,
    ! [A: int] :
      ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ one_one_int ) @ A )
      = A ) ).

% and.left_neutral
thf(fact_4547_and_Oright__neutral,axiom,
    ! [A: code_integer] :
      ( ( bit_se3949692690581998587nteger @ A @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = A ) ).

% and.right_neutral
thf(fact_4548_and_Oright__neutral,axiom,
    ! [A: int] :
      ( ( bit_se725231765392027082nd_int @ A @ ( uminus_uminus_int @ one_one_int ) )
      = A ) ).

% and.right_neutral
thf(fact_4549_bit_Oconj__one__right,axiom,
    ! [X4: code_integer] :
      ( ( bit_se3949692690581998587nteger @ X4 @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = X4 ) ).

% bit.conj_one_right
thf(fact_4550_bit_Oconj__one__right,axiom,
    ! [X4: int] :
      ( ( bit_se725231765392027082nd_int @ X4 @ ( uminus_uminus_int @ one_one_int ) )
      = X4 ) ).

% bit.conj_one_right
thf(fact_4551_of__int__eq__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( ( ( ring_17405671764205052669omplex @ Z )
        = ( numera6690914467698888265omplex @ N ) )
      = ( Z
        = ( numeral_numeral_int @ N ) ) ) ).

% of_int_eq_numeral_iff
thf(fact_4552_of__int__eq__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( ( ( ring_1_of_int_real @ Z )
        = ( numeral_numeral_real @ N ) )
      = ( Z
        = ( numeral_numeral_int @ N ) ) ) ).

% of_int_eq_numeral_iff
thf(fact_4553_of__int__eq__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( ( ( ring_1_of_int_rat @ Z )
        = ( numeral_numeral_rat @ N ) )
      = ( Z
        = ( numeral_numeral_int @ N ) ) ) ).

% of_int_eq_numeral_iff
thf(fact_4554_of__int__eq__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( ( ( ring_1_of_int_int @ Z )
        = ( numeral_numeral_int @ N ) )
      = ( Z
        = ( numeral_numeral_int @ N ) ) ) ).

% of_int_eq_numeral_iff
thf(fact_4555_of__int__numeral,axiom,
    ! [K: num] :
      ( ( ring_17405671764205052669omplex @ ( numeral_numeral_int @ K ) )
      = ( numera6690914467698888265omplex @ K ) ) ).

% of_int_numeral
thf(fact_4556_of__int__numeral,axiom,
    ! [K: num] :
      ( ( ring_1_of_int_real @ ( numeral_numeral_int @ K ) )
      = ( numeral_numeral_real @ K ) ) ).

% of_int_numeral
thf(fact_4557_of__int__numeral,axiom,
    ! [K: num] :
      ( ( ring_1_of_int_rat @ ( numeral_numeral_int @ K ) )
      = ( numeral_numeral_rat @ K ) ) ).

% of_int_numeral
thf(fact_4558_of__int__numeral,axiom,
    ! [K: num] :
      ( ( ring_1_of_int_int @ ( numeral_numeral_int @ K ) )
      = ( numeral_numeral_int @ K ) ) ).

% of_int_numeral
thf(fact_4559_of__int__le__iff,axiom,
    ! [W: int,Z: int] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) )
      = ( ord_less_eq_int @ W @ Z ) ) ).

% of_int_le_iff
thf(fact_4560_of__int__le__iff,axiom,
    ! [W: int,Z: int] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) )
      = ( ord_less_eq_int @ W @ Z ) ) ).

% of_int_le_iff
thf(fact_4561_of__int__le__iff,axiom,
    ! [W: int,Z: int] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) )
      = ( ord_less_eq_int @ W @ Z ) ) ).

% of_int_le_iff
thf(fact_4562_take__bit__numeral__1,axiom,
    ! [L2: num] :
      ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L2 ) @ one_one_int )
      = one_one_int ) ).

% take_bit_numeral_1
thf(fact_4563_take__bit__numeral__1,axiom,
    ! [L2: num] :
      ( ( bit_se2925701944663578781it_nat @ ( numeral_numeral_nat @ L2 ) @ one_one_nat )
      = one_one_nat ) ).

% take_bit_numeral_1
thf(fact_4564_of__int__less__iff,axiom,
    ! [W: int,Z: int] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) )
      = ( ord_less_int @ W @ Z ) ) ).

% of_int_less_iff
thf(fact_4565_of__int__less__iff,axiom,
    ! [W: int,Z: int] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) )
      = ( ord_less_int @ W @ Z ) ) ).

% of_int_less_iff
thf(fact_4566_of__int__less__iff,axiom,
    ! [W: int,Z: int] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) )
      = ( ord_less_int @ W @ Z ) ) ).

% of_int_less_iff
thf(fact_4567_of__int__1,axiom,
    ( ( ring_17405671764205052669omplex @ one_one_int )
    = one_one_complex ) ).

% of_int_1
thf(fact_4568_of__int__1,axiom,
    ( ( ring_1_of_int_int @ one_one_int )
    = one_one_int ) ).

% of_int_1
thf(fact_4569_of__int__1,axiom,
    ( ( ring_1_of_int_real @ one_one_int )
    = one_one_real ) ).

% of_int_1
thf(fact_4570_of__int__1,axiom,
    ( ( ring_1_of_int_rat @ one_one_int )
    = one_one_rat ) ).

% of_int_1
thf(fact_4571_of__int__eq__1__iff,axiom,
    ! [Z: int] :
      ( ( ( ring_17405671764205052669omplex @ Z )
        = one_one_complex )
      = ( Z = one_one_int ) ) ).

% of_int_eq_1_iff
thf(fact_4572_of__int__eq__1__iff,axiom,
    ! [Z: int] :
      ( ( ( ring_1_of_int_int @ Z )
        = one_one_int )
      = ( Z = one_one_int ) ) ).

% of_int_eq_1_iff
thf(fact_4573_of__int__eq__1__iff,axiom,
    ! [Z: int] :
      ( ( ( ring_1_of_int_real @ Z )
        = one_one_real )
      = ( Z = one_one_int ) ) ).

% of_int_eq_1_iff
thf(fact_4574_of__int__eq__1__iff,axiom,
    ! [Z: int] :
      ( ( ( ring_1_of_int_rat @ Z )
        = one_one_rat )
      = ( Z = one_one_int ) ) ).

% of_int_eq_1_iff
thf(fact_4575_of__int__mult,axiom,
    ! [W: int,Z: int] :
      ( ( ring_1_of_int_real @ ( times_times_int @ W @ Z ) )
      = ( times_times_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) ) ) ).

% of_int_mult
thf(fact_4576_of__int__mult,axiom,
    ! [W: int,Z: int] :
      ( ( ring_1_of_int_rat @ ( times_times_int @ W @ Z ) )
      = ( times_times_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) ) ) ).

% of_int_mult
thf(fact_4577_of__int__mult,axiom,
    ! [W: int,Z: int] :
      ( ( ring_1_of_int_int @ ( times_times_int @ W @ Z ) )
      = ( times_times_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) ) ) ).

% of_int_mult
thf(fact_4578_of__int__add,axiom,
    ! [W: int,Z: int] :
      ( ( ring_1_of_int_int @ ( plus_plus_int @ W @ Z ) )
      = ( plus_plus_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) ) ) ).

% of_int_add
thf(fact_4579_of__int__add,axiom,
    ! [W: int,Z: int] :
      ( ( ring_1_of_int_real @ ( plus_plus_int @ W @ Z ) )
      = ( plus_plus_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) ) ) ).

% of_int_add
thf(fact_4580_of__int__add,axiom,
    ! [W: int,Z: int] :
      ( ( ring_1_of_int_rat @ ( plus_plus_int @ W @ Z ) )
      = ( plus_plus_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) ) ) ).

% of_int_add
thf(fact_4581_real__sqrt__lt__0__iff,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ ( sqrt @ X4 ) @ zero_zero_real )
      = ( ord_less_real @ X4 @ zero_zero_real ) ) ).

% real_sqrt_lt_0_iff
thf(fact_4582_real__sqrt__gt__0__iff,axiom,
    ! [Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ ( sqrt @ Y3 ) )
      = ( ord_less_real @ zero_zero_real @ Y3 ) ) ).

% real_sqrt_gt_0_iff
thf(fact_4583_real__sqrt__ge__0__iff,axiom,
    ! [Y3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ Y3 ) )
      = ( ord_less_eq_real @ zero_zero_real @ Y3 ) ) ).

% real_sqrt_ge_0_iff
thf(fact_4584_real__sqrt__le__0__iff,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ ( sqrt @ X4 ) @ zero_zero_real )
      = ( ord_less_eq_real @ X4 @ zero_zero_real ) ) ).

% real_sqrt_le_0_iff
thf(fact_4585_mask__eq__0__iff,axiom,
    ! [N: nat] :
      ( ( ( bit_se2002935070580805687sk_nat @ N )
        = zero_zero_nat )
      = ( N = zero_zero_nat ) ) ).

% mask_eq_0_iff
thf(fact_4586_mask__eq__0__iff,axiom,
    ! [N: nat] :
      ( ( ( bit_se2000444600071755411sk_int @ N )
        = zero_zero_int )
      = ( N = zero_zero_nat ) ) ).

% mask_eq_0_iff
thf(fact_4587_mask__0,axiom,
    ( ( bit_se2002935070580805687sk_nat @ zero_zero_nat )
    = zero_zero_nat ) ).

% mask_0
thf(fact_4588_mask__0,axiom,
    ( ( bit_se2000444600071755411sk_int @ zero_zero_nat )
    = zero_zero_int ) ).

% mask_0
thf(fact_4589_of__int__diff,axiom,
    ! [W: int,Z: int] :
      ( ( ring_1_of_int_real @ ( minus_minus_int @ W @ Z ) )
      = ( minus_minus_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) ) ) ).

% of_int_diff
thf(fact_4590_of__int__diff,axiom,
    ! [W: int,Z: int] :
      ( ( ring_1_of_int_rat @ ( minus_minus_int @ W @ Z ) )
      = ( minus_minus_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) ) ) ).

% of_int_diff
thf(fact_4591_of__int__diff,axiom,
    ! [W: int,Z: int] :
      ( ( ring_1_of_int_int @ ( minus_minus_int @ W @ Z ) )
      = ( minus_minus_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) ) ) ).

% of_int_diff
thf(fact_4592_real__sqrt__lt__1__iff,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ ( sqrt @ X4 ) @ one_one_real )
      = ( ord_less_real @ X4 @ one_one_real ) ) ).

% real_sqrt_lt_1_iff
thf(fact_4593_real__sqrt__gt__1__iff,axiom,
    ! [Y3: real] :
      ( ( ord_less_real @ one_one_real @ ( sqrt @ Y3 ) )
      = ( ord_less_real @ one_one_real @ Y3 ) ) ).

% real_sqrt_gt_1_iff
thf(fact_4594_real__sqrt__le__1__iff,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ ( sqrt @ X4 ) @ one_one_real )
      = ( ord_less_eq_real @ X4 @ one_one_real ) ) ).

% real_sqrt_le_1_iff
thf(fact_4595_real__sqrt__ge__1__iff,axiom,
    ! [Y3: real] :
      ( ( ord_less_eq_real @ one_one_real @ ( sqrt @ Y3 ) )
      = ( ord_less_eq_real @ one_one_real @ Y3 ) ) ).

% real_sqrt_ge_1_iff
thf(fact_4596_of__int__power,axiom,
    ! [Z: int,N: nat] :
      ( ( ring_1_of_int_rat @ ( power_power_int @ Z @ N ) )
      = ( power_power_rat @ ( ring_1_of_int_rat @ Z ) @ N ) ) ).

% of_int_power
thf(fact_4597_of__int__power,axiom,
    ! [Z: int,N: nat] :
      ( ( ring_1_of_int_real @ ( power_power_int @ Z @ N ) )
      = ( power_power_real @ ( ring_1_of_int_real @ Z ) @ N ) ) ).

% of_int_power
thf(fact_4598_of__int__power,axiom,
    ! [Z: int,N: nat] :
      ( ( ring_1_of_int_int @ ( power_power_int @ Z @ N ) )
      = ( power_power_int @ ( ring_1_of_int_int @ Z ) @ N ) ) ).

% of_int_power
thf(fact_4599_of__int__power,axiom,
    ! [Z: int,N: nat] :
      ( ( ring_17405671764205052669omplex @ ( power_power_int @ Z @ N ) )
      = ( power_power_complex @ ( ring_17405671764205052669omplex @ Z ) @ N ) ) ).

% of_int_power
thf(fact_4600_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B2: int,W: nat,X4: int] :
      ( ( ( power_power_rat @ ( ring_1_of_int_rat @ B2 ) @ W )
        = ( ring_1_of_int_rat @ X4 ) )
      = ( ( power_power_int @ B2 @ W )
        = X4 ) ) ).

% of_int_eq_of_int_power_cancel_iff
thf(fact_4601_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B2: int,W: nat,X4: int] :
      ( ( ( power_power_real @ ( ring_1_of_int_real @ B2 ) @ W )
        = ( ring_1_of_int_real @ X4 ) )
      = ( ( power_power_int @ B2 @ W )
        = X4 ) ) ).

% of_int_eq_of_int_power_cancel_iff
thf(fact_4602_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B2: int,W: nat,X4: int] :
      ( ( ( power_power_int @ ( ring_1_of_int_int @ B2 ) @ W )
        = ( ring_1_of_int_int @ X4 ) )
      = ( ( power_power_int @ B2 @ W )
        = X4 ) ) ).

% of_int_eq_of_int_power_cancel_iff
thf(fact_4603_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B2: int,W: nat,X4: int] :
      ( ( ( power_power_complex @ ( ring_17405671764205052669omplex @ B2 ) @ W )
        = ( ring_17405671764205052669omplex @ X4 ) )
      = ( ( power_power_int @ B2 @ W )
        = X4 ) ) ).

% of_int_eq_of_int_power_cancel_iff
thf(fact_4604_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X4: int,B2: int,W: nat] :
      ( ( ( ring_1_of_int_rat @ X4 )
        = ( power_power_rat @ ( ring_1_of_int_rat @ B2 ) @ W ) )
      = ( X4
        = ( power_power_int @ B2 @ W ) ) ) ).

% of_int_power_eq_of_int_cancel_iff
thf(fact_4605_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X4: int,B2: int,W: nat] :
      ( ( ( ring_1_of_int_real @ X4 )
        = ( power_power_real @ ( ring_1_of_int_real @ B2 ) @ W ) )
      = ( X4
        = ( power_power_int @ B2 @ W ) ) ) ).

% of_int_power_eq_of_int_cancel_iff
thf(fact_4606_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X4: int,B2: int,W: nat] :
      ( ( ( ring_1_of_int_int @ X4 )
        = ( power_power_int @ ( ring_1_of_int_int @ B2 ) @ W ) )
      = ( X4
        = ( power_power_int @ B2 @ W ) ) ) ).

% of_int_power_eq_of_int_cancel_iff
thf(fact_4607_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X4: int,B2: int,W: nat] :
      ( ( ( ring_17405671764205052669omplex @ X4 )
        = ( power_power_complex @ ( ring_17405671764205052669omplex @ B2 ) @ W ) )
      = ( X4
        = ( power_power_int @ B2 @ W ) ) ) ).

% of_int_power_eq_of_int_cancel_iff
thf(fact_4608_and__nonnegative__int__iff,axiom,
    ! [K: int,L2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se725231765392027082nd_int @ K @ L2 ) )
      = ( ( ord_less_eq_int @ zero_zero_int @ K )
        | ( ord_less_eq_int @ zero_zero_int @ L2 ) ) ) ).

% and_nonnegative_int_iff
thf(fact_4609_and__negative__int__iff,axiom,
    ! [K: int,L2: int] :
      ( ( ord_less_int @ ( bit_se725231765392027082nd_int @ K @ L2 ) @ zero_zero_int )
      = ( ( ord_less_int @ K @ zero_zero_int )
        & ( ord_less_int @ L2 @ zero_zero_int ) ) ) ).

% and_negative_int_iff
thf(fact_4610_take__bit__of__1__eq__0__iff,axiom,
    ! [N: nat] :
      ( ( ( bit_se2923211474154528505it_int @ N @ one_one_int )
        = zero_zero_int )
      = ( N = zero_zero_nat ) ) ).

% take_bit_of_1_eq_0_iff
thf(fact_4611_take__bit__of__1__eq__0__iff,axiom,
    ! [N: nat] :
      ( ( ( bit_se2925701944663578781it_nat @ N @ one_one_nat )
        = zero_zero_nat )
      = ( N = zero_zero_nat ) ) ).

% take_bit_of_1_eq_0_iff
thf(fact_4612_and__numerals_I2_J,axiom,
    ! [Y3: num] :
      ( ( bit_se725231765392027082nd_int @ one_one_int @ ( numeral_numeral_int @ ( bit1 @ Y3 ) ) )
      = one_one_int ) ).

% and_numerals(2)
thf(fact_4613_and__numerals_I2_J,axiom,
    ! [Y3: num] :
      ( ( bit_se727722235901077358nd_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) )
      = one_one_nat ) ).

% and_numerals(2)
thf(fact_4614_and__numerals_I8_J,axiom,
    ! [X4: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ X4 ) ) @ one_one_int )
      = one_one_int ) ).

% and_numerals(8)
thf(fact_4615_and__numerals_I8_J,axiom,
    ! [X4: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X4 ) ) @ one_one_nat )
      = one_one_nat ) ).

% and_numerals(8)
thf(fact_4616_real__sqrt__four,axiom,
    ( ( sqrt @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) )
    = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).

% real_sqrt_four
thf(fact_4617_mask__Suc__0,axiom,
    ( ( bit_se2002935070580805687sk_nat @ ( suc @ zero_zero_nat ) )
    = one_one_nat ) ).

% mask_Suc_0
thf(fact_4618_mask__Suc__0,axiom,
    ( ( bit_se2000444600071755411sk_int @ ( suc @ zero_zero_nat ) )
    = one_one_int ) ).

% mask_Suc_0
thf(fact_4619_take__bit__minus__one__eq__mask,axiom,
    ! [N: nat] :
      ( ( bit_se1745604003318907178nteger @ N @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( bit_se2119862282449309892nteger @ N ) ) ).

% take_bit_minus_one_eq_mask
thf(fact_4620_take__bit__minus__one__eq__mask,axiom,
    ! [N: nat] :
      ( ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
      = ( bit_se2000444600071755411sk_int @ N ) ) ).

% take_bit_minus_one_eq_mask
thf(fact_4621_take__bit__of__Suc__0,axiom,
    ! [N: nat] :
      ( ( bit_se2925701944663578781it_nat @ N @ ( suc @ zero_zero_nat ) )
      = ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% take_bit_of_Suc_0
thf(fact_4622_and__numerals_I1_J,axiom,
    ! [Y3: num] :
      ( ( bit_se725231765392027082nd_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ Y3 ) ) )
      = zero_zero_int ) ).

% and_numerals(1)
thf(fact_4623_and__numerals_I1_J,axiom,
    ! [Y3: num] :
      ( ( bit_se727722235901077358nd_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) )
      = zero_zero_nat ) ).

% and_numerals(1)
thf(fact_4624_and__numerals_I5_J,axiom,
    ! [X4: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ X4 ) ) @ one_one_int )
      = zero_zero_int ) ).

% and_numerals(5)
thf(fact_4625_and__numerals_I5_J,axiom,
    ! [X4: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X4 ) ) @ one_one_nat )
      = zero_zero_nat ) ).

% and_numerals(5)
thf(fact_4626_and__numerals_I3_J,axiom,
    ! [X4: num,Y3: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ X4 ) ) @ ( numeral_numeral_int @ ( bit0 @ Y3 ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ X4 ) @ ( numeral_numeral_int @ Y3 ) ) ) ) ).

% and_numerals(3)
thf(fact_4627_and__numerals_I3_J,axiom,
    ! [X4: num,Y3: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X4 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) )
      = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ X4 ) @ ( numeral_numeral_nat @ Y3 ) ) ) ) ).

% and_numerals(3)
thf(fact_4628_of__int__le__0__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ zero_zero_real )
      = ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).

% of_int_le_0_iff
thf(fact_4629_of__int__le__0__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ zero_zero_rat )
      = ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).

% of_int_le_0_iff
thf(fact_4630_of__int__le__0__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ zero_zero_int )
      = ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).

% of_int_le_0_iff
thf(fact_4631_of__int__0__le__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) )
      = ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).

% of_int_0_le_iff
thf(fact_4632_of__int__0__le__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) )
      = ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).

% of_int_0_le_iff
thf(fact_4633_of__int__0__le__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) )
      = ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).

% of_int_0_le_iff
thf(fact_4634_of__int__0__less__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) )
      = ( ord_less_int @ zero_zero_int @ Z ) ) ).

% of_int_0_less_iff
thf(fact_4635_of__int__0__less__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) )
      = ( ord_less_int @ zero_zero_int @ Z ) ) ).

% of_int_0_less_iff
thf(fact_4636_of__int__0__less__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) )
      = ( ord_less_int @ zero_zero_int @ Z ) ) ).

% of_int_0_less_iff
thf(fact_4637_of__int__less__0__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ zero_zero_real )
      = ( ord_less_int @ Z @ zero_zero_int ) ) ).

% of_int_less_0_iff
thf(fact_4638_of__int__less__0__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ zero_zero_rat )
      = ( ord_less_int @ Z @ zero_zero_int ) ) ).

% of_int_less_0_iff
thf(fact_4639_of__int__less__0__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ zero_zero_int )
      = ( ord_less_int @ Z @ zero_zero_int ) ) ).

% of_int_less_0_iff
thf(fact_4640_of__int__le__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ ( numeral_numeral_real @ N ) )
      = ( ord_less_eq_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).

% of_int_le_numeral_iff
thf(fact_4641_of__int__le__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ ( numeral_numeral_rat @ N ) )
      = ( ord_less_eq_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).

% of_int_le_numeral_iff
thf(fact_4642_of__int__le__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ ( numeral_numeral_int @ N ) )
      = ( ord_less_eq_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).

% of_int_le_numeral_iff
thf(fact_4643_of__int__numeral__le__iff,axiom,
    ! [N: num,Z: int] :
      ( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ ( ring_1_of_int_real @ Z ) )
      = ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).

% of_int_numeral_le_iff
thf(fact_4644_of__int__numeral__le__iff,axiom,
    ! [N: num,Z: int] :
      ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ N ) @ ( ring_1_of_int_rat @ Z ) )
      = ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).

% of_int_numeral_le_iff
thf(fact_4645_of__int__numeral__le__iff,axiom,
    ! [N: num,Z: int] :
      ( ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ ( ring_1_of_int_int @ Z ) )
      = ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).

% of_int_numeral_le_iff
thf(fact_4646_of__int__less__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ ( numeral_numeral_real @ N ) )
      = ( ord_less_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).

% of_int_less_numeral_iff
thf(fact_4647_of__int__less__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ ( numeral_numeral_rat @ N ) )
      = ( ord_less_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).

% of_int_less_numeral_iff
thf(fact_4648_of__int__less__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ ( numeral_numeral_int @ N ) )
      = ( ord_less_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).

% of_int_less_numeral_iff
thf(fact_4649_of__int__numeral__less__iff,axiom,
    ! [N: num,Z: int] :
      ( ( ord_less_real @ ( numeral_numeral_real @ N ) @ ( ring_1_of_int_real @ Z ) )
      = ( ord_less_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).

% of_int_numeral_less_iff
thf(fact_4650_of__int__numeral__less__iff,axiom,
    ! [N: num,Z: int] :
      ( ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ ( ring_1_of_int_rat @ Z ) )
      = ( ord_less_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).

% of_int_numeral_less_iff
thf(fact_4651_of__int__numeral__less__iff,axiom,
    ! [N: num,Z: int] :
      ( ( ord_less_int @ ( numeral_numeral_int @ N ) @ ( ring_1_of_int_int @ Z ) )
      = ( ord_less_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).

% of_int_numeral_less_iff
thf(fact_4652_of__int__le__1__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ one_one_real )
      = ( ord_less_eq_int @ Z @ one_one_int ) ) ).

% of_int_le_1_iff
thf(fact_4653_of__int__le__1__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat )
      = ( ord_less_eq_int @ Z @ one_one_int ) ) ).

% of_int_le_1_iff
thf(fact_4654_of__int__le__1__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ one_one_int )
      = ( ord_less_eq_int @ Z @ one_one_int ) ) ).

% of_int_le_1_iff
thf(fact_4655_of__int__1__le__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_real @ one_one_real @ ( ring_1_of_int_real @ Z ) )
      = ( ord_less_eq_int @ one_one_int @ Z ) ) ).

% of_int_1_le_iff
thf(fact_4656_of__int__1__le__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_rat @ one_one_rat @ ( ring_1_of_int_rat @ Z ) )
      = ( ord_less_eq_int @ one_one_int @ Z ) ) ).

% of_int_1_le_iff
thf(fact_4657_of__int__1__le__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ one_one_int @ ( ring_1_of_int_int @ Z ) )
      = ( ord_less_eq_int @ one_one_int @ Z ) ) ).

% of_int_1_le_iff
thf(fact_4658_of__int__less__1__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ one_one_real )
      = ( ord_less_int @ Z @ one_one_int ) ) ).

% of_int_less_1_iff
thf(fact_4659_of__int__less__1__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat )
      = ( ord_less_int @ Z @ one_one_int ) ) ).

% of_int_less_1_iff
thf(fact_4660_of__int__less__1__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ one_one_int )
      = ( ord_less_int @ Z @ one_one_int ) ) ).

% of_int_less_1_iff
thf(fact_4661_of__int__1__less__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_real @ one_one_real @ ( ring_1_of_int_real @ Z ) )
      = ( ord_less_int @ one_one_int @ Z ) ) ).

% of_int_1_less_iff
thf(fact_4662_of__int__1__less__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_rat @ one_one_rat @ ( ring_1_of_int_rat @ Z ) )
      = ( ord_less_int @ one_one_int @ Z ) ) ).

% of_int_1_less_iff
thf(fact_4663_of__int__1__less__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_int @ one_one_int @ ( ring_1_of_int_int @ Z ) )
      = ( ord_less_int @ one_one_int @ Z ) ) ).

% of_int_1_less_iff
thf(fact_4664_take__bit__of__1,axiom,
    ! [N: nat] :
      ( ( bit_se1745604003318907178nteger @ N @ one_one_Code_integer )
      = ( zero_n356916108424825756nteger @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% take_bit_of_1
thf(fact_4665_take__bit__of__1,axiom,
    ! [N: nat] :
      ( ( bit_se2923211474154528505it_int @ N @ one_one_int )
      = ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% take_bit_of_1
thf(fact_4666_take__bit__of__1,axiom,
    ! [N: nat] :
      ( ( bit_se2925701944663578781it_nat @ N @ one_one_nat )
      = ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% take_bit_of_1
thf(fact_4667_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X4: num,N: nat,Y3: int] :
      ( ( ( power_power_complex @ ( numera6690914467698888265omplex @ X4 ) @ N )
        = ( ring_17405671764205052669omplex @ Y3 ) )
      = ( ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N )
        = Y3 ) ) ).

% numeral_power_eq_of_int_cancel_iff
thf(fact_4668_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X4: num,N: nat,Y3: int] :
      ( ( ( power_power_real @ ( numeral_numeral_real @ X4 ) @ N )
        = ( ring_1_of_int_real @ Y3 ) )
      = ( ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N )
        = Y3 ) ) ).

% numeral_power_eq_of_int_cancel_iff
thf(fact_4669_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X4: num,N: nat,Y3: int] :
      ( ( ( power_power_rat @ ( numeral_numeral_rat @ X4 ) @ N )
        = ( ring_1_of_int_rat @ Y3 ) )
      = ( ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N )
        = Y3 ) ) ).

% numeral_power_eq_of_int_cancel_iff
thf(fact_4670_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X4: num,N: nat,Y3: int] :
      ( ( ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N )
        = ( ring_1_of_int_int @ Y3 ) )
      = ( ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N )
        = Y3 ) ) ).

% numeral_power_eq_of_int_cancel_iff
thf(fact_4671_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y3: int,X4: num,N: nat] :
      ( ( ( ring_17405671764205052669omplex @ Y3 )
        = ( power_power_complex @ ( numera6690914467698888265omplex @ X4 ) @ N ) )
      = ( Y3
        = ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N ) ) ) ).

% of_int_eq_numeral_power_cancel_iff
thf(fact_4672_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y3: int,X4: num,N: nat] :
      ( ( ( ring_1_of_int_real @ Y3 )
        = ( power_power_real @ ( numeral_numeral_real @ X4 ) @ N ) )
      = ( Y3
        = ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N ) ) ) ).

% of_int_eq_numeral_power_cancel_iff
thf(fact_4673_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y3: int,X4: num,N: nat] :
      ( ( ( ring_1_of_int_rat @ Y3 )
        = ( power_power_rat @ ( numeral_numeral_rat @ X4 ) @ N ) )
      = ( Y3
        = ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N ) ) ) ).

% of_int_eq_numeral_power_cancel_iff
thf(fact_4674_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y3: int,X4: num,N: nat] :
      ( ( ( ring_1_of_int_int @ Y3 )
        = ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N ) )
      = ( Y3
        = ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N ) ) ) ).

% of_int_eq_numeral_power_cancel_iff
thf(fact_4675_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B2: int,W: nat,X4: int] :
      ( ( ord_less_eq_real @ ( power_power_real @ ( ring_1_of_int_real @ B2 ) @ W ) @ ( ring_1_of_int_real @ X4 ) )
      = ( ord_less_eq_int @ ( power_power_int @ B2 @ W ) @ X4 ) ) ).

% of_int_le_of_int_power_cancel_iff
thf(fact_4676_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B2: int,W: nat,X4: int] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ ( ring_1_of_int_rat @ B2 ) @ W ) @ ( ring_1_of_int_rat @ X4 ) )
      = ( ord_less_eq_int @ ( power_power_int @ B2 @ W ) @ X4 ) ) ).

% of_int_le_of_int_power_cancel_iff
thf(fact_4677_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B2: int,W: nat,X4: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( ring_1_of_int_int @ B2 ) @ W ) @ ( ring_1_of_int_int @ X4 ) )
      = ( ord_less_eq_int @ ( power_power_int @ B2 @ W ) @ X4 ) ) ).

% of_int_le_of_int_power_cancel_iff
thf(fact_4678_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X4: int,B2: int,W: nat] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ X4 ) @ ( power_power_real @ ( ring_1_of_int_real @ B2 ) @ W ) )
      = ( ord_less_eq_int @ X4 @ ( power_power_int @ B2 @ W ) ) ) ).

% of_int_power_le_of_int_cancel_iff
thf(fact_4679_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X4: int,B2: int,W: nat] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ X4 ) @ ( power_power_rat @ ( ring_1_of_int_rat @ B2 ) @ W ) )
      = ( ord_less_eq_int @ X4 @ ( power_power_int @ B2 @ W ) ) ) ).

% of_int_power_le_of_int_cancel_iff
thf(fact_4680_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X4: int,B2: int,W: nat] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ X4 ) @ ( power_power_int @ ( ring_1_of_int_int @ B2 ) @ W ) )
      = ( ord_less_eq_int @ X4 @ ( power_power_int @ B2 @ W ) ) ) ).

% of_int_power_le_of_int_cancel_iff
thf(fact_4681_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B2: int,W: nat,X4: int] :
      ( ( ord_less_real @ ( power_power_real @ ( ring_1_of_int_real @ B2 ) @ W ) @ ( ring_1_of_int_real @ X4 ) )
      = ( ord_less_int @ ( power_power_int @ B2 @ W ) @ X4 ) ) ).

% of_int_less_of_int_power_cancel_iff
thf(fact_4682_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B2: int,W: nat,X4: int] :
      ( ( ord_less_rat @ ( power_power_rat @ ( ring_1_of_int_rat @ B2 ) @ W ) @ ( ring_1_of_int_rat @ X4 ) )
      = ( ord_less_int @ ( power_power_int @ B2 @ W ) @ X4 ) ) ).

% of_int_less_of_int_power_cancel_iff
thf(fact_4683_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B2: int,W: nat,X4: int] :
      ( ( ord_less_int @ ( power_power_int @ ( ring_1_of_int_int @ B2 ) @ W ) @ ( ring_1_of_int_int @ X4 ) )
      = ( ord_less_int @ ( power_power_int @ B2 @ W ) @ X4 ) ) ).

% of_int_less_of_int_power_cancel_iff
thf(fact_4684_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X4: int,B2: int,W: nat] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ X4 ) @ ( power_power_real @ ( ring_1_of_int_real @ B2 ) @ W ) )
      = ( ord_less_int @ X4 @ ( power_power_int @ B2 @ W ) ) ) ).

% of_int_power_less_of_int_cancel_iff
thf(fact_4685_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X4: int,B2: int,W: nat] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ X4 ) @ ( power_power_rat @ ( ring_1_of_int_rat @ B2 ) @ W ) )
      = ( ord_less_int @ X4 @ ( power_power_int @ B2 @ W ) ) ) ).

% of_int_power_less_of_int_cancel_iff
thf(fact_4686_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X4: int,B2: int,W: nat] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ X4 ) @ ( power_power_int @ ( ring_1_of_int_int @ B2 ) @ W ) )
      = ( ord_less_int @ X4 @ ( power_power_int @ B2 @ W ) ) ) ).

% of_int_power_less_of_int_cancel_iff
thf(fact_4687_and__minus__numerals_I6_J,axiom,
    ! [N: num] :
      ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ one_one_int )
      = one_one_int ) ).

% and_minus_numerals(6)
thf(fact_4688_and__minus__numerals_I2_J,axiom,
    ! [N: num] :
      ( ( bit_se725231765392027082nd_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
      = one_one_int ) ).

% and_minus_numerals(2)
thf(fact_4689_and__numerals_I4_J,axiom,
    ! [X4: num,Y3: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ X4 ) ) @ ( numeral_numeral_int @ ( bit1 @ Y3 ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ X4 ) @ ( numeral_numeral_int @ Y3 ) ) ) ) ).

% and_numerals(4)
thf(fact_4690_and__numerals_I4_J,axiom,
    ! [X4: num,Y3: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X4 ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) )
      = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ X4 ) @ ( numeral_numeral_nat @ Y3 ) ) ) ) ).

% and_numerals(4)
thf(fact_4691_and__numerals_I6_J,axiom,
    ! [X4: num,Y3: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ X4 ) ) @ ( numeral_numeral_int @ ( bit0 @ Y3 ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ X4 ) @ ( numeral_numeral_int @ Y3 ) ) ) ) ).

% and_numerals(6)
thf(fact_4692_and__numerals_I6_J,axiom,
    ! [X4: num,Y3: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X4 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) )
      = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ X4 ) @ ( numeral_numeral_nat @ Y3 ) ) ) ) ).

% and_numerals(6)
thf(fact_4693_even__take__bit__eq,axiom,
    ! [N: nat,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1745604003318907178nteger @ N @ A ) )
      = ( ( N = zero_zero_nat )
        | ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).

% even_take_bit_eq
thf(fact_4694_even__take__bit__eq,axiom,
    ! [N: nat,A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2923211474154528505it_int @ N @ A ) )
      = ( ( N = zero_zero_nat )
        | ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).

% even_take_bit_eq
thf(fact_4695_even__take__bit__eq,axiom,
    ! [N: nat,A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2925701944663578781it_nat @ N @ A ) )
      = ( ( N = zero_zero_nat )
        | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).

% even_take_bit_eq
thf(fact_4696_and__minus__numerals_I5_J,axiom,
    ! [N: num] :
      ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ one_one_int )
      = zero_zero_int ) ).

% and_minus_numerals(5)
thf(fact_4697_and__minus__numerals_I1_J,axiom,
    ! [N: num] :
      ( ( bit_se725231765392027082nd_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
      = zero_zero_int ) ).

% and_minus_numerals(1)
thf(fact_4698_take__bit__Suc__0,axiom,
    ! [A: code_integer] :
      ( ( bit_se1745604003318907178nteger @ ( suc @ zero_zero_nat ) @ A )
      = ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% take_bit_Suc_0
thf(fact_4699_take__bit__Suc__0,axiom,
    ! [A: int] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ zero_zero_nat ) @ A )
      = ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% take_bit_Suc_0
thf(fact_4700_take__bit__Suc__0,axiom,
    ! [A: nat] :
      ( ( bit_se2925701944663578781it_nat @ ( suc @ zero_zero_nat ) @ A )
      = ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% take_bit_Suc_0
thf(fact_4701_real__sqrt__pow2__iff,axiom,
    ! [X4: real] :
      ( ( ( power_power_real @ ( sqrt @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = X4 )
      = ( ord_less_eq_real @ zero_zero_real @ X4 ) ) ).

% real_sqrt_pow2_iff
thf(fact_4702_real__sqrt__pow2,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( power_power_real @ ( sqrt @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = X4 ) ) ).

% real_sqrt_pow2
thf(fact_4703_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X4: num,N: nat,A: int] :
      ( ( ord_less_eq_real @ ( power_power_real @ ( numeral_numeral_real @ X4 ) @ N ) @ ( ring_1_of_int_real @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N ) @ A ) ) ).

% numeral_power_le_of_int_cancel_iff
thf(fact_4704_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X4: num,N: nat,A: int] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X4 ) @ N ) @ ( ring_1_of_int_rat @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N ) @ A ) ) ).

% numeral_power_le_of_int_cancel_iff
thf(fact_4705_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X4: num,N: nat,A: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N ) @ ( ring_1_of_int_int @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N ) @ A ) ) ).

% numeral_power_le_of_int_cancel_iff
thf(fact_4706_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X4: num,N: nat] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( numeral_numeral_real @ X4 ) @ N ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N ) ) ) ).

% of_int_le_numeral_power_cancel_iff
thf(fact_4707_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X4: num,N: nat] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( numeral_numeral_rat @ X4 ) @ N ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N ) ) ) ).

% of_int_le_numeral_power_cancel_iff
thf(fact_4708_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X4: num,N: nat] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N ) ) ) ).

% of_int_le_numeral_power_cancel_iff
thf(fact_4709_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X4: num,N: nat,A: int] :
      ( ( ord_less_real @ ( power_power_real @ ( numeral_numeral_real @ X4 ) @ N ) @ ( ring_1_of_int_real @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N ) @ A ) ) ).

% numeral_power_less_of_int_cancel_iff
thf(fact_4710_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X4: num,N: nat,A: int] :
      ( ( ord_less_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X4 ) @ N ) @ ( ring_1_of_int_rat @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N ) @ A ) ) ).

% numeral_power_less_of_int_cancel_iff
thf(fact_4711_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X4: num,N: nat,A: int] :
      ( ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N ) @ ( ring_1_of_int_int @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N ) @ A ) ) ).

% numeral_power_less_of_int_cancel_iff
thf(fact_4712_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X4: num,N: nat] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( numeral_numeral_real @ X4 ) @ N ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N ) ) ) ).

% of_int_less_numeral_power_cancel_iff
thf(fact_4713_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X4: num,N: nat] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( numeral_numeral_rat @ X4 ) @ N ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N ) ) ) ).

% of_int_less_numeral_power_cancel_iff
thf(fact_4714_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X4: num,N: nat] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N ) ) ) ).

% of_int_less_numeral_power_cancel_iff
thf(fact_4715_real__sqrt__sum__squares__mult__squared__eq,axiom,
    ! [X4: real,Y3: real,Xa: real,Ya: real] :
      ( ( power_power_real @ ( sqrt @ ( times_times_real @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_real @ ( power_power_real @ Xa @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Ya @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( times_times_real @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_real @ ( power_power_real @ Xa @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Ya @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% real_sqrt_sum_squares_mult_squared_eq
thf(fact_4716_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y3: int,X4: num,N: nat] :
      ( ( ( ring_1_of_int_real @ Y3 )
        = ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X4 ) ) @ N ) )
      = ( Y3
        = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N ) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_4717_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y3: int,X4: num,N: nat] :
      ( ( ( ring_1_of_int_int @ Y3 )
        = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N ) )
      = ( Y3
        = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N ) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_4718_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y3: int,X4: num,N: nat] :
      ( ( ( ring_17405671764205052669omplex @ Y3 )
        = ( power_power_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ X4 ) ) @ N ) )
      = ( Y3
        = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N ) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_4719_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y3: int,X4: num,N: nat] :
      ( ( ( ring_18347121197199848620nteger @ Y3 )
        = ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X4 ) ) @ N ) )
      = ( Y3
        = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N ) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_4720_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y3: int,X4: num,N: nat] :
      ( ( ( ring_1_of_int_rat @ Y3 )
        = ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X4 ) ) @ N ) )
      = ( Y3
        = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N ) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_4721_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X4: num,N: nat,Y3: int] :
      ( ( ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X4 ) ) @ N )
        = ( ring_1_of_int_real @ Y3 ) )
      = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N )
        = Y3 ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_4722_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X4: num,N: nat,Y3: int] :
      ( ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N )
        = ( ring_1_of_int_int @ Y3 ) )
      = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N )
        = Y3 ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_4723_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X4: num,N: nat,Y3: int] :
      ( ( ( power_power_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ X4 ) ) @ N )
        = ( ring_17405671764205052669omplex @ Y3 ) )
      = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N )
        = Y3 ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_4724_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X4: num,N: nat,Y3: int] :
      ( ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X4 ) ) @ N )
        = ( ring_18347121197199848620nteger @ Y3 ) )
      = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N )
        = Y3 ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_4725_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X4: num,N: nat,Y3: int] :
      ( ( ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X4 ) ) @ N )
        = ( ring_1_of_int_rat @ Y3 ) )
      = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N )
        = Y3 ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_4726_and__numerals_I7_J,axiom,
    ! [X4: num,Y3: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ X4 ) ) @ ( numeral_numeral_int @ ( bit1 @ Y3 ) ) )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ X4 ) @ ( numeral_numeral_int @ Y3 ) ) ) ) ) ).

% and_numerals(7)
thf(fact_4727_and__numerals_I7_J,axiom,
    ! [X4: num,Y3: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X4 ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) )
      = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ X4 ) @ ( numeral_numeral_nat @ Y3 ) ) ) ) ) ).

% and_numerals(7)
thf(fact_4728_take__bit__of__exp,axiom,
    ! [M: nat,N: nat] :
      ( ( bit_se1745604003318907178nteger @ M @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
      = ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ ( ord_less_nat @ N @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ).

% take_bit_of_exp
thf(fact_4729_take__bit__of__exp,axiom,
    ! [M: nat,N: nat] :
      ( ( bit_se2923211474154528505it_int @ M @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
      = ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ N @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).

% take_bit_of_exp
thf(fact_4730_take__bit__of__exp,axiom,
    ! [M: nat,N: nat] :
      ( ( bit_se2925701944663578781it_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ N @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% take_bit_of_exp
thf(fact_4731_take__bit__of__2,axiom,
    ! [N: nat] :
      ( ( bit_se1745604003318907178nteger @ N @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
      = ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% take_bit_of_2
thf(fact_4732_take__bit__of__2,axiom,
    ! [N: nat] :
      ( ( bit_se2923211474154528505it_int @ N @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
      = ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% take_bit_of_2
thf(fact_4733_take__bit__of__2,axiom,
    ! [N: nat] :
      ( ( bit_se2925701944663578781it_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% take_bit_of_2
thf(fact_4734_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X4: num,N: nat,A: int] :
      ( ( ord_less_eq_real @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X4 ) ) @ N ) @ ( ring_1_of_int_real @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N ) @ A ) ) ).

% neg_numeral_power_le_of_int_cancel_iff
thf(fact_4735_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X4: num,N: nat,A: int] :
      ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X4 ) ) @ N ) @ ( ring_18347121197199848620nteger @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N ) @ A ) ) ).

% neg_numeral_power_le_of_int_cancel_iff
thf(fact_4736_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X4: num,N: nat,A: int] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X4 ) ) @ N ) @ ( ring_1_of_int_rat @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N ) @ A ) ) ).

% neg_numeral_power_le_of_int_cancel_iff
thf(fact_4737_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X4: num,N: nat,A: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N ) @ ( ring_1_of_int_int @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N ) @ A ) ) ).

% neg_numeral_power_le_of_int_cancel_iff
thf(fact_4738_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X4: num,N: nat] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X4 ) ) @ N ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N ) ) ) ).

% of_int_le_neg_numeral_power_cancel_iff
thf(fact_4739_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X4: num,N: nat] :
      ( ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ A ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X4 ) ) @ N ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N ) ) ) ).

% of_int_le_neg_numeral_power_cancel_iff
thf(fact_4740_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X4: num,N: nat] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X4 ) ) @ N ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N ) ) ) ).

% of_int_le_neg_numeral_power_cancel_iff
thf(fact_4741_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X4: num,N: nat] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N ) ) ) ).

% of_int_le_neg_numeral_power_cancel_iff
thf(fact_4742_take__bit__eq__mask,axiom,
    ( bit_se2923211474154528505it_int
    = ( ^ [N2: nat,A4: int] : ( bit_se725231765392027082nd_int @ A4 @ ( bit_se2000444600071755411sk_int @ N2 ) ) ) ) ).

% take_bit_eq_mask
thf(fact_4743_take__bit__eq__mask,axiom,
    ( bit_se2925701944663578781it_nat
    = ( ^ [N2: nat,A4: nat] : ( bit_se727722235901077358nd_nat @ A4 @ ( bit_se2002935070580805687sk_nat @ N2 ) ) ) ) ).

% take_bit_eq_mask
thf(fact_4744_and_Oleft__commute,axiom,
    ! [B2: int,A: int,C: int] :
      ( ( bit_se725231765392027082nd_int @ B2 @ ( bit_se725231765392027082nd_int @ A @ C ) )
      = ( bit_se725231765392027082nd_int @ A @ ( bit_se725231765392027082nd_int @ B2 @ C ) ) ) ).

% and.left_commute
thf(fact_4745_and_Oleft__commute,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( ( bit_se727722235901077358nd_nat @ B2 @ ( bit_se727722235901077358nd_nat @ A @ C ) )
      = ( bit_se727722235901077358nd_nat @ A @ ( bit_se727722235901077358nd_nat @ B2 @ C ) ) ) ).

% and.left_commute
thf(fact_4746_and_Ocommute,axiom,
    ( bit_se725231765392027082nd_int
    = ( ^ [A4: int,B3: int] : ( bit_se725231765392027082nd_int @ B3 @ A4 ) ) ) ).

% and.commute
thf(fact_4747_and_Ocommute,axiom,
    ( bit_se727722235901077358nd_nat
    = ( ^ [A4: nat,B3: nat] : ( bit_se727722235901077358nd_nat @ B3 @ A4 ) ) ) ).

% and.commute
thf(fact_4748_take__bit__of__int,axiom,
    ! [N: nat,K: int] :
      ( ( bit_se2923211474154528505it_int @ N @ ( ring_1_of_int_int @ K ) )
      = ( ring_1_of_int_int @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).

% take_bit_of_int
thf(fact_4749_of__int__mask__eq,axiom,
    ! [N: nat] :
      ( ( ring_1_of_int_int @ ( bit_se2000444600071755411sk_int @ N ) )
      = ( bit_se2000444600071755411sk_int @ N ) ) ).

% of_int_mask_eq
thf(fact_4750_and_Oassoc,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( bit_se725231765392027082nd_int @ ( bit_se725231765392027082nd_int @ A @ B2 ) @ C )
      = ( bit_se725231765392027082nd_int @ A @ ( bit_se725231765392027082nd_int @ B2 @ C ) ) ) ).

% and.assoc
thf(fact_4751_and_Oassoc,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( bit_se727722235901077358nd_nat @ ( bit_se727722235901077358nd_nat @ A @ B2 ) @ C )
      = ( bit_se727722235901077358nd_nat @ A @ ( bit_se727722235901077358nd_nat @ B2 @ C ) ) ) ).

% and.assoc
thf(fact_4752_of__int__and__eq,axiom,
    ! [K: int,L2: int] :
      ( ( ring_1_of_int_int @ ( bit_se725231765392027082nd_int @ K @ L2 ) )
      = ( bit_se725231765392027082nd_int @ ( ring_1_of_int_int @ K ) @ ( ring_1_of_int_int @ L2 ) ) ) ).

% of_int_and_eq
thf(fact_4753_take__bit__add,axiom,
    ! [N: nat,A: int,B2: int] :
      ( ( bit_se2923211474154528505it_int @ N @ ( plus_plus_int @ ( bit_se2923211474154528505it_int @ N @ A ) @ ( bit_se2923211474154528505it_int @ N @ B2 ) ) )
      = ( bit_se2923211474154528505it_int @ N @ ( plus_plus_int @ A @ B2 ) ) ) ).

% take_bit_add
thf(fact_4754_take__bit__add,axiom,
    ! [N: nat,A: nat,B2: nat] :
      ( ( bit_se2925701944663578781it_nat @ N @ ( plus_plus_nat @ ( bit_se2925701944663578781it_nat @ N @ A ) @ ( bit_se2925701944663578781it_nat @ N @ B2 ) ) )
      = ( bit_se2925701944663578781it_nat @ N @ ( plus_plus_nat @ A @ B2 ) ) ) ).

% take_bit_add
thf(fact_4755_take__bit__tightened,axiom,
    ! [N: nat,A: int,B2: int,M: nat] :
      ( ( ( bit_se2923211474154528505it_int @ N @ A )
        = ( bit_se2923211474154528505it_int @ N @ B2 ) )
     => ( ( ord_less_eq_nat @ M @ N )
       => ( ( bit_se2923211474154528505it_int @ M @ A )
          = ( bit_se2923211474154528505it_int @ M @ B2 ) ) ) ) ).

% take_bit_tightened
thf(fact_4756_take__bit__tightened,axiom,
    ! [N: nat,A: nat,B2: nat,M: nat] :
      ( ( ( bit_se2925701944663578781it_nat @ N @ A )
        = ( bit_se2925701944663578781it_nat @ N @ B2 ) )
     => ( ( ord_less_eq_nat @ M @ N )
       => ( ( bit_se2925701944663578781it_nat @ M @ A )
          = ( bit_se2925701944663578781it_nat @ M @ B2 ) ) ) ) ).

% take_bit_tightened
thf(fact_4757_take__bit__nat__less__eq__self,axiom,
    ! [N: nat,M: nat] : ( ord_less_eq_nat @ ( bit_se2925701944663578781it_nat @ N @ M ) @ M ) ).

% take_bit_nat_less_eq_self
thf(fact_4758_take__bit__tightened__less__eq__nat,axiom,
    ! [M: nat,N: nat,Q2: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_nat @ ( bit_se2925701944663578781it_nat @ M @ Q2 ) @ ( bit_se2925701944663578781it_nat @ N @ Q2 ) ) ) ).

% take_bit_tightened_less_eq_nat
thf(fact_4759_real__sqrt__less__mono,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_real @ X4 @ Y3 )
     => ( ord_less_real @ ( sqrt @ X4 ) @ ( sqrt @ Y3 ) ) ) ).

% real_sqrt_less_mono
thf(fact_4760_real__sqrt__le__mono,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_eq_real @ X4 @ Y3 )
     => ( ord_less_eq_real @ ( sqrt @ X4 ) @ ( sqrt @ Y3 ) ) ) ).

% real_sqrt_le_mono
thf(fact_4761_mult__of__int__commute,axiom,
    ! [X4: int,Y3: real] :
      ( ( times_times_real @ ( ring_1_of_int_real @ X4 ) @ Y3 )
      = ( times_times_real @ Y3 @ ( ring_1_of_int_real @ X4 ) ) ) ).

% mult_of_int_commute
thf(fact_4762_mult__of__int__commute,axiom,
    ! [X4: int,Y3: rat] :
      ( ( times_times_rat @ ( ring_1_of_int_rat @ X4 ) @ Y3 )
      = ( times_times_rat @ Y3 @ ( ring_1_of_int_rat @ X4 ) ) ) ).

% mult_of_int_commute
thf(fact_4763_mult__of__int__commute,axiom,
    ! [X4: int,Y3: int] :
      ( ( times_times_int @ ( ring_1_of_int_int @ X4 ) @ Y3 )
      = ( times_times_int @ Y3 @ ( ring_1_of_int_int @ X4 ) ) ) ).

% mult_of_int_commute
thf(fact_4764_real__sqrt__divide,axiom,
    ! [X4: real,Y3: real] :
      ( ( sqrt @ ( divide_divide_real @ X4 @ Y3 ) )
      = ( divide_divide_real @ ( sqrt @ X4 ) @ ( sqrt @ Y3 ) ) ) ).

% real_sqrt_divide
thf(fact_4765_real__sqrt__mult,axiom,
    ! [X4: real,Y3: real] :
      ( ( sqrt @ ( times_times_real @ X4 @ Y3 ) )
      = ( times_times_real @ ( sqrt @ X4 ) @ ( sqrt @ Y3 ) ) ) ).

% real_sqrt_mult
thf(fact_4766_real__sqrt__power,axiom,
    ! [X4: real,K: nat] :
      ( ( sqrt @ ( power_power_real @ X4 @ K ) )
      = ( power_power_real @ ( sqrt @ X4 ) @ K ) ) ).

% real_sqrt_power
thf(fact_4767_real__sqrt__minus,axiom,
    ! [X4: real] :
      ( ( sqrt @ ( uminus_uminus_real @ X4 ) )
      = ( uminus_uminus_real @ ( sqrt @ X4 ) ) ) ).

% real_sqrt_minus
thf(fact_4768_take__bit__minus,axiom,
    ! [N: nat,K: int] :
      ( ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ ( bit_se2923211474154528505it_int @ N @ K ) ) )
      = ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ K ) ) ) ).

% take_bit_minus
thf(fact_4769_take__bit__mult,axiom,
    ! [N: nat,K: int,L2: int] :
      ( ( bit_se2923211474154528505it_int @ N @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ ( bit_se2923211474154528505it_int @ N @ L2 ) ) )
      = ( bit_se2923211474154528505it_int @ N @ ( times_times_int @ K @ L2 ) ) ) ).

% take_bit_mult
thf(fact_4770_take__bit__diff,axiom,
    ! [N: nat,K: int,L2: int] :
      ( ( bit_se2923211474154528505it_int @ N @ ( minus_minus_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ ( bit_se2923211474154528505it_int @ N @ L2 ) ) )
      = ( bit_se2923211474154528505it_int @ N @ ( minus_minus_int @ K @ L2 ) ) ) ).

% take_bit_diff
thf(fact_4771_concat__bit__take__bit__eq,axiom,
    ! [N: nat,B2: int] :
      ( ( bit_concat_bit @ N @ ( bit_se2923211474154528505it_int @ N @ B2 ) )
      = ( bit_concat_bit @ N @ B2 ) ) ).

% concat_bit_take_bit_eq
thf(fact_4772_concat__bit__eq__iff,axiom,
    ! [N: nat,K: int,L2: int,R2: int,S: int] :
      ( ( ( bit_concat_bit @ N @ K @ L2 )
        = ( bit_concat_bit @ N @ R2 @ S ) )
      = ( ( ( bit_se2923211474154528505it_int @ N @ K )
          = ( bit_se2923211474154528505it_int @ N @ R2 ) )
        & ( L2 = S ) ) ) ).

% concat_bit_eq_iff
thf(fact_4773_less__eq__mask,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ N @ ( bit_se2002935070580805687sk_nat @ N ) ) ).

% less_eq_mask
thf(fact_4774_take__bit__eq__mask__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ( bit_se2923211474154528505it_int @ N @ K )
        = ( bit_se2000444600071755411sk_int @ N ) )
      = ( ( bit_se2923211474154528505it_int @ N @ ( plus_plus_int @ K @ one_one_int ) )
        = zero_zero_int ) ) ).

% take_bit_eq_mask_iff
thf(fact_4775_and__eq__minus__1__iff,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( ( bit_se3949692690581998587nteger @ A @ B2 )
        = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( ( A
          = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
        & ( B2
          = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).

% and_eq_minus_1_iff
thf(fact_4776_and__eq__minus__1__iff,axiom,
    ! [A: int,B2: int] :
      ( ( ( bit_se725231765392027082nd_int @ A @ B2 )
        = ( uminus_uminus_int @ one_one_int ) )
      = ( ( A
          = ( uminus_uminus_int @ one_one_int ) )
        & ( B2
          = ( uminus_uminus_int @ one_one_int ) ) ) ) ).

% and_eq_minus_1_iff
thf(fact_4777_real__sqrt__gt__zero,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ord_less_real @ zero_zero_real @ ( sqrt @ X4 ) ) ) ).

% real_sqrt_gt_zero
thf(fact_4778_real__sqrt__ge__zero,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ X4 ) ) ) ).

% real_sqrt_ge_zero
thf(fact_4779_real__sqrt__eq__zero__cancel,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ( sqrt @ X4 )
          = zero_zero_real )
       => ( X4 = zero_zero_real ) ) ) ).

% real_sqrt_eq_zero_cancel
thf(fact_4780_real__sqrt__ge__one,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ one_one_real @ X4 )
     => ( ord_less_eq_real @ one_one_real @ ( sqrt @ X4 ) ) ) ).

% real_sqrt_ge_one
thf(fact_4781_take__bit__tightened__less__eq__int,axiom,
    ! [M: nat,N: nat,K: int] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ M @ K ) @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).

% take_bit_tightened_less_eq_int
thf(fact_4782_AND__upper2_H,axiom,
    ! [Y3: int,Z: int,X4: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
     => ( ( ord_less_eq_int @ Y3 @ Z )
       => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X4 @ Y3 ) @ Z ) ) ) ).

% AND_upper2'
thf(fact_4783_AND__upper1_H,axiom,
    ! [Y3: int,Z: int,Ya: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
     => ( ( ord_less_eq_int @ Y3 @ Z )
       => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ Y3 @ Ya ) @ Z ) ) ) ).

% AND_upper1'
thf(fact_4784_AND__upper2,axiom,
    ! [Y3: int,X4: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
     => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X4 @ Y3 ) @ Y3 ) ) ).

% AND_upper2
thf(fact_4785_AND__upper1,axiom,
    ! [X4: int,Y3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X4 )
     => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X4 @ Y3 ) @ X4 ) ) ).

% AND_upper1
thf(fact_4786_AND__lower,axiom,
    ! [X4: int,Y3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X4 )
     => ( ord_less_eq_int @ zero_zero_int @ ( bit_se725231765392027082nd_int @ X4 @ Y3 ) ) ) ).

% AND_lower
thf(fact_4787_take__bit__int__less__eq__self__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ K )
      = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).

% take_bit_int_less_eq_self_iff
thf(fact_4788_take__bit__nonnegative,axiom,
    ! [N: nat,K: int] : ( ord_less_eq_int @ zero_zero_int @ ( bit_se2923211474154528505it_int @ N @ K ) ) ).

% take_bit_nonnegative
thf(fact_4789_signed__take__bit__eq__iff__take__bit__eq,axiom,
    ! [N: nat,A: int,B2: int] :
      ( ( ( bit_ri631733984087533419it_int @ N @ A )
        = ( bit_ri631733984087533419it_int @ N @ B2 ) )
      = ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ A )
        = ( bit_se2923211474154528505it_int @ ( suc @ N ) @ B2 ) ) ) ).

% signed_take_bit_eq_iff_take_bit_eq
thf(fact_4790_not__take__bit__negative,axiom,
    ! [N: nat,K: int] :
      ~ ( ord_less_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ zero_zero_int ) ).

% not_take_bit_negative
thf(fact_4791_take__bit__int__greater__self__iff,axiom,
    ! [K: int,N: nat] :
      ( ( ord_less_int @ K @ ( bit_se2923211474154528505it_int @ N @ K ) )
      = ( ord_less_int @ K @ zero_zero_int ) ) ).

% take_bit_int_greater_self_iff
thf(fact_4792_signed__take__bit__take__bit,axiom,
    ! [M: nat,N: nat,A: int] :
      ( ( bit_ri631733984087533419it_int @ M @ ( bit_se2923211474154528505it_int @ N @ A ) )
      = ( if_int_int @ ( ord_less_eq_nat @ N @ M ) @ ( bit_se2923211474154528505it_int @ N ) @ ( bit_ri631733984087533419it_int @ M ) @ A ) ) ).

% signed_take_bit_take_bit
thf(fact_4793_take__bit__unset__bit__eq,axiom,
    ! [N: nat,M: nat,A: int] :
      ( ( ( ord_less_eq_nat @ N @ M )
       => ( ( bit_se2923211474154528505it_int @ N @ ( bit_se4203085406695923979it_int @ M @ A ) )
          = ( bit_se2923211474154528505it_int @ N @ A ) ) )
      & ( ~ ( ord_less_eq_nat @ N @ M )
       => ( ( bit_se2923211474154528505it_int @ N @ ( bit_se4203085406695923979it_int @ M @ A ) )
          = ( bit_se4203085406695923979it_int @ M @ ( bit_se2923211474154528505it_int @ N @ A ) ) ) ) ) ).

% take_bit_unset_bit_eq
thf(fact_4794_take__bit__unset__bit__eq,axiom,
    ! [N: nat,M: nat,A: nat] :
      ( ( ( ord_less_eq_nat @ N @ M )
       => ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se4205575877204974255it_nat @ M @ A ) )
          = ( bit_se2925701944663578781it_nat @ N @ A ) ) )
      & ( ~ ( ord_less_eq_nat @ N @ M )
       => ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se4205575877204974255it_nat @ M @ A ) )
          = ( bit_se4205575877204974255it_nat @ M @ ( bit_se2925701944663578781it_nat @ N @ A ) ) ) ) ) ).

% take_bit_unset_bit_eq
thf(fact_4795_take__bit__set__bit__eq,axiom,
    ! [N: nat,M: nat,A: int] :
      ( ( ( ord_less_eq_nat @ N @ M )
       => ( ( bit_se2923211474154528505it_int @ N @ ( bit_se7879613467334960850it_int @ M @ A ) )
          = ( bit_se2923211474154528505it_int @ N @ A ) ) )
      & ( ~ ( ord_less_eq_nat @ N @ M )
       => ( ( bit_se2923211474154528505it_int @ N @ ( bit_se7879613467334960850it_int @ M @ A ) )
          = ( bit_se7879613467334960850it_int @ M @ ( bit_se2923211474154528505it_int @ N @ A ) ) ) ) ) ).

% take_bit_set_bit_eq
thf(fact_4796_take__bit__set__bit__eq,axiom,
    ! [N: nat,M: nat,A: nat] :
      ( ( ( ord_less_eq_nat @ N @ M )
       => ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se7882103937844011126it_nat @ M @ A ) )
          = ( bit_se2925701944663578781it_nat @ N @ A ) ) )
      & ( ~ ( ord_less_eq_nat @ N @ M )
       => ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se7882103937844011126it_nat @ M @ A ) )
          = ( bit_se7882103937844011126it_nat @ M @ ( bit_se2925701944663578781it_nat @ N @ A ) ) ) ) ) ).

% take_bit_set_bit_eq
thf(fact_4797_take__bit__flip__bit__eq,axiom,
    ! [N: nat,M: nat,A: int] :
      ( ( ( ord_less_eq_nat @ N @ M )
       => ( ( bit_se2923211474154528505it_int @ N @ ( bit_se2159334234014336723it_int @ M @ A ) )
          = ( bit_se2923211474154528505it_int @ N @ A ) ) )
      & ( ~ ( ord_less_eq_nat @ N @ M )
       => ( ( bit_se2923211474154528505it_int @ N @ ( bit_se2159334234014336723it_int @ M @ A ) )
          = ( bit_se2159334234014336723it_int @ M @ ( bit_se2923211474154528505it_int @ N @ A ) ) ) ) ) ).

% take_bit_flip_bit_eq
thf(fact_4798_take__bit__flip__bit__eq,axiom,
    ! [N: nat,M: nat,A: nat] :
      ( ( ( ord_less_eq_nat @ N @ M )
       => ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se2161824704523386999it_nat @ M @ A ) )
          = ( bit_se2925701944663578781it_nat @ N @ A ) ) )
      & ( ~ ( ord_less_eq_nat @ N @ M )
       => ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se2161824704523386999it_nat @ M @ A ) )
          = ( bit_se2161824704523386999it_nat @ M @ ( bit_se2925701944663578781it_nat @ N @ A ) ) ) ) ) ).

% take_bit_flip_bit_eq
thf(fact_4799_mask__nonnegative__int,axiom,
    ! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( bit_se2000444600071755411sk_int @ N ) ) ).

% mask_nonnegative_int
thf(fact_4800_not__mask__negative__int,axiom,
    ! [N: nat] :
      ~ ( ord_less_int @ ( bit_se2000444600071755411sk_int @ N ) @ zero_zero_int ) ).

% not_mask_negative_int
thf(fact_4801_real__div__sqrt,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( divide_divide_real @ X4 @ ( sqrt @ X4 ) )
        = ( sqrt @ X4 ) ) ) ).

% real_div_sqrt
thf(fact_4802_sqrt__add__le__add__sqrt,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
       => ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ X4 @ Y3 ) ) @ ( plus_plus_real @ ( sqrt @ X4 ) @ ( sqrt @ Y3 ) ) ) ) ) ).

% sqrt_add_le_add_sqrt
thf(fact_4803_take__bit__signed__take__bit,axiom,
    ! [M: nat,N: nat,A: int] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
     => ( ( bit_se2923211474154528505it_int @ M @ ( bit_ri631733984087533419it_int @ N @ A ) )
        = ( bit_se2923211474154528505it_int @ M @ A ) ) ) ).

% take_bit_signed_take_bit
thf(fact_4804_le__real__sqrt__sumsq,axiom,
    ! [X4: real,Y3: real] : ( ord_less_eq_real @ X4 @ ( sqrt @ ( plus_plus_real @ ( times_times_real @ X4 @ X4 ) @ ( times_times_real @ Y3 @ Y3 ) ) ) ) ).

% le_real_sqrt_sumsq
thf(fact_4805_and__less__eq,axiom,
    ! [L2: int,K: int] :
      ( ( ord_less_int @ L2 @ zero_zero_int )
     => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ K @ L2 ) @ K ) ) ).

% and_less_eq
thf(fact_4806_AND__upper1_H_H,axiom,
    ! [Y3: int,Z: int,Ya: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
     => ( ( ord_less_int @ Y3 @ Z )
       => ( ord_less_int @ ( bit_se725231765392027082nd_int @ Y3 @ Ya ) @ Z ) ) ) ).

% AND_upper1''
thf(fact_4807_AND__upper2_H_H,axiom,
    ! [Y3: int,Z: int,X4: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
     => ( ( ord_less_int @ Y3 @ Z )
       => ( ord_less_int @ ( bit_se725231765392027082nd_int @ X4 @ Y3 ) @ Z ) ) ) ).

% AND_upper2''
thf(fact_4808_take__bit__decr__eq,axiom,
    ! [N: nat,K: int] :
      ( ( ( bit_se2923211474154528505it_int @ N @ K )
       != zero_zero_int )
     => ( ( bit_se2923211474154528505it_int @ N @ ( minus_minus_int @ K @ one_one_int ) )
        = ( minus_minus_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ one_one_int ) ) ) ).

% take_bit_decr_eq
thf(fact_4809_real__of__int__div4,axiom,
    ! [N: int,X4: int] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( divide_divide_int @ N @ X4 ) ) @ ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ X4 ) ) ) ).

% real_of_int_div4
thf(fact_4810_real__of__int__div,axiom,
    ! [D: int,N: int] :
      ( ( dvd_dvd_int @ D @ N )
     => ( ( ring_1_of_int_real @ ( divide_divide_int @ N @ D ) )
        = ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ D ) ) ) ) ).

% real_of_int_div
thf(fact_4811_less__mask,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
     => ( ord_less_nat @ N @ ( bit_se2002935070580805687sk_nat @ N ) ) ) ).

% less_mask
thf(fact_4812_take__bit__eq__mask__iff__exp__dvd,axiom,
    ! [N: nat,K: int] :
      ( ( ( bit_se2923211474154528505it_int @ N @ K )
        = ( bit_se2000444600071755411sk_int @ N ) )
      = ( dvd_dvd_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ ( plus_plus_int @ K @ one_one_int ) ) ) ).

% take_bit_eq_mask_iff_exp_dvd
thf(fact_4813_even__and__iff,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se3949692690581998587nteger @ A @ B2 ) )
      = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
        | ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) ) ) ).

% even_and_iff
thf(fact_4814_even__and__iff,axiom,
    ! [A: int,B2: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ A @ B2 ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
        | ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) ) ).

% even_and_iff
thf(fact_4815_even__and__iff,axiom,
    ! [A: nat,B2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ A @ B2 ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
        | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) ) ).

% even_and_iff
thf(fact_4816_sqrt2__less__2,axiom,
    ord_less_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).

% sqrt2_less_2
thf(fact_4817_of__int__nonneg,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) ) ) ).

% of_int_nonneg
thf(fact_4818_of__int__nonneg,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) ) ) ).

% of_int_nonneg
thf(fact_4819_of__int__nonneg,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) ) ) ).

% of_int_nonneg
thf(fact_4820_of__int__pos,axiom,
    ! [Z: int] :
      ( ( ord_less_int @ zero_zero_int @ Z )
     => ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) ) ) ).

% of_int_pos
thf(fact_4821_of__int__pos,axiom,
    ! [Z: int] :
      ( ( ord_less_int @ zero_zero_int @ Z )
     => ( ord_less_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) ) ) ).

% of_int_pos
thf(fact_4822_of__int__pos,axiom,
    ! [Z: int] :
      ( ( ord_less_int @ zero_zero_int @ Z )
     => ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) ) ) ).

% of_int_pos
thf(fact_4823_even__and__iff__int,axiom,
    ! [K: int,L2: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ K @ L2 ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
        | ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) ) ).

% even_and_iff_int
thf(fact_4824_of__int__neg__numeral,axiom,
    ! [K: num] :
      ( ( ring_1_of_int_real @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
      = ( uminus_uminus_real @ ( numeral_numeral_real @ K ) ) ) ).

% of_int_neg_numeral
thf(fact_4825_of__int__neg__numeral,axiom,
    ! [K: num] :
      ( ( ring_1_of_int_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) ).

% of_int_neg_numeral
thf(fact_4826_of__int__neg__numeral,axiom,
    ! [K: num] :
      ( ( ring_17405671764205052669omplex @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) ) ).

% of_int_neg_numeral
thf(fact_4827_of__int__neg__numeral,axiom,
    ! [K: num] :
      ( ( ring_18347121197199848620nteger @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) ) ).

% of_int_neg_numeral
thf(fact_4828_of__int__neg__numeral,axiom,
    ! [K: num] :
      ( ( ring_1_of_int_rat @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) ) ).

% of_int_neg_numeral
thf(fact_4829_int__le__real__less,axiom,
    ( ord_less_eq_int
    = ( ^ [N2: int,M6: int] : ( ord_less_real @ ( ring_1_of_int_real @ N2 ) @ ( plus_plus_real @ ( ring_1_of_int_real @ M6 ) @ one_one_real ) ) ) ) ).

% int_le_real_less
thf(fact_4830_int__less__real__le,axiom,
    ( ord_less_int
    = ( ^ [N2: int,M6: int] : ( ord_less_eq_real @ ( plus_plus_real @ ( ring_1_of_int_real @ N2 ) @ one_one_real ) @ ( ring_1_of_int_real @ M6 ) ) ) ) ).

% int_less_real_le
thf(fact_4831_real__of__int__div__aux,axiom,
    ! [X4: int,D: int] :
      ( ( divide_divide_real @ ( ring_1_of_int_real @ X4 ) @ ( ring_1_of_int_real @ D ) )
      = ( plus_plus_real @ ( ring_1_of_int_real @ ( divide_divide_int @ X4 @ D ) ) @ ( divide_divide_real @ ( ring_1_of_int_real @ ( modulo_modulo_int @ X4 @ D ) ) @ ( ring_1_of_int_real @ D ) ) ) ) ).

% real_of_int_div_aux
thf(fact_4832_take__bit__Suc__bit0,axiom,
    ! [N: nat,K: num] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
      = ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% take_bit_Suc_bit0
thf(fact_4833_take__bit__Suc__bit0,axiom,
    ! [N: nat,K: num] :
      ( ( bit_se2925701944663578781it_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
      = ( times_times_nat @ ( bit_se2925701944663578781it_nat @ N @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% take_bit_Suc_bit0
thf(fact_4834_take__bit__eq__mod,axiom,
    ( bit_se1745604003318907178nteger
    = ( ^ [N2: nat,A4: code_integer] : ( modulo364778990260209775nteger @ A4 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% take_bit_eq_mod
thf(fact_4835_take__bit__eq__mod,axiom,
    ( bit_se2923211474154528505it_int
    = ( ^ [N2: nat,A4: int] : ( modulo_modulo_int @ A4 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% take_bit_eq_mod
thf(fact_4836_take__bit__eq__mod,axiom,
    ( bit_se2925701944663578781it_nat
    = ( ^ [N2: nat,A4: nat] : ( modulo_modulo_nat @ A4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% take_bit_eq_mod
thf(fact_4837_one__and__eq,axiom,
    ! [A: code_integer] :
      ( ( bit_se3949692690581998587nteger @ one_one_Code_integer @ A )
      = ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% one_and_eq
thf(fact_4838_one__and__eq,axiom,
    ! [A: int] :
      ( ( bit_se725231765392027082nd_int @ one_one_int @ A )
      = ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% one_and_eq
thf(fact_4839_one__and__eq,axiom,
    ! [A: nat] :
      ( ( bit_se727722235901077358nd_nat @ one_one_nat @ A )
      = ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% one_and_eq
thf(fact_4840_and__one__eq,axiom,
    ! [A: code_integer] :
      ( ( bit_se3949692690581998587nteger @ A @ one_one_Code_integer )
      = ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% and_one_eq
thf(fact_4841_and__one__eq,axiom,
    ! [A: int] :
      ( ( bit_se725231765392027082nd_int @ A @ one_one_int )
      = ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% and_one_eq
thf(fact_4842_and__one__eq,axiom,
    ! [A: nat] :
      ( ( bit_se727722235901077358nd_nat @ A @ one_one_nat )
      = ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% and_one_eq
thf(fact_4843_take__bit__nat__eq__self__iff,axiom,
    ! [N: nat,M: nat] :
      ( ( ( bit_se2925701944663578781it_nat @ N @ M )
        = M )
      = ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% take_bit_nat_eq_self_iff
thf(fact_4844_take__bit__nat__less__exp,axiom,
    ! [N: nat,M: nat] : ( ord_less_nat @ ( bit_se2925701944663578781it_nat @ N @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).

% take_bit_nat_less_exp
thf(fact_4845_take__bit__nat__eq__self,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
     => ( ( bit_se2925701944663578781it_nat @ N @ M )
        = M ) ) ).

% take_bit_nat_eq_self
thf(fact_4846_real__less__rsqrt,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y3 )
     => ( ord_less_real @ X4 @ ( sqrt @ Y3 ) ) ) ).

% real_less_rsqrt
thf(fact_4847_real__le__rsqrt,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_eq_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y3 )
     => ( ord_less_eq_real @ X4 @ ( sqrt @ Y3 ) ) ) ).

% real_le_rsqrt
thf(fact_4848_sqrt__le__D,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_eq_real @ ( sqrt @ X4 ) @ Y3 )
     => ( ord_less_eq_real @ X4 @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sqrt_le_D
thf(fact_4849_take__bit__nat__def,axiom,
    ( bit_se2925701944663578781it_nat
    = ( ^ [N2: nat,M6: nat] : ( modulo_modulo_nat @ M6 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% take_bit_nat_def
thf(fact_4850_take__bit__int__less__exp,axiom,
    ! [N: nat,K: int] : ( ord_less_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).

% take_bit_int_less_exp
thf(fact_4851_take__bit__int__def,axiom,
    ( bit_se2923211474154528505it_int
    = ( ^ [N2: nat,K3: int] : ( modulo_modulo_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% take_bit_int_def
thf(fact_4852_real__of__int__div2,axiom,
    ! [N: int,X4: int] : ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ X4 ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ N @ X4 ) ) ) ) ).

% real_of_int_div2
thf(fact_4853_real__of__int__div3,axiom,
    ! [N: int,X4: int] : ( ord_less_eq_real @ ( minus_minus_real @ ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ X4 ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ N @ X4 ) ) ) @ one_one_real ) ).

% real_of_int_div3
thf(fact_4854_take__bit__eq__0__iff,axiom,
    ! [N: nat,A: code_integer] :
      ( ( ( bit_se1745604003318907178nteger @ N @ A )
        = zero_z3403309356797280102nteger )
      = ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) @ A ) ) ).

% take_bit_eq_0_iff
thf(fact_4855_take__bit__eq__0__iff,axiom,
    ! [N: nat,A: int] :
      ( ( ( bit_se2923211474154528505it_int @ N @ A )
        = zero_zero_int )
      = ( dvd_dvd_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ A ) ) ).

% take_bit_eq_0_iff
thf(fact_4856_take__bit__eq__0__iff,axiom,
    ! [N: nat,A: nat] :
      ( ( ( bit_se2925701944663578781it_nat @ N @ A )
        = zero_zero_nat )
      = ( dvd_dvd_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ A ) ) ).

% take_bit_eq_0_iff
thf(fact_4857_take__bit__numeral__bit0,axiom,
    ! [L2: num,K: num] :
      ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L2 ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
      = ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L2 ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% take_bit_numeral_bit0
thf(fact_4858_take__bit__numeral__bit0,axiom,
    ! [L2: num,K: num] :
      ( ( bit_se2925701944663578781it_nat @ ( numeral_numeral_nat @ L2 ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
      = ( times_times_nat @ ( bit_se2925701944663578781it_nat @ ( pred_numeral @ L2 ) @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% take_bit_numeral_bit0
thf(fact_4859_take__bit__nat__less__self__iff,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ ( bit_se2925701944663578781it_nat @ N @ M ) @ M )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ M ) ) ).

% take_bit_nat_less_self_iff
thf(fact_4860_real__sqrt__unique,axiom,
    ! [Y3: real,X4: real] :
      ( ( ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = X4 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
       => ( ( sqrt @ X4 )
          = Y3 ) ) ) ).

% real_sqrt_unique
thf(fact_4861_real__le__lsqrt,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
       => ( ( ord_less_eq_real @ X4 @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
         => ( ord_less_eq_real @ ( sqrt @ X4 ) @ Y3 ) ) ) ) ).

% real_le_lsqrt
thf(fact_4862_lemma__real__divide__sqrt__less,axiom,
    ! [U: real] :
      ( ( ord_less_real @ zero_zero_real @ U )
     => ( ord_less_real @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ U ) ) ).

% lemma_real_divide_sqrt_less
thf(fact_4863_Suc__mask__eq__exp,axiom,
    ! [N: nat] :
      ( ( suc @ ( bit_se2002935070580805687sk_nat @ N ) )
      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).

% Suc_mask_eq_exp
thf(fact_4864_real__sqrt__sum__squares__eq__cancel2,axiom,
    ! [X4: real,Y3: real] :
      ( ( ( sqrt @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
        = Y3 )
     => ( X4 = zero_zero_real ) ) ).

% real_sqrt_sum_squares_eq_cancel2
thf(fact_4865_real__sqrt__sum__squares__eq__cancel,axiom,
    ! [X4: real,Y3: real] :
      ( ( ( sqrt @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
        = X4 )
     => ( Y3 = zero_zero_real ) ) ).

% real_sqrt_sum_squares_eq_cancel
thf(fact_4866_mask__nat__less__exp,axiom,
    ! [N: nat] : ( ord_less_nat @ ( bit_se2002935070580805687sk_nat @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).

% mask_nat_less_exp
thf(fact_4867_real__sqrt__sum__squares__triangle__ineq,axiom,
    ! [A: real,C: real,B2: real,D: real] : ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ ( plus_plus_real @ A @ C ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( plus_plus_real @ B2 @ D ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ B2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ C @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ D @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% real_sqrt_sum_squares_triangle_ineq
thf(fact_4868_real__sqrt__sum__squares__ge2,axiom,
    ! [Y3: real,X4: real] : ( ord_less_eq_real @ Y3 @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% real_sqrt_sum_squares_ge2
thf(fact_4869_real__sqrt__sum__squares__ge1,axiom,
    ! [X4: real,Y3: real] : ( ord_less_eq_real @ X4 @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% real_sqrt_sum_squares_ge1
thf(fact_4870_take__bit__Suc__minus__bit0,axiom,
    ! [N: nat,K: num] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
      = ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% take_bit_Suc_minus_bit0
thf(fact_4871_even__of__int__iff,axiom,
    ! [K: int] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( ring_18347121197199848620nteger @ K ) )
      = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ).

% even_of_int_iff
thf(fact_4872_even__of__int__iff,axiom,
    ! [K: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( ring_1_of_int_int @ K ) )
      = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ).

% even_of_int_iff
thf(fact_4873_take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,N: nat] :
      ( ( ord_less_eq_int @ K @ ( bit_se2923211474154528505it_int @ N @ K ) )
      = ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).

% take_bit_int_greater_eq_self_iff
thf(fact_4874_take__bit__int__less__self__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ K )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K ) ) ).

% take_bit_int_less_self_iff
thf(fact_4875_semiring__bit__operations__class_Oeven__mask__iff,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se2119862282449309892nteger @ N ) )
      = ( N = zero_zero_nat ) ) ).

% semiring_bit_operations_class.even_mask_iff
thf(fact_4876_semiring__bit__operations__class_Oeven__mask__iff,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2002935070580805687sk_nat @ N ) )
      = ( N = zero_zero_nat ) ) ).

% semiring_bit_operations_class.even_mask_iff
thf(fact_4877_semiring__bit__operations__class_Oeven__mask__iff,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2000444600071755411sk_int @ N ) )
      = ( N = zero_zero_nat ) ) ).

% semiring_bit_operations_class.even_mask_iff
thf(fact_4878_real__less__lsqrt,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
       => ( ( ord_less_real @ X4 @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
         => ( ord_less_real @ ( sqrt @ X4 ) @ Y3 ) ) ) ) ).

% real_less_lsqrt
thf(fact_4879_sqrt__sum__squares__le__sum,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
       => ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ X4 @ Y3 ) ) ) ) ).

% sqrt_sum_squares_le_sum
thf(fact_4880_sqrt__even__pow2,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( sqrt @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) )
        = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% sqrt_even_pow2
thf(fact_4881_take__bit__int__eq__self__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ( bit_se2923211474154528505it_int @ N @ K )
        = K )
      = ( ( ord_less_eq_int @ zero_zero_int @ K )
        & ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% take_bit_int_eq_self_iff
thf(fact_4882_take__bit__int__eq__self,axiom,
    ! [K: int,N: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ( ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
       => ( ( bit_se2923211474154528505it_int @ N @ K )
          = K ) ) ) ).

% take_bit_int_eq_self
thf(fact_4883_and__int__rec,axiom,
    ( bit_se725231765392027082nd_int
    = ( ^ [K3: int,L: int] :
          ( plus_plus_int
          @ ( zero_n2684676970156552555ol_int
            @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 )
              & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) )
          @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% and_int_rec
thf(fact_4884_mask__nat__def,axiom,
    ( bit_se2002935070580805687sk_nat
    = ( ^ [N2: nat] : ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) ) ).

% mask_nat_def
thf(fact_4885_take__bit__numeral__minus__bit0,axiom,
    ! [L2: num,K: num] :
      ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
      = ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% take_bit_numeral_minus_bit0
thf(fact_4886_mask__half__int,axiom,
    ! [N: nat] :
      ( ( divide_divide_int @ ( bit_se2000444600071755411sk_int @ N ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
      = ( bit_se2000444600071755411sk_int @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ).

% mask_half_int
thf(fact_4887_take__bit__incr__eq,axiom,
    ! [N: nat,K: int] :
      ( ( ( bit_se2923211474154528505it_int @ N @ K )
       != ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int ) )
     => ( ( bit_se2923211474154528505it_int @ N @ ( plus_plus_int @ K @ one_one_int ) )
        = ( plus_plus_int @ one_one_int @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ) ).

% take_bit_incr_eq
thf(fact_4888_mask__int__def,axiom,
    ( bit_se2000444600071755411sk_int
    = ( ^ [N2: nat] : ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ one_one_int ) ) ) ).

% mask_int_def
thf(fact_4889_take__bit__Suc__minus__1__eq,axiom,
    ! [N: nat] :
      ( ( bit_se1745604003318907178nteger @ ( suc @ N ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( suc @ N ) ) @ one_one_Code_integer ) ) ).

% take_bit_Suc_minus_1_eq
thf(fact_4890_take__bit__Suc__minus__1__eq,axiom,
    ! [N: nat] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( uminus_uminus_int @ one_one_int ) )
      = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N ) ) @ one_one_int ) ) ).

% take_bit_Suc_minus_1_eq
thf(fact_4891_take__bit__Suc__bit1,axiom,
    ! [N: nat,K: num] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% take_bit_Suc_bit1
thf(fact_4892_take__bit__Suc__bit1,axiom,
    ! [N: nat,K: num] :
      ( ( bit_se2925701944663578781it_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ N @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ).

% take_bit_Suc_bit1
thf(fact_4893_take__bit__numeral__minus__1__eq,axiom,
    ! [K: num] :
      ( ( bit_se1745604003318907178nteger @ ( numeral_numeral_nat @ K ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ K ) ) @ one_one_Code_integer ) ) ).

% take_bit_numeral_minus_1_eq
thf(fact_4894_take__bit__numeral__minus__1__eq,axiom,
    ! [K: num] :
      ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ K ) @ ( uminus_uminus_int @ one_one_int ) )
      = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ K ) ) @ one_one_int ) ) ).

% take_bit_numeral_minus_1_eq
thf(fact_4895_take__bit__Suc,axiom,
    ! [N: nat,A: code_integer] :
      ( ( bit_se1745604003318907178nteger @ ( suc @ N ) @ A )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( bit_se1745604003318907178nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).

% take_bit_Suc
thf(fact_4896_take__bit__Suc,axiom,
    ! [N: nat,A: int] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ A )
      = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).

% take_bit_Suc
thf(fact_4897_take__bit__Suc,axiom,
    ! [N: nat,A: nat] :
      ( ( bit_se2925701944663578781it_nat @ ( suc @ N ) @ A )
      = ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ N @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% take_bit_Suc
thf(fact_4898_mask__eq__exp__minus__1,axiom,
    ( bit_se2002935070580805687sk_nat
    = ( ^ [N2: nat] : ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) ) ).

% mask_eq_exp_minus_1
thf(fact_4899_mask__eq__exp__minus__1,axiom,
    ( bit_se2000444600071755411sk_int
    = ( ^ [N2: nat] : ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ one_one_int ) ) ) ).

% mask_eq_exp_minus_1
thf(fact_4900_real__sqrt__power__even,axiom,
    ! [N: nat,X4: real] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( ord_less_eq_real @ zero_zero_real @ X4 )
       => ( ( power_power_real @ ( sqrt @ X4 ) @ N )
          = ( power_power_real @ X4 @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% real_sqrt_power_even
thf(fact_4901_real__sqrt__sum__squares__mult__ge__zero,axiom,
    ! [X4: real,Y3: real,Xa: real,Ya: real] : ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ ( times_times_real @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_real @ ( power_power_real @ Xa @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Ya @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% real_sqrt_sum_squares_mult_ge_zero
thf(fact_4902_arith__geo__mean__sqrt,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
       => ( ord_less_eq_real @ ( sqrt @ ( times_times_real @ X4 @ Y3 ) ) @ ( divide_divide_real @ ( plus_plus_real @ X4 @ Y3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% arith_geo_mean_sqrt
thf(fact_4903_take__bit__int__less__eq,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ ( minus_minus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).

% take_bit_int_less_eq
thf(fact_4904_take__bit__int__greater__eq,axiom,
    ! [K: int,N: nat] :
      ( ( ord_less_int @ K @ zero_zero_int )
     => ( ord_less_eq_int @ ( plus_plus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).

% take_bit_int_greater_eq
thf(fact_4905_signed__take__bit__eq__take__bit__shift,axiom,
    ( bit_ri631733984087533419it_int
    = ( ^ [N2: nat,K3: int] : ( minus_minus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ ( plus_plus_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% signed_take_bit_eq_take_bit_shift
thf(fact_4906_stable__imp__take__bit__eq,axiom,
    ! [A: code_integer,N: nat] :
      ( ( ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = A )
     => ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
         => ( ( bit_se1745604003318907178nteger @ N @ A )
            = zero_z3403309356797280102nteger ) )
        & ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
         => ( ( bit_se1745604003318907178nteger @ N @ A )
            = ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) @ one_one_Code_integer ) ) ) ) ) ).

% stable_imp_take_bit_eq
thf(fact_4907_stable__imp__take__bit__eq,axiom,
    ! [A: int,N: nat] :
      ( ( ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = A )
     => ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
         => ( ( bit_se2923211474154528505it_int @ N @ A )
            = zero_zero_int ) )
        & ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
         => ( ( bit_se2923211474154528505it_int @ N @ A )
            = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int ) ) ) ) ) ).

% stable_imp_take_bit_eq
thf(fact_4908_stable__imp__take__bit__eq,axiom,
    ! [A: nat,N: nat] :
      ( ( ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = A )
     => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
         => ( ( bit_se2925701944663578781it_nat @ N @ A )
            = zero_zero_nat ) )
        & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
         => ( ( bit_se2925701944663578781it_nat @ N @ A )
            = ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ) ).

% stable_imp_take_bit_eq
thf(fact_4909_take__bit__numeral__bit1,axiom,
    ! [L2: num,K: num] :
      ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L2 ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L2 ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% take_bit_numeral_bit1
thf(fact_4910_take__bit__numeral__bit1,axiom,
    ! [L2: num,K: num] :
      ( ( bit_se2925701944663578781it_nat @ ( numeral_numeral_nat @ L2 ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ ( pred_numeral @ L2 ) @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ).

% take_bit_numeral_bit1
thf(fact_4911_take__bit__minus__small__eq,axiom,
    ! [K: int,N: nat] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ( ( ord_less_eq_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
       => ( ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ K ) )
          = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K ) ) ) ) ).

% take_bit_minus_small_eq
thf(fact_4912_one__le__exp__iff,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ one_one_real @ ( exp_real @ X4 ) )
      = ( ord_less_eq_real @ zero_zero_real @ X4 ) ) ).

% one_le_exp_iff
thf(fact_4913_exp__le__one__iff,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ ( exp_real @ X4 ) @ one_one_real )
      = ( ord_less_eq_real @ X4 @ zero_zero_real ) ) ).

% exp_le_one_iff
thf(fact_4914_exp__less__one__iff,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ ( exp_real @ X4 ) @ one_one_real )
      = ( ord_less_real @ X4 @ zero_zero_real ) ) ).

% exp_less_one_iff
thf(fact_4915_one__less__exp__iff,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ one_one_real @ ( exp_real @ X4 ) )
      = ( ord_less_real @ zero_zero_real @ X4 ) ) ).

% one_less_exp_iff
thf(fact_4916_real__exp__bound__lemma,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_eq_real @ X4 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_eq_real @ ( exp_real @ X4 ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X4 ) ) ) ) ) ).

% real_exp_bound_lemma
thf(fact_4917_exp__bound,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_eq_real @ X4 @ one_one_real )
       => ( ord_less_eq_real @ ( exp_real @ X4 ) @ ( plus_plus_real @ ( plus_plus_real @ one_one_real @ X4 ) @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% exp_bound
thf(fact_4918_arsinh__real__aux,axiom,
    ! [X4: real] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X4 @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ).

% arsinh_real_aux
thf(fact_4919_exp__less__cancel__iff,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_real @ ( exp_real @ X4 ) @ ( exp_real @ Y3 ) )
      = ( ord_less_real @ X4 @ Y3 ) ) ).

% exp_less_cancel_iff
thf(fact_4920_exp__less__mono,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_real @ X4 @ Y3 )
     => ( ord_less_real @ ( exp_real @ X4 ) @ ( exp_real @ Y3 ) ) ) ).

% exp_less_mono
thf(fact_4921_exp__le__cancel__iff,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_eq_real @ ( exp_real @ X4 ) @ ( exp_real @ Y3 ) )
      = ( ord_less_eq_real @ X4 @ Y3 ) ) ).

% exp_le_cancel_iff
thf(fact_4922_exp__zero,axiom,
    ( ( exp_complex @ zero_zero_complex )
    = one_one_complex ) ).

% exp_zero
thf(fact_4923_exp__zero,axiom,
    ( ( exp_real @ zero_zero_real )
    = one_one_real ) ).

% exp_zero
thf(fact_4924_exp__eq__one__iff,axiom,
    ! [X4: real] :
      ( ( ( exp_real @ X4 )
        = one_one_real )
      = ( X4 = zero_zero_real ) ) ).

% exp_eq_one_iff
thf(fact_4925_and__nat__numerals_I3_J,axiom,
    ! [X4: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X4 ) ) @ ( suc @ zero_zero_nat ) )
      = zero_zero_nat ) ).

% and_nat_numerals(3)
thf(fact_4926_and__nat__numerals_I1_J,axiom,
    ! [Y3: num] :
      ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) )
      = zero_zero_nat ) ).

% and_nat_numerals(1)
thf(fact_4927_and__nat__numerals_I2_J,axiom,
    ! [Y3: num] :
      ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) )
      = one_one_nat ) ).

% and_nat_numerals(2)
thf(fact_4928_and__nat__numerals_I4_J,axiom,
    ! [X4: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X4 ) ) @ ( suc @ zero_zero_nat ) )
      = one_one_nat ) ).

% and_nat_numerals(4)
thf(fact_4929_Suc__0__and__eq,axiom,
    ! [N: nat] :
      ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ N )
      = ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% Suc_0_and_eq
thf(fact_4930_and__Suc__0__eq,axiom,
    ! [N: nat] :
      ( ( bit_se727722235901077358nd_nat @ N @ ( suc @ zero_zero_nat ) )
      = ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% and_Suc_0_eq
thf(fact_4931_exp__times__arg__commute,axiom,
    ! [A3: complex] :
      ( ( times_times_complex @ ( exp_complex @ A3 ) @ A3 )
      = ( times_times_complex @ A3 @ ( exp_complex @ A3 ) ) ) ).

% exp_times_arg_commute
thf(fact_4932_exp__times__arg__commute,axiom,
    ! [A3: real] :
      ( ( times_times_real @ ( exp_real @ A3 ) @ A3 )
      = ( times_times_real @ A3 @ ( exp_real @ A3 ) ) ) ).

% exp_times_arg_commute
thf(fact_4933_exp__less__cancel,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_real @ ( exp_real @ X4 ) @ ( exp_real @ Y3 ) )
     => ( ord_less_real @ X4 @ Y3 ) ) ).

% exp_less_cancel
thf(fact_4934_and__nat__unfold,axiom,
    ( bit_se727722235901077358nd_nat
    = ( ^ [M6: nat,N2: nat] :
          ( if_nat
          @ ( ( M6 = zero_zero_nat )
            | ( N2 = zero_zero_nat ) )
          @ zero_zero_nat
          @ ( plus_plus_nat @ ( times_times_nat @ ( modulo_modulo_nat @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( divide_divide_nat @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% and_nat_unfold
thf(fact_4935_and__nat__rec,axiom,
    ( bit_se727722235901077358nd_nat
    = ( ^ [M6: nat,N2: nat] :
          ( plus_plus_nat
          @ ( zero_n2687167440665602831ol_nat
            @ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M6 )
              & ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
          @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( divide_divide_nat @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% and_nat_rec
thf(fact_4936_mult__exp__exp,axiom,
    ! [X4: complex,Y3: complex] :
      ( ( times_times_complex @ ( exp_complex @ X4 ) @ ( exp_complex @ Y3 ) )
      = ( exp_complex @ ( plus_plus_complex @ X4 @ Y3 ) ) ) ).

% mult_exp_exp
thf(fact_4937_mult__exp__exp,axiom,
    ! [X4: real,Y3: real] :
      ( ( times_times_real @ ( exp_real @ X4 ) @ ( exp_real @ Y3 ) )
      = ( exp_real @ ( plus_plus_real @ X4 @ Y3 ) ) ) ).

% mult_exp_exp
thf(fact_4938_exp__add__commuting,axiom,
    ! [X4: complex,Y3: complex] :
      ( ( ( times_times_complex @ X4 @ Y3 )
        = ( times_times_complex @ Y3 @ X4 ) )
     => ( ( exp_complex @ ( plus_plus_complex @ X4 @ Y3 ) )
        = ( times_times_complex @ ( exp_complex @ X4 ) @ ( exp_complex @ Y3 ) ) ) ) ).

% exp_add_commuting
thf(fact_4939_exp__add__commuting,axiom,
    ! [X4: real,Y3: real] :
      ( ( ( times_times_real @ X4 @ Y3 )
        = ( times_times_real @ Y3 @ X4 ) )
     => ( ( exp_real @ ( plus_plus_real @ X4 @ Y3 ) )
        = ( times_times_real @ ( exp_real @ X4 ) @ ( exp_real @ Y3 ) ) ) ) ).

% exp_add_commuting
thf(fact_4940_exp__diff,axiom,
    ! [X4: complex,Y3: complex] :
      ( ( exp_complex @ ( minus_minus_complex @ X4 @ Y3 ) )
      = ( divide1717551699836669952omplex @ ( exp_complex @ X4 ) @ ( exp_complex @ Y3 ) ) ) ).

% exp_diff
thf(fact_4941_exp__diff,axiom,
    ! [X4: real,Y3: real] :
      ( ( exp_real @ ( minus_minus_real @ X4 @ Y3 ) )
      = ( divide_divide_real @ ( exp_real @ X4 ) @ ( exp_real @ Y3 ) ) ) ).

% exp_diff
thf(fact_4942_not__exp__less__zero,axiom,
    ! [X4: real] :
      ~ ( ord_less_real @ ( exp_real @ X4 ) @ zero_zero_real ) ).

% not_exp_less_zero
thf(fact_4943_exp__gt__zero,axiom,
    ! [X4: real] : ( ord_less_real @ zero_zero_real @ ( exp_real @ X4 ) ) ).

% exp_gt_zero
thf(fact_4944_exp__total,axiom,
    ! [Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ Y3 )
     => ? [X3: real] :
          ( ( exp_real @ X3 )
          = Y3 ) ) ).

% exp_total
thf(fact_4945_exp__ge__zero,axiom,
    ! [X4: real] : ( ord_less_eq_real @ zero_zero_real @ ( exp_real @ X4 ) ) ).

% exp_ge_zero
thf(fact_4946_not__exp__le__zero,axiom,
    ! [X4: real] :
      ~ ( ord_less_eq_real @ ( exp_real @ X4 ) @ zero_zero_real ) ).

% not_exp_le_zero
thf(fact_4947_exp__minus__inverse,axiom,
    ! [X4: real] :
      ( ( times_times_real @ ( exp_real @ X4 ) @ ( exp_real @ ( uminus_uminus_real @ X4 ) ) )
      = one_one_real ) ).

% exp_minus_inverse
thf(fact_4948_exp__minus__inverse,axiom,
    ! [X4: complex] :
      ( ( times_times_complex @ ( exp_complex @ X4 ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ X4 ) ) )
      = one_one_complex ) ).

% exp_minus_inverse
thf(fact_4949_exp__gt__one,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ord_less_real @ one_one_real @ ( exp_real @ X4 ) ) ) ).

% exp_gt_one
thf(fact_4950_exp__ge__add__one__self,axiom,
    ! [X4: real] : ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ X4 ) @ ( exp_real @ X4 ) ) ).

% exp_ge_add_one_self
thf(fact_4951_exp__ge__add__one__self__aux,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ X4 ) @ ( exp_real @ X4 ) ) ) ).

% exp_ge_add_one_self_aux
thf(fact_4952_lemma__exp__total,axiom,
    ! [Y3: real] :
      ( ( ord_less_eq_real @ one_one_real @ Y3 )
     => ? [X3: real] :
          ( ( ord_less_eq_real @ zero_zero_real @ X3 )
          & ( ord_less_eq_real @ X3 @ ( minus_minus_real @ Y3 @ one_one_real ) )
          & ( ( exp_real @ X3 )
            = Y3 ) ) ) ).

% lemma_exp_total
thf(fact_4953_exp__le,axiom,
    ord_less_eq_real @ ( exp_real @ one_one_real ) @ ( numeral_numeral_real @ ( bit1 @ one ) ) ).

% exp_le
thf(fact_4954_exp__double,axiom,
    ! [Z: complex] :
      ( ( exp_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ Z ) )
      = ( power_power_complex @ ( exp_complex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% exp_double
thf(fact_4955_exp__double,axiom,
    ! [Z: real] :
      ( ( exp_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z ) )
      = ( power_power_real @ ( exp_real @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% exp_double
thf(fact_4956_exp__half__le2,axiom,
    ord_less_eq_real @ ( exp_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).

% exp_half_le2
thf(fact_4957_arcosh__1,axiom,
    ( ( arcosh_real @ one_one_real )
    = zero_zero_real ) ).

% arcosh_1
thf(fact_4958_floor__exists1,axiom,
    ! [X4: real] :
    ? [X3: int] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ X3 ) @ X4 )
      & ( ord_less_real @ X4 @ ( ring_1_of_int_real @ ( plus_plus_int @ X3 @ one_one_int ) ) )
      & ! [Y5: int] :
          ( ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Y5 ) @ X4 )
            & ( ord_less_real @ X4 @ ( ring_1_of_int_real @ ( plus_plus_int @ Y5 @ one_one_int ) ) ) )
         => ( Y5 = X3 ) ) ) ).

% floor_exists1
thf(fact_4959_floor__exists1,axiom,
    ! [X4: rat] :
    ? [X3: int] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ X3 ) @ X4 )
      & ( ord_less_rat @ X4 @ ( ring_1_of_int_rat @ ( plus_plus_int @ X3 @ one_one_int ) ) )
      & ! [Y5: int] :
          ( ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Y5 ) @ X4 )
            & ( ord_less_rat @ X4 @ ( ring_1_of_int_rat @ ( plus_plus_int @ Y5 @ one_one_int ) ) ) )
         => ( Y5 = X3 ) ) ) ).

% floor_exists1
thf(fact_4960_floor__exists,axiom,
    ! [X4: real] :
    ? [Z2: int] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z2 ) @ X4 )
      & ( ord_less_real @ X4 @ ( ring_1_of_int_real @ ( plus_plus_int @ Z2 @ one_one_int ) ) ) ) ).

% floor_exists
thf(fact_4961_floor__exists,axiom,
    ! [X4: rat] :
    ? [Z2: int] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z2 ) @ X4 )
      & ( ord_less_rat @ X4 @ ( ring_1_of_int_rat @ ( plus_plus_int @ Z2 @ one_one_int ) ) ) ) ).

% floor_exists
thf(fact_4962_take__bit__numeral__minus__bit1,axiom,
    ! [L2: num,K: num] :
      ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% take_bit_numeral_minus_bit1
thf(fact_4963_take__bit__Suc__minus__bit1,axiom,
    ! [N: nat,K: num] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% take_bit_Suc_minus_bit1
thf(fact_4964_tanh__real__altdef,axiom,
    ( tanh_real
    = ( ^ [X: real] : ( divide_divide_real @ ( minus_minus_real @ one_one_real @ ( exp_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X ) ) ) @ ( plus_plus_real @ one_one_real @ ( exp_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X ) ) ) ) ) ) ).

% tanh_real_altdef
thf(fact_4965_ln__one__minus__pos__lower__bound,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_eq_real @ X4 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_eq_real @ ( minus_minus_real @ ( uminus_uminus_real @ X4 ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( ln_ln_real @ ( minus_minus_real @ one_one_real @ X4 ) ) ) ) ) ).

% ln_one_minus_pos_lower_bound
thf(fact_4966_tanh__real__less__iff,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_real @ ( tanh_real @ X4 ) @ ( tanh_real @ Y3 ) )
      = ( ord_less_real @ X4 @ Y3 ) ) ).

% tanh_real_less_iff
thf(fact_4967_tanh__real__le__iff,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_eq_real @ ( tanh_real @ X4 ) @ ( tanh_real @ Y3 ) )
      = ( ord_less_eq_real @ X4 @ Y3 ) ) ).

% tanh_real_le_iff
thf(fact_4968_ln__one,axiom,
    ( ( ln_ln_real @ one_one_real )
    = zero_zero_real ) ).

% ln_one
thf(fact_4969_ln__inj__iff,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( ord_less_real @ zero_zero_real @ Y3 )
       => ( ( ( ln_ln_real @ X4 )
            = ( ln_ln_real @ Y3 ) )
          = ( X4 = Y3 ) ) ) ) ).

% ln_inj_iff
thf(fact_4970_ln__less__cancel__iff,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( ord_less_real @ zero_zero_real @ Y3 )
       => ( ( ord_less_real @ ( ln_ln_real @ X4 ) @ ( ln_ln_real @ Y3 ) )
          = ( ord_less_real @ X4 @ Y3 ) ) ) ) ).

% ln_less_cancel_iff
thf(fact_4971_tanh__real__neg__iff,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ ( tanh_real @ X4 ) @ zero_zero_real )
      = ( ord_less_real @ X4 @ zero_zero_real ) ) ).

% tanh_real_neg_iff
thf(fact_4972_tanh__real__pos__iff,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ ( tanh_real @ X4 ) )
      = ( ord_less_real @ zero_zero_real @ X4 ) ) ).

% tanh_real_pos_iff
thf(fact_4973_tanh__real__nonpos__iff,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ ( tanh_real @ X4 ) @ zero_zero_real )
      = ( ord_less_eq_real @ X4 @ zero_zero_real ) ) ).

% tanh_real_nonpos_iff
thf(fact_4974_tanh__real__nonneg__iff,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( tanh_real @ X4 ) )
      = ( ord_less_eq_real @ zero_zero_real @ X4 ) ) ).

% tanh_real_nonneg_iff
thf(fact_4975_pred__numeral__inc,axiom,
    ! [K: num] :
      ( ( pred_numeral @ ( inc @ K ) )
      = ( numeral_numeral_nat @ K ) ) ).

% pred_numeral_inc
thf(fact_4976_ln__le__cancel__iff,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( ord_less_real @ zero_zero_real @ Y3 )
       => ( ( ord_less_eq_real @ ( ln_ln_real @ X4 ) @ ( ln_ln_real @ Y3 ) )
          = ( ord_less_eq_real @ X4 @ Y3 ) ) ) ) ).

% ln_le_cancel_iff
thf(fact_4977_ln__eq__zero__iff,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( ( ln_ln_real @ X4 )
          = zero_zero_real )
        = ( X4 = one_one_real ) ) ) ).

% ln_eq_zero_iff
thf(fact_4978_ln__gt__zero__iff,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X4 ) )
        = ( ord_less_real @ one_one_real @ X4 ) ) ) ).

% ln_gt_zero_iff
thf(fact_4979_ln__less__zero__iff,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( ord_less_real @ ( ln_ln_real @ X4 ) @ zero_zero_real )
        = ( ord_less_real @ X4 @ one_one_real ) ) ) ).

% ln_less_zero_iff
thf(fact_4980_exp__ln,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( exp_real @ ( ln_ln_real @ X4 ) )
        = X4 ) ) ).

% exp_ln
thf(fact_4981_exp__ln__iff,axiom,
    ! [X4: real] :
      ( ( ( exp_real @ ( ln_ln_real @ X4 ) )
        = X4 )
      = ( ord_less_real @ zero_zero_real @ X4 ) ) ).

% exp_ln_iff
thf(fact_4982_ln__le__zero__iff,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( ord_less_eq_real @ ( ln_ln_real @ X4 ) @ zero_zero_real )
        = ( ord_less_eq_real @ X4 @ one_one_real ) ) ) ).

% ln_le_zero_iff
thf(fact_4983_ln__ge__zero__iff,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X4 ) )
        = ( ord_less_eq_real @ one_one_real @ X4 ) ) ) ).

% ln_ge_zero_iff
thf(fact_4984_add__neg__numeral__special_I5_J,axiom,
    ! [N: num] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
      = ( uminus_uminus_real @ ( numeral_numeral_real @ ( inc @ N ) ) ) ) ).

% add_neg_numeral_special(5)
thf(fact_4985_add__neg__numeral__special_I5_J,axiom,
    ! [N: num] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ N ) ) ) ) ).

% add_neg_numeral_special(5)
thf(fact_4986_add__neg__numeral__special_I5_J,axiom,
    ! [N: num] :
      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( inc @ N ) ) ) ) ).

% add_neg_numeral_special(5)
thf(fact_4987_add__neg__numeral__special_I5_J,axiom,
    ! [N: num] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( inc @ N ) ) ) ) ).

% add_neg_numeral_special(5)
thf(fact_4988_add__neg__numeral__special_I5_J,axiom,
    ! [N: num] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( inc @ N ) ) ) ) ).

% add_neg_numeral_special(5)
thf(fact_4989_add__neg__numeral__special_I6_J,axiom,
    ! [M: num] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ one_one_real ) )
      = ( uminus_uminus_real @ ( numeral_numeral_real @ ( inc @ M ) ) ) ) ).

% add_neg_numeral_special(6)
thf(fact_4990_add__neg__numeral__special_I6_J,axiom,
    ! [M: num] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ M ) ) ) ) ).

% add_neg_numeral_special(6)
thf(fact_4991_add__neg__numeral__special_I6_J,axiom,
    ! [M: num] :
      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( inc @ M ) ) ) ) ).

% add_neg_numeral_special(6)
thf(fact_4992_add__neg__numeral__special_I6_J,axiom,
    ! [M: num] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( inc @ M ) ) ) ) ).

% add_neg_numeral_special(6)
thf(fact_4993_add__neg__numeral__special_I6_J,axiom,
    ! [M: num] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ one_one_rat ) )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( inc @ M ) ) ) ) ).

% add_neg_numeral_special(6)
thf(fact_4994_diff__numeral__special_I5_J,axiom,
    ! [N: num] :
      ( ( minus_minus_real @ ( uminus_uminus_real @ one_one_real ) @ ( numeral_numeral_real @ N ) )
      = ( uminus_uminus_real @ ( numeral_numeral_real @ ( inc @ N ) ) ) ) ).

% diff_numeral_special(5)
thf(fact_4995_diff__numeral__special_I5_J,axiom,
    ! [N: num] :
      ( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ N ) ) ) ) ).

% diff_numeral_special(5)
thf(fact_4996_diff__numeral__special_I5_J,axiom,
    ! [N: num] :
      ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( numera6690914467698888265omplex @ N ) )
      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( inc @ N ) ) ) ) ).

% diff_numeral_special(5)
thf(fact_4997_diff__numeral__special_I5_J,axiom,
    ! [N: num] :
      ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ N ) )
      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( inc @ N ) ) ) ) ).

% diff_numeral_special(5)
thf(fact_4998_diff__numeral__special_I5_J,axiom,
    ! [N: num] :
      ( ( minus_minus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ N ) )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( inc @ N ) ) ) ) ).

% diff_numeral_special(5)
thf(fact_4999_diff__numeral__special_I6_J,axiom,
    ! [M: num] :
      ( ( minus_minus_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ one_one_real ) )
      = ( numeral_numeral_real @ ( inc @ M ) ) ) ).

% diff_numeral_special(6)
thf(fact_5000_diff__numeral__special_I6_J,axiom,
    ! [M: num] :
      ( ( minus_minus_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) )
      = ( numeral_numeral_int @ ( inc @ M ) ) ) ).

% diff_numeral_special(6)
thf(fact_5001_diff__numeral__special_I6_J,axiom,
    ! [M: num] :
      ( ( minus_minus_complex @ ( numera6690914467698888265omplex @ M ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
      = ( numera6690914467698888265omplex @ ( inc @ M ) ) ) ).

% diff_numeral_special(6)
thf(fact_5002_diff__numeral__special_I6_J,axiom,
    ! [M: num] :
      ( ( minus_8373710615458151222nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( numera6620942414471956472nteger @ ( inc @ M ) ) ) ).

% diff_numeral_special(6)
thf(fact_5003_diff__numeral__special_I6_J,axiom,
    ! [M: num] :
      ( ( minus_minus_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ one_one_rat ) )
      = ( numeral_numeral_rat @ ( inc @ M ) ) ) ).

% diff_numeral_special(6)
thf(fact_5004_num__induct,axiom,
    ! [P: num > $o,X4: num] :
      ( ( P @ one )
     => ( ! [X3: num] :
            ( ( P @ X3 )
           => ( P @ ( inc @ X3 ) ) )
       => ( P @ X4 ) ) ) ).

% num_induct
thf(fact_5005_add__inc,axiom,
    ! [X4: num,Y3: num] :
      ( ( plus_plus_num @ X4 @ ( inc @ Y3 ) )
      = ( inc @ ( plus_plus_num @ X4 @ Y3 ) ) ) ).

% add_inc
thf(fact_5006_ln__less__self,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ord_less_real @ ( ln_ln_real @ X4 ) @ X4 ) ) ).

% ln_less_self
thf(fact_5007_tanh__real__lt__1,axiom,
    ! [X4: real] : ( ord_less_real @ ( tanh_real @ X4 ) @ one_one_real ) ).

% tanh_real_lt_1
thf(fact_5008_inc_Osimps_I1_J,axiom,
    ( ( inc @ one )
    = ( bit0 @ one ) ) ).

% inc.simps(1)
thf(fact_5009_inc_Osimps_I2_J,axiom,
    ! [X4: num] :
      ( ( inc @ ( bit0 @ X4 ) )
      = ( bit1 @ X4 ) ) ).

% inc.simps(2)
thf(fact_5010_inc_Osimps_I3_J,axiom,
    ! [X4: num] :
      ( ( inc @ ( bit1 @ X4 ) )
      = ( bit0 @ ( inc @ X4 ) ) ) ).

% inc.simps(3)
thf(fact_5011_ln__bound,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ord_less_eq_real @ ( ln_ln_real @ X4 ) @ X4 ) ) ).

% ln_bound
thf(fact_5012_ln__gt__zero,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ one_one_real @ X4 )
     => ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X4 ) ) ) ).

% ln_gt_zero
thf(fact_5013_ln__less__zero,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( ord_less_real @ X4 @ one_one_real )
       => ( ord_less_real @ ( ln_ln_real @ X4 ) @ zero_zero_real ) ) ) ).

% ln_less_zero
thf(fact_5014_ln__gt__zero__imp__gt__one,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X4 ) )
     => ( ( ord_less_real @ zero_zero_real @ X4 )
       => ( ord_less_real @ one_one_real @ X4 ) ) ) ).

% ln_gt_zero_imp_gt_one
thf(fact_5015_ln__ge__zero,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ one_one_real @ X4 )
     => ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X4 ) ) ) ).

% ln_ge_zero
thf(fact_5016_add__One,axiom,
    ! [X4: num] :
      ( ( plus_plus_num @ X4 @ one )
      = ( inc @ X4 ) ) ).

% add_One
thf(fact_5017_mult__inc,axiom,
    ! [X4: num,Y3: num] :
      ( ( times_times_num @ X4 @ ( inc @ Y3 ) )
      = ( plus_plus_num @ ( times_times_num @ X4 @ Y3 ) @ X4 ) ) ).

% mult_inc
thf(fact_5018_tanh__real__gt__neg1,axiom,
    ! [X4: real] : ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( tanh_real @ X4 ) ) ).

% tanh_real_gt_neg1
thf(fact_5019_ln__ge__zero__imp__ge__one,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X4 ) )
     => ( ( ord_less_real @ zero_zero_real @ X4 )
       => ( ord_less_eq_real @ one_one_real @ X4 ) ) ) ).

% ln_ge_zero_imp_ge_one
thf(fact_5020_ln__add__one__self__le__self,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ord_less_eq_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X4 ) ) @ X4 ) ) ).

% ln_add_one_self_le_self
thf(fact_5021_ln__mult,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( ord_less_real @ zero_zero_real @ Y3 )
       => ( ( ln_ln_real @ ( times_times_real @ X4 @ Y3 ) )
          = ( plus_plus_real @ ( ln_ln_real @ X4 ) @ ( ln_ln_real @ Y3 ) ) ) ) ) ).

% ln_mult
thf(fact_5022_ln__eq__minus__one,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( ( ln_ln_real @ X4 )
          = ( minus_minus_real @ X4 @ one_one_real ) )
       => ( X4 = one_one_real ) ) ) ).

% ln_eq_minus_one
thf(fact_5023_ln__div,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( ord_less_real @ zero_zero_real @ Y3 )
       => ( ( ln_ln_real @ ( divide_divide_real @ X4 @ Y3 ) )
          = ( minus_minus_real @ ( ln_ln_real @ X4 ) @ ( ln_ln_real @ Y3 ) ) ) ) ) ).

% ln_div
thf(fact_5024_ln__ge__iff,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( ord_less_eq_real @ Y3 @ ( ln_ln_real @ X4 ) )
        = ( ord_less_eq_real @ ( exp_real @ Y3 ) @ X4 ) ) ) ).

% ln_ge_iff
thf(fact_5025_ln__x__over__x__mono,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_eq_real @ ( exp_real @ one_one_real ) @ X4 )
     => ( ( ord_less_eq_real @ X4 @ Y3 )
       => ( ord_less_eq_real @ ( divide_divide_real @ ( ln_ln_real @ Y3 ) @ Y3 ) @ ( divide_divide_real @ ( ln_ln_real @ X4 ) @ X4 ) ) ) ) ).

% ln_x_over_x_mono
thf(fact_5026_numeral__inc,axiom,
    ! [X4: num] :
      ( ( numera6690914467698888265omplex @ ( inc @ X4 ) )
      = ( plus_plus_complex @ ( numera6690914467698888265omplex @ X4 ) @ one_one_complex ) ) ).

% numeral_inc
thf(fact_5027_numeral__inc,axiom,
    ! [X4: num] :
      ( ( numeral_numeral_real @ ( inc @ X4 ) )
      = ( plus_plus_real @ ( numeral_numeral_real @ X4 ) @ one_one_real ) ) ).

% numeral_inc
thf(fact_5028_numeral__inc,axiom,
    ! [X4: num] :
      ( ( numeral_numeral_rat @ ( inc @ X4 ) )
      = ( plus_plus_rat @ ( numeral_numeral_rat @ X4 ) @ one_one_rat ) ) ).

% numeral_inc
thf(fact_5029_numeral__inc,axiom,
    ! [X4: num] :
      ( ( numeral_numeral_nat @ ( inc @ X4 ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ X4 ) @ one_one_nat ) ) ).

% numeral_inc
thf(fact_5030_numeral__inc,axiom,
    ! [X4: num] :
      ( ( numeral_numeral_int @ ( inc @ X4 ) )
      = ( plus_plus_int @ ( numeral_numeral_int @ X4 ) @ one_one_int ) ) ).

% numeral_inc
thf(fact_5031_ln__2__less__1,axiom,
    ord_less_real @ ( ln_ln_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ one_one_real ).

% ln_2_less_1
thf(fact_5032_ln__le__minus__one,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ord_less_eq_real @ ( ln_ln_real @ X4 ) @ ( minus_minus_real @ X4 @ one_one_real ) ) ) ).

% ln_le_minus_one
thf(fact_5033_ln__diff__le,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( ord_less_real @ zero_zero_real @ Y3 )
       => ( ord_less_eq_real @ ( minus_minus_real @ ( ln_ln_real @ X4 ) @ ( ln_ln_real @ Y3 ) ) @ ( divide_divide_real @ ( minus_minus_real @ X4 @ Y3 ) @ Y3 ) ) ) ) ).

% ln_diff_le
thf(fact_5034_ln__add__one__self__le__self2,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X4 )
     => ( ord_less_eq_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X4 ) ) @ X4 ) ) ).

% ln_add_one_self_le_self2
thf(fact_5035_tanh__ln__real,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( tanh_real @ ( ln_ln_real @ X4 ) )
        = ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ).

% tanh_ln_real
thf(fact_5036_arcosh__real__def,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ one_one_real @ X4 )
     => ( ( arcosh_real @ X4 )
        = ( ln_ln_real @ ( plus_plus_real @ X4 @ ( sqrt @ ( minus_minus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ) ) ).

% arcosh_real_def
thf(fact_5037_ln__one__minus__pos__upper__bound,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_real @ X4 @ one_one_real )
       => ( ord_less_eq_real @ ( ln_ln_real @ ( minus_minus_real @ one_one_real @ X4 ) ) @ ( uminus_uminus_real @ X4 ) ) ) ) ).

% ln_one_minus_pos_upper_bound
thf(fact_5038_ln__sqrt,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( ln_ln_real @ ( sqrt @ X4 ) )
        = ( divide_divide_real @ ( ln_ln_real @ X4 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% ln_sqrt
thf(fact_5039_exists__least__lemma,axiom,
    ! [P: nat > $o] :
      ( ~ ( P @ zero_zero_nat )
     => ( ? [X_12: nat] : ( P @ X_12 )
       => ? [N4: nat] :
            ( ~ ( P @ N4 )
            & ( P @ ( suc @ N4 ) ) ) ) ) ).

% exists_least_lemma
thf(fact_5040_ex__le__of__int,axiom,
    ! [X4: real] :
    ? [Z2: int] : ( ord_less_eq_real @ X4 @ ( ring_1_of_int_real @ Z2 ) ) ).

% ex_le_of_int
thf(fact_5041_ex__le__of__int,axiom,
    ! [X4: rat] :
    ? [Z2: int] : ( ord_less_eq_rat @ X4 @ ( ring_1_of_int_rat @ Z2 ) ) ).

% ex_le_of_int
thf(fact_5042_ex__of__int__less,axiom,
    ! [X4: real] :
    ? [Z2: int] : ( ord_less_real @ ( ring_1_of_int_real @ Z2 ) @ X4 ) ).

% ex_of_int_less
thf(fact_5043_ex__of__int__less,axiom,
    ! [X4: rat] :
    ? [Z2: int] : ( ord_less_rat @ ( ring_1_of_int_rat @ Z2 ) @ X4 ) ).

% ex_of_int_less
thf(fact_5044_ex__less__of__int,axiom,
    ! [X4: real] :
    ? [Z2: int] : ( ord_less_real @ X4 @ ( ring_1_of_int_real @ Z2 ) ) ).

% ex_less_of_int
thf(fact_5045_ex__less__of__int,axiom,
    ! [X4: rat] :
    ? [Z2: int] : ( ord_less_rat @ X4 @ ( ring_1_of_int_rat @ Z2 ) ) ).

% ex_less_of_int
thf(fact_5046_tanh__altdef,axiom,
    ( tanh_real
    = ( ^ [X: real] : ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ X ) @ ( exp_real @ ( uminus_uminus_real @ X ) ) ) @ ( plus_plus_real @ ( exp_real @ X ) @ ( exp_real @ ( uminus_uminus_real @ X ) ) ) ) ) ) ).

% tanh_altdef
thf(fact_5047_tanh__altdef,axiom,
    ( tanh_complex
    = ( ^ [X: complex] : ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( exp_complex @ X ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ X ) ) ) @ ( plus_plus_complex @ ( exp_complex @ X ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ X ) ) ) ) ) ) ).

% tanh_altdef
thf(fact_5048_ln__one__plus__pos__lower__bound,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_eq_real @ X4 @ one_one_real )
       => ( ord_less_eq_real @ ( minus_minus_real @ X4 @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X4 ) ) ) ) ) ).

% ln_one_plus_pos_lower_bound
thf(fact_5049_artanh__def,axiom,
    ( artanh_real
    = ( ^ [X: real] : ( divide_divide_real @ ( ln_ln_real @ ( divide_divide_real @ ( plus_plus_real @ one_one_real @ X ) @ ( minus_minus_real @ one_one_real @ X ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% artanh_def
thf(fact_5050_abs__ln__one__plus__x__minus__x__bound__nonpos,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X4 )
     => ( ( ord_less_eq_real @ X4 @ zero_zero_real )
       => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X4 ) ) @ X4 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonpos
thf(fact_5051_round__unique,axiom,
    ! [X4: real,Y3: int] :
      ( ( ord_less_real @ ( minus_minus_real @ X4 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ Y3 ) )
     => ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Y3 ) @ ( plus_plus_real @ X4 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
       => ( ( archim8280529875227126926d_real @ X4 )
          = Y3 ) ) ) ).

% round_unique
thf(fact_5052_round__unique,axiom,
    ! [X4: rat,Y3: int] :
      ( ( ord_less_rat @ ( minus_minus_rat @ X4 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ Y3 ) )
     => ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Y3 ) @ ( plus_plus_rat @ X4 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) )
       => ( ( archim7778729529865785530nd_rat @ X4 )
          = Y3 ) ) ) ).

% round_unique
thf(fact_5053_arsinh__real__def,axiom,
    ( arsinh_real
    = ( ^ [X: real] : ( ln_ln_real @ ( plus_plus_real @ X @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ) ) ).

% arsinh_real_def
thf(fact_5054_abs__ln__one__plus__x__minus__x__bound,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X4 ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
     => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X4 ) ) @ X4 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% abs_ln_one_plus_x_minus_x_bound
thf(fact_5055_of__int__round__gt,axiom,
    ! [X4: real] : ( ord_less_real @ ( minus_minus_real @ X4 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X4 ) ) ) ).

% of_int_round_gt
thf(fact_5056_of__int__round__gt,axiom,
    ! [X4: rat] : ( ord_less_rat @ ( minus_minus_rat @ X4 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X4 ) ) ) ).

% of_int_round_gt
thf(fact_5057_of__int__round__ge,axiom,
    ! [X4: real] : ( ord_less_eq_real @ ( minus_minus_real @ X4 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X4 ) ) ) ).

% of_int_round_ge
thf(fact_5058_of__int__round__ge,axiom,
    ! [X4: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ X4 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X4 ) ) ) ).

% of_int_round_ge
thf(fact_5059_abs__abs,axiom,
    ! [A: real] :
      ( ( abs_abs_real @ ( abs_abs_real @ A ) )
      = ( abs_abs_real @ A ) ) ).

% abs_abs
thf(fact_5060_abs__abs,axiom,
    ! [A: int] :
      ( ( abs_abs_int @ ( abs_abs_int @ A ) )
      = ( abs_abs_int @ A ) ) ).

% abs_abs
thf(fact_5061_abs__abs,axiom,
    ! [A: code_integer] :
      ( ( abs_abs_Code_integer @ ( abs_abs_Code_integer @ A ) )
      = ( abs_abs_Code_integer @ A ) ) ).

% abs_abs
thf(fact_5062_abs__abs,axiom,
    ! [A: rat] :
      ( ( abs_abs_rat @ ( abs_abs_rat @ A ) )
      = ( abs_abs_rat @ A ) ) ).

% abs_abs
thf(fact_5063_abs__idempotent,axiom,
    ! [A: real] :
      ( ( abs_abs_real @ ( abs_abs_real @ A ) )
      = ( abs_abs_real @ A ) ) ).

% abs_idempotent
thf(fact_5064_abs__idempotent,axiom,
    ! [A: int] :
      ( ( abs_abs_int @ ( abs_abs_int @ A ) )
      = ( abs_abs_int @ A ) ) ).

% abs_idempotent
thf(fact_5065_abs__idempotent,axiom,
    ! [A: code_integer] :
      ( ( abs_abs_Code_integer @ ( abs_abs_Code_integer @ A ) )
      = ( abs_abs_Code_integer @ A ) ) ).

% abs_idempotent
thf(fact_5066_abs__idempotent,axiom,
    ! [A: rat] :
      ( ( abs_abs_rat @ ( abs_abs_rat @ A ) )
      = ( abs_abs_rat @ A ) ) ).

% abs_idempotent
thf(fact_5067_abs__0,axiom,
    ( ( abs_abs_Code_integer @ zero_z3403309356797280102nteger )
    = zero_z3403309356797280102nteger ) ).

% abs_0
thf(fact_5068_abs__0,axiom,
    ( ( abs_abs_real @ zero_zero_real )
    = zero_zero_real ) ).

% abs_0
thf(fact_5069_abs__0,axiom,
    ( ( abs_abs_rat @ zero_zero_rat )
    = zero_zero_rat ) ).

% abs_0
thf(fact_5070_abs__0,axiom,
    ( ( abs_abs_int @ zero_zero_int )
    = zero_zero_int ) ).

% abs_0
thf(fact_5071_abs__0__eq,axiom,
    ! [A: code_integer] :
      ( ( zero_z3403309356797280102nteger
        = ( abs_abs_Code_integer @ A ) )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% abs_0_eq
thf(fact_5072_abs__0__eq,axiom,
    ! [A: real] :
      ( ( zero_zero_real
        = ( abs_abs_real @ A ) )
      = ( A = zero_zero_real ) ) ).

% abs_0_eq
thf(fact_5073_abs__0__eq,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat
        = ( abs_abs_rat @ A ) )
      = ( A = zero_zero_rat ) ) ).

% abs_0_eq
thf(fact_5074_abs__0__eq,axiom,
    ! [A: int] :
      ( ( zero_zero_int
        = ( abs_abs_int @ A ) )
      = ( A = zero_zero_int ) ) ).

% abs_0_eq
thf(fact_5075_abs__eq__0,axiom,
    ! [A: code_integer] :
      ( ( ( abs_abs_Code_integer @ A )
        = zero_z3403309356797280102nteger )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% abs_eq_0
thf(fact_5076_abs__eq__0,axiom,
    ! [A: real] :
      ( ( ( abs_abs_real @ A )
        = zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% abs_eq_0
thf(fact_5077_abs__eq__0,axiom,
    ! [A: rat] :
      ( ( ( abs_abs_rat @ A )
        = zero_zero_rat )
      = ( A = zero_zero_rat ) ) ).

% abs_eq_0
thf(fact_5078_abs__eq__0,axiom,
    ! [A: int] :
      ( ( ( abs_abs_int @ A )
        = zero_zero_int )
      = ( A = zero_zero_int ) ) ).

% abs_eq_0
thf(fact_5079_abs__zero,axiom,
    ( ( abs_abs_Code_integer @ zero_z3403309356797280102nteger )
    = zero_z3403309356797280102nteger ) ).

% abs_zero
thf(fact_5080_abs__zero,axiom,
    ( ( abs_abs_real @ zero_zero_real )
    = zero_zero_real ) ).

% abs_zero
thf(fact_5081_abs__zero,axiom,
    ( ( abs_abs_rat @ zero_zero_rat )
    = zero_zero_rat ) ).

% abs_zero
thf(fact_5082_abs__zero,axiom,
    ( ( abs_abs_int @ zero_zero_int )
    = zero_zero_int ) ).

% abs_zero
thf(fact_5083_abs__numeral,axiom,
    ! [N: num] :
      ( ( abs_abs_Code_integer @ ( numera6620942414471956472nteger @ N ) )
      = ( numera6620942414471956472nteger @ N ) ) ).

% abs_numeral
thf(fact_5084_abs__numeral,axiom,
    ! [N: num] :
      ( ( abs_abs_real @ ( numeral_numeral_real @ N ) )
      = ( numeral_numeral_real @ N ) ) ).

% abs_numeral
thf(fact_5085_abs__numeral,axiom,
    ! [N: num] :
      ( ( abs_abs_rat @ ( numeral_numeral_rat @ N ) )
      = ( numeral_numeral_rat @ N ) ) ).

% abs_numeral
thf(fact_5086_abs__numeral,axiom,
    ! [N: num] :
      ( ( abs_abs_int @ ( numeral_numeral_int @ N ) )
      = ( numeral_numeral_int @ N ) ) ).

% abs_numeral
thf(fact_5087_abs__mult__self__eq,axiom,
    ! [A: code_integer] :
      ( ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ A ) )
      = ( times_3573771949741848930nteger @ A @ A ) ) ).

% abs_mult_self_eq
thf(fact_5088_abs__mult__self__eq,axiom,
    ! [A: real] :
      ( ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ A ) )
      = ( times_times_real @ A @ A ) ) ).

% abs_mult_self_eq
thf(fact_5089_abs__mult__self__eq,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ A ) )
      = ( times_times_rat @ A @ A ) ) ).

% abs_mult_self_eq
thf(fact_5090_abs__mult__self__eq,axiom,
    ! [A: int] :
      ( ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ A ) )
      = ( times_times_int @ A @ A ) ) ).

% abs_mult_self_eq
thf(fact_5091_abs__1,axiom,
    ( ( abs_abs_Code_integer @ one_one_Code_integer )
    = one_one_Code_integer ) ).

% abs_1
thf(fact_5092_abs__1,axiom,
    ( ( abs_abs_complex @ one_one_complex )
    = one_one_complex ) ).

% abs_1
thf(fact_5093_abs__1,axiom,
    ( ( abs_abs_real @ one_one_real )
    = one_one_real ) ).

% abs_1
thf(fact_5094_abs__1,axiom,
    ( ( abs_abs_rat @ one_one_rat )
    = one_one_rat ) ).

% abs_1
thf(fact_5095_abs__1,axiom,
    ( ( abs_abs_int @ one_one_int )
    = one_one_int ) ).

% abs_1
thf(fact_5096_abs__add__abs,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( abs_abs_Code_integer @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B2 ) ) )
      = ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B2 ) ) ) ).

% abs_add_abs
thf(fact_5097_abs__add__abs,axiom,
    ! [A: real,B2: real] :
      ( ( abs_abs_real @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B2 ) ) )
      = ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B2 ) ) ) ).

% abs_add_abs
thf(fact_5098_abs__add__abs,axiom,
    ! [A: rat,B2: rat] :
      ( ( abs_abs_rat @ ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B2 ) ) )
      = ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B2 ) ) ) ).

% abs_add_abs
thf(fact_5099_abs__add__abs,axiom,
    ! [A: int,B2: int] :
      ( ( abs_abs_int @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B2 ) ) )
      = ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B2 ) ) ) ).

% abs_add_abs
thf(fact_5100_abs__divide,axiom,
    ! [A: complex,B2: complex] :
      ( ( abs_abs_complex @ ( divide1717551699836669952omplex @ A @ B2 ) )
      = ( divide1717551699836669952omplex @ ( abs_abs_complex @ A ) @ ( abs_abs_complex @ B2 ) ) ) ).

% abs_divide
thf(fact_5101_abs__divide,axiom,
    ! [A: real,B2: real] :
      ( ( abs_abs_real @ ( divide_divide_real @ A @ B2 ) )
      = ( divide_divide_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B2 ) ) ) ).

% abs_divide
thf(fact_5102_abs__divide,axiom,
    ! [A: rat,B2: rat] :
      ( ( abs_abs_rat @ ( divide_divide_rat @ A @ B2 ) )
      = ( divide_divide_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B2 ) ) ) ).

% abs_divide
thf(fact_5103_abs__minus,axiom,
    ! [A: real] :
      ( ( abs_abs_real @ ( uminus_uminus_real @ A ) )
      = ( abs_abs_real @ A ) ) ).

% abs_minus
thf(fact_5104_abs__minus,axiom,
    ! [A: int] :
      ( ( abs_abs_int @ ( uminus_uminus_int @ A ) )
      = ( abs_abs_int @ A ) ) ).

% abs_minus
thf(fact_5105_abs__minus,axiom,
    ! [A: complex] :
      ( ( abs_abs_complex @ ( uminus1482373934393186551omplex @ A ) )
      = ( abs_abs_complex @ A ) ) ).

% abs_minus
thf(fact_5106_abs__minus,axiom,
    ! [A: code_integer] :
      ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ A ) )
      = ( abs_abs_Code_integer @ A ) ) ).

% abs_minus
thf(fact_5107_abs__minus,axiom,
    ! [A: rat] :
      ( ( abs_abs_rat @ ( uminus_uminus_rat @ A ) )
      = ( abs_abs_rat @ A ) ) ).

% abs_minus
thf(fact_5108_abs__minus__cancel,axiom,
    ! [A: real] :
      ( ( abs_abs_real @ ( uminus_uminus_real @ A ) )
      = ( abs_abs_real @ A ) ) ).

% abs_minus_cancel
thf(fact_5109_abs__minus__cancel,axiom,
    ! [A: int] :
      ( ( abs_abs_int @ ( uminus_uminus_int @ A ) )
      = ( abs_abs_int @ A ) ) ).

% abs_minus_cancel
thf(fact_5110_abs__minus__cancel,axiom,
    ! [A: code_integer] :
      ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ A ) )
      = ( abs_abs_Code_integer @ A ) ) ).

% abs_minus_cancel
thf(fact_5111_abs__minus__cancel,axiom,
    ! [A: rat] :
      ( ( abs_abs_rat @ ( uminus_uminus_rat @ A ) )
      = ( abs_abs_rat @ A ) ) ).

% abs_minus_cancel
thf(fact_5112_abs__dvd__iff,axiom,
    ! [M: real,K: real] :
      ( ( dvd_dvd_real @ ( abs_abs_real @ M ) @ K )
      = ( dvd_dvd_real @ M @ K ) ) ).

% abs_dvd_iff
thf(fact_5113_abs__dvd__iff,axiom,
    ! [M: int,K: int] :
      ( ( dvd_dvd_int @ ( abs_abs_int @ M ) @ K )
      = ( dvd_dvd_int @ M @ K ) ) ).

% abs_dvd_iff
thf(fact_5114_abs__dvd__iff,axiom,
    ! [M: code_integer,K: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( abs_abs_Code_integer @ M ) @ K )
      = ( dvd_dvd_Code_integer @ M @ K ) ) ).

% abs_dvd_iff
thf(fact_5115_abs__dvd__iff,axiom,
    ! [M: rat,K: rat] :
      ( ( dvd_dvd_rat @ ( abs_abs_rat @ M ) @ K )
      = ( dvd_dvd_rat @ M @ K ) ) ).

% abs_dvd_iff
thf(fact_5116_dvd__abs__iff,axiom,
    ! [M: real,K: real] :
      ( ( dvd_dvd_real @ M @ ( abs_abs_real @ K ) )
      = ( dvd_dvd_real @ M @ K ) ) ).

% dvd_abs_iff
thf(fact_5117_dvd__abs__iff,axiom,
    ! [M: int,K: int] :
      ( ( dvd_dvd_int @ M @ ( abs_abs_int @ K ) )
      = ( dvd_dvd_int @ M @ K ) ) ).

% dvd_abs_iff
thf(fact_5118_dvd__abs__iff,axiom,
    ! [M: code_integer,K: code_integer] :
      ( ( dvd_dvd_Code_integer @ M @ ( abs_abs_Code_integer @ K ) )
      = ( dvd_dvd_Code_integer @ M @ K ) ) ).

% dvd_abs_iff
thf(fact_5119_dvd__abs__iff,axiom,
    ! [M: rat,K: rat] :
      ( ( dvd_dvd_rat @ M @ ( abs_abs_rat @ K ) )
      = ( dvd_dvd_rat @ M @ K ) ) ).

% dvd_abs_iff
thf(fact_5120_abs__bool__eq,axiom,
    ! [P: $o] :
      ( ( abs_abs_real @ ( zero_n3304061248610475627l_real @ P ) )
      = ( zero_n3304061248610475627l_real @ P ) ) ).

% abs_bool_eq
thf(fact_5121_abs__bool__eq,axiom,
    ! [P: $o] :
      ( ( abs_abs_rat @ ( zero_n2052037380579107095ol_rat @ P ) )
      = ( zero_n2052037380579107095ol_rat @ P ) ) ).

% abs_bool_eq
thf(fact_5122_abs__bool__eq,axiom,
    ! [P: $o] :
      ( ( abs_abs_int @ ( zero_n2684676970156552555ol_int @ P ) )
      = ( zero_n2684676970156552555ol_int @ P ) ) ).

% abs_bool_eq
thf(fact_5123_abs__bool__eq,axiom,
    ! [P: $o] :
      ( ( abs_abs_Code_integer @ ( zero_n356916108424825756nteger @ P ) )
      = ( zero_n356916108424825756nteger @ P ) ) ).

% abs_bool_eq
thf(fact_5124_abs__le__zero__iff,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ zero_z3403309356797280102nteger )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% abs_le_zero_iff
thf(fact_5125_abs__le__zero__iff,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% abs_le_zero_iff
thf(fact_5126_abs__le__zero__iff,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ zero_zero_rat )
      = ( A = zero_zero_rat ) ) ).

% abs_le_zero_iff
thf(fact_5127_abs__le__zero__iff,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ zero_zero_int )
      = ( A = zero_zero_int ) ) ).

% abs_le_zero_iff
thf(fact_5128_abs__le__self__iff,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ A )
      = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).

% abs_le_self_iff
thf(fact_5129_abs__le__self__iff,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ A )
      = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).

% abs_le_self_iff
thf(fact_5130_abs__le__self__iff,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ A )
      = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).

% abs_le_self_iff
thf(fact_5131_abs__le__self__iff,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ A )
      = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).

% abs_le_self_iff
thf(fact_5132_abs__of__nonneg,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( abs_abs_Code_integer @ A )
        = A ) ) ).

% abs_of_nonneg
thf(fact_5133_abs__of__nonneg,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( abs_abs_real @ A )
        = A ) ) ).

% abs_of_nonneg
thf(fact_5134_abs__of__nonneg,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( abs_abs_rat @ A )
        = A ) ) ).

% abs_of_nonneg
thf(fact_5135_abs__of__nonneg,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( abs_abs_int @ A )
        = A ) ) ).

% abs_of_nonneg
thf(fact_5136_zero__less__abs__iff,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ A ) )
      = ( A != zero_z3403309356797280102nteger ) ) ).

% zero_less_abs_iff
thf(fact_5137_zero__less__abs__iff,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ A ) )
      = ( A != zero_zero_real ) ) ).

% zero_less_abs_iff
thf(fact_5138_zero__less__abs__iff,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( abs_abs_rat @ A ) )
      = ( A != zero_zero_rat ) ) ).

% zero_less_abs_iff
thf(fact_5139_zero__less__abs__iff,axiom,
    ! [A: int] :
      ( ( ord_less_int @ zero_zero_int @ ( abs_abs_int @ A ) )
      = ( A != zero_zero_int ) ) ).

% zero_less_abs_iff
thf(fact_5140_abs__neg__numeral,axiom,
    ! [N: num] :
      ( ( abs_abs_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
      = ( numeral_numeral_real @ N ) ) ).

% abs_neg_numeral
thf(fact_5141_abs__neg__numeral,axiom,
    ! [N: num] :
      ( ( abs_abs_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( numeral_numeral_int @ N ) ) ).

% abs_neg_numeral
thf(fact_5142_abs__neg__numeral,axiom,
    ! [N: num] :
      ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
      = ( numera6620942414471956472nteger @ N ) ) ).

% abs_neg_numeral
thf(fact_5143_abs__neg__numeral,axiom,
    ! [N: num] :
      ( ( abs_abs_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
      = ( numeral_numeral_rat @ N ) ) ).

% abs_neg_numeral
thf(fact_5144_abs__neg__one,axiom,
    ( ( abs_abs_real @ ( uminus_uminus_real @ one_one_real ) )
    = one_one_real ) ).

% abs_neg_one
thf(fact_5145_abs__neg__one,axiom,
    ( ( abs_abs_int @ ( uminus_uminus_int @ one_one_int ) )
    = one_one_int ) ).

% abs_neg_one
thf(fact_5146_abs__neg__one,axiom,
    ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = one_one_Code_integer ) ).

% abs_neg_one
thf(fact_5147_abs__neg__one,axiom,
    ( ( abs_abs_rat @ ( uminus_uminus_rat @ one_one_rat ) )
    = one_one_rat ) ).

% abs_neg_one
thf(fact_5148_abs__power__minus,axiom,
    ! [A: real,N: nat] :
      ( ( abs_abs_real @ ( power_power_real @ ( uminus_uminus_real @ A ) @ N ) )
      = ( abs_abs_real @ ( power_power_real @ A @ N ) ) ) ).

% abs_power_minus
thf(fact_5149_abs__power__minus,axiom,
    ! [A: int,N: nat] :
      ( ( abs_abs_int @ ( power_power_int @ ( uminus_uminus_int @ A ) @ N ) )
      = ( abs_abs_int @ ( power_power_int @ A @ N ) ) ) ).

% abs_power_minus
thf(fact_5150_abs__power__minus,axiom,
    ! [A: code_integer,N: nat] :
      ( ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N ) )
      = ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) ) ) ).

% abs_power_minus
thf(fact_5151_abs__power__minus,axiom,
    ! [A: rat,N: nat] :
      ( ( abs_abs_rat @ ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N ) )
      = ( abs_abs_rat @ ( power_power_rat @ A @ N ) ) ) ).

% abs_power_minus
thf(fact_5152_real__sqrt__abs2,axiom,
    ! [X4: real] :
      ( ( sqrt @ ( times_times_real @ X4 @ X4 ) )
      = ( abs_abs_real @ X4 ) ) ).

% real_sqrt_abs2
thf(fact_5153_real__sqrt__mult__self,axiom,
    ! [A: real] :
      ( ( times_times_real @ ( sqrt @ A ) @ ( sqrt @ A ) )
      = ( abs_abs_real @ A ) ) ).

% real_sqrt_mult_self
thf(fact_5154_round__numeral,axiom,
    ! [N: num] :
      ( ( archim8280529875227126926d_real @ ( numeral_numeral_real @ N ) )
      = ( numeral_numeral_int @ N ) ) ).

% round_numeral
thf(fact_5155_round__numeral,axiom,
    ! [N: num] :
      ( ( archim7778729529865785530nd_rat @ ( numeral_numeral_rat @ N ) )
      = ( numeral_numeral_int @ N ) ) ).

% round_numeral
thf(fact_5156_round__1,axiom,
    ( ( archim8280529875227126926d_real @ one_one_real )
    = one_one_int ) ).

% round_1
thf(fact_5157_round__1,axiom,
    ( ( archim7778729529865785530nd_rat @ one_one_rat )
    = one_one_int ) ).

% round_1
thf(fact_5158_zero__le__divide__abs__iff,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A @ ( abs_abs_real @ B2 ) ) )
      = ( ( ord_less_eq_real @ zero_zero_real @ A )
        | ( B2 = zero_zero_real ) ) ) ).

% zero_le_divide_abs_iff
thf(fact_5159_zero__le__divide__abs__iff,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ ( abs_abs_rat @ B2 ) ) )
      = ( ( ord_less_eq_rat @ zero_zero_rat @ A )
        | ( B2 = zero_zero_rat ) ) ) ).

% zero_le_divide_abs_iff
thf(fact_5160_divide__le__0__abs__iff,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ A @ ( abs_abs_real @ B2 ) ) @ zero_zero_real )
      = ( ( ord_less_eq_real @ A @ zero_zero_real )
        | ( B2 = zero_zero_real ) ) ) ).

% divide_le_0_abs_iff
thf(fact_5161_divide__le__0__abs__iff,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ ( abs_abs_rat @ B2 ) ) @ zero_zero_rat )
      = ( ( ord_less_eq_rat @ A @ zero_zero_rat )
        | ( B2 = zero_zero_rat ) ) ) ).

% divide_le_0_abs_iff
thf(fact_5162_abs__of__nonpos,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ A @ zero_zero_real )
     => ( ( abs_abs_real @ A )
        = ( uminus_uminus_real @ A ) ) ) ).

% abs_of_nonpos
thf(fact_5163_abs__of__nonpos,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger )
     => ( ( abs_abs_Code_integer @ A )
        = ( uminus1351360451143612070nteger @ A ) ) ) ).

% abs_of_nonpos
thf(fact_5164_abs__of__nonpos,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ A @ zero_zero_rat )
     => ( ( abs_abs_rat @ A )
        = ( uminus_uminus_rat @ A ) ) ) ).

% abs_of_nonpos
thf(fact_5165_abs__of__nonpos,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( abs_abs_int @ A )
        = ( uminus_uminus_int @ A ) ) ) ).

% abs_of_nonpos
thf(fact_5166_artanh__minus__real,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X4 ) @ one_one_real )
     => ( ( artanh_real @ ( uminus_uminus_real @ X4 ) )
        = ( uminus_uminus_real @ ( artanh_real @ X4 ) ) ) ) ).

% artanh_minus_real
thf(fact_5167_zero__less__power__abs__iff,axiom,
    ! [A: code_integer,N: nat] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N ) )
      = ( ( A != zero_z3403309356797280102nteger )
        | ( N = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
thf(fact_5168_zero__less__power__abs__iff,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( abs_abs_real @ A ) @ N ) )
      = ( ( A != zero_zero_real )
        | ( N = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
thf(fact_5169_zero__less__power__abs__iff,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ ( abs_abs_rat @ A ) @ N ) )
      = ( ( A != zero_zero_rat )
        | ( N = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
thf(fact_5170_zero__less__power__abs__iff,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( abs_abs_int @ A ) @ N ) )
      = ( ( A != zero_zero_int )
        | ( N = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
thf(fact_5171_abs__power2,axiom,
    ! [A: code_integer] :
      ( ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% abs_power2
thf(fact_5172_abs__power2,axiom,
    ! [A: rat] :
      ( ( abs_abs_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% abs_power2
thf(fact_5173_abs__power2,axiom,
    ! [A: real] :
      ( ( abs_abs_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% abs_power2
thf(fact_5174_abs__power2,axiom,
    ! [A: int] :
      ( ( abs_abs_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% abs_power2
thf(fact_5175_power2__abs,axiom,
    ! [A: code_integer] :
      ( ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_abs
thf(fact_5176_power2__abs,axiom,
    ! [A: rat] :
      ( ( power_power_rat @ ( abs_abs_rat @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_abs
thf(fact_5177_power2__abs,axiom,
    ! [A: real] :
      ( ( power_power_real @ ( abs_abs_real @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_abs
thf(fact_5178_power2__abs,axiom,
    ! [A: int] :
      ( ( power_power_int @ ( abs_abs_int @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_abs
thf(fact_5179_round__neg__numeral,axiom,
    ! [N: num] :
      ( ( archim8280529875227126926d_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).

% round_neg_numeral
thf(fact_5180_round__neg__numeral,axiom,
    ! [N: num] :
      ( ( archim7778729529865785530nd_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).

% round_neg_numeral
thf(fact_5181_power__even__abs__numeral,axiom,
    ! [W: num,A: code_integer] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
     => ( ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ ( numeral_numeral_nat @ W ) )
        = ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_even_abs_numeral
thf(fact_5182_power__even__abs__numeral,axiom,
    ! [W: num,A: rat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
     => ( ( power_power_rat @ ( abs_abs_rat @ A ) @ ( numeral_numeral_nat @ W ) )
        = ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_even_abs_numeral
thf(fact_5183_power__even__abs__numeral,axiom,
    ! [W: num,A: real] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
     => ( ( power_power_real @ ( abs_abs_real @ A ) @ ( numeral_numeral_nat @ W ) )
        = ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_even_abs_numeral
thf(fact_5184_power__even__abs__numeral,axiom,
    ! [W: num,A: int] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
     => ( ( power_power_int @ ( abs_abs_int @ A ) @ ( numeral_numeral_nat @ W ) )
        = ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_even_abs_numeral
thf(fact_5185_real__sqrt__abs,axiom,
    ! [X4: real] :
      ( ( sqrt @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( abs_abs_real @ X4 ) ) ).

% real_sqrt_abs
thf(fact_5186_abs__ge__self,axiom,
    ! [A: real] : ( ord_less_eq_real @ A @ ( abs_abs_real @ A ) ) ).

% abs_ge_self
thf(fact_5187_abs__ge__self,axiom,
    ! [A: code_integer] : ( ord_le3102999989581377725nteger @ A @ ( abs_abs_Code_integer @ A ) ) ).

% abs_ge_self
thf(fact_5188_abs__ge__self,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ A @ ( abs_abs_rat @ A ) ) ).

% abs_ge_self
thf(fact_5189_abs__ge__self,axiom,
    ! [A: int] : ( ord_less_eq_int @ A @ ( abs_abs_int @ A ) ) ).

% abs_ge_self
thf(fact_5190_abs__le__D1,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B2 )
     => ( ord_less_eq_real @ A @ B2 ) ) ).

% abs_le_D1
thf(fact_5191_abs__le__D1,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B2 )
     => ( ord_le3102999989581377725nteger @ A @ B2 ) ) ).

% abs_le_D1
thf(fact_5192_abs__le__D1,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B2 )
     => ( ord_less_eq_rat @ A @ B2 ) ) ).

% abs_le_D1
thf(fact_5193_abs__le__D1,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B2 )
     => ( ord_less_eq_int @ A @ B2 ) ) ).

% abs_le_D1
thf(fact_5194_abs__eq__0__iff,axiom,
    ! [A: code_integer] :
      ( ( ( abs_abs_Code_integer @ A )
        = zero_z3403309356797280102nteger )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% abs_eq_0_iff
thf(fact_5195_abs__eq__0__iff,axiom,
    ! [A: real] :
      ( ( ( abs_abs_real @ A )
        = zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% abs_eq_0_iff
thf(fact_5196_abs__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( ( abs_abs_rat @ A )
        = zero_zero_rat )
      = ( A = zero_zero_rat ) ) ).

% abs_eq_0_iff
thf(fact_5197_abs__eq__0__iff,axiom,
    ! [A: int] :
      ( ( ( abs_abs_int @ A )
        = zero_zero_int )
      = ( A = zero_zero_int ) ) ).

% abs_eq_0_iff
thf(fact_5198_abs__mult,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ A @ B2 ) )
      = ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B2 ) ) ) ).

% abs_mult
thf(fact_5199_abs__mult,axiom,
    ! [A: real,B2: real] :
      ( ( abs_abs_real @ ( times_times_real @ A @ B2 ) )
      = ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B2 ) ) ) ).

% abs_mult
thf(fact_5200_abs__mult,axiom,
    ! [A: rat,B2: rat] :
      ( ( abs_abs_rat @ ( times_times_rat @ A @ B2 ) )
      = ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B2 ) ) ) ).

% abs_mult
thf(fact_5201_abs__mult,axiom,
    ! [A: int,B2: int] :
      ( ( abs_abs_int @ ( times_times_int @ A @ B2 ) )
      = ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B2 ) ) ) ).

% abs_mult
thf(fact_5202_abs__one,axiom,
    ( ( abs_abs_Code_integer @ one_one_Code_integer )
    = one_one_Code_integer ) ).

% abs_one
thf(fact_5203_abs__one,axiom,
    ( ( abs_abs_real @ one_one_real )
    = one_one_real ) ).

% abs_one
thf(fact_5204_abs__one,axiom,
    ( ( abs_abs_rat @ one_one_rat )
    = one_one_rat ) ).

% abs_one
thf(fact_5205_abs__one,axiom,
    ( ( abs_abs_int @ one_one_int )
    = one_one_int ) ).

% abs_one
thf(fact_5206_abs__minus__commute,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B2 ) )
      = ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B2 @ A ) ) ) ).

% abs_minus_commute
thf(fact_5207_abs__minus__commute,axiom,
    ! [A: real,B2: real] :
      ( ( abs_abs_real @ ( minus_minus_real @ A @ B2 ) )
      = ( abs_abs_real @ ( minus_minus_real @ B2 @ A ) ) ) ).

% abs_minus_commute
thf(fact_5208_abs__minus__commute,axiom,
    ! [A: rat,B2: rat] :
      ( ( abs_abs_rat @ ( minus_minus_rat @ A @ B2 ) )
      = ( abs_abs_rat @ ( minus_minus_rat @ B2 @ A ) ) ) ).

% abs_minus_commute
thf(fact_5209_abs__minus__commute,axiom,
    ! [A: int,B2: int] :
      ( ( abs_abs_int @ ( minus_minus_int @ A @ B2 ) )
      = ( abs_abs_int @ ( minus_minus_int @ B2 @ A ) ) ) ).

% abs_minus_commute
thf(fact_5210_abs__eq__iff,axiom,
    ! [X4: real,Y3: real] :
      ( ( ( abs_abs_real @ X4 )
        = ( abs_abs_real @ Y3 ) )
      = ( ( X4 = Y3 )
        | ( X4
          = ( uminus_uminus_real @ Y3 ) ) ) ) ).

% abs_eq_iff
thf(fact_5211_abs__eq__iff,axiom,
    ! [X4: int,Y3: int] :
      ( ( ( abs_abs_int @ X4 )
        = ( abs_abs_int @ Y3 ) )
      = ( ( X4 = Y3 )
        | ( X4
          = ( uminus_uminus_int @ Y3 ) ) ) ) ).

% abs_eq_iff
thf(fact_5212_abs__eq__iff,axiom,
    ! [X4: code_integer,Y3: code_integer] :
      ( ( ( abs_abs_Code_integer @ X4 )
        = ( abs_abs_Code_integer @ Y3 ) )
      = ( ( X4 = Y3 )
        | ( X4
          = ( uminus1351360451143612070nteger @ Y3 ) ) ) ) ).

% abs_eq_iff
thf(fact_5213_abs__eq__iff,axiom,
    ! [X4: rat,Y3: rat] :
      ( ( ( abs_abs_rat @ X4 )
        = ( abs_abs_rat @ Y3 ) )
      = ( ( X4 = Y3 )
        | ( X4
          = ( uminus_uminus_rat @ Y3 ) ) ) ) ).

% abs_eq_iff
thf(fact_5214_power__abs,axiom,
    ! [A: code_integer,N: nat] :
      ( ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) )
      = ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N ) ) ).

% power_abs
thf(fact_5215_power__abs,axiom,
    ! [A: rat,N: nat] :
      ( ( abs_abs_rat @ ( power_power_rat @ A @ N ) )
      = ( power_power_rat @ ( abs_abs_rat @ A ) @ N ) ) ).

% power_abs
thf(fact_5216_power__abs,axiom,
    ! [A: real,N: nat] :
      ( ( abs_abs_real @ ( power_power_real @ A @ N ) )
      = ( power_power_real @ ( abs_abs_real @ A ) @ N ) ) ).

% power_abs
thf(fact_5217_power__abs,axiom,
    ! [A: int,N: nat] :
      ( ( abs_abs_int @ ( power_power_int @ A @ N ) )
      = ( power_power_int @ ( abs_abs_int @ A ) @ N ) ) ).

% power_abs
thf(fact_5218_dvd__if__abs__eq,axiom,
    ! [L2: real,K: real] :
      ( ( ( abs_abs_real @ L2 )
        = ( abs_abs_real @ K ) )
     => ( dvd_dvd_real @ L2 @ K ) ) ).

% dvd_if_abs_eq
thf(fact_5219_dvd__if__abs__eq,axiom,
    ! [L2: int,K: int] :
      ( ( ( abs_abs_int @ L2 )
        = ( abs_abs_int @ K ) )
     => ( dvd_dvd_int @ L2 @ K ) ) ).

% dvd_if_abs_eq
thf(fact_5220_dvd__if__abs__eq,axiom,
    ! [L2: code_integer,K: code_integer] :
      ( ( ( abs_abs_Code_integer @ L2 )
        = ( abs_abs_Code_integer @ K ) )
     => ( dvd_dvd_Code_integer @ L2 @ K ) ) ).

% dvd_if_abs_eq
thf(fact_5221_dvd__if__abs__eq,axiom,
    ! [L2: rat,K: rat] :
      ( ( ( abs_abs_rat @ L2 )
        = ( abs_abs_rat @ K ) )
     => ( dvd_dvd_rat @ L2 @ K ) ) ).

% dvd_if_abs_eq
thf(fact_5222_round__diff__minimal,axiom,
    ! [Z: real,M: int] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ Z ) ) ) ) @ ( abs_abs_real @ ( minus_minus_real @ Z @ ( ring_1_of_int_real @ M ) ) ) ) ).

% round_diff_minimal
thf(fact_5223_round__diff__minimal,axiom,
    ! [Z: rat,M: int] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ Z @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ Z ) ) ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ Z @ ( ring_1_of_int_rat @ M ) ) ) ) ).

% round_diff_minimal
thf(fact_5224_abs__ge__zero,axiom,
    ! [A: code_integer] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ A ) ) ).

% abs_ge_zero
thf(fact_5225_abs__ge__zero,axiom,
    ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( abs_abs_real @ A ) ) ).

% abs_ge_zero
thf(fact_5226_abs__ge__zero,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( abs_abs_rat @ A ) ) ).

% abs_ge_zero
thf(fact_5227_abs__ge__zero,axiom,
    ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( abs_abs_int @ A ) ) ).

% abs_ge_zero
thf(fact_5228_abs__not__less__zero,axiom,
    ! [A: code_integer] :
      ~ ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ zero_z3403309356797280102nteger ) ).

% abs_not_less_zero
thf(fact_5229_abs__not__less__zero,axiom,
    ! [A: real] :
      ~ ( ord_less_real @ ( abs_abs_real @ A ) @ zero_zero_real ) ).

% abs_not_less_zero
thf(fact_5230_abs__not__less__zero,axiom,
    ! [A: rat] :
      ~ ( ord_less_rat @ ( abs_abs_rat @ A ) @ zero_zero_rat ) ).

% abs_not_less_zero
thf(fact_5231_abs__not__less__zero,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ ( abs_abs_int @ A ) @ zero_zero_int ) ).

% abs_not_less_zero
thf(fact_5232_abs__of__pos,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( abs_abs_Code_integer @ A )
        = A ) ) ).

% abs_of_pos
thf(fact_5233_abs__of__pos,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( abs_abs_real @ A )
        = A ) ) ).

% abs_of_pos
thf(fact_5234_abs__of__pos,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( abs_abs_rat @ A )
        = A ) ) ).

% abs_of_pos
thf(fact_5235_abs__of__pos,axiom,
    ! [A: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( abs_abs_int @ A )
        = A ) ) ).

% abs_of_pos
thf(fact_5236_abs__triangle__ineq,axiom,
    ! [A: code_integer,B2: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( plus_p5714425477246183910nteger @ A @ B2 ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B2 ) ) ) ).

% abs_triangle_ineq
thf(fact_5237_abs__triangle__ineq,axiom,
    ! [A: real,B2: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( plus_plus_real @ A @ B2 ) ) @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B2 ) ) ) ).

% abs_triangle_ineq
thf(fact_5238_abs__triangle__ineq,axiom,
    ! [A: rat,B2: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( plus_plus_rat @ A @ B2 ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B2 ) ) ) ).

% abs_triangle_ineq
thf(fact_5239_abs__triangle__ineq,axiom,
    ! [A: int,B2: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( plus_plus_int @ A @ B2 ) ) @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B2 ) ) ) ).

% abs_triangle_ineq
thf(fact_5240_abs__mult__less,axiom,
    ! [A: code_integer,C: code_integer,B2: code_integer,D: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ C )
     => ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ B2 ) @ D )
       => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B2 ) ) @ ( times_3573771949741848930nteger @ C @ D ) ) ) ) ).

% abs_mult_less
thf(fact_5241_abs__mult__less,axiom,
    ! [A: real,C: real,B2: real,D: real] :
      ( ( ord_less_real @ ( abs_abs_real @ A ) @ C )
     => ( ( ord_less_real @ ( abs_abs_real @ B2 ) @ D )
       => ( ord_less_real @ ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B2 ) ) @ ( times_times_real @ C @ D ) ) ) ) ).

% abs_mult_less
thf(fact_5242_abs__mult__less,axiom,
    ! [A: rat,C: rat,B2: rat,D: rat] :
      ( ( ord_less_rat @ ( abs_abs_rat @ A ) @ C )
     => ( ( ord_less_rat @ ( abs_abs_rat @ B2 ) @ D )
       => ( ord_less_rat @ ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B2 ) ) @ ( times_times_rat @ C @ D ) ) ) ) ).

% abs_mult_less
thf(fact_5243_abs__mult__less,axiom,
    ! [A: int,C: int,B2: int,D: int] :
      ( ( ord_less_int @ ( abs_abs_int @ A ) @ C )
     => ( ( ord_less_int @ ( abs_abs_int @ B2 ) @ D )
       => ( ord_less_int @ ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B2 ) ) @ ( times_times_int @ C @ D ) ) ) ) ).

% abs_mult_less
thf(fact_5244_abs__triangle__ineq2,axiom,
    ! [A: code_integer,B2: code_integer] : ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B2 ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B2 ) ) ) ).

% abs_triangle_ineq2
thf(fact_5245_abs__triangle__ineq2,axiom,
    ! [A: real,B2: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B2 ) ) @ ( abs_abs_real @ ( minus_minus_real @ A @ B2 ) ) ) ).

% abs_triangle_ineq2
thf(fact_5246_abs__triangle__ineq2,axiom,
    ! [A: rat,B2: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B2 ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ A @ B2 ) ) ) ).

% abs_triangle_ineq2
thf(fact_5247_abs__triangle__ineq2,axiom,
    ! [A: int,B2: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B2 ) ) @ ( abs_abs_int @ ( minus_minus_int @ A @ B2 ) ) ) ).

% abs_triangle_ineq2
thf(fact_5248_abs__triangle__ineq3,axiom,
    ! [A: code_integer,B2: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B2 ) ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B2 ) ) ) ).

% abs_triangle_ineq3
thf(fact_5249_abs__triangle__ineq3,axiom,
    ! [A: real,B2: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B2 ) ) ) @ ( abs_abs_real @ ( minus_minus_real @ A @ B2 ) ) ) ).

% abs_triangle_ineq3
thf(fact_5250_abs__triangle__ineq3,axiom,
    ! [A: rat,B2: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B2 ) ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ A @ B2 ) ) ) ).

% abs_triangle_ineq3
thf(fact_5251_abs__triangle__ineq3,axiom,
    ! [A: int,B2: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B2 ) ) ) @ ( abs_abs_int @ ( minus_minus_int @ A @ B2 ) ) ) ).

% abs_triangle_ineq3
thf(fact_5252_abs__triangle__ineq2__sym,axiom,
    ! [A: code_integer,B2: code_integer] : ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B2 ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B2 @ A ) ) ) ).

% abs_triangle_ineq2_sym
thf(fact_5253_abs__triangle__ineq2__sym,axiom,
    ! [A: real,B2: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B2 ) ) @ ( abs_abs_real @ ( minus_minus_real @ B2 @ A ) ) ) ).

% abs_triangle_ineq2_sym
thf(fact_5254_abs__triangle__ineq2__sym,axiom,
    ! [A: rat,B2: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B2 ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ B2 @ A ) ) ) ).

% abs_triangle_ineq2_sym
thf(fact_5255_abs__triangle__ineq2__sym,axiom,
    ! [A: int,B2: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B2 ) ) @ ( abs_abs_int @ ( minus_minus_int @ B2 @ A ) ) ) ).

% abs_triangle_ineq2_sym
thf(fact_5256_nonzero__abs__divide,axiom,
    ! [B2: real,A: real] :
      ( ( B2 != zero_zero_real )
     => ( ( abs_abs_real @ ( divide_divide_real @ A @ B2 ) )
        = ( divide_divide_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B2 ) ) ) ) ).

% nonzero_abs_divide
thf(fact_5257_nonzero__abs__divide,axiom,
    ! [B2: rat,A: rat] :
      ( ( B2 != zero_zero_rat )
     => ( ( abs_abs_rat @ ( divide_divide_rat @ A @ B2 ) )
        = ( divide_divide_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B2 ) ) ) ) ).

% nonzero_abs_divide
thf(fact_5258_abs__ge__minus__self,axiom,
    ! [A: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ ( abs_abs_real @ A ) ) ).

% abs_ge_minus_self
thf(fact_5259_abs__ge__minus__self,axiom,
    ! [A: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ ( abs_abs_Code_integer @ A ) ) ).

% abs_ge_minus_self
thf(fact_5260_abs__ge__minus__self,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ ( abs_abs_rat @ A ) ) ).

% abs_ge_minus_self
thf(fact_5261_abs__ge__minus__self,axiom,
    ! [A: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ ( abs_abs_int @ A ) ) ).

% abs_ge_minus_self
thf(fact_5262_abs__le__iff,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B2 )
      = ( ( ord_less_eq_real @ A @ B2 )
        & ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B2 ) ) ) ).

% abs_le_iff
thf(fact_5263_abs__le__iff,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B2 )
      = ( ( ord_le3102999989581377725nteger @ A @ B2 )
        & ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B2 ) ) ) ).

% abs_le_iff
thf(fact_5264_abs__le__iff,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B2 )
      = ( ( ord_less_eq_rat @ A @ B2 )
        & ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B2 ) ) ) ).

% abs_le_iff
thf(fact_5265_abs__le__iff,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B2 )
      = ( ( ord_less_eq_int @ A @ B2 )
        & ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B2 ) ) ) ).

% abs_le_iff
thf(fact_5266_abs__le__D2,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B2 )
     => ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B2 ) ) ).

% abs_le_D2
thf(fact_5267_abs__le__D2,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B2 )
     => ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B2 ) ) ).

% abs_le_D2
thf(fact_5268_abs__le__D2,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B2 )
     => ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B2 ) ) ).

% abs_le_D2
thf(fact_5269_abs__le__D2,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B2 )
     => ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B2 ) ) ).

% abs_le_D2
thf(fact_5270_abs__leI,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ A @ B2 )
     => ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B2 )
       => ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B2 ) ) ) ).

% abs_leI
thf(fact_5271_abs__leI,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ B2 )
     => ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B2 )
       => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B2 ) ) ) ).

% abs_leI
thf(fact_5272_abs__leI,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_eq_rat @ A @ B2 )
     => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B2 )
       => ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B2 ) ) ) ).

% abs_leI
thf(fact_5273_abs__leI,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ A @ B2 )
     => ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B2 )
       => ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B2 ) ) ) ).

% abs_leI
thf(fact_5274_abs__less__iff,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ ( abs_abs_real @ A ) @ B2 )
      = ( ( ord_less_real @ A @ B2 )
        & ( ord_less_real @ ( uminus_uminus_real @ A ) @ B2 ) ) ) ).

% abs_less_iff
thf(fact_5275_abs__less__iff,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_int @ ( abs_abs_int @ A ) @ B2 )
      = ( ( ord_less_int @ A @ B2 )
        & ( ord_less_int @ ( uminus_uminus_int @ A ) @ B2 ) ) ) ).

% abs_less_iff
thf(fact_5276_abs__less__iff,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ B2 )
      = ( ( ord_le6747313008572928689nteger @ A @ B2 )
        & ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ B2 ) ) ) ).

% abs_less_iff
thf(fact_5277_abs__less__iff,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_rat @ ( abs_abs_rat @ A ) @ B2 )
      = ( ( ord_less_rat @ A @ B2 )
        & ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ B2 ) ) ) ).

% abs_less_iff
thf(fact_5278_dense__eq0__I,axiom,
    ! [X4: real] :
      ( ! [E2: real] :
          ( ( ord_less_real @ zero_zero_real @ E2 )
         => ( ord_less_eq_real @ ( abs_abs_real @ X4 ) @ E2 ) )
     => ( X4 = zero_zero_real ) ) ).

% dense_eq0_I
thf(fact_5279_dense__eq0__I,axiom,
    ! [X4: rat] :
      ( ! [E2: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ E2 )
         => ( ord_less_eq_rat @ ( abs_abs_rat @ X4 ) @ E2 ) )
     => ( X4 = zero_zero_rat ) ) ).

% dense_eq0_I
thf(fact_5280_abs__eq__mult,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
          | ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) )
        & ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B2 )
          | ( ord_le3102999989581377725nteger @ B2 @ zero_z3403309356797280102nteger ) ) )
     => ( ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ A @ B2 ) )
        = ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B2 ) ) ) ) ).

% abs_eq_mult
thf(fact_5281_abs__eq__mult,axiom,
    ! [A: real,B2: real] :
      ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
          | ( ord_less_eq_real @ A @ zero_zero_real ) )
        & ( ( ord_less_eq_real @ zero_zero_real @ B2 )
          | ( ord_less_eq_real @ B2 @ zero_zero_real ) ) )
     => ( ( abs_abs_real @ ( times_times_real @ A @ B2 ) )
        = ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B2 ) ) ) ) ).

% abs_eq_mult
thf(fact_5282_abs__eq__mult,axiom,
    ! [A: rat,B2: rat] :
      ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
          | ( ord_less_eq_rat @ A @ zero_zero_rat ) )
        & ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
          | ( ord_less_eq_rat @ B2 @ zero_zero_rat ) ) )
     => ( ( abs_abs_rat @ ( times_times_rat @ A @ B2 ) )
        = ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B2 ) ) ) ) ).

% abs_eq_mult
thf(fact_5283_abs__eq__mult,axiom,
    ! [A: int,B2: int] :
      ( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
          | ( ord_less_eq_int @ A @ zero_zero_int ) )
        & ( ( ord_less_eq_int @ zero_zero_int @ B2 )
          | ( ord_less_eq_int @ B2 @ zero_zero_int ) ) )
     => ( ( abs_abs_int @ ( times_times_int @ A @ B2 ) )
        = ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B2 ) ) ) ) ).

% abs_eq_mult
thf(fact_5284_abs__mult__pos,axiom,
    ! [X4: code_integer,Y3: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X4 )
     => ( ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ Y3 ) @ X4 )
        = ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ Y3 @ X4 ) ) ) ) ).

% abs_mult_pos
thf(fact_5285_abs__mult__pos,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( times_times_real @ ( abs_abs_real @ Y3 ) @ X4 )
        = ( abs_abs_real @ ( times_times_real @ Y3 @ X4 ) ) ) ) ).

% abs_mult_pos
thf(fact_5286_abs__mult__pos,axiom,
    ! [X4: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X4 )
     => ( ( times_times_rat @ ( abs_abs_rat @ Y3 ) @ X4 )
        = ( abs_abs_rat @ ( times_times_rat @ Y3 @ X4 ) ) ) ) ).

% abs_mult_pos
thf(fact_5287_abs__mult__pos,axiom,
    ! [X4: int,Y3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X4 )
     => ( ( times_times_int @ ( abs_abs_int @ Y3 ) @ X4 )
        = ( abs_abs_int @ ( times_times_int @ Y3 @ X4 ) ) ) ) ).

% abs_mult_pos
thf(fact_5288_abs__minus__le__zero,axiom,
    ! [A: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( abs_abs_real @ A ) ) @ zero_zero_real ) ).

% abs_minus_le_zero
thf(fact_5289_abs__minus__le__zero,axiom,
    ! [A: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( abs_abs_Code_integer @ A ) ) @ zero_z3403309356797280102nteger ) ).

% abs_minus_le_zero
thf(fact_5290_abs__minus__le__zero,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( abs_abs_rat @ A ) ) @ zero_zero_rat ) ).

% abs_minus_le_zero
thf(fact_5291_abs__minus__le__zero,axiom,
    ! [A: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( abs_abs_int @ A ) ) @ zero_zero_int ) ).

% abs_minus_le_zero
thf(fact_5292_eq__abs__iff_H,axiom,
    ! [A: real,B2: real] :
      ( ( A
        = ( abs_abs_real @ B2 ) )
      = ( ( ord_less_eq_real @ zero_zero_real @ A )
        & ( ( B2 = A )
          | ( B2
            = ( uminus_uminus_real @ A ) ) ) ) ) ).

% eq_abs_iff'
thf(fact_5293_eq__abs__iff_H,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( A
        = ( abs_abs_Code_integer @ B2 ) )
      = ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
        & ( ( B2 = A )
          | ( B2
            = ( uminus1351360451143612070nteger @ A ) ) ) ) ) ).

% eq_abs_iff'
thf(fact_5294_eq__abs__iff_H,axiom,
    ! [A: rat,B2: rat] :
      ( ( A
        = ( abs_abs_rat @ B2 ) )
      = ( ( ord_less_eq_rat @ zero_zero_rat @ A )
        & ( ( B2 = A )
          | ( B2
            = ( uminus_uminus_rat @ A ) ) ) ) ) ).

% eq_abs_iff'
thf(fact_5295_eq__abs__iff_H,axiom,
    ! [A: int,B2: int] :
      ( ( A
        = ( abs_abs_int @ B2 ) )
      = ( ( ord_less_eq_int @ zero_zero_int @ A )
        & ( ( B2 = A )
          | ( B2
            = ( uminus_uminus_int @ A ) ) ) ) ) ).

% eq_abs_iff'
thf(fact_5296_abs__eq__iff_H,axiom,
    ! [A: real,B2: real] :
      ( ( ( abs_abs_real @ A )
        = B2 )
      = ( ( ord_less_eq_real @ zero_zero_real @ B2 )
        & ( ( A = B2 )
          | ( A
            = ( uminus_uminus_real @ B2 ) ) ) ) ) ).

% abs_eq_iff'
thf(fact_5297_abs__eq__iff_H,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( ( abs_abs_Code_integer @ A )
        = B2 )
      = ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B2 )
        & ( ( A = B2 )
          | ( A
            = ( uminus1351360451143612070nteger @ B2 ) ) ) ) ) ).

% abs_eq_iff'
thf(fact_5298_abs__eq__iff_H,axiom,
    ! [A: rat,B2: rat] :
      ( ( ( abs_abs_rat @ A )
        = B2 )
      = ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
        & ( ( A = B2 )
          | ( A
            = ( uminus_uminus_rat @ B2 ) ) ) ) ) ).

% abs_eq_iff'
thf(fact_5299_abs__eq__iff_H,axiom,
    ! [A: int,B2: int] :
      ( ( ( abs_abs_int @ A )
        = B2 )
      = ( ( ord_less_eq_int @ zero_zero_int @ B2 )
        & ( ( A = B2 )
          | ( A
            = ( uminus_uminus_int @ B2 ) ) ) ) ) ).

% abs_eq_iff'
thf(fact_5300_abs__div__pos,axiom,
    ! [Y3: real,X4: real] :
      ( ( ord_less_real @ zero_zero_real @ Y3 )
     => ( ( divide_divide_real @ ( abs_abs_real @ X4 ) @ Y3 )
        = ( abs_abs_real @ ( divide_divide_real @ X4 @ Y3 ) ) ) ) ).

% abs_div_pos
thf(fact_5301_abs__div__pos,axiom,
    ! [Y3: rat,X4: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Y3 )
     => ( ( divide_divide_rat @ ( abs_abs_rat @ X4 ) @ Y3 )
        = ( abs_abs_rat @ ( divide_divide_rat @ X4 @ Y3 ) ) ) ) ).

% abs_div_pos
thf(fact_5302_zero__le__power__abs,axiom,
    ! [A: code_integer,N: nat] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N ) ) ).

% zero_le_power_abs
thf(fact_5303_zero__le__power__abs,axiom,
    ! [A: real,N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ ( abs_abs_real @ A ) @ N ) ) ).

% zero_le_power_abs
thf(fact_5304_zero__le__power__abs,axiom,
    ! [A: rat,N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ ( abs_abs_rat @ A ) @ N ) ) ).

% zero_le_power_abs
thf(fact_5305_zero__le__power__abs,axiom,
    ! [A: int,N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ ( abs_abs_int @ A ) @ N ) ) ).

% zero_le_power_abs
thf(fact_5306_abs__if,axiom,
    ( abs_abs_real
    = ( ^ [A4: real] : ( if_real @ ( ord_less_real @ A4 @ zero_zero_real ) @ ( uminus_uminus_real @ A4 ) @ A4 ) ) ) ).

% abs_if
thf(fact_5307_abs__if,axiom,
    ( abs_abs_int
    = ( ^ [A4: int] : ( if_int @ ( ord_less_int @ A4 @ zero_zero_int ) @ ( uminus_uminus_int @ A4 ) @ A4 ) ) ) ).

% abs_if
thf(fact_5308_abs__if,axiom,
    ( abs_abs_Code_integer
    = ( ^ [A4: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ A4 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ A4 ) @ A4 ) ) ) ).

% abs_if
thf(fact_5309_abs__if,axiom,
    ( abs_abs_rat
    = ( ^ [A4: rat] : ( if_rat @ ( ord_less_rat @ A4 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A4 ) @ A4 ) ) ) ).

% abs_if
thf(fact_5310_abs__if__raw,axiom,
    ( abs_abs_real
    = ( ^ [A4: real] : ( if_real @ ( ord_less_real @ A4 @ zero_zero_real ) @ ( uminus_uminus_real @ A4 ) @ A4 ) ) ) ).

% abs_if_raw
thf(fact_5311_abs__if__raw,axiom,
    ( abs_abs_int
    = ( ^ [A4: int] : ( if_int @ ( ord_less_int @ A4 @ zero_zero_int ) @ ( uminus_uminus_int @ A4 ) @ A4 ) ) ) ).

% abs_if_raw
thf(fact_5312_abs__if__raw,axiom,
    ( abs_abs_Code_integer
    = ( ^ [A4: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ A4 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ A4 ) @ A4 ) ) ) ).

% abs_if_raw
thf(fact_5313_abs__if__raw,axiom,
    ( abs_abs_rat
    = ( ^ [A4: rat] : ( if_rat @ ( ord_less_rat @ A4 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A4 ) @ A4 ) ) ) ).

% abs_if_raw
thf(fact_5314_abs__of__neg,axiom,
    ! [A: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( abs_abs_real @ A )
        = ( uminus_uminus_real @ A ) ) ) ).

% abs_of_neg
thf(fact_5315_abs__of__neg,axiom,
    ! [A: int] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ( abs_abs_int @ A )
        = ( uminus_uminus_int @ A ) ) ) ).

% abs_of_neg
thf(fact_5316_abs__of__neg,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger )
     => ( ( abs_abs_Code_integer @ A )
        = ( uminus1351360451143612070nteger @ A ) ) ) ).

% abs_of_neg
thf(fact_5317_abs__of__neg,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( abs_abs_rat @ A )
        = ( uminus_uminus_rat @ A ) ) ) ).

% abs_of_neg
thf(fact_5318_abs__triangle__ineq4,axiom,
    ! [A: code_integer,B2: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B2 ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B2 ) ) ) ).

% abs_triangle_ineq4
thf(fact_5319_abs__triangle__ineq4,axiom,
    ! [A: real,B2: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ A @ B2 ) ) @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B2 ) ) ) ).

% abs_triangle_ineq4
thf(fact_5320_abs__triangle__ineq4,axiom,
    ! [A: rat,B2: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ A @ B2 ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B2 ) ) ) ).

% abs_triangle_ineq4
thf(fact_5321_abs__triangle__ineq4,axiom,
    ! [A: int,B2: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ A @ B2 ) ) @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B2 ) ) ) ).

% abs_triangle_ineq4
thf(fact_5322_abs__diff__triangle__ineq,axiom,
    ! [A: code_integer,B2: code_integer,C: code_integer,D: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A @ B2 ) @ ( plus_p5714425477246183910nteger @ C @ D ) ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ C ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B2 @ D ) ) ) ) ).

% abs_diff_triangle_ineq
thf(fact_5323_abs__diff__triangle__ineq,axiom,
    ! [A: real,B2: real,C: real,D: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( plus_plus_real @ A @ B2 ) @ ( plus_plus_real @ C @ D ) ) ) @ ( plus_plus_real @ ( abs_abs_real @ ( minus_minus_real @ A @ C ) ) @ ( abs_abs_real @ ( minus_minus_real @ B2 @ D ) ) ) ) ).

% abs_diff_triangle_ineq
thf(fact_5324_abs__diff__triangle__ineq,axiom,
    ! [A: rat,B2: rat,C: rat,D: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( plus_plus_rat @ A @ B2 ) @ ( plus_plus_rat @ C @ D ) ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ ( minus_minus_rat @ A @ C ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ B2 @ D ) ) ) ) ).

% abs_diff_triangle_ineq
thf(fact_5325_abs__diff__triangle__ineq,axiom,
    ! [A: int,B2: int,C: int,D: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( plus_plus_int @ A @ B2 ) @ ( plus_plus_int @ C @ D ) ) ) @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ A @ C ) ) @ ( abs_abs_int @ ( minus_minus_int @ B2 @ D ) ) ) ) ).

% abs_diff_triangle_ineq
thf(fact_5326_abs__diff__le__iff,axiom,
    ! [X4: code_integer,A: code_integer,R2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X4 @ A ) ) @ R2 )
      = ( ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ A @ R2 ) @ X4 )
        & ( ord_le3102999989581377725nteger @ X4 @ ( plus_p5714425477246183910nteger @ A @ R2 ) ) ) ) ).

% abs_diff_le_iff
thf(fact_5327_abs__diff__le__iff,axiom,
    ! [X4: real,A: real,R2: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ X4 @ A ) ) @ R2 )
      = ( ( ord_less_eq_real @ ( minus_minus_real @ A @ R2 ) @ X4 )
        & ( ord_less_eq_real @ X4 @ ( plus_plus_real @ A @ R2 ) ) ) ) ).

% abs_diff_le_iff
thf(fact_5328_abs__diff__le__iff,axiom,
    ! [X4: rat,A: rat,R2: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X4 @ A ) ) @ R2 )
      = ( ( ord_less_eq_rat @ ( minus_minus_rat @ A @ R2 ) @ X4 )
        & ( ord_less_eq_rat @ X4 @ ( plus_plus_rat @ A @ R2 ) ) ) ) ).

% abs_diff_le_iff
thf(fact_5329_abs__diff__le__iff,axiom,
    ! [X4: int,A: int,R2: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ X4 @ A ) ) @ R2 )
      = ( ( ord_less_eq_int @ ( minus_minus_int @ A @ R2 ) @ X4 )
        & ( ord_less_eq_int @ X4 @ ( plus_plus_int @ A @ R2 ) ) ) ) ).

% abs_diff_le_iff
thf(fact_5330_abs__diff__less__iff,axiom,
    ! [X4: code_integer,A: code_integer,R2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X4 @ A ) ) @ R2 )
      = ( ( ord_le6747313008572928689nteger @ ( minus_8373710615458151222nteger @ A @ R2 ) @ X4 )
        & ( ord_le6747313008572928689nteger @ X4 @ ( plus_p5714425477246183910nteger @ A @ R2 ) ) ) ) ).

% abs_diff_less_iff
thf(fact_5331_abs__diff__less__iff,axiom,
    ! [X4: real,A: real,R2: real] :
      ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X4 @ A ) ) @ R2 )
      = ( ( ord_less_real @ ( minus_minus_real @ A @ R2 ) @ X4 )
        & ( ord_less_real @ X4 @ ( plus_plus_real @ A @ R2 ) ) ) ) ).

% abs_diff_less_iff
thf(fact_5332_abs__diff__less__iff,axiom,
    ! [X4: rat,A: rat,R2: rat] :
      ( ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X4 @ A ) ) @ R2 )
      = ( ( ord_less_rat @ ( minus_minus_rat @ A @ R2 ) @ X4 )
        & ( ord_less_rat @ X4 @ ( plus_plus_rat @ A @ R2 ) ) ) ) ).

% abs_diff_less_iff
thf(fact_5333_abs__diff__less__iff,axiom,
    ! [X4: int,A: int,R2: int] :
      ( ( ord_less_int @ ( abs_abs_int @ ( minus_minus_int @ X4 @ A ) ) @ R2 )
      = ( ( ord_less_int @ ( minus_minus_int @ A @ R2 ) @ X4 )
        & ( ord_less_int @ X4 @ ( plus_plus_int @ A @ R2 ) ) ) ) ).

% abs_diff_less_iff
thf(fact_5334_round__mono,axiom,
    ! [X4: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ X4 @ Y3 )
     => ( ord_less_eq_int @ ( archim7778729529865785530nd_rat @ X4 ) @ ( archim7778729529865785530nd_rat @ Y3 ) ) ) ).

% round_mono
thf(fact_5335_abs__real__def,axiom,
    ( abs_abs_real
    = ( ^ [A4: real] : ( if_real @ ( ord_less_real @ A4 @ zero_zero_real ) @ ( uminus_uminus_real @ A4 ) @ A4 ) ) ) ).

% abs_real_def
thf(fact_5336_lemma__interval__lt,axiom,
    ! [A: real,X4: real,B2: real] :
      ( ( ord_less_real @ A @ X4 )
     => ( ( ord_less_real @ X4 @ B2 )
       => ? [D3: real] :
            ( ( ord_less_real @ zero_zero_real @ D3 )
            & ! [Y5: real] :
                ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X4 @ Y5 ) ) @ D3 )
               => ( ( ord_less_real @ A @ Y5 )
                  & ( ord_less_real @ Y5 @ B2 ) ) ) ) ) ) ).

% lemma_interval_lt
thf(fact_5337_sin__bound__lemma,axiom,
    ! [X4: real,Y3: real,U: real,V: real] :
      ( ( X4 = Y3 )
     => ( ( ord_less_eq_real @ ( abs_abs_real @ U ) @ V )
       => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( plus_plus_real @ X4 @ U ) @ Y3 ) ) @ V ) ) ) ).

% sin_bound_lemma
thf(fact_5338_abs__add__one__gt__zero,axiom,
    ! [X4: code_integer] : ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( abs_abs_Code_integer @ X4 ) ) ) ).

% abs_add_one_gt_zero
thf(fact_5339_abs__add__one__gt__zero,axiom,
    ! [X4: real] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ ( abs_abs_real @ X4 ) ) ) ).

% abs_add_one_gt_zero
thf(fact_5340_abs__add__one__gt__zero,axiom,
    ! [X4: rat] : ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ ( abs_abs_rat @ X4 ) ) ) ).

% abs_add_one_gt_zero
thf(fact_5341_abs__add__one__gt__zero,axiom,
    ! [X4: int] : ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ ( abs_abs_int @ X4 ) ) ) ).

% abs_add_one_gt_zero
thf(fact_5342_of__int__leD,axiom,
    ! [N: int,X4: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( ring_18347121197199848620nteger @ N ) ) @ X4 )
     => ( ( N = zero_zero_int )
        | ( ord_le3102999989581377725nteger @ one_one_Code_integer @ X4 ) ) ) ).

% of_int_leD
thf(fact_5343_of__int__leD,axiom,
    ! [N: int,X4: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ ( ring_1_of_int_real @ N ) ) @ X4 )
     => ( ( N = zero_zero_int )
        | ( ord_less_eq_real @ one_one_real @ X4 ) ) ) ).

% of_int_leD
thf(fact_5344_of__int__leD,axiom,
    ! [N: int,X4: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ ( ring_1_of_int_rat @ N ) ) @ X4 )
     => ( ( N = zero_zero_int )
        | ( ord_less_eq_rat @ one_one_rat @ X4 ) ) ) ).

% of_int_leD
thf(fact_5345_of__int__leD,axiom,
    ! [N: int,X4: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N ) ) @ X4 )
     => ( ( N = zero_zero_int )
        | ( ord_less_eq_int @ one_one_int @ X4 ) ) ) ).

% of_int_leD
thf(fact_5346_of__int__lessD,axiom,
    ! [N: int,X4: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( ring_18347121197199848620nteger @ N ) ) @ X4 )
     => ( ( N = zero_zero_int )
        | ( ord_le6747313008572928689nteger @ one_one_Code_integer @ X4 ) ) ) ).

% of_int_lessD
thf(fact_5347_of__int__lessD,axiom,
    ! [N: int,X4: real] :
      ( ( ord_less_real @ ( abs_abs_real @ ( ring_1_of_int_real @ N ) ) @ X4 )
     => ( ( N = zero_zero_int )
        | ( ord_less_real @ one_one_real @ X4 ) ) ) ).

% of_int_lessD
thf(fact_5348_of__int__lessD,axiom,
    ! [N: int,X4: rat] :
      ( ( ord_less_rat @ ( abs_abs_rat @ ( ring_1_of_int_rat @ N ) ) @ X4 )
     => ( ( N = zero_zero_int )
        | ( ord_less_rat @ one_one_rat @ X4 ) ) ) ).

% of_int_lessD
thf(fact_5349_of__int__lessD,axiom,
    ! [N: int,X4: int] :
      ( ( ord_less_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N ) ) @ X4 )
     => ( ( N = zero_zero_int )
        | ( ord_less_int @ one_one_int @ X4 ) ) ) ).

% of_int_lessD
thf(fact_5350_lemma__interval,axiom,
    ! [A: real,X4: real,B2: real] :
      ( ( ord_less_real @ A @ X4 )
     => ( ( ord_less_real @ X4 @ B2 )
       => ? [D3: real] :
            ( ( ord_less_real @ zero_zero_real @ D3 )
            & ! [Y5: real] :
                ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X4 @ Y5 ) ) @ D3 )
               => ( ( ord_less_eq_real @ A @ Y5 )
                  & ( ord_less_eq_real @ Y5 @ B2 ) ) ) ) ) ) ).

% lemma_interval
thf(fact_5351_of__int__round__abs__le,axiom,
    ! [X4: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X4 ) ) @ X4 ) ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% of_int_round_abs_le
thf(fact_5352_of__int__round__abs__le,axiom,
    ! [X4: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X4 ) ) @ X4 ) ) @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).

% of_int_round_abs_le
thf(fact_5353_round__unique_H,axiom,
    ! [X4: real,N: int] :
      ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X4 @ ( ring_1_of_int_real @ N ) ) ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
     => ( ( archim8280529875227126926d_real @ X4 )
        = N ) ) ).

% round_unique'
thf(fact_5354_round__unique_H,axiom,
    ! [X4: rat,N: int] :
      ( ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X4 @ ( ring_1_of_int_rat @ N ) ) ) @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
     => ( ( archim7778729529865785530nd_rat @ X4 )
        = N ) ) ).

% round_unique'
thf(fact_5355_abs__le__square__iff,axiom,
    ! [X4: code_integer,Y3: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X4 ) @ ( abs_abs_Code_integer @ Y3 ) )
      = ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_le_square_iff
thf(fact_5356_abs__le__square__iff,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X4 ) @ ( abs_abs_real @ Y3 ) )
      = ( ord_less_eq_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_le_square_iff
thf(fact_5357_abs__le__square__iff,axiom,
    ! [X4: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ X4 ) @ ( abs_abs_rat @ Y3 ) )
      = ( ord_less_eq_rat @ ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_le_square_iff
thf(fact_5358_abs__le__square__iff,axiom,
    ! [X4: int,Y3: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ X4 ) @ ( abs_abs_int @ Y3 ) )
      = ( ord_less_eq_int @ ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_le_square_iff
thf(fact_5359_abs__square__eq__1,axiom,
    ! [X4: code_integer] :
      ( ( ( power_8256067586552552935nteger @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_Code_integer )
      = ( ( abs_abs_Code_integer @ X4 )
        = one_one_Code_integer ) ) ).

% abs_square_eq_1
thf(fact_5360_abs__square__eq__1,axiom,
    ! [X4: rat] :
      ( ( ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_rat )
      = ( ( abs_abs_rat @ X4 )
        = one_one_rat ) ) ).

% abs_square_eq_1
thf(fact_5361_abs__square__eq__1,axiom,
    ! [X4: real] :
      ( ( ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_real )
      = ( ( abs_abs_real @ X4 )
        = one_one_real ) ) ).

% abs_square_eq_1
thf(fact_5362_abs__square__eq__1,axiom,
    ! [X4: int] :
      ( ( ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_int )
      = ( ( abs_abs_int @ X4 )
        = one_one_int ) ) ).

% abs_square_eq_1
thf(fact_5363_power__even__abs,axiom,
    ! [N: nat,A: code_integer] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N )
        = ( power_8256067586552552935nteger @ A @ N ) ) ) ).

% power_even_abs
thf(fact_5364_power__even__abs,axiom,
    ! [N: nat,A: rat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_rat @ ( abs_abs_rat @ A ) @ N )
        = ( power_power_rat @ A @ N ) ) ) ).

% power_even_abs
thf(fact_5365_power__even__abs,axiom,
    ! [N: nat,A: real] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_real @ ( abs_abs_real @ A ) @ N )
        = ( power_power_real @ A @ N ) ) ) ).

% power_even_abs
thf(fact_5366_power__even__abs,axiom,
    ! [N: nat,A: int] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_int @ ( abs_abs_int @ A ) @ N )
        = ( power_power_int @ A @ N ) ) ) ).

% power_even_abs
thf(fact_5367_power2__le__iff__abs__le,axiom,
    ! [Y3: code_integer,X4: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ Y3 )
     => ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X4 ) @ Y3 ) ) ) ).

% power2_le_iff_abs_le
thf(fact_5368_power2__le__iff__abs__le,axiom,
    ! [Y3: real,X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
     => ( ( ord_less_eq_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = ( ord_less_eq_real @ ( abs_abs_real @ X4 ) @ Y3 ) ) ) ).

% power2_le_iff_abs_le
thf(fact_5369_power2__le__iff__abs__le,axiom,
    ! [Y3: rat,X4: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
     => ( ( ord_less_eq_rat @ ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = ( ord_less_eq_rat @ ( abs_abs_rat @ X4 ) @ Y3 ) ) ) ).

% power2_le_iff_abs_le
thf(fact_5370_power2__le__iff__abs__le,axiom,
    ! [Y3: int,X4: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
     => ( ( ord_less_eq_int @ ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = ( ord_less_eq_int @ ( abs_abs_int @ X4 ) @ Y3 ) ) ) ).

% power2_le_iff_abs_le
thf(fact_5371_abs__square__le__1,axiom,
    ! [X4: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer )
      = ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X4 ) @ one_one_Code_integer ) ) ).

% abs_square_le_1
thf(fact_5372_abs__square__le__1,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real )
      = ( ord_less_eq_real @ ( abs_abs_real @ X4 ) @ one_one_real ) ) ).

% abs_square_le_1
thf(fact_5373_abs__square__le__1,axiom,
    ! [X4: rat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat )
      = ( ord_less_eq_rat @ ( abs_abs_rat @ X4 ) @ one_one_rat ) ) ).

% abs_square_le_1
thf(fact_5374_abs__square__le__1,axiom,
    ! [X4: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int )
      = ( ord_less_eq_int @ ( abs_abs_int @ X4 ) @ one_one_int ) ) ).

% abs_square_le_1
thf(fact_5375_abs__square__less__1,axiom,
    ! [X4: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer )
      = ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ X4 ) @ one_one_Code_integer ) ) ).

% abs_square_less_1
thf(fact_5376_abs__square__less__1,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real )
      = ( ord_less_real @ ( abs_abs_real @ X4 ) @ one_one_real ) ) ).

% abs_square_less_1
thf(fact_5377_abs__square__less__1,axiom,
    ! [X4: rat] :
      ( ( ord_less_rat @ ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat )
      = ( ord_less_rat @ ( abs_abs_rat @ X4 ) @ one_one_rat ) ) ).

% abs_square_less_1
thf(fact_5378_abs__square__less__1,axiom,
    ! [X4: int] :
      ( ( ord_less_int @ ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int )
      = ( ord_less_int @ ( abs_abs_int @ X4 ) @ one_one_int ) ) ).

% abs_square_less_1
thf(fact_5379_power__mono__even,axiom,
    ! [N: nat,A: code_integer,B2: code_integer] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B2 ) )
       => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A @ N ) @ ( power_8256067586552552935nteger @ B2 @ N ) ) ) ) ).

% power_mono_even
thf(fact_5380_power__mono__even,axiom,
    ! [N: nat,A: real,B2: real] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B2 ) )
       => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B2 @ N ) ) ) ) ).

% power_mono_even
thf(fact_5381_power__mono__even,axiom,
    ! [N: nat,A: rat,B2: rat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B2 ) )
       => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B2 @ N ) ) ) ) ).

% power_mono_even
thf(fact_5382_power__mono__even,axiom,
    ! [N: nat,A: int,B2: int] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B2 ) )
       => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B2 @ N ) ) ) ) ).

% power_mono_even
thf(fact_5383_sqrt__ge__absD,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X4 ) @ ( sqrt @ Y3 ) )
     => ( ord_less_eq_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y3 ) ) ).

% sqrt_ge_absD
thf(fact_5384_real__sqrt__ge__abs1,axiom,
    ! [X4: real,Y3: real] : ( ord_less_eq_real @ ( abs_abs_real @ X4 ) @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% real_sqrt_ge_abs1
thf(fact_5385_real__sqrt__ge__abs2,axiom,
    ! [Y3: real,X4: real] : ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% real_sqrt_ge_abs2
thf(fact_5386_sqrt__sum__squares__le__sum__abs,axiom,
    ! [X4: real,Y3: real] : ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ ( abs_abs_real @ X4 ) @ ( abs_abs_real @ Y3 ) ) ) ).

% sqrt_sum_squares_le_sum_abs
thf(fact_5387_cos__x__y__le__one,axiom,
    ! [X4: real,Y3: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( divide_divide_real @ X4 @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ one_one_real ) ).

% cos_x_y_le_one
thf(fact_5388_real__sqrt__sum__squares__less,axiom,
    ! [X4: real,U: real,Y3: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X4 ) @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
     => ( ( ord_less_real @ ( abs_abs_real @ Y3 ) @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
       => ( ord_less_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ U ) ) ) ).

% real_sqrt_sum_squares_less
thf(fact_5389_abs__ln__one__plus__x__minus__x__bound__nonneg,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_eq_real @ X4 @ one_one_real )
       => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X4 ) ) @ X4 ) ) @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonneg
thf(fact_5390_of__int__round__le,axiom,
    ! [X4: real] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X4 ) ) @ ( plus_plus_real @ X4 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% of_int_round_le
thf(fact_5391_of__int__round__le,axiom,
    ! [X4: rat] : ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X4 ) ) @ ( plus_plus_rat @ X4 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).

% of_int_round_le
thf(fact_5392_abs__sqrt__wlog,axiom,
    ! [P: code_integer > code_integer > $o,X4: code_integer] :
      ( ! [X3: code_integer] :
          ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X3 )
         => ( P @ X3 @ ( power_8256067586552552935nteger @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
     => ( P @ ( abs_abs_Code_integer @ X4 ) @ ( power_8256067586552552935nteger @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_sqrt_wlog
thf(fact_5393_abs__sqrt__wlog,axiom,
    ! [P: real > real > $o,X4: real] :
      ( ! [X3: real] :
          ( ( ord_less_eq_real @ zero_zero_real @ X3 )
         => ( P @ X3 @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
     => ( P @ ( abs_abs_real @ X4 ) @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_sqrt_wlog
thf(fact_5394_abs__sqrt__wlog,axiom,
    ! [P: rat > rat > $o,X4: rat] :
      ( ! [X3: rat] :
          ( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
         => ( P @ X3 @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
     => ( P @ ( abs_abs_rat @ X4 ) @ ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_sqrt_wlog
thf(fact_5395_abs__sqrt__wlog,axiom,
    ! [P: int > int > $o,X4: int] :
      ( ! [X3: int] :
          ( ( ord_less_eq_int @ zero_zero_int @ X3 )
         => ( P @ X3 @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
     => ( P @ ( abs_abs_int @ X4 ) @ ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_sqrt_wlog
thf(fact_5396_arctan__double,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X4 ) @ one_one_real )
     => ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( arctan @ X4 ) )
        = ( arctan @ ( divide_divide_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X4 ) @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% arctan_double
thf(fact_5397_arctan__half,axiom,
    ( arctan
    = ( ^ [X: real] : ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( arctan @ ( divide_divide_real @ X @ ( plus_plus_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).

% arctan_half
thf(fact_5398_log__base__10__eq1,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ X4 )
        = ( times_times_real @ ( divide_divide_real @ ( ln_ln_real @ ( exp_real @ one_one_real ) ) @ ( ln_ln_real @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) ) @ ( ln_ln_real @ X4 ) ) ) ) ).

% log_base_10_eq1
thf(fact_5399_signed__take__bit__eq__take__bit__minus,axiom,
    ( bit_ri631733984087533419it_int
    = ( ^ [N2: nat,K3: int] : ( minus_minus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ K3 ) @ ( times_times_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K3 @ N2 ) ) ) ) ) ) ).

% signed_take_bit_eq_take_bit_minus
thf(fact_5400_log__base__10__eq2,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ X4 )
        = ( times_times_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( exp_real @ one_one_real ) ) @ ( ln_ln_real @ X4 ) ) ) ) ).

% log_base_10_eq2
thf(fact_5401_exp__ge__one__minus__x__over__n__power__n,axiom,
    ! [X4: real,N: nat] :
      ( ( ord_less_eq_real @ X4 @ ( semiri5074537144036343181t_real @ N ) )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_eq_real @ ( power_power_real @ ( minus_minus_real @ one_one_real @ ( divide_divide_real @ X4 @ ( semiri5074537144036343181t_real @ N ) ) ) @ N ) @ ( exp_real @ ( uminus_uminus_real @ X4 ) ) ) ) ) ).

% exp_ge_one_minus_x_over_n_power_n
thf(fact_5402_of__nat__eq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ( semiri8010041392384452111omplex @ M )
        = ( semiri8010041392384452111omplex @ N ) )
      = ( M = N ) ) ).

% of_nat_eq_iff
thf(fact_5403_of__nat__eq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ( semiri5074537144036343181t_real @ M )
        = ( semiri5074537144036343181t_real @ N ) )
      = ( M = N ) ) ).

% of_nat_eq_iff
thf(fact_5404_of__nat__eq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ( semiri681578069525770553at_rat @ M )
        = ( semiri681578069525770553at_rat @ N ) )
      = ( M = N ) ) ).

% of_nat_eq_iff
thf(fact_5405_of__nat__eq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ( semiri1316708129612266289at_nat @ M )
        = ( semiri1316708129612266289at_nat @ N ) )
      = ( M = N ) ) ).

% of_nat_eq_iff
thf(fact_5406_of__nat__eq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ( semiri1314217659103216013at_int @ M )
        = ( semiri1314217659103216013at_int @ N ) )
      = ( M = N ) ) ).

% of_nat_eq_iff
thf(fact_5407_abs__of__nat,axiom,
    ! [N: nat] :
      ( ( abs_abs_Code_integer @ ( semiri4939895301339042750nteger @ N ) )
      = ( semiri4939895301339042750nteger @ N ) ) ).

% abs_of_nat
thf(fact_5408_abs__of__nat,axiom,
    ! [N: nat] :
      ( ( abs_abs_real @ ( semiri5074537144036343181t_real @ N ) )
      = ( semiri5074537144036343181t_real @ N ) ) ).

% abs_of_nat
thf(fact_5409_abs__of__nat,axiom,
    ! [N: nat] :
      ( ( abs_abs_rat @ ( semiri681578069525770553at_rat @ N ) )
      = ( semiri681578069525770553at_rat @ N ) ) ).

% abs_of_nat
thf(fact_5410_abs__of__nat,axiom,
    ! [N: nat] :
      ( ( abs_abs_int @ ( semiri1314217659103216013at_int @ N ) )
      = ( semiri1314217659103216013at_int @ N ) ) ).

% abs_of_nat
thf(fact_5411_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( ( semiri8010041392384452111omplex @ M )
        = zero_zero_complex )
      = ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
thf(fact_5412_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( ( semiri5074537144036343181t_real @ M )
        = zero_zero_real )
      = ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
thf(fact_5413_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( ( semiri681578069525770553at_rat @ M )
        = zero_zero_rat )
      = ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
thf(fact_5414_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( ( semiri1316708129612266289at_nat @ M )
        = zero_zero_nat )
      = ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
thf(fact_5415_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( ( semiri1314217659103216013at_int @ M )
        = zero_zero_int )
      = ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
thf(fact_5416_of__nat__0__eq__iff,axiom,
    ! [N: nat] :
      ( ( zero_zero_complex
        = ( semiri8010041392384452111omplex @ N ) )
      = ( zero_zero_nat = N ) ) ).

% of_nat_0_eq_iff
thf(fact_5417_of__nat__0__eq__iff,axiom,
    ! [N: nat] :
      ( ( zero_zero_real
        = ( semiri5074537144036343181t_real @ N ) )
      = ( zero_zero_nat = N ) ) ).

% of_nat_0_eq_iff
thf(fact_5418_of__nat__0__eq__iff,axiom,
    ! [N: nat] :
      ( ( zero_zero_rat
        = ( semiri681578069525770553at_rat @ N ) )
      = ( zero_zero_nat = N ) ) ).

% of_nat_0_eq_iff
thf(fact_5419_of__nat__0__eq__iff,axiom,
    ! [N: nat] :
      ( ( zero_zero_nat
        = ( semiri1316708129612266289at_nat @ N ) )
      = ( zero_zero_nat = N ) ) ).

% of_nat_0_eq_iff
thf(fact_5420_of__nat__0__eq__iff,axiom,
    ! [N: nat] :
      ( ( zero_zero_int
        = ( semiri1314217659103216013at_int @ N ) )
      = ( zero_zero_nat = N ) ) ).

% of_nat_0_eq_iff
thf(fact_5421_of__nat__0,axiom,
    ( ( semiri8010041392384452111omplex @ zero_zero_nat )
    = zero_zero_complex ) ).

% of_nat_0
thf(fact_5422_of__nat__0,axiom,
    ( ( semiri5074537144036343181t_real @ zero_zero_nat )
    = zero_zero_real ) ).

% of_nat_0
thf(fact_5423_of__nat__0,axiom,
    ( ( semiri681578069525770553at_rat @ zero_zero_nat )
    = zero_zero_rat ) ).

% of_nat_0
thf(fact_5424_of__nat__0,axiom,
    ( ( semiri1316708129612266289at_nat @ zero_zero_nat )
    = zero_zero_nat ) ).

% of_nat_0
thf(fact_5425_of__nat__0,axiom,
    ( ( semiri1314217659103216013at_int @ zero_zero_nat )
    = zero_zero_int ) ).

% of_nat_0
thf(fact_5426_of__nat__less__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% of_nat_less_iff
thf(fact_5427_of__nat__less__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% of_nat_less_iff
thf(fact_5428_of__nat__less__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% of_nat_less_iff
thf(fact_5429_of__nat__less__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% of_nat_less_iff
thf(fact_5430_of__nat__numeral,axiom,
    ! [N: num] :
      ( ( semiri8010041392384452111omplex @ ( numeral_numeral_nat @ N ) )
      = ( numera6690914467698888265omplex @ N ) ) ).

% of_nat_numeral
thf(fact_5431_of__nat__numeral,axiom,
    ! [N: num] :
      ( ( semiri5074537144036343181t_real @ ( numeral_numeral_nat @ N ) )
      = ( numeral_numeral_real @ N ) ) ).

% of_nat_numeral
thf(fact_5432_of__nat__numeral,axiom,
    ! [N: num] :
      ( ( semiri681578069525770553at_rat @ ( numeral_numeral_nat @ N ) )
      = ( numeral_numeral_rat @ N ) ) ).

% of_nat_numeral
thf(fact_5433_of__nat__numeral,axiom,
    ! [N: num] :
      ( ( semiri1316708129612266289at_nat @ ( numeral_numeral_nat @ N ) )
      = ( numeral_numeral_nat @ N ) ) ).

% of_nat_numeral
thf(fact_5434_of__nat__numeral,axiom,
    ! [N: num] :
      ( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N ) )
      = ( numeral_numeral_int @ N ) ) ).

% of_nat_numeral
thf(fact_5435_of__nat__le__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% of_nat_le_iff
thf(fact_5436_of__nat__le__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% of_nat_le_iff
thf(fact_5437_of__nat__le__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% of_nat_le_iff
thf(fact_5438_of__nat__le__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% of_nat_le_iff
thf(fact_5439_of__nat__add,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri8010041392384452111omplex @ ( plus_plus_nat @ M @ N ) )
      = ( plus_plus_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ N ) ) ) ).

% of_nat_add
thf(fact_5440_of__nat__add,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri5074537144036343181t_real @ ( plus_plus_nat @ M @ N ) )
      = ( plus_plus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).

% of_nat_add
thf(fact_5441_of__nat__add,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri681578069525770553at_rat @ ( plus_plus_nat @ M @ N ) )
      = ( plus_plus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).

% of_nat_add
thf(fact_5442_of__nat__add,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ M @ N ) )
      = ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).

% of_nat_add
thf(fact_5443_of__nat__add,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).

% of_nat_add
thf(fact_5444_of__nat__mult,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri8010041392384452111omplex @ ( times_times_nat @ M @ N ) )
      = ( times_times_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ N ) ) ) ).

% of_nat_mult
thf(fact_5445_of__nat__mult,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri5074537144036343181t_real @ ( times_times_nat @ M @ N ) )
      = ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).

% of_nat_mult
thf(fact_5446_of__nat__mult,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri681578069525770553at_rat @ ( times_times_nat @ M @ N ) )
      = ( times_times_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).

% of_nat_mult
thf(fact_5447_of__nat__mult,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M @ N ) )
      = ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).

% of_nat_mult
thf(fact_5448_of__nat__mult,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri1314217659103216013at_int @ ( times_times_nat @ M @ N ) )
      = ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).

% of_nat_mult
thf(fact_5449_of__nat__1,axiom,
    ( ( semiri8010041392384452111omplex @ one_one_nat )
    = one_one_complex ) ).

% of_nat_1
thf(fact_5450_of__nat__1,axiom,
    ( ( semiri5074537144036343181t_real @ one_one_nat )
    = one_one_real ) ).

% of_nat_1
thf(fact_5451_of__nat__1,axiom,
    ( ( semiri681578069525770553at_rat @ one_one_nat )
    = one_one_rat ) ).

% of_nat_1
thf(fact_5452_of__nat__1,axiom,
    ( ( semiri1316708129612266289at_nat @ one_one_nat )
    = one_one_nat ) ).

% of_nat_1
thf(fact_5453_of__nat__1,axiom,
    ( ( semiri1314217659103216013at_int @ one_one_nat )
    = one_one_int ) ).

% of_nat_1
thf(fact_5454_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_complex
        = ( semiri8010041392384452111omplex @ N ) )
      = ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_5455_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_real
        = ( semiri5074537144036343181t_real @ N ) )
      = ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_5456_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_rat
        = ( semiri681578069525770553at_rat @ N ) )
      = ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_5457_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_nat
        = ( semiri1316708129612266289at_nat @ N ) )
      = ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_5458_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_int
        = ( semiri1314217659103216013at_int @ N ) )
      = ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_5459_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( ( semiri8010041392384452111omplex @ N )
        = one_one_complex )
      = ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
thf(fact_5460_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( ( semiri5074537144036343181t_real @ N )
        = one_one_real )
      = ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
thf(fact_5461_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( ( semiri681578069525770553at_rat @ N )
        = one_one_rat )
      = ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
thf(fact_5462_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( ( semiri1316708129612266289at_nat @ N )
        = one_one_nat )
      = ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
thf(fact_5463_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( ( semiri1314217659103216013at_int @ N )
        = one_one_int )
      = ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
thf(fact_5464_of__nat__power,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri8010041392384452111omplex @ ( power_power_nat @ M @ N ) )
      = ( power_power_complex @ ( semiri8010041392384452111omplex @ M ) @ N ) ) ).

% of_nat_power
thf(fact_5465_of__nat__power,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri5074537144036343181t_real @ ( power_power_nat @ M @ N ) )
      = ( power_power_real @ ( semiri5074537144036343181t_real @ M ) @ N ) ) ).

% of_nat_power
thf(fact_5466_of__nat__power,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri681578069525770553at_rat @ ( power_power_nat @ M @ N ) )
      = ( power_power_rat @ ( semiri681578069525770553at_rat @ M ) @ N ) ) ).

% of_nat_power
thf(fact_5467_of__nat__power,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri1316708129612266289at_nat @ ( power_power_nat @ M @ N ) )
      = ( power_power_nat @ ( semiri1316708129612266289at_nat @ M ) @ N ) ) ).

% of_nat_power
thf(fact_5468_of__nat__power,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri1314217659103216013at_int @ ( power_power_nat @ M @ N ) )
      = ( power_power_int @ ( semiri1314217659103216013at_int @ M ) @ N ) ) ).

% of_nat_power
thf(fact_5469_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W: nat,X4: nat] :
      ( ( ( power_power_complex @ ( semiri8010041392384452111omplex @ B2 ) @ W )
        = ( semiri8010041392384452111omplex @ X4 ) )
      = ( ( power_power_nat @ B2 @ W )
        = X4 ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
thf(fact_5470_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W: nat,X4: nat] :
      ( ( ( power_power_real @ ( semiri5074537144036343181t_real @ B2 ) @ W )
        = ( semiri5074537144036343181t_real @ X4 ) )
      = ( ( power_power_nat @ B2 @ W )
        = X4 ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
thf(fact_5471_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W: nat,X4: nat] :
      ( ( ( power_power_rat @ ( semiri681578069525770553at_rat @ B2 ) @ W )
        = ( semiri681578069525770553at_rat @ X4 ) )
      = ( ( power_power_nat @ B2 @ W )
        = X4 ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
thf(fact_5472_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W: nat,X4: nat] :
      ( ( ( power_power_nat @ ( semiri1316708129612266289at_nat @ B2 ) @ W )
        = ( semiri1316708129612266289at_nat @ X4 ) )
      = ( ( power_power_nat @ B2 @ W )
        = X4 ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
thf(fact_5473_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W: nat,X4: nat] :
      ( ( ( power_power_int @ ( semiri1314217659103216013at_int @ B2 ) @ W )
        = ( semiri1314217659103216013at_int @ X4 ) )
      = ( ( power_power_nat @ B2 @ W )
        = X4 ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
thf(fact_5474_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [X4: nat,B2: nat,W: nat] :
      ( ( ( semiri8010041392384452111omplex @ X4 )
        = ( power_power_complex @ ( semiri8010041392384452111omplex @ B2 ) @ W ) )
      = ( X4
        = ( power_power_nat @ B2 @ W ) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
thf(fact_5475_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [X4: nat,B2: nat,W: nat] :
      ( ( ( semiri5074537144036343181t_real @ X4 )
        = ( power_power_real @ ( semiri5074537144036343181t_real @ B2 ) @ W ) )
      = ( X4
        = ( power_power_nat @ B2 @ W ) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
thf(fact_5476_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [X4: nat,B2: nat,W: nat] :
      ( ( ( semiri681578069525770553at_rat @ X4 )
        = ( power_power_rat @ ( semiri681578069525770553at_rat @ B2 ) @ W ) )
      = ( X4
        = ( power_power_nat @ B2 @ W ) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
thf(fact_5477_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [X4: nat,B2: nat,W: nat] :
      ( ( ( semiri1316708129612266289at_nat @ X4 )
        = ( power_power_nat @ ( semiri1316708129612266289at_nat @ B2 ) @ W ) )
      = ( X4
        = ( power_power_nat @ B2 @ W ) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
thf(fact_5478_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [X4: nat,B2: nat,W: nat] :
      ( ( ( semiri1314217659103216013at_int @ X4 )
        = ( power_power_int @ ( semiri1314217659103216013at_int @ B2 ) @ W ) )
      = ( X4
        = ( power_power_nat @ B2 @ W ) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
thf(fact_5479_zabs__less__one__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_int @ ( abs_abs_int @ Z ) @ one_one_int )
      = ( Z = zero_zero_int ) ) ).

% zabs_less_one_iff
thf(fact_5480_log__one,axiom,
    ! [A: real] :
      ( ( log @ A @ one_one_real )
      = zero_zero_real ) ).

% log_one
thf(fact_5481_arctan__less__zero__iff,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ ( arctan @ X4 ) @ zero_zero_real )
      = ( ord_less_real @ X4 @ zero_zero_real ) ) ).

% arctan_less_zero_iff
thf(fact_5482_zero__less__arctan__iff,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ ( arctan @ X4 ) )
      = ( ord_less_real @ zero_zero_real @ X4 ) ) ).

% zero_less_arctan_iff
thf(fact_5483_zero__le__arctan__iff,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( arctan @ X4 ) )
      = ( ord_less_eq_real @ zero_zero_real @ X4 ) ) ).

% zero_le_arctan_iff
thf(fact_5484_arctan__le__zero__iff,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ ( arctan @ X4 ) @ zero_zero_real )
      = ( ord_less_eq_real @ X4 @ zero_zero_real ) ) ).

% arctan_le_zero_iff
thf(fact_5485_of__nat__of__bool,axiom,
    ! [P: $o] :
      ( ( semiri8010041392384452111omplex @ ( zero_n2687167440665602831ol_nat @ P ) )
      = ( zero_n1201886186963655149omplex @ P ) ) ).

% of_nat_of_bool
thf(fact_5486_of__nat__of__bool,axiom,
    ! [P: $o] :
      ( ( semiri5074537144036343181t_real @ ( zero_n2687167440665602831ol_nat @ P ) )
      = ( zero_n3304061248610475627l_real @ P ) ) ).

% of_nat_of_bool
thf(fact_5487_of__nat__of__bool,axiom,
    ! [P: $o] :
      ( ( semiri681578069525770553at_rat @ ( zero_n2687167440665602831ol_nat @ P ) )
      = ( zero_n2052037380579107095ol_rat @ P ) ) ).

% of_nat_of_bool
thf(fact_5488_of__nat__of__bool,axiom,
    ! [P: $o] :
      ( ( semiri1316708129612266289at_nat @ ( zero_n2687167440665602831ol_nat @ P ) )
      = ( zero_n2687167440665602831ol_nat @ P ) ) ).

% of_nat_of_bool
thf(fact_5489_of__nat__of__bool,axiom,
    ! [P: $o] :
      ( ( semiri1314217659103216013at_int @ ( zero_n2687167440665602831ol_nat @ P ) )
      = ( zero_n2684676970156552555ol_int @ P ) ) ).

% of_nat_of_bool
thf(fact_5490_of__nat__of__bool,axiom,
    ! [P: $o] :
      ( ( semiri4939895301339042750nteger @ ( zero_n2687167440665602831ol_nat @ P ) )
      = ( zero_n356916108424825756nteger @ P ) ) ).

% of_nat_of_bool
thf(fact_5491_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real )
      = ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
thf(fact_5492_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ M ) @ zero_zero_rat )
      = ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
thf(fact_5493_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat )
      = ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
thf(fact_5494_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int )
      = ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
thf(fact_5495_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri8010041392384452111omplex @ ( suc @ M ) )
      = ( plus_plus_complex @ one_one_complex @ ( semiri8010041392384452111omplex @ M ) ) ) ).

% of_nat_Suc
thf(fact_5496_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri5074537144036343181t_real @ ( suc @ M ) )
      = ( plus_plus_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) ) ).

% of_nat_Suc
thf(fact_5497_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri681578069525770553at_rat @ ( suc @ M ) )
      = ( plus_plus_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ M ) ) ) ).

% of_nat_Suc
thf(fact_5498_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri1316708129612266289at_nat @ ( suc @ M ) )
      = ( plus_plus_nat @ one_one_nat @ ( semiri1316708129612266289at_nat @ M ) ) ) ).

% of_nat_Suc
thf(fact_5499_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri1314217659103216013at_int @ ( suc @ M ) )
      = ( plus_plus_int @ one_one_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).

% of_nat_Suc
thf(fact_5500_bit__numeral__Bit0__Suc__iff,axiom,
    ! [M: num,N: nat] :
      ( ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( suc @ N ) )
      = ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ M ) @ N ) ) ).

% bit_numeral_Bit0_Suc_iff
thf(fact_5501_bit__numeral__Bit0__Suc__iff,axiom,
    ! [M: num,N: nat] :
      ( ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( suc @ N ) )
      = ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ M ) @ N ) ) ).

% bit_numeral_Bit0_Suc_iff
thf(fact_5502_bit__numeral__Bit1__Suc__iff,axiom,
    ! [M: num,N: nat] :
      ( ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( suc @ N ) )
      = ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ M ) @ N ) ) ).

% bit_numeral_Bit1_Suc_iff
thf(fact_5503_bit__numeral__Bit1__Suc__iff,axiom,
    ! [M: num,N: nat] :
      ( ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( suc @ N ) )
      = ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ M ) @ N ) ) ).

% bit_numeral_Bit1_Suc_iff
thf(fact_5504_log__eq__one,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( log @ A @ A )
          = one_one_real ) ) ) ).

% log_eq_one
thf(fact_5505_log__less__cancel__iff,axiom,
    ! [A: real,X4: real,Y3: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X4 )
       => ( ( ord_less_real @ zero_zero_real @ Y3 )
         => ( ( ord_less_real @ ( log @ A @ X4 ) @ ( log @ A @ Y3 ) )
            = ( ord_less_real @ X4 @ Y3 ) ) ) ) ) ).

% log_less_cancel_iff
thf(fact_5506_log__less__one__cancel__iff,axiom,
    ! [A: real,X4: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X4 )
       => ( ( ord_less_real @ ( log @ A @ X4 ) @ one_one_real )
          = ( ord_less_real @ X4 @ A ) ) ) ) ).

% log_less_one_cancel_iff
thf(fact_5507_one__less__log__cancel__iff,axiom,
    ! [A: real,X4: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X4 )
       => ( ( ord_less_real @ one_one_real @ ( log @ A @ X4 ) )
          = ( ord_less_real @ A @ X4 ) ) ) ) ).

% one_less_log_cancel_iff
thf(fact_5508_log__less__zero__cancel__iff,axiom,
    ! [A: real,X4: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X4 )
       => ( ( ord_less_real @ ( log @ A @ X4 ) @ zero_zero_real )
          = ( ord_less_real @ X4 @ one_one_real ) ) ) ) ).

% log_less_zero_cancel_iff
thf(fact_5509_zero__less__log__cancel__iff,axiom,
    ! [A: real,X4: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X4 )
       => ( ( ord_less_real @ zero_zero_real @ ( log @ A @ X4 ) )
          = ( ord_less_real @ one_one_real @ X4 ) ) ) ) ).

% zero_less_log_cancel_iff
thf(fact_5510_signed__take__bit__nonnegative__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_ri631733984087533419it_int @ N @ K ) )
      = ( ~ ( bit_se1146084159140164899it_int @ K @ N ) ) ) ).

% signed_take_bit_nonnegative_iff
thf(fact_5511_signed__take__bit__negative__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ zero_zero_int )
      = ( bit_se1146084159140164899it_int @ K @ N ) ) ).

% signed_take_bit_negative_iff
thf(fact_5512_of__nat__0__less__iff,axiom,
    ! [N: nat] :
      ( ( ord_less_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) )
      = ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% of_nat_0_less_iff
thf(fact_5513_of__nat__0__less__iff,axiom,
    ! [N: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( semiri681578069525770553at_rat @ N ) )
      = ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% of_nat_0_less_iff
thf(fact_5514_of__nat__0__less__iff,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) )
      = ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% of_nat_0_less_iff
thf(fact_5515_of__nat__0__less__iff,axiom,
    ! [N: nat] :
      ( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) )
      = ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% of_nat_0_less_iff
thf(fact_5516_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W: nat,X4: nat] :
      ( ( ord_less_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B2 ) @ W ) @ ( semiri5074537144036343181t_real @ X4 ) )
      = ( ord_less_nat @ ( power_power_nat @ B2 @ W ) @ X4 ) ) ).

% of_nat_less_of_nat_power_cancel_iff
thf(fact_5517_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W: nat,X4: nat] :
      ( ( ord_less_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B2 ) @ W ) @ ( semiri681578069525770553at_rat @ X4 ) )
      = ( ord_less_nat @ ( power_power_nat @ B2 @ W ) @ X4 ) ) ).

% of_nat_less_of_nat_power_cancel_iff
thf(fact_5518_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W: nat,X4: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B2 ) @ W ) @ ( semiri1316708129612266289at_nat @ X4 ) )
      = ( ord_less_nat @ ( power_power_nat @ B2 @ W ) @ X4 ) ) ).

% of_nat_less_of_nat_power_cancel_iff
thf(fact_5519_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W: nat,X4: nat] :
      ( ( ord_less_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B2 ) @ W ) @ ( semiri1314217659103216013at_int @ X4 ) )
      = ( ord_less_nat @ ( power_power_nat @ B2 @ W ) @ X4 ) ) ).

% of_nat_less_of_nat_power_cancel_iff
thf(fact_5520_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X4: nat,B2: nat,W: nat] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ X4 ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B2 ) @ W ) )
      = ( ord_less_nat @ X4 @ ( power_power_nat @ B2 @ W ) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
thf(fact_5521_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X4: nat,B2: nat,W: nat] :
      ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ X4 ) @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B2 ) @ W ) )
      = ( ord_less_nat @ X4 @ ( power_power_nat @ B2 @ W ) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
thf(fact_5522_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X4: nat,B2: nat,W: nat] :
      ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X4 ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B2 ) @ W ) )
      = ( ord_less_nat @ X4 @ ( power_power_nat @ B2 @ W ) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
thf(fact_5523_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X4: nat,B2: nat,W: nat] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ X4 ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B2 ) @ W ) )
      = ( ord_less_nat @ X4 @ ( power_power_nat @ B2 @ W ) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
thf(fact_5524_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y3: nat,X4: num,N: nat] :
      ( ( ( semiri8010041392384452111omplex @ Y3 )
        = ( power_power_complex @ ( numera6690914467698888265omplex @ X4 ) @ N ) )
      = ( Y3
        = ( power_power_nat @ ( numeral_numeral_nat @ X4 ) @ N ) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_5525_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y3: nat,X4: num,N: nat] :
      ( ( ( semiri5074537144036343181t_real @ Y3 )
        = ( power_power_real @ ( numeral_numeral_real @ X4 ) @ N ) )
      = ( Y3
        = ( power_power_nat @ ( numeral_numeral_nat @ X4 ) @ N ) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_5526_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y3: nat,X4: num,N: nat] :
      ( ( ( semiri681578069525770553at_rat @ Y3 )
        = ( power_power_rat @ ( numeral_numeral_rat @ X4 ) @ N ) )
      = ( Y3
        = ( power_power_nat @ ( numeral_numeral_nat @ X4 ) @ N ) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_5527_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y3: nat,X4: num,N: nat] :
      ( ( ( semiri1316708129612266289at_nat @ Y3 )
        = ( power_power_nat @ ( numeral_numeral_nat @ X4 ) @ N ) )
      = ( Y3
        = ( power_power_nat @ ( numeral_numeral_nat @ X4 ) @ N ) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_5528_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y3: nat,X4: num,N: nat] :
      ( ( ( semiri1314217659103216013at_int @ Y3 )
        = ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N ) )
      = ( Y3
        = ( power_power_nat @ ( numeral_numeral_nat @ X4 ) @ N ) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_5529_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X4: num,N: nat,Y3: nat] :
      ( ( ( power_power_complex @ ( numera6690914467698888265omplex @ X4 ) @ N )
        = ( semiri8010041392384452111omplex @ Y3 ) )
      = ( ( power_power_nat @ ( numeral_numeral_nat @ X4 ) @ N )
        = Y3 ) ) ).

% numeral_power_eq_of_nat_cancel_iff
thf(fact_5530_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X4: num,N: nat,Y3: nat] :
      ( ( ( power_power_real @ ( numeral_numeral_real @ X4 ) @ N )
        = ( semiri5074537144036343181t_real @ Y3 ) )
      = ( ( power_power_nat @ ( numeral_numeral_nat @ X4 ) @ N )
        = Y3 ) ) ).

% numeral_power_eq_of_nat_cancel_iff
thf(fact_5531_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X4: num,N: nat,Y3: nat] :
      ( ( ( power_power_rat @ ( numeral_numeral_rat @ X4 ) @ N )
        = ( semiri681578069525770553at_rat @ Y3 ) )
      = ( ( power_power_nat @ ( numeral_numeral_nat @ X4 ) @ N )
        = Y3 ) ) ).

% numeral_power_eq_of_nat_cancel_iff
thf(fact_5532_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X4: num,N: nat,Y3: nat] :
      ( ( ( power_power_nat @ ( numeral_numeral_nat @ X4 ) @ N )
        = ( semiri1316708129612266289at_nat @ Y3 ) )
      = ( ( power_power_nat @ ( numeral_numeral_nat @ X4 ) @ N )
        = Y3 ) ) ).

% numeral_power_eq_of_nat_cancel_iff
thf(fact_5533_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X4: num,N: nat,Y3: nat] :
      ( ( ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N )
        = ( semiri1314217659103216013at_int @ Y3 ) )
      = ( ( power_power_nat @ ( numeral_numeral_nat @ X4 ) @ N )
        = Y3 ) ) ).

% numeral_power_eq_of_nat_cancel_iff
thf(fact_5534_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W: nat,X4: nat] :
      ( ( ord_less_eq_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B2 ) @ W ) @ ( semiri5074537144036343181t_real @ X4 ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ B2 @ W ) @ X4 ) ) ).

% of_nat_le_of_nat_power_cancel_iff
thf(fact_5535_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W: nat,X4: nat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B2 ) @ W ) @ ( semiri681578069525770553at_rat @ X4 ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ B2 @ W ) @ X4 ) ) ).

% of_nat_le_of_nat_power_cancel_iff
thf(fact_5536_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W: nat,X4: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B2 ) @ W ) @ ( semiri1316708129612266289at_nat @ X4 ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ B2 @ W ) @ X4 ) ) ).

% of_nat_le_of_nat_power_cancel_iff
thf(fact_5537_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W: nat,X4: nat] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B2 ) @ W ) @ ( semiri1314217659103216013at_int @ X4 ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ B2 @ W ) @ X4 ) ) ).

% of_nat_le_of_nat_power_cancel_iff
thf(fact_5538_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X4: nat,B2: nat,W: nat] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X4 ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B2 ) @ W ) )
      = ( ord_less_eq_nat @ X4 @ ( power_power_nat @ B2 @ W ) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
thf(fact_5539_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X4: nat,B2: nat,W: nat] :
      ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ X4 ) @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B2 ) @ W ) )
      = ( ord_less_eq_nat @ X4 @ ( power_power_nat @ B2 @ W ) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
thf(fact_5540_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X4: nat,B2: nat,W: nat] :
      ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X4 ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B2 ) @ W ) )
      = ( ord_less_eq_nat @ X4 @ ( power_power_nat @ B2 @ W ) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
thf(fact_5541_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X4: nat,B2: nat,W: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X4 ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B2 ) @ W ) )
      = ( ord_less_eq_nat @ X4 @ ( power_power_nat @ B2 @ W ) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
thf(fact_5542_numeral__less__real__of__nat__iff,axiom,
    ! [W: num,N: nat] :
      ( ( ord_less_real @ ( numeral_numeral_real @ W ) @ ( semiri5074537144036343181t_real @ N ) )
      = ( ord_less_nat @ ( numeral_numeral_nat @ W ) @ N ) ) ).

% numeral_less_real_of_nat_iff
thf(fact_5543_real__of__nat__less__numeral__iff,axiom,
    ! [N: nat,W: num] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( numeral_numeral_real @ W ) )
      = ( ord_less_nat @ N @ ( numeral_numeral_nat @ W ) ) ) ).

% real_of_nat_less_numeral_iff
thf(fact_5544_numeral__le__real__of__nat__iff,axiom,
    ! [N: num,M: nat] :
      ( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ ( semiri5074537144036343181t_real @ M ) )
      = ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ M ) ) ).

% numeral_le_real_of_nat_iff
thf(fact_5545_zero__le__log__cancel__iff,axiom,
    ! [A: real,X4: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X4 )
       => ( ( ord_less_eq_real @ zero_zero_real @ ( log @ A @ X4 ) )
          = ( ord_less_eq_real @ one_one_real @ X4 ) ) ) ) ).

% zero_le_log_cancel_iff
thf(fact_5546_log__le__zero__cancel__iff,axiom,
    ! [A: real,X4: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X4 )
       => ( ( ord_less_eq_real @ ( log @ A @ X4 ) @ zero_zero_real )
          = ( ord_less_eq_real @ X4 @ one_one_real ) ) ) ) ).

% log_le_zero_cancel_iff
thf(fact_5547_one__le__log__cancel__iff,axiom,
    ! [A: real,X4: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X4 )
       => ( ( ord_less_eq_real @ one_one_real @ ( log @ A @ X4 ) )
          = ( ord_less_eq_real @ A @ X4 ) ) ) ) ).

% one_le_log_cancel_iff
thf(fact_5548_log__le__one__cancel__iff,axiom,
    ! [A: real,X4: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X4 )
       => ( ( ord_less_eq_real @ ( log @ A @ X4 ) @ one_one_real )
          = ( ord_less_eq_real @ X4 @ A ) ) ) ) ).

% log_le_one_cancel_iff
thf(fact_5549_log__le__cancel__iff,axiom,
    ! [A: real,X4: real,Y3: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X4 )
       => ( ( ord_less_real @ zero_zero_real @ Y3 )
         => ( ( ord_less_eq_real @ ( log @ A @ X4 ) @ ( log @ A @ Y3 ) )
            = ( ord_less_eq_real @ X4 @ Y3 ) ) ) ) ) ).

% log_le_cancel_iff
thf(fact_5550_bit__numeral__simps_I2_J,axiom,
    ! [W: num,N: num] :
      ( ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ ( bit0 @ W ) ) @ ( numeral_numeral_nat @ N ) )
      = ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W ) @ ( pred_numeral @ N ) ) ) ).

% bit_numeral_simps(2)
thf(fact_5551_bit__numeral__simps_I2_J,axiom,
    ! [W: num,N: num] :
      ( ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ ( bit0 @ W ) ) @ ( numeral_numeral_nat @ N ) )
      = ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ W ) @ ( pred_numeral @ N ) ) ) ).

% bit_numeral_simps(2)
thf(fact_5552_bit__minus__numeral__Bit0__Suc__iff,axiom,
    ! [W: num,N: nat] :
      ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ W ) ) ) @ ( suc @ N ) )
      = ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ N ) ) ).

% bit_minus_numeral_Bit0_Suc_iff
thf(fact_5553_bit__numeral__simps_I3_J,axiom,
    ! [W: num,N: num] :
      ( ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ ( bit1 @ W ) ) @ ( numeral_numeral_nat @ N ) )
      = ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W ) @ ( pred_numeral @ N ) ) ) ).

% bit_numeral_simps(3)
thf(fact_5554_bit__numeral__simps_I3_J,axiom,
    ! [W: num,N: num] :
      ( ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ ( bit1 @ W ) ) @ ( numeral_numeral_nat @ N ) )
      = ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ W ) @ ( pred_numeral @ N ) ) ) ).

% bit_numeral_simps(3)
thf(fact_5555_bit__minus__numeral__Bit1__Suc__iff,axiom,
    ! [W: num,N: nat] :
      ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ W ) ) ) @ ( suc @ N ) )
      = ( ~ ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W ) @ N ) ) ) ).

% bit_minus_numeral_Bit1_Suc_iff
thf(fact_5556_of__nat__zero__less__power__iff,axiom,
    ! [X4: nat,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ X4 ) @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X4 )
        | ( N = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
thf(fact_5557_of__nat__zero__less__power__iff,axiom,
    ! [X4: nat,N: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ X4 ) @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X4 )
        | ( N = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
thf(fact_5558_of__nat__zero__less__power__iff,axiom,
    ! [X4: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ X4 ) @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X4 )
        | ( N = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
thf(fact_5559_of__nat__zero__less__power__iff,axiom,
    ! [X4: nat,N: nat] :
      ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ X4 ) @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X4 )
        | ( N = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
thf(fact_5560_bit__0,axiom,
    ! [A: code_integer] :
      ( ( bit_se9216721137139052372nteger @ A @ zero_zero_nat )
      = ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).

% bit_0
thf(fact_5561_bit__0,axiom,
    ! [A: int] :
      ( ( bit_se1146084159140164899it_int @ A @ zero_zero_nat )
      = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).

% bit_0
thf(fact_5562_bit__0,axiom,
    ! [A: nat] :
      ( ( bit_se1148574629649215175it_nat @ A @ zero_zero_nat )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).

% bit_0
thf(fact_5563_log__pow__cancel,axiom,
    ! [A: real,B2: nat] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( log @ A @ ( power_power_real @ A @ B2 ) )
          = ( semiri5074537144036343181t_real @ B2 ) ) ) ) ).

% log_pow_cancel
thf(fact_5564_bit__minus__numeral__int_I1_J,axiom,
    ! [W: num,N: num] :
      ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ W ) ) ) @ ( numeral_numeral_nat @ N ) )
      = ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ ( pred_numeral @ N ) ) ) ).

% bit_minus_numeral_int(1)
thf(fact_5565_bit__minus__numeral__int_I2_J,axiom,
    ! [W: num,N: num] :
      ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ W ) ) ) @ ( numeral_numeral_nat @ N ) )
      = ( ~ ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W ) @ ( pred_numeral @ N ) ) ) ) ).

% bit_minus_numeral_int(2)
thf(fact_5566_even__of__nat,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( semiri4939895301339042750nteger @ N ) )
      = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).

% even_of_nat
thf(fact_5567_even__of__nat,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( semiri1316708129612266289at_nat @ N ) )
      = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).

% even_of_nat
thf(fact_5568_even__of__nat,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( semiri1314217659103216013at_int @ N ) )
      = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).

% even_of_nat
thf(fact_5569_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X4: nat] :
      ( ( ord_less_real @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) @ ( semiri5074537144036343181t_real @ X4 ) )
      = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X4 ) ) ).

% numeral_power_less_of_nat_cancel_iff
thf(fact_5570_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X4: nat] :
      ( ( ord_less_rat @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N ) @ ( semiri681578069525770553at_rat @ X4 ) )
      = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X4 ) ) ).

% numeral_power_less_of_nat_cancel_iff
thf(fact_5571_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X4: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ ( semiri1316708129612266289at_nat @ X4 ) )
      = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X4 ) ) ).

% numeral_power_less_of_nat_cancel_iff
thf(fact_5572_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X4: nat] :
      ( ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) @ ( semiri1314217659103216013at_int @ X4 ) )
      = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X4 ) ) ).

% numeral_power_less_of_nat_cancel_iff
thf(fact_5573_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X4: nat,I: num,N: nat] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ X4 ) @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) )
      = ( ord_less_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).

% of_nat_less_numeral_power_cancel_iff
thf(fact_5574_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X4: nat,I: num,N: nat] :
      ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ X4 ) @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N ) )
      = ( ord_less_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).

% of_nat_less_numeral_power_cancel_iff
thf(fact_5575_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X4: nat,I: num,N: nat] :
      ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X4 ) @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) )
      = ( ord_less_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).

% of_nat_less_numeral_power_cancel_iff
thf(fact_5576_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X4: nat,I: num,N: nat] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ X4 ) @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) )
      = ( ord_less_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).

% of_nat_less_numeral_power_cancel_iff
thf(fact_5577_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X4: nat] :
      ( ( ord_less_eq_real @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) @ ( semiri5074537144036343181t_real @ X4 ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X4 ) ) ).

% numeral_power_le_of_nat_cancel_iff
thf(fact_5578_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X4: nat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N ) @ ( semiri681578069525770553at_rat @ X4 ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X4 ) ) ).

% numeral_power_le_of_nat_cancel_iff
thf(fact_5579_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X4: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ ( semiri1316708129612266289at_nat @ X4 ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X4 ) ) ).

% numeral_power_le_of_nat_cancel_iff
thf(fact_5580_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X4: nat] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) @ ( semiri1314217659103216013at_int @ X4 ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X4 ) ) ).

% numeral_power_le_of_nat_cancel_iff
thf(fact_5581_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X4: nat,I: num,N: nat] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X4 ) @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) )
      = ( ord_less_eq_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).

% of_nat_le_numeral_power_cancel_iff
thf(fact_5582_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X4: nat,I: num,N: nat] :
      ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ X4 ) @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N ) )
      = ( ord_less_eq_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).

% of_nat_le_numeral_power_cancel_iff
thf(fact_5583_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X4: nat,I: num,N: nat] :
      ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X4 ) @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) )
      = ( ord_less_eq_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).

% of_nat_le_numeral_power_cancel_iff
thf(fact_5584_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X4: nat,I: num,N: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X4 ) @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) )
      = ( ord_less_eq_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).

% of_nat_le_numeral_power_cancel_iff
thf(fact_5585_bit__mod__2__iff,axiom,
    ! [A: code_integer,N: nat] :
      ( ( bit_se9216721137139052372nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ N )
      = ( ( N = zero_zero_nat )
        & ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).

% bit_mod_2_iff
thf(fact_5586_bit__mod__2__iff,axiom,
    ! [A: int,N: nat] :
      ( ( bit_se1146084159140164899it_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ N )
      = ( ( N = zero_zero_nat )
        & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).

% bit_mod_2_iff
thf(fact_5587_bit__mod__2__iff,axiom,
    ! [A: nat,N: nat] :
      ( ( bit_se1148574629649215175it_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N )
      = ( ( N = zero_zero_nat )
        & ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).

% bit_mod_2_iff
thf(fact_5588_bit__numeral__iff,axiom,
    ! [M: num,N: nat] :
      ( ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ M ) @ N )
      = ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ M ) @ N ) ) ).

% bit_numeral_iff
thf(fact_5589_bit__numeral__iff,axiom,
    ! [M: num,N: nat] :
      ( ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ M ) @ N )
      = ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ M ) @ N ) ) ).

% bit_numeral_iff
thf(fact_5590_bit__of__nat__iff__bit,axiom,
    ! [M: nat,N: nat] :
      ( ( bit_se1146084159140164899it_int @ ( semiri1314217659103216013at_int @ M ) @ N )
      = ( bit_se1148574629649215175it_nat @ M @ N ) ) ).

% bit_of_nat_iff_bit
thf(fact_5591_bit__of__nat__iff__bit,axiom,
    ! [M: nat,N: nat] :
      ( ( bit_se1148574629649215175it_nat @ ( semiri1316708129612266289at_nat @ M ) @ N )
      = ( bit_se1148574629649215175it_nat @ M @ N ) ) ).

% bit_of_nat_iff_bit
thf(fact_5592_bit__disjunctive__add__iff,axiom,
    ! [A: int,B2: int,N: nat] :
      ( ! [N4: nat] :
          ( ~ ( bit_se1146084159140164899it_int @ A @ N4 )
          | ~ ( bit_se1146084159140164899it_int @ B2 @ N4 ) )
     => ( ( bit_se1146084159140164899it_int @ ( plus_plus_int @ A @ B2 ) @ N )
        = ( ( bit_se1146084159140164899it_int @ A @ N )
          | ( bit_se1146084159140164899it_int @ B2 @ N ) ) ) ) ).

% bit_disjunctive_add_iff
thf(fact_5593_bit__disjunctive__add__iff,axiom,
    ! [A: nat,B2: nat,N: nat] :
      ( ! [N4: nat] :
          ( ~ ( bit_se1148574629649215175it_nat @ A @ N4 )
          | ~ ( bit_se1148574629649215175it_nat @ B2 @ N4 ) )
     => ( ( bit_se1148574629649215175it_nat @ ( plus_plus_nat @ A @ B2 ) @ N )
        = ( ( bit_se1148574629649215175it_nat @ A @ N )
          | ( bit_se1148574629649215175it_nat @ B2 @ N ) ) ) ) ).

% bit_disjunctive_add_iff
thf(fact_5594_real__arch__simple,axiom,
    ! [X4: real] :
    ? [N4: nat] : ( ord_less_eq_real @ X4 @ ( semiri5074537144036343181t_real @ N4 ) ) ).

% real_arch_simple
thf(fact_5595_real__arch__simple,axiom,
    ! [X4: rat] :
    ? [N4: nat] : ( ord_less_eq_rat @ X4 @ ( semiri681578069525770553at_rat @ N4 ) ) ).

% real_arch_simple
thf(fact_5596_reals__Archimedean2,axiom,
    ! [X4: real] :
    ? [N4: nat] : ( ord_less_real @ X4 @ ( semiri5074537144036343181t_real @ N4 ) ) ).

% reals_Archimedean2
thf(fact_5597_reals__Archimedean2,axiom,
    ! [X4: rat] :
    ? [N4: nat] : ( ord_less_rat @ X4 @ ( semiri681578069525770553at_rat @ N4 ) ) ).

% reals_Archimedean2
thf(fact_5598_mult__of__nat__commute,axiom,
    ! [X4: nat,Y3: complex] :
      ( ( times_times_complex @ ( semiri8010041392384452111omplex @ X4 ) @ Y3 )
      = ( times_times_complex @ Y3 @ ( semiri8010041392384452111omplex @ X4 ) ) ) ).

% mult_of_nat_commute
thf(fact_5599_mult__of__nat__commute,axiom,
    ! [X4: nat,Y3: real] :
      ( ( times_times_real @ ( semiri5074537144036343181t_real @ X4 ) @ Y3 )
      = ( times_times_real @ Y3 @ ( semiri5074537144036343181t_real @ X4 ) ) ) ).

% mult_of_nat_commute
thf(fact_5600_mult__of__nat__commute,axiom,
    ! [X4: nat,Y3: rat] :
      ( ( times_times_rat @ ( semiri681578069525770553at_rat @ X4 ) @ Y3 )
      = ( times_times_rat @ Y3 @ ( semiri681578069525770553at_rat @ X4 ) ) ) ).

% mult_of_nat_commute
thf(fact_5601_mult__of__nat__commute,axiom,
    ! [X4: nat,Y3: nat] :
      ( ( times_times_nat @ ( semiri1316708129612266289at_nat @ X4 ) @ Y3 )
      = ( times_times_nat @ Y3 @ ( semiri1316708129612266289at_nat @ X4 ) ) ) ).

% mult_of_nat_commute
thf(fact_5602_mult__of__nat__commute,axiom,
    ! [X4: nat,Y3: int] :
      ( ( times_times_int @ ( semiri1314217659103216013at_int @ X4 ) @ Y3 )
      = ( times_times_int @ Y3 @ ( semiri1314217659103216013at_int @ X4 ) ) ) ).

% mult_of_nat_commute
thf(fact_5603_bit__and__iff,axiom,
    ! [A: int,B2: int,N: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_se725231765392027082nd_int @ A @ B2 ) @ N )
      = ( ( bit_se1146084159140164899it_int @ A @ N )
        & ( bit_se1146084159140164899it_int @ B2 @ N ) ) ) ).

% bit_and_iff
thf(fact_5604_bit__and__iff,axiom,
    ! [A: nat,B2: nat,N: nat] :
      ( ( bit_se1148574629649215175it_nat @ ( bit_se727722235901077358nd_nat @ A @ B2 ) @ N )
      = ( ( bit_se1148574629649215175it_nat @ A @ N )
        & ( bit_se1148574629649215175it_nat @ B2 @ N ) ) ) ).

% bit_and_iff
thf(fact_5605_bit__and__int__iff,axiom,
    ! [K: int,L2: int,N: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_se725231765392027082nd_int @ K @ L2 ) @ N )
      = ( ( bit_se1146084159140164899it_int @ K @ N )
        & ( bit_se1146084159140164899it_int @ L2 @ N ) ) ) ).

% bit_and_int_iff
thf(fact_5606_arctan__less__iff,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_real @ ( arctan @ X4 ) @ ( arctan @ Y3 ) )
      = ( ord_less_real @ X4 @ Y3 ) ) ).

% arctan_less_iff
thf(fact_5607_arctan__monotone,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_real @ X4 @ Y3 )
     => ( ord_less_real @ ( arctan @ X4 ) @ ( arctan @ Y3 ) ) ) ).

% arctan_monotone
thf(fact_5608_arctan__le__iff,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_eq_real @ ( arctan @ X4 ) @ ( arctan @ Y3 ) )
      = ( ord_less_eq_real @ X4 @ Y3 ) ) ).

% arctan_le_iff
thf(fact_5609_arctan__monotone_H,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_eq_real @ X4 @ Y3 )
     => ( ord_less_eq_real @ ( arctan @ X4 ) @ ( arctan @ Y3 ) ) ) ).

% arctan_monotone'
thf(fact_5610_bit__unset__bit__iff,axiom,
    ! [M: nat,A: int,N: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_se4203085406695923979it_int @ M @ A ) @ N )
      = ( ( bit_se1146084159140164899it_int @ A @ N )
        & ( M != N ) ) ) ).

% bit_unset_bit_iff
thf(fact_5611_bit__unset__bit__iff,axiom,
    ! [M: nat,A: nat,N: nat] :
      ( ( bit_se1148574629649215175it_nat @ ( bit_se4205575877204974255it_nat @ M @ A ) @ N )
      = ( ( bit_se1148574629649215175it_nat @ A @ N )
        & ( M != N ) ) ) ).

% bit_unset_bit_iff
thf(fact_5612_less__log__of__power,axiom,
    ! [B2: real,N: nat,M: real] :
      ( ( ord_less_real @ ( power_power_real @ B2 @ N ) @ M )
     => ( ( ord_less_real @ one_one_real @ B2 )
       => ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B2 @ M ) ) ) ) ).

% less_log_of_power
thf(fact_5613_log__of__power__eq,axiom,
    ! [M: nat,B2: real,N: nat] :
      ( ( ( semiri5074537144036343181t_real @ M )
        = ( power_power_real @ B2 @ N ) )
     => ( ( ord_less_real @ one_one_real @ B2 )
       => ( ( semiri5074537144036343181t_real @ N )
          = ( log @ B2 @ ( semiri5074537144036343181t_real @ M ) ) ) ) ) ).

% log_of_power_eq
thf(fact_5614_of__nat__less__of__int__iff,axiom,
    ! [N: nat,X4: int] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( ring_1_of_int_real @ X4 ) )
      = ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X4 ) ) ).

% of_nat_less_of_int_iff
thf(fact_5615_of__nat__less__of__int__iff,axiom,
    ! [N: nat,X4: int] :
      ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ N ) @ ( ring_1_of_int_rat @ X4 ) )
      = ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X4 ) ) ).

% of_nat_less_of_int_iff
thf(fact_5616_of__nat__less__of__int__iff,axiom,
    ! [N: nat,X4: int] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( ring_1_of_int_int @ X4 ) )
      = ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X4 ) ) ).

% of_nat_less_of_int_iff
thf(fact_5617_le__log__of__power,axiom,
    ! [B2: real,N: nat,M: real] :
      ( ( ord_less_eq_real @ ( power_power_real @ B2 @ N ) @ M )
     => ( ( ord_less_real @ one_one_real @ B2 )
       => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B2 @ M ) ) ) ) ).

% le_log_of_power
thf(fact_5618_log__base__pow,axiom,
    ! [A: real,N: nat,X4: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( log @ ( power_power_real @ A @ N ) @ X4 )
        = ( divide_divide_real @ ( log @ A @ X4 ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).

% log_base_pow
thf(fact_5619_log__nat__power,axiom,
    ! [X4: real,B2: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( log @ B2 @ ( power_power_real @ X4 @ N ) )
        = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B2 @ X4 ) ) ) ) ).

% log_nat_power
thf(fact_5620_not__bit__1__Suc,axiom,
    ! [N: nat] :
      ~ ( bit_se1146084159140164899it_int @ one_one_int @ ( suc @ N ) ) ).

% not_bit_1_Suc
thf(fact_5621_not__bit__1__Suc,axiom,
    ! [N: nat] :
      ~ ( bit_se1148574629649215175it_nat @ one_one_nat @ ( suc @ N ) ) ).

% not_bit_1_Suc
thf(fact_5622_bit__1__iff,axiom,
    ! [N: nat] :
      ( ( bit_se1146084159140164899it_int @ one_one_int @ N )
      = ( N = zero_zero_nat ) ) ).

% bit_1_iff
thf(fact_5623_bit__1__iff,axiom,
    ! [N: nat] :
      ( ( bit_se1148574629649215175it_nat @ one_one_nat @ N )
      = ( N = zero_zero_nat ) ) ).

% bit_1_iff
thf(fact_5624_bit__numeral__simps_I1_J,axiom,
    ! [N: num] :
      ~ ( bit_se1146084159140164899it_int @ one_one_int @ ( numeral_numeral_nat @ N ) ) ).

% bit_numeral_simps(1)
thf(fact_5625_bit__numeral__simps_I1_J,axiom,
    ! [N: num] :
      ~ ( bit_se1148574629649215175it_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) ) ).

% bit_numeral_simps(1)
thf(fact_5626_of__nat__0__le__iff,axiom,
    ! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) ) ).

% of_nat_0_le_iff
thf(fact_5627_of__nat__0__le__iff,axiom,
    ! [N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( semiri681578069525770553at_rat @ N ) ) ).

% of_nat_0_le_iff
thf(fact_5628_of__nat__0__le__iff,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) ) ).

% of_nat_0_le_iff
thf(fact_5629_of__nat__0__le__iff,axiom,
    ! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) ) ).

% of_nat_0_le_iff
thf(fact_5630_of__nat__less__0__iff,axiom,
    ! [M: nat] :
      ~ ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real ) ).

% of_nat_less_0_iff
thf(fact_5631_of__nat__less__0__iff,axiom,
    ! [M: nat] :
      ~ ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ zero_zero_rat ) ).

% of_nat_less_0_iff
thf(fact_5632_of__nat__less__0__iff,axiom,
    ! [M: nat] :
      ~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat ) ).

% of_nat_less_0_iff
thf(fact_5633_of__nat__less__0__iff,axiom,
    ! [M: nat] :
      ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int ) ).

% of_nat_less_0_iff
thf(fact_5634_bit__take__bit__iff,axiom,
    ! [M: nat,A: int,N: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_se2923211474154528505it_int @ M @ A ) @ N )
      = ( ( ord_less_nat @ N @ M )
        & ( bit_se1146084159140164899it_int @ A @ N ) ) ) ).

% bit_take_bit_iff
thf(fact_5635_bit__take__bit__iff,axiom,
    ! [M: nat,A: nat,N: nat] :
      ( ( bit_se1148574629649215175it_nat @ ( bit_se2925701944663578781it_nat @ M @ A ) @ N )
      = ( ( ord_less_nat @ N @ M )
        & ( bit_se1148574629649215175it_nat @ A @ N ) ) ) ).

% bit_take_bit_iff
thf(fact_5636_of__nat__neq__0,axiom,
    ! [N: nat] :
      ( ( semiri8010041392384452111omplex @ ( suc @ N ) )
     != zero_zero_complex ) ).

% of_nat_neq_0
thf(fact_5637_of__nat__neq__0,axiom,
    ! [N: nat] :
      ( ( semiri5074537144036343181t_real @ ( suc @ N ) )
     != zero_zero_real ) ).

% of_nat_neq_0
thf(fact_5638_of__nat__neq__0,axiom,
    ! [N: nat] :
      ( ( semiri681578069525770553at_rat @ ( suc @ N ) )
     != zero_zero_rat ) ).

% of_nat_neq_0
thf(fact_5639_of__nat__neq__0,axiom,
    ! [N: nat] :
      ( ( semiri1316708129612266289at_nat @ ( suc @ N ) )
     != zero_zero_nat ) ).

% of_nat_neq_0
thf(fact_5640_of__nat__neq__0,axiom,
    ! [N: nat] :
      ( ( semiri1314217659103216013at_int @ ( suc @ N ) )
     != zero_zero_int ) ).

% of_nat_neq_0
thf(fact_5641_bit__of__bool__iff,axiom,
    ! [B2: $o,N: nat] :
      ( ( bit_se9216721137139052372nteger @ ( zero_n356916108424825756nteger @ B2 ) @ N )
      = ( B2
        & ( N = zero_zero_nat ) ) ) ).

% bit_of_bool_iff
thf(fact_5642_bit__of__bool__iff,axiom,
    ! [B2: $o,N: nat] :
      ( ( bit_se1146084159140164899it_int @ ( zero_n2684676970156552555ol_int @ B2 ) @ N )
      = ( B2
        & ( N = zero_zero_nat ) ) ) ).

% bit_of_bool_iff
thf(fact_5643_bit__of__bool__iff,axiom,
    ! [B2: $o,N: nat] :
      ( ( bit_se1148574629649215175it_nat @ ( zero_n2687167440665602831ol_nat @ B2 ) @ N )
      = ( B2
        & ( N = zero_zero_nat ) ) ) ).

% bit_of_bool_iff
thf(fact_5644_div__mult2__eq_H,axiom,
    ! [A: nat,M: nat,N: nat] :
      ( ( divide_divide_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) )
      = ( divide_divide_nat @ ( divide_divide_nat @ A @ ( semiri1316708129612266289at_nat @ M ) ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).

% div_mult2_eq'
thf(fact_5645_div__mult2__eq_H,axiom,
    ! [A: int,M: nat,N: nat] :
      ( ( divide_divide_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) )
      = ( divide_divide_int @ ( divide_divide_int @ A @ ( semiri1314217659103216013at_int @ M ) ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).

% div_mult2_eq'
thf(fact_5646_less__imp__of__nat__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).

% less_imp_of_nat_less
thf(fact_5647_less__imp__of__nat__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).

% less_imp_of_nat_less
thf(fact_5648_less__imp__of__nat__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).

% less_imp_of_nat_less
thf(fact_5649_less__imp__of__nat__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).

% less_imp_of_nat_less
thf(fact_5650_of__nat__less__imp__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
     => ( ord_less_nat @ M @ N ) ) ).

% of_nat_less_imp_less
thf(fact_5651_of__nat__less__imp__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) )
     => ( ord_less_nat @ M @ N ) ) ).

% of_nat_less_imp_less
thf(fact_5652_of__nat__less__imp__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
     => ( ord_less_nat @ M @ N ) ) ).

% of_nat_less_imp_less
thf(fact_5653_of__nat__less__imp__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
     => ( ord_less_nat @ M @ N ) ) ).

% of_nat_less_imp_less
thf(fact_5654_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ I ) @ ( semiri5074537144036343181t_real @ J ) ) ) ).

% of_nat_mono
thf(fact_5655_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ I ) @ ( semiri681578069525770553at_rat @ J ) ) ) ).

% of_nat_mono
thf(fact_5656_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I ) @ ( semiri1316708129612266289at_nat @ J ) ) ) ).

% of_nat_mono
thf(fact_5657_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J ) ) ) ).

% of_nat_mono
thf(fact_5658_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri1316708129612266289at_nat @ ( divide_divide_nat @ M @ N ) )
      = ( divide_divide_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).

% unique_euclidean_semiring_with_nat_class.of_nat_div
thf(fact_5659_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N ) )
      = ( divide_divide_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).

% unique_euclidean_semiring_with_nat_class.of_nat_div
thf(fact_5660_log__def,axiom,
    ( log
    = ( ^ [A4: real,X: real] : ( divide_divide_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ A4 ) ) ) ) ).

% log_def
thf(fact_5661_of__nat__dvd__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( dvd_dvd_Code_integer @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) )
      = ( dvd_dvd_nat @ M @ N ) ) ).

% of_nat_dvd_iff
thf(fact_5662_of__nat__dvd__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( dvd_dvd_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
      = ( dvd_dvd_nat @ M @ N ) ) ).

% of_nat_dvd_iff
thf(fact_5663_of__nat__dvd__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( dvd_dvd_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
      = ( dvd_dvd_nat @ M @ N ) ) ).

% of_nat_dvd_iff
thf(fact_5664_signed__take__bit__eq__if__positive,axiom,
    ! [A: int,N: nat] :
      ( ~ ( bit_se1146084159140164899it_int @ A @ N )
     => ( ( bit_ri631733984087533419it_int @ N @ A )
        = ( bit_se2923211474154528505it_int @ N @ A ) ) ) ).

% signed_take_bit_eq_if_positive
thf(fact_5665_abs__zmult__eq__1,axiom,
    ! [M: int,N: int] :
      ( ( ( abs_abs_int @ ( times_times_int @ M @ N ) )
        = one_one_int )
     => ( ( abs_abs_int @ M )
        = one_one_int ) ) ).

% abs_zmult_eq_1
thf(fact_5666_of__nat__mod,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri4939895301339042750nteger @ ( modulo_modulo_nat @ M @ N ) )
      = ( modulo364778990260209775nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) ) ) ).

% of_nat_mod
thf(fact_5667_of__nat__mod,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri1316708129612266289at_nat @ ( modulo_modulo_nat @ M @ N ) )
      = ( modulo_modulo_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).

% of_nat_mod
thf(fact_5668_of__nat__mod,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N ) )
      = ( modulo_modulo_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).

% of_nat_mod
thf(fact_5669_abs__div,axiom,
    ! [Y3: int,X4: int] :
      ( ( dvd_dvd_int @ Y3 @ X4 )
     => ( ( abs_abs_int @ ( divide_divide_int @ X4 @ Y3 ) )
        = ( divide_divide_int @ ( abs_abs_int @ X4 ) @ ( abs_abs_int @ Y3 ) ) ) ) ).

% abs_div
thf(fact_5670_log2__of__power__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( M
        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
     => ( ( semiri5074537144036343181t_real @ N )
        = ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).

% log2_of_power_eq
thf(fact_5671_log__of__power__less,axiom,
    ! [M: nat,B2: real,N: nat] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( power_power_real @ B2 @ N ) )
     => ( ( ord_less_real @ one_one_real @ B2 )
       => ( ( ord_less_nat @ zero_zero_nat @ M )
         => ( ord_less_real @ ( log @ B2 @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).

% log_of_power_less
thf(fact_5672_take__bit__of__nat,axiom,
    ! [N: nat,M: nat] :
      ( ( bit_se2923211474154528505it_int @ N @ ( semiri1314217659103216013at_int @ M ) )
      = ( semiri1314217659103216013at_int @ ( bit_se2925701944663578781it_nat @ N @ M ) ) ) ).

% take_bit_of_nat
thf(fact_5673_take__bit__of__nat,axiom,
    ! [N: nat,M: nat] :
      ( ( bit_se2925701944663578781it_nat @ N @ ( semiri1316708129612266289at_nat @ M ) )
      = ( semiri1316708129612266289at_nat @ ( bit_se2925701944663578781it_nat @ N @ M ) ) ) ).

% take_bit_of_nat
thf(fact_5674_of__nat__and__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri1314217659103216013at_int @ ( bit_se727722235901077358nd_nat @ M @ N ) )
      = ( bit_se725231765392027082nd_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).

% of_nat_and_eq
thf(fact_5675_of__nat__and__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri1316708129612266289at_nat @ ( bit_se727722235901077358nd_nat @ M @ N ) )
      = ( bit_se727722235901077358nd_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).

% of_nat_and_eq
thf(fact_5676_of__nat__mask__eq,axiom,
    ! [N: nat] :
      ( ( semiri1316708129612266289at_nat @ ( bit_se2002935070580805687sk_nat @ N ) )
      = ( bit_se2002935070580805687sk_nat @ N ) ) ).

% of_nat_mask_eq
thf(fact_5677_of__nat__mask__eq,axiom,
    ! [N: nat] :
      ( ( semiri1314217659103216013at_int @ ( bit_se2002935070580805687sk_nat @ N ) )
      = ( bit_se2000444600071755411sk_int @ N ) ) ).

% of_nat_mask_eq
thf(fact_5678_log__of__power__le,axiom,
    ! [M: nat,B2: real,N: nat] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( power_power_real @ B2 @ N ) )
     => ( ( ord_less_real @ one_one_real @ B2 )
       => ( ( ord_less_nat @ zero_zero_nat @ M )
         => ( ord_less_eq_real @ ( log @ B2 @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).

% log_of_power_le
thf(fact_5679_ex__less__of__nat__mult,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ? [N4: nat] : ( ord_less_real @ Y3 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N4 ) @ X4 ) ) ) ).

% ex_less_of_nat_mult
thf(fact_5680_ex__less__of__nat__mult,axiom,
    ! [X4: rat,Y3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ X4 )
     => ? [N4: nat] : ( ord_less_rat @ Y3 @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N4 ) @ X4 ) ) ) ).

% ex_less_of_nat_mult
thf(fact_5681_of__nat__diff,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( semiri8010041392384452111omplex @ ( minus_minus_nat @ M @ N ) )
        = ( minus_minus_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ N ) ) ) ) ).

% of_nat_diff
thf(fact_5682_of__nat__diff,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( semiri5074537144036343181t_real @ ( minus_minus_nat @ M @ N ) )
        = ( minus_minus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).

% of_nat_diff
thf(fact_5683_of__nat__diff,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( semiri681578069525770553at_rat @ ( minus_minus_nat @ M @ N ) )
        = ( minus_minus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ) ).

% of_nat_diff
thf(fact_5684_of__nat__diff,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ M @ N ) )
        = ( minus_minus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ) ).

% of_nat_diff
thf(fact_5685_of__nat__diff,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ M @ N ) )
        = ( minus_minus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).

% of_nat_diff
thf(fact_5686_bit__not__int__iff_H,axiom,
    ! [K: int,N: nat] :
      ( ( bit_se1146084159140164899it_int @ ( minus_minus_int @ ( uminus_uminus_int @ K ) @ one_one_int ) @ N )
      = ( ~ ( bit_se1146084159140164899it_int @ K @ N ) ) ) ).

% bit_not_int_iff'
thf(fact_5687_exp__of__nat__mult,axiom,
    ! [N: nat,X4: complex] :
      ( ( exp_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ X4 ) )
      = ( power_power_complex @ ( exp_complex @ X4 ) @ N ) ) ).

% exp_of_nat_mult
thf(fact_5688_exp__of__nat__mult,axiom,
    ! [N: nat,X4: real] :
      ( ( exp_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X4 ) )
      = ( power_power_real @ ( exp_real @ X4 ) @ N ) ) ).

% exp_of_nat_mult
thf(fact_5689_exp__of__nat2__mult,axiom,
    ! [X4: complex,N: nat] :
      ( ( exp_complex @ ( times_times_complex @ X4 @ ( semiri8010041392384452111omplex @ N ) ) )
      = ( power_power_complex @ ( exp_complex @ X4 ) @ N ) ) ).

% exp_of_nat2_mult
thf(fact_5690_exp__of__nat2__mult,axiom,
    ! [X4: real,N: nat] :
      ( ( exp_real @ ( times_times_real @ X4 @ ( semiri5074537144036343181t_real @ N ) ) )
      = ( power_power_real @ ( exp_real @ X4 ) @ N ) ) ).

% exp_of_nat2_mult
thf(fact_5691_zabs__def,axiom,
    ( abs_abs_int
    = ( ^ [I2: int] : ( if_int @ ( ord_less_int @ I2 @ zero_zero_int ) @ ( uminus_uminus_int @ I2 ) @ I2 ) ) ) ).

% zabs_def
thf(fact_5692_reals__Archimedean3,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ! [Y5: real] :
        ? [N4: nat] : ( ord_less_real @ Y5 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N4 ) @ X4 ) ) ) ).

% reals_Archimedean3
thf(fact_5693_log__ln,axiom,
    ( ln_ln_real
    = ( log @ ( exp_real @ one_one_real ) ) ) ).

% log_ln
thf(fact_5694_real__of__nat__div4,axiom,
    ! [N: nat,X4: nat] : ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X4 ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X4 ) ) ) ).

% real_of_nat_div4
thf(fact_5695_abs__mod__less,axiom,
    ! [L2: int,K: int] :
      ( ( L2 != zero_zero_int )
     => ( ord_less_int @ ( abs_abs_int @ ( modulo_modulo_int @ K @ L2 ) ) @ ( abs_abs_int @ L2 ) ) ) ).

% abs_mod_less
thf(fact_5696_real__of__nat__div,axiom,
    ! [D: nat,N: nat] :
      ( ( dvd_dvd_nat @ D @ N )
     => ( ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ D ) )
        = ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ D ) ) ) ) ).

% real_of_nat_div
thf(fact_5697_less__log2__of__power,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ M )
     => ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).

% less_log2_of_power
thf(fact_5698_le__log2__of__power,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ M )
     => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).

% le_log2_of_power
thf(fact_5699_flip__bit__eq__if,axiom,
    ( bit_se2159334234014336723it_int
    = ( ^ [N2: nat,A4: int] : ( if_nat_int_int @ ( bit_se1146084159140164899it_int @ A4 @ N2 ) @ bit_se4203085406695923979it_int @ bit_se7879613467334960850it_int @ N2 @ A4 ) ) ) ).

% flip_bit_eq_if
thf(fact_5700_flip__bit__eq__if,axiom,
    ( bit_se2161824704523386999it_nat
    = ( ^ [N2: nat,A4: nat] : ( if_nat_nat_nat @ ( bit_se1148574629649215175it_nat @ A4 @ N2 ) @ bit_se4205575877204974255it_nat @ bit_se7882103937844011126it_nat @ N2 @ A4 ) ) ) ).

% flip_bit_eq_if
thf(fact_5701_log2__of__power__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
     => ( ( ord_less_nat @ zero_zero_nat @ M )
       => ( ord_less_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).

% log2_of_power_less
thf(fact_5702_log__base__change,axiom,
    ! [A: real,B2: real,X4: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( log @ B2 @ X4 )
          = ( divide_divide_real @ ( log @ A @ X4 ) @ ( log @ A @ B2 ) ) ) ) ) ).

% log_base_change
thf(fact_5703_mod__mult2__eq_H,axiom,
    ! [A: code_integer,M: nat,N: nat] :
      ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) ) )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M ) @ ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ ( semiri4939895301339042750nteger @ M ) ) @ ( semiri4939895301339042750nteger @ N ) ) ) @ ( modulo364778990260209775nteger @ A @ ( semiri4939895301339042750nteger @ M ) ) ) ) ).

% mod_mult2_eq'
thf(fact_5704_mod__mult2__eq_H,axiom,
    ! [A: nat,M: nat,N: nat] :
      ( ( modulo_modulo_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( modulo_modulo_nat @ ( divide_divide_nat @ A @ ( semiri1316708129612266289at_nat @ M ) ) @ ( semiri1316708129612266289at_nat @ N ) ) ) @ ( modulo_modulo_nat @ A @ ( semiri1316708129612266289at_nat @ M ) ) ) ) ).

% mod_mult2_eq'
thf(fact_5705_mod__mult2__eq_H,axiom,
    ! [A: int,M: nat,N: nat] :
      ( ( modulo_modulo_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( modulo_modulo_int @ ( divide_divide_int @ A @ ( semiri1314217659103216013at_int @ M ) ) @ ( semiri1314217659103216013at_int @ N ) ) ) @ ( modulo_modulo_int @ A @ ( semiri1314217659103216013at_int @ M ) ) ) ) ).

% mod_mult2_eq'
thf(fact_5706_bit__imp__take__bit__positive,axiom,
    ! [N: nat,M: nat,K: int] :
      ( ( ord_less_nat @ N @ M )
     => ( ( bit_se1146084159140164899it_int @ K @ N )
       => ( ord_less_int @ zero_zero_int @ ( bit_se2923211474154528505it_int @ M @ K ) ) ) ) ).

% bit_imp_take_bit_positive
thf(fact_5707_field__char__0__class_Oof__nat__div,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri8010041392384452111omplex @ ( divide_divide_nat @ M @ N ) )
      = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ ( modulo_modulo_nat @ M @ N ) ) ) @ ( semiri8010041392384452111omplex @ N ) ) ) ).

% field_char_0_class.of_nat_div
thf(fact_5708_field__char__0__class_Oof__nat__div,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri5074537144036343181t_real @ ( divide_divide_nat @ M @ N ) )
      = ( divide_divide_real @ ( minus_minus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ ( modulo_modulo_nat @ M @ N ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).

% field_char_0_class.of_nat_div
thf(fact_5709_field__char__0__class_Oof__nat__div,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri681578069525770553at_rat @ ( divide_divide_nat @ M @ N ) )
      = ( divide_divide_rat @ ( minus_minus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ ( modulo_modulo_nat @ M @ N ) ) ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).

% field_char_0_class.of_nat_div
thf(fact_5710_bit__concat__bit__iff,axiom,
    ! [M: nat,K: int,L2: int,N: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_concat_bit @ M @ K @ L2 ) @ N )
      = ( ( ( ord_less_nat @ N @ M )
          & ( bit_se1146084159140164899it_int @ K @ N ) )
        | ( ( ord_less_eq_nat @ M @ N )
          & ( bit_se1146084159140164899it_int @ L2 @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).

% bit_concat_bit_iff
thf(fact_5711_nat__less__real__le,axiom,
    ( ord_less_nat
    = ( ^ [N2: nat,M6: nat] : ( ord_less_eq_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N2 ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ M6 ) ) ) ) ).

% nat_less_real_le
thf(fact_5712_nat__le__real__less,axiom,
    ( ord_less_eq_nat
    = ( ^ [N2: nat,M6: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M6 ) @ one_one_real ) ) ) ) ).

% nat_le_real_less
thf(fact_5713_zdvd__mult__cancel1,axiom,
    ! [M: int,N: int] :
      ( ( M != zero_zero_int )
     => ( ( dvd_dvd_int @ ( times_times_int @ M @ N ) @ M )
        = ( ( abs_abs_int @ N )
          = one_one_int ) ) ) ).

% zdvd_mult_cancel1
thf(fact_5714_log2__of__power__le,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
     => ( ( ord_less_nat @ zero_zero_nat @ M )
       => ( ord_less_eq_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).

% log2_of_power_le
thf(fact_5715_real__of__nat__div__aux,axiom,
    ! [X4: nat,D: nat] :
      ( ( divide_divide_real @ ( semiri5074537144036343181t_real @ X4 ) @ ( semiri5074537144036343181t_real @ D ) )
      = ( plus_plus_real @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ X4 @ D ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ ( modulo_modulo_nat @ X4 @ D ) ) @ ( semiri5074537144036343181t_real @ D ) ) ) ) ).

% real_of_nat_div_aux
thf(fact_5716_signed__take__bit__eq__concat__bit,axiom,
    ( bit_ri631733984087533419it_int
    = ( ^ [N2: nat,K3: int] : ( bit_concat_bit @ N2 @ K3 @ ( uminus_uminus_int @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K3 @ N2 ) ) ) ) ) ) ).

% signed_take_bit_eq_concat_bit
thf(fact_5717_exp__eq__0__imp__not__bit,axiom,
    ! [N: nat,A: int] :
      ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
        = zero_zero_int )
     => ~ ( bit_se1146084159140164899it_int @ A @ N ) ) ).

% exp_eq_0_imp_not_bit
thf(fact_5718_exp__eq__0__imp__not__bit,axiom,
    ! [N: nat,A: nat] :
      ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
        = zero_zero_nat )
     => ~ ( bit_se1148574629649215175it_nat @ A @ N ) ) ).

% exp_eq_0_imp_not_bit
thf(fact_5719_bit__Suc,axiom,
    ! [A: int,N: nat] :
      ( ( bit_se1146084159140164899it_int @ A @ ( suc @ N ) )
      = ( bit_se1146084159140164899it_int @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ N ) ) ).

% bit_Suc
thf(fact_5720_bit__Suc,axiom,
    ! [A: nat,N: nat] :
      ( ( bit_se1148574629649215175it_nat @ A @ ( suc @ N ) )
      = ( bit_se1148574629649215175it_nat @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N ) ) ).

% bit_Suc
thf(fact_5721_stable__imp__bit__iff__odd,axiom,
    ! [A: code_integer,N: nat] :
      ( ( ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = A )
     => ( ( bit_se9216721137139052372nteger @ A @ N )
        = ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ) ).

% stable_imp_bit_iff_odd
thf(fact_5722_stable__imp__bit__iff__odd,axiom,
    ! [A: int,N: nat] :
      ( ( ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = A )
     => ( ( bit_se1146084159140164899it_int @ A @ N )
        = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ) ).

% stable_imp_bit_iff_odd
thf(fact_5723_stable__imp__bit__iff__odd,axiom,
    ! [A: nat,N: nat] :
      ( ( ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = A )
     => ( ( bit_se1148574629649215175it_nat @ A @ N )
        = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ) ).

% stable_imp_bit_iff_odd
thf(fact_5724_bit__iff__idd__imp__stable,axiom,
    ! [A: code_integer] :
      ( ! [N4: nat] :
          ( ( bit_se9216721137139052372nteger @ A @ N4 )
          = ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) )
     => ( ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = A ) ) ).

% bit_iff_idd_imp_stable
thf(fact_5725_bit__iff__idd__imp__stable,axiom,
    ! [A: int] :
      ( ! [N4: nat] :
          ( ( bit_se1146084159140164899it_int @ A @ N4 )
          = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) )
     => ( ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = A ) ) ).

% bit_iff_idd_imp_stable
thf(fact_5726_bit__iff__idd__imp__stable,axiom,
    ! [A: nat] :
      ( ! [N4: nat] :
          ( ( bit_se1148574629649215175it_nat @ A @ N4 )
          = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) )
     => ( ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = A ) ) ).

% bit_iff_idd_imp_stable
thf(fact_5727_log__mult,axiom,
    ! [A: real,X4: real,Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X4 )
         => ( ( ord_less_real @ zero_zero_real @ Y3 )
           => ( ( log @ A @ ( times_times_real @ X4 @ Y3 ) )
              = ( plus_plus_real @ ( log @ A @ X4 ) @ ( log @ A @ Y3 ) ) ) ) ) ) ) ).

% log_mult
thf(fact_5728_nat__approx__posE,axiom,
    ! [E: real] :
      ( ( ord_less_real @ zero_zero_real @ E )
     => ~ ! [N4: nat] :
            ~ ( ord_less_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( suc @ N4 ) ) ) @ E ) ) ).

% nat_approx_posE
thf(fact_5729_nat__approx__posE,axiom,
    ! [E: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ E )
     => ~ ! [N4: nat] :
            ~ ( ord_less_rat @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ ( suc @ N4 ) ) ) @ E ) ) ).

% nat_approx_posE
thf(fact_5730_of__nat__less__two__power,axiom,
    ! [N: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ).

% of_nat_less_two_power
thf(fact_5731_of__nat__less__two__power,axiom,
    ! [N: nat] : ( ord_less_rat @ ( semiri681578069525770553at_rat @ N ) @ ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ N ) ) ).

% of_nat_less_two_power
thf(fact_5732_of__nat__less__two__power,axiom,
    ! [N: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).

% of_nat_less_two_power
thf(fact_5733_int__bit__bound,axiom,
    ! [K: int] :
      ~ ! [N4: nat] :
          ( ! [M5: nat] :
              ( ( ord_less_eq_nat @ N4 @ M5 )
             => ( ( bit_se1146084159140164899it_int @ K @ M5 )
                = ( bit_se1146084159140164899it_int @ K @ N4 ) ) )
         => ~ ( ( ord_less_nat @ zero_zero_nat @ N4 )
             => ( ( bit_se1146084159140164899it_int @ K @ ( minus_minus_nat @ N4 @ one_one_nat ) )
                = ( ~ ( bit_se1146084159140164899it_int @ K @ N4 ) ) ) ) ) ).

% int_bit_bound
thf(fact_5734_log__divide,axiom,
    ! [A: real,X4: real,Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X4 )
         => ( ( ord_less_real @ zero_zero_real @ Y3 )
           => ( ( log @ A @ ( divide_divide_real @ X4 @ Y3 ) )
              = ( minus_minus_real @ ( log @ A @ X4 ) @ ( log @ A @ Y3 ) ) ) ) ) ) ) ).

% log_divide
thf(fact_5735_inverse__of__nat__le,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( N != zero_zero_nat )
       => ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).

% inverse_of_nat_le
thf(fact_5736_inverse__of__nat__le,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( N != zero_zero_nat )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ M ) ) @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ N ) ) ) ) ) ).

% inverse_of_nat_le
thf(fact_5737_exp__divide__power__eq,axiom,
    ! [N: nat,X4: complex] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( power_power_complex @ ( exp_complex @ ( divide1717551699836669952omplex @ X4 @ ( semiri8010041392384452111omplex @ N ) ) ) @ N )
        = ( exp_complex @ X4 ) ) ) ).

% exp_divide_power_eq
thf(fact_5738_exp__divide__power__eq,axiom,
    ! [N: nat,X4: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( power_power_real @ ( exp_real @ ( divide_divide_real @ X4 @ ( semiri5074537144036343181t_real @ N ) ) ) @ N )
        = ( exp_real @ X4 ) ) ) ).

% exp_divide_power_eq
thf(fact_5739_even__abs__add__iff,axiom,
    ! [K: int,L2: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ ( abs_abs_int @ K ) @ L2 ) )
      = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ L2 ) ) ) ).

% even_abs_add_iff
thf(fact_5740_even__add__abs__iff,axiom,
    ! [K: int,L2: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ ( abs_abs_int @ L2 ) ) )
      = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ L2 ) ) ) ).

% even_add_abs_iff
thf(fact_5741_real__archimedian__rdiv__eq__0,axiom,
    ! [X4: real,C: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ! [M4: nat] :
              ( ( ord_less_nat @ zero_zero_nat @ M4 )
             => ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M4 ) @ X4 ) @ C ) )
         => ( X4 = zero_zero_real ) ) ) ) ).

% real_archimedian_rdiv_eq_0
thf(fact_5742_real__of__nat__div2,axiom,
    ! [N: nat,X4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X4 ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X4 ) ) ) ) ).

% real_of_nat_div2
thf(fact_5743_real__of__nat__div3,axiom,
    ! [N: nat,X4: nat] : ( ord_less_eq_real @ ( minus_minus_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X4 ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X4 ) ) ) @ one_one_real ) ).

% real_of_nat_div3
thf(fact_5744_ln__realpow,axiom,
    ! [X4: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( ln_ln_real @ ( power_power_real @ X4 @ N ) )
        = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( ln_ln_real @ X4 ) ) ) ) ).

% ln_realpow
thf(fact_5745_bit__iff__odd,axiom,
    ( bit_se9216721137139052372nteger
    = ( ^ [A4: code_integer,N2: nat] :
          ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A4 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).

% bit_iff_odd
thf(fact_5746_bit__iff__odd,axiom,
    ( bit_se1146084159140164899it_int
    = ( ^ [A4: int,N2: nat] :
          ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A4 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).

% bit_iff_odd
thf(fact_5747_bit__iff__odd,axiom,
    ( bit_se1148574629649215175it_nat
    = ( ^ [A4: nat,N2: nat] :
          ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).

% bit_iff_odd
thf(fact_5748_and__exp__eq__0__iff__not__bit,axiom,
    ! [A: int,N: nat] :
      ( ( ( bit_se725231765392027082nd_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
        = zero_zero_int )
      = ( ~ ( bit_se1146084159140164899it_int @ A @ N ) ) ) ).

% and_exp_eq_0_iff_not_bit
thf(fact_5749_and__exp__eq__0__iff__not__bit,axiom,
    ! [A: nat,N: nat] :
      ( ( ( bit_se727722235901077358nd_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
        = zero_zero_nat )
      = ( ~ ( bit_se1148574629649215175it_nat @ A @ N ) ) ) ).

% and_exp_eq_0_iff_not_bit
thf(fact_5750_log__eq__div__ln__mult__log,axiom,
    ! [A: real,B2: real,X4: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ B2 )
         => ( ( B2 != one_one_real )
           => ( ( ord_less_real @ zero_zero_real @ X4 )
             => ( ( log @ A @ X4 )
                = ( times_times_real @ ( divide_divide_real @ ( ln_ln_real @ B2 ) @ ( ln_ln_real @ A ) ) @ ( log @ B2 @ X4 ) ) ) ) ) ) ) ) ).

% log_eq_div_ln_mult_log
thf(fact_5751_bit__int__def,axiom,
    ( bit_se1146084159140164899it_int
    = ( ^ [K3: int,N2: nat] :
          ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).

% bit_int_def
thf(fact_5752_nat__intermed__int__val,axiom,
    ! [M: nat,N: nat,F: nat > int,K: int] :
      ( ! [I3: nat] :
          ( ( ( ord_less_eq_nat @ M @ I3 )
            & ( ord_less_nat @ I3 @ N ) )
         => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) ) ) @ one_one_int ) )
     => ( ( ord_less_eq_nat @ M @ N )
       => ( ( ord_less_eq_int @ ( F @ M ) @ K )
         => ( ( ord_less_eq_int @ K @ ( F @ N ) )
           => ? [I3: nat] :
                ( ( ord_less_eq_nat @ M @ I3 )
                & ( ord_less_eq_nat @ I3 @ N )
                & ( ( F @ I3 )
                  = K ) ) ) ) ) ) ).

% nat_intermed_int_val
thf(fact_5753_decr__lemma,axiom,
    ! [D: int,X4: int,Z: int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ord_less_int @ ( minus_minus_int @ X4 @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X4 @ Z ) ) @ one_one_int ) @ D ) ) @ Z ) ) ).

% decr_lemma
thf(fact_5754_incr__lemma,axiom,
    ! [D: int,Z: int,X4: int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ord_less_int @ Z @ ( plus_plus_int @ X4 @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X4 @ Z ) ) @ one_one_int ) @ D ) ) ) ) ).

% incr_lemma
thf(fact_5755_linear__plus__1__le__power,axiom,
    ! [X4: real,N: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X4 ) @ one_one_real ) @ ( power_power_real @ ( plus_plus_real @ X4 @ one_one_real ) @ N ) ) ) ).

% linear_plus_1_le_power
thf(fact_5756_Bernoulli__inequality,axiom,
    ! [X4: real,N: nat] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X4 )
     => ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X4 ) ) @ ( power_power_real @ ( plus_plus_real @ one_one_real @ X4 ) @ N ) ) ) ).

% Bernoulli_inequality
thf(fact_5757_even__bit__succ__iff,axiom,
    ! [A: code_integer,N: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ( ( bit_se9216721137139052372nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ N )
        = ( ( bit_se9216721137139052372nteger @ A @ N )
          | ( N = zero_zero_nat ) ) ) ) ).

% even_bit_succ_iff
thf(fact_5758_even__bit__succ__iff,axiom,
    ! [A: int,N: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ( ( bit_se1146084159140164899it_int @ ( plus_plus_int @ one_one_int @ A ) @ N )
        = ( ( bit_se1146084159140164899it_int @ A @ N )
          | ( N = zero_zero_nat ) ) ) ) ).

% even_bit_succ_iff
thf(fact_5759_even__bit__succ__iff,axiom,
    ! [A: nat,N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ( ( bit_se1148574629649215175it_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ N )
        = ( ( bit_se1148574629649215175it_nat @ A @ N )
          | ( N = zero_zero_nat ) ) ) ) ).

% even_bit_succ_iff
thf(fact_5760_odd__bit__iff__bit__pred,axiom,
    ! [A: code_integer,N: nat] :
      ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ( ( bit_se9216721137139052372nteger @ A @ N )
        = ( ( bit_se9216721137139052372nteger @ ( minus_8373710615458151222nteger @ A @ one_one_Code_integer ) @ N )
          | ( N = zero_zero_nat ) ) ) ) ).

% odd_bit_iff_bit_pred
thf(fact_5761_odd__bit__iff__bit__pred,axiom,
    ! [A: int,N: nat] :
      ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ( ( bit_se1146084159140164899it_int @ A @ N )
        = ( ( bit_se1146084159140164899it_int @ ( minus_minus_int @ A @ one_one_int ) @ N )
          | ( N = zero_zero_nat ) ) ) ) ).

% odd_bit_iff_bit_pred
thf(fact_5762_odd__bit__iff__bit__pred,axiom,
    ! [A: nat,N: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ( ( bit_se1148574629649215175it_nat @ A @ N )
        = ( ( bit_se1148574629649215175it_nat @ ( minus_minus_nat @ A @ one_one_nat ) @ N )
          | ( N = zero_zero_nat ) ) ) ) ).

% odd_bit_iff_bit_pred
thf(fact_5763_nat__ivt__aux,axiom,
    ! [N: nat,F: nat > int,K: int] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ N )
         => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) ) ) @ one_one_int ) )
     => ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
       => ( ( ord_less_eq_int @ K @ ( F @ N ) )
         => ? [I3: nat] :
              ( ( ord_less_eq_nat @ I3 @ N )
              & ( ( F @ I3 )
                = K ) ) ) ) ) ).

% nat_ivt_aux
thf(fact_5764_bit__sum__mult__2__cases,axiom,
    ! [A: code_integer,B2: code_integer,N: nat] :
      ( ! [J2: nat] :
          ~ ( bit_se9216721137139052372nteger @ A @ ( suc @ J2 ) )
     => ( ( bit_se9216721137139052372nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) ) @ N )
        = ( ( ( N = zero_zero_nat )
           => ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) )
          & ( ( N != zero_zero_nat )
           => ( bit_se9216721137139052372nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) @ N ) ) ) ) ) ).

% bit_sum_mult_2_cases
thf(fact_5765_bit__sum__mult__2__cases,axiom,
    ! [A: int,B2: int,N: nat] :
      ( ! [J2: nat] :
          ~ ( bit_se1146084159140164899it_int @ A @ ( suc @ J2 ) )
     => ( ( bit_se1146084159140164899it_int @ ( plus_plus_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) @ N )
        = ( ( ( N = zero_zero_nat )
           => ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
          & ( ( N != zero_zero_nat )
           => ( bit_se1146084159140164899it_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) @ N ) ) ) ) ) ).

% bit_sum_mult_2_cases
thf(fact_5766_bit__sum__mult__2__cases,axiom,
    ! [A: nat,B2: nat,N: nat] :
      ( ! [J2: nat] :
          ~ ( bit_se1148574629649215175it_nat @ A @ ( suc @ J2 ) )
     => ( ( bit_se1148574629649215175it_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) @ N )
        = ( ( ( N = zero_zero_nat )
           => ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
          & ( ( N != zero_zero_nat )
           => ( bit_se1148574629649215175it_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) @ N ) ) ) ) ) ).

% bit_sum_mult_2_cases
thf(fact_5767_bit__rec,axiom,
    ( bit_se9216721137139052372nteger
    = ( ^ [A4: code_integer,N2: nat] :
          ( ( ( N2 = zero_zero_nat )
           => ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A4 ) )
          & ( ( N2 != zero_zero_nat )
           => ( bit_se9216721137139052372nteger @ ( divide6298287555418463151nteger @ A4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ) ) ).

% bit_rec
thf(fact_5768_bit__rec,axiom,
    ( bit_se1146084159140164899it_int
    = ( ^ [A4: int,N2: nat] :
          ( ( ( N2 = zero_zero_nat )
           => ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A4 ) )
          & ( ( N2 != zero_zero_nat )
           => ( bit_se1146084159140164899it_int @ ( divide_divide_int @ A4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ) ) ).

% bit_rec
thf(fact_5769_bit__rec,axiom,
    ( bit_se1148574629649215175it_nat
    = ( ^ [A4: nat,N2: nat] :
          ( ( ( N2 = zero_zero_nat )
           => ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A4 ) )
          & ( ( N2 != zero_zero_nat )
           => ( bit_se1148574629649215175it_nat @ ( divide_divide_nat @ A4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ) ) ).

% bit_rec
thf(fact_5770_nat0__intermed__int__val,axiom,
    ! [N: nat,F: nat > int,K: int] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ N )
         => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I3 @ one_one_nat ) ) @ ( F @ I3 ) ) ) @ one_one_int ) )
     => ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
       => ( ( ord_less_eq_int @ K @ ( F @ N ) )
         => ? [I3: nat] :
              ( ( ord_less_eq_nat @ I3 @ N )
              & ( ( F @ I3 )
                = K ) ) ) ) ) ).

% nat0_intermed_int_val
thf(fact_5771_set__bit__eq,axiom,
    ( bit_se7879613467334960850it_int
    = ( ^ [N2: nat,K3: int] :
          ( plus_plus_int @ K3
          @ ( times_times_int
            @ ( zero_n2684676970156552555ol_int
              @ ~ ( bit_se1146084159140164899it_int @ K3 @ N2 ) )
            @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).

% set_bit_eq
thf(fact_5772_unset__bit__eq,axiom,
    ( bit_se4203085406695923979it_int
    = ( ^ [N2: nat,K3: int] : ( minus_minus_int @ K3 @ ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K3 @ N2 ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).

% unset_bit_eq
thf(fact_5773_arctan__add,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X4 ) @ one_one_real )
     => ( ( ord_less_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
       => ( ( plus_plus_real @ ( arctan @ X4 ) @ ( arctan @ Y3 ) )
          = ( arctan @ ( divide_divide_real @ ( plus_plus_real @ X4 @ Y3 ) @ ( minus_minus_real @ one_one_real @ ( times_times_real @ X4 @ Y3 ) ) ) ) ) ) ) ).

% arctan_add
thf(fact_5774_take__bit__Suc__from__most,axiom,
    ! [N: nat,K: int] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ K )
      = ( plus_plus_int @ ( times_times_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K @ N ) ) ) @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).

% take_bit_Suc_from_most
thf(fact_5775_Bernoulli__inequality__even,axiom,
    ! [N: nat,X4: real] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X4 ) ) @ ( power_power_real @ ( plus_plus_real @ one_one_real @ X4 ) @ N ) ) ) ).

% Bernoulli_inequality_even
thf(fact_5776_exp__ge__one__plus__x__over__n__power__n,axiom,
    ! [N: nat,X4: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ X4 )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_eq_real @ ( power_power_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X4 @ ( semiri5074537144036343181t_real @ N ) ) ) @ N ) @ ( exp_real @ X4 ) ) ) ) ).

% exp_ge_one_plus_x_over_n_power_n
thf(fact_5777_lemma__termdiff3,axiom,
    ! [H2: real,Z: real,K5: real,N: nat] :
      ( ( H2 != zero_zero_real )
     => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ K5 )
       => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ Z @ H2 ) ) @ K5 )
         => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z @ H2 ) @ N ) @ ( power_power_real @ Z @ N ) ) @ H2 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ Z @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) @ ( power_power_real @ K5 @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( real_V7735802525324610683m_real @ H2 ) ) ) ) ) ) ).

% lemma_termdiff3
thf(fact_5778_lemma__termdiff3,axiom,
    ! [H2: complex,Z: complex,K5: real,N: nat] :
      ( ( H2 != zero_zero_complex )
     => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ K5 )
       => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ Z @ H2 ) ) @ K5 )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H2 ) @ N ) @ ( power_power_complex @ Z @ N ) ) @ H2 ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( power_power_complex @ Z @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) @ ( power_power_real @ K5 @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( real_V1022390504157884413omplex @ H2 ) ) ) ) ) ) ).

% lemma_termdiff3
thf(fact_5779_ceiling__log2__div2,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) )
        = ( plus_plus_int @ ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( divide_divide_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) @ one_one_int ) ) ) ).

% ceiling_log2_div2
thf(fact_5780_pochhammer__double,axiom,
    ! [Z: complex,N: nat] :
      ( ( comm_s2602460028002588243omplex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ Z ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( times_times_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ ( comm_s2602460028002588243omplex @ Z @ N ) ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z @ ( divide1717551699836669952omplex @ one_one_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) @ N ) ) ) ).

% pochhammer_double
thf(fact_5781_pochhammer__double,axiom,
    ! [Z: real,N: nat] :
      ( ( comm_s7457072308508201937r_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ ( comm_s7457072308508201937r_real @ Z @ N ) ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ N ) ) ) ).

% pochhammer_double
thf(fact_5782_pochhammer__double,axiom,
    ! [Z: rat,N: nat] :
      ( ( comm_s4028243227959126397er_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ Z ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( times_times_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ ( comm_s4028243227959126397er_rat @ Z @ N ) ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ N ) ) ) ).

% pochhammer_double
thf(fact_5783_central__binomial__lower__bound,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ord_less_eq_real @ ( divide_divide_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ N ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) @ ( semiri5074537144036343181t_real @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ N ) ) ) ) ).

% central_binomial_lower_bound
thf(fact_5784_floor__log2__div2,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( archim6058952711729229775r_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) )
        = ( plus_plus_int @ ( archim6058952711729229775r_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ one_one_int ) ) ) ).

% floor_log2_div2
thf(fact_5785_machin,axiom,
    ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) )
    = ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ ( arctan @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit1 @ ( bit0 @ one ) ) ) ) ) ) @ ( arctan @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ).

% machin
thf(fact_5786_of__int__floor__cancel,axiom,
    ! [X4: real] :
      ( ( ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X4 ) )
        = X4 )
      = ( ? [N2: int] :
            ( X4
            = ( ring_1_of_int_real @ N2 ) ) ) ) ).

% of_int_floor_cancel
thf(fact_5787_of__int__floor__cancel,axiom,
    ! [X4: rat] :
      ( ( ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X4 ) )
        = X4 )
      = ( ? [N2: int] :
            ( X4
            = ( ring_1_of_int_rat @ N2 ) ) ) ) ).

% of_int_floor_cancel
thf(fact_5788_of__int__ceiling__cancel,axiom,
    ! [X4: rat] :
      ( ( ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X4 ) )
        = X4 )
      = ( ? [N2: int] :
            ( X4
            = ( ring_1_of_int_rat @ N2 ) ) ) ) ).

% of_int_ceiling_cancel
thf(fact_5789_of__int__ceiling__cancel,axiom,
    ! [X4: real] :
      ( ( ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X4 ) )
        = X4 )
      = ( ? [N2: int] :
            ( X4
            = ( ring_1_of_int_real @ N2 ) ) ) ) ).

% of_int_ceiling_cancel
thf(fact_5790_int__eq__iff__numeral,axiom,
    ! [M: nat,V: num] :
      ( ( ( semiri1314217659103216013at_int @ M )
        = ( numeral_numeral_int @ V ) )
      = ( M
        = ( numeral_numeral_nat @ V ) ) ) ).

% int_eq_iff_numeral
thf(fact_5791_floor__numeral,axiom,
    ! [V: num] :
      ( ( archim6058952711729229775r_real @ ( numeral_numeral_real @ V ) )
      = ( numeral_numeral_int @ V ) ) ).

% floor_numeral
thf(fact_5792_floor__numeral,axiom,
    ! [V: num] :
      ( ( archim3151403230148437115or_rat @ ( numeral_numeral_rat @ V ) )
      = ( numeral_numeral_int @ V ) ) ).

% floor_numeral
thf(fact_5793_floor__one,axiom,
    ( ( archim6058952711729229775r_real @ one_one_real )
    = one_one_int ) ).

% floor_one
thf(fact_5794_floor__one,axiom,
    ( ( archim3151403230148437115or_rat @ one_one_rat )
    = one_one_int ) ).

% floor_one
thf(fact_5795_ceiling__numeral,axiom,
    ! [V: num] :
      ( ( archim7802044766580827645g_real @ ( numeral_numeral_real @ V ) )
      = ( numeral_numeral_int @ V ) ) ).

% ceiling_numeral
thf(fact_5796_ceiling__numeral,axiom,
    ! [V: num] :
      ( ( archim2889992004027027881ng_rat @ ( numeral_numeral_rat @ V ) )
      = ( numeral_numeral_int @ V ) ) ).

% ceiling_numeral
thf(fact_5797_pochhammer__0,axiom,
    ! [A: complex] :
      ( ( comm_s2602460028002588243omplex @ A @ zero_zero_nat )
      = one_one_complex ) ).

% pochhammer_0
thf(fact_5798_pochhammer__0,axiom,
    ! [A: real] :
      ( ( comm_s7457072308508201937r_real @ A @ zero_zero_nat )
      = one_one_real ) ).

% pochhammer_0
thf(fact_5799_pochhammer__0,axiom,
    ! [A: rat] :
      ( ( comm_s4028243227959126397er_rat @ A @ zero_zero_nat )
      = one_one_rat ) ).

% pochhammer_0
thf(fact_5800_pochhammer__0,axiom,
    ! [A: nat] :
      ( ( comm_s4663373288045622133er_nat @ A @ zero_zero_nat )
      = one_one_nat ) ).

% pochhammer_0
thf(fact_5801_pochhammer__0,axiom,
    ! [A: int] :
      ( ( comm_s4660882817536571857er_int @ A @ zero_zero_nat )
      = one_one_int ) ).

% pochhammer_0
thf(fact_5802_ceiling__one,axiom,
    ( ( archim2889992004027027881ng_rat @ one_one_rat )
    = one_one_int ) ).

% ceiling_one
thf(fact_5803_ceiling__one,axiom,
    ( ( archim7802044766580827645g_real @ one_one_real )
    = one_one_int ) ).

% ceiling_one
thf(fact_5804_negative__zless,axiom,
    ! [N: nat,M: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).

% negative_zless
thf(fact_5805_ceiling__add__of__int,axiom,
    ! [X4: rat,Z: int] :
      ( ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X4 @ ( ring_1_of_int_rat @ Z ) ) )
      = ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X4 ) @ Z ) ) ).

% ceiling_add_of_int
thf(fact_5806_ceiling__add__of__int,axiom,
    ! [X4: real,Z: int] :
      ( ( archim7802044766580827645g_real @ ( plus_plus_real @ X4 @ ( ring_1_of_int_real @ Z ) ) )
      = ( plus_plus_int @ ( archim7802044766580827645g_real @ X4 ) @ Z ) ) ).

% ceiling_add_of_int
thf(fact_5807_floor__diff__of__int,axiom,
    ! [X4: real,Z: int] :
      ( ( archim6058952711729229775r_real @ ( minus_minus_real @ X4 @ ( ring_1_of_int_real @ Z ) ) )
      = ( minus_minus_int @ ( archim6058952711729229775r_real @ X4 ) @ Z ) ) ).

% floor_diff_of_int
thf(fact_5808_floor__diff__of__int,axiom,
    ! [X4: rat,Z: int] :
      ( ( archim3151403230148437115or_rat @ ( minus_minus_rat @ X4 @ ( ring_1_of_int_rat @ Z ) ) )
      = ( minus_minus_int @ ( archim3151403230148437115or_rat @ X4 ) @ Z ) ) ).

% floor_diff_of_int
thf(fact_5809_ceiling__diff__of__int,axiom,
    ! [X4: rat,Z: int] :
      ( ( archim2889992004027027881ng_rat @ ( minus_minus_rat @ X4 @ ( ring_1_of_int_rat @ Z ) ) )
      = ( minus_minus_int @ ( archim2889992004027027881ng_rat @ X4 ) @ Z ) ) ).

% ceiling_diff_of_int
thf(fact_5810_ceiling__diff__of__int,axiom,
    ! [X4: real,Z: int] :
      ( ( archim7802044766580827645g_real @ ( minus_minus_real @ X4 @ ( ring_1_of_int_real @ Z ) ) )
      = ( minus_minus_int @ ( archim7802044766580827645g_real @ X4 ) @ Z ) ) ).

% ceiling_diff_of_int
thf(fact_5811_zero__le__floor,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( archim6058952711729229775r_real @ X4 ) )
      = ( ord_less_eq_real @ zero_zero_real @ X4 ) ) ).

% zero_le_floor
thf(fact_5812_zero__le__floor,axiom,
    ! [X4: rat] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( archim3151403230148437115or_rat @ X4 ) )
      = ( ord_less_eq_rat @ zero_zero_rat @ X4 ) ) ).

% zero_le_floor
thf(fact_5813_floor__less__zero,axiom,
    ! [X4: real] :
      ( ( ord_less_int @ ( archim6058952711729229775r_real @ X4 ) @ zero_zero_int )
      = ( ord_less_real @ X4 @ zero_zero_real ) ) ).

% floor_less_zero
thf(fact_5814_floor__less__zero,axiom,
    ! [X4: rat] :
      ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X4 ) @ zero_zero_int )
      = ( ord_less_rat @ X4 @ zero_zero_rat ) ) ).

% floor_less_zero
thf(fact_5815_numeral__le__floor,axiom,
    ! [V: num,X4: real] :
      ( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim6058952711729229775r_real @ X4 ) )
      = ( ord_less_eq_real @ ( numeral_numeral_real @ V ) @ X4 ) ) ).

% numeral_le_floor
thf(fact_5816_numeral__le__floor,axiom,
    ! [V: num,X4: rat] :
      ( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim3151403230148437115or_rat @ X4 ) )
      = ( ord_less_eq_rat @ ( numeral_numeral_rat @ V ) @ X4 ) ) ).

% numeral_le_floor
thf(fact_5817_zero__less__floor,axiom,
    ! [X4: real] :
      ( ( ord_less_int @ zero_zero_int @ ( archim6058952711729229775r_real @ X4 ) )
      = ( ord_less_eq_real @ one_one_real @ X4 ) ) ).

% zero_less_floor
thf(fact_5818_zero__less__floor,axiom,
    ! [X4: rat] :
      ( ( ord_less_int @ zero_zero_int @ ( archim3151403230148437115or_rat @ X4 ) )
      = ( ord_less_eq_rat @ one_one_rat @ X4 ) ) ).

% zero_less_floor
thf(fact_5819_floor__le__zero,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X4 ) @ zero_zero_int )
      = ( ord_less_real @ X4 @ one_one_real ) ) ).

% floor_le_zero
thf(fact_5820_floor__le__zero,axiom,
    ! [X4: rat] :
      ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X4 ) @ zero_zero_int )
      = ( ord_less_rat @ X4 @ one_one_rat ) ) ).

% floor_le_zero
thf(fact_5821_ceiling__le__zero,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X4 ) @ zero_zero_int )
      = ( ord_less_eq_real @ X4 @ zero_zero_real ) ) ).

% ceiling_le_zero
thf(fact_5822_ceiling__le__zero,axiom,
    ! [X4: rat] :
      ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X4 ) @ zero_zero_int )
      = ( ord_less_eq_rat @ X4 @ zero_zero_rat ) ) ).

% ceiling_le_zero
thf(fact_5823_floor__less__numeral,axiom,
    ! [X4: real,V: num] :
      ( ( ord_less_int @ ( archim6058952711729229775r_real @ X4 ) @ ( numeral_numeral_int @ V ) )
      = ( ord_less_real @ X4 @ ( numeral_numeral_real @ V ) ) ) ).

% floor_less_numeral
thf(fact_5824_floor__less__numeral,axiom,
    ! [X4: rat,V: num] :
      ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X4 ) @ ( numeral_numeral_int @ V ) )
      = ( ord_less_rat @ X4 @ ( numeral_numeral_rat @ V ) ) ) ).

% floor_less_numeral
thf(fact_5825_zero__less__ceiling,axiom,
    ! [X4: rat] :
      ( ( ord_less_int @ zero_zero_int @ ( archim2889992004027027881ng_rat @ X4 ) )
      = ( ord_less_rat @ zero_zero_rat @ X4 ) ) ).

% zero_less_ceiling
thf(fact_5826_zero__less__ceiling,axiom,
    ! [X4: real] :
      ( ( ord_less_int @ zero_zero_int @ ( archim7802044766580827645g_real @ X4 ) )
      = ( ord_less_real @ zero_zero_real @ X4 ) ) ).

% zero_less_ceiling
thf(fact_5827_one__le__floor,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_int @ one_one_int @ ( archim6058952711729229775r_real @ X4 ) )
      = ( ord_less_eq_real @ one_one_real @ X4 ) ) ).

% one_le_floor
thf(fact_5828_one__le__floor,axiom,
    ! [X4: rat] :
      ( ( ord_less_eq_int @ one_one_int @ ( archim3151403230148437115or_rat @ X4 ) )
      = ( ord_less_eq_rat @ one_one_rat @ X4 ) ) ).

% one_le_floor
thf(fact_5829_ceiling__le__numeral,axiom,
    ! [X4: real,V: num] :
      ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X4 ) @ ( numeral_numeral_int @ V ) )
      = ( ord_less_eq_real @ X4 @ ( numeral_numeral_real @ V ) ) ) ).

% ceiling_le_numeral
thf(fact_5830_ceiling__le__numeral,axiom,
    ! [X4: rat,V: num] :
      ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X4 ) @ ( numeral_numeral_int @ V ) )
      = ( ord_less_eq_rat @ X4 @ ( numeral_numeral_rat @ V ) ) ) ).

% ceiling_le_numeral
thf(fact_5831_floor__less__one,axiom,
    ! [X4: real] :
      ( ( ord_less_int @ ( archim6058952711729229775r_real @ X4 ) @ one_one_int )
      = ( ord_less_real @ X4 @ one_one_real ) ) ).

% floor_less_one
thf(fact_5832_floor__less__one,axiom,
    ! [X4: rat] :
      ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X4 ) @ one_one_int )
      = ( ord_less_rat @ X4 @ one_one_rat ) ) ).

% floor_less_one
thf(fact_5833_ceiling__less__one,axiom,
    ! [X4: real] :
      ( ( ord_less_int @ ( archim7802044766580827645g_real @ X4 ) @ one_one_int )
      = ( ord_less_eq_real @ X4 @ zero_zero_real ) ) ).

% ceiling_less_one
thf(fact_5834_ceiling__less__one,axiom,
    ! [X4: rat] :
      ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X4 ) @ one_one_int )
      = ( ord_less_eq_rat @ X4 @ zero_zero_rat ) ) ).

% ceiling_less_one
thf(fact_5835_one__le__ceiling,axiom,
    ! [X4: rat] :
      ( ( ord_less_eq_int @ one_one_int @ ( archim2889992004027027881ng_rat @ X4 ) )
      = ( ord_less_rat @ zero_zero_rat @ X4 ) ) ).

% one_le_ceiling
thf(fact_5836_one__le__ceiling,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_int @ one_one_int @ ( archim7802044766580827645g_real @ X4 ) )
      = ( ord_less_real @ zero_zero_real @ X4 ) ) ).

% one_le_ceiling
thf(fact_5837_numeral__less__ceiling,axiom,
    ! [V: num,X4: real] :
      ( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim7802044766580827645g_real @ X4 ) )
      = ( ord_less_real @ ( numeral_numeral_real @ V ) @ X4 ) ) ).

% numeral_less_ceiling
thf(fact_5838_numeral__less__ceiling,axiom,
    ! [V: num,X4: rat] :
      ( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim2889992004027027881ng_rat @ X4 ) )
      = ( ord_less_rat @ ( numeral_numeral_rat @ V ) @ X4 ) ) ).

% numeral_less_ceiling
thf(fact_5839_floor__neg__numeral,axiom,
    ! [V: num] :
      ( ( archim6058952711729229775r_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) ).

% floor_neg_numeral
thf(fact_5840_floor__neg__numeral,axiom,
    ! [V: num] :
      ( ( archim3151403230148437115or_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) ).

% floor_neg_numeral
thf(fact_5841_ceiling__le__one,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X4 ) @ one_one_int )
      = ( ord_less_eq_real @ X4 @ one_one_real ) ) ).

% ceiling_le_one
thf(fact_5842_ceiling__le__one,axiom,
    ! [X4: rat] :
      ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X4 ) @ one_one_int )
      = ( ord_less_eq_rat @ X4 @ one_one_rat ) ) ).

% ceiling_le_one
thf(fact_5843_one__less__ceiling,axiom,
    ! [X4: rat] :
      ( ( ord_less_int @ one_one_int @ ( archim2889992004027027881ng_rat @ X4 ) )
      = ( ord_less_rat @ one_one_rat @ X4 ) ) ).

% one_less_ceiling
thf(fact_5844_one__less__ceiling,axiom,
    ! [X4: real] :
      ( ( ord_less_int @ one_one_int @ ( archim7802044766580827645g_real @ X4 ) )
      = ( ord_less_real @ one_one_real @ X4 ) ) ).

% one_less_ceiling
thf(fact_5845_ceiling__add__numeral,axiom,
    ! [X4: real,V: num] :
      ( ( archim7802044766580827645g_real @ ( plus_plus_real @ X4 @ ( numeral_numeral_real @ V ) ) )
      = ( plus_plus_int @ ( archim7802044766580827645g_real @ X4 ) @ ( numeral_numeral_int @ V ) ) ) ).

% ceiling_add_numeral
thf(fact_5846_ceiling__add__numeral,axiom,
    ! [X4: rat,V: num] :
      ( ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X4 @ ( numeral_numeral_rat @ V ) ) )
      = ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X4 ) @ ( numeral_numeral_int @ V ) ) ) ).

% ceiling_add_numeral
thf(fact_5847_floor__diff__numeral,axiom,
    ! [X4: real,V: num] :
      ( ( archim6058952711729229775r_real @ ( minus_minus_real @ X4 @ ( numeral_numeral_real @ V ) ) )
      = ( minus_minus_int @ ( archim6058952711729229775r_real @ X4 ) @ ( numeral_numeral_int @ V ) ) ) ).

% floor_diff_numeral
thf(fact_5848_floor__diff__numeral,axiom,
    ! [X4: rat,V: num] :
      ( ( archim3151403230148437115or_rat @ ( minus_minus_rat @ X4 @ ( numeral_numeral_rat @ V ) ) )
      = ( minus_minus_int @ ( archim3151403230148437115or_rat @ X4 ) @ ( numeral_numeral_int @ V ) ) ) ).

% floor_diff_numeral
thf(fact_5849_ceiling__neg__numeral,axiom,
    ! [V: num] :
      ( ( archim7802044766580827645g_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) ).

% ceiling_neg_numeral
thf(fact_5850_ceiling__neg__numeral,axiom,
    ! [V: num] :
      ( ( archim2889992004027027881ng_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) ).

% ceiling_neg_numeral
thf(fact_5851_ceiling__add__one,axiom,
    ! [X4: rat] :
      ( ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X4 @ one_one_rat ) )
      = ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X4 ) @ one_one_int ) ) ).

% ceiling_add_one
thf(fact_5852_ceiling__add__one,axiom,
    ! [X4: real] :
      ( ( archim7802044766580827645g_real @ ( plus_plus_real @ X4 @ one_one_real ) )
      = ( plus_plus_int @ ( archim7802044766580827645g_real @ X4 ) @ one_one_int ) ) ).

% ceiling_add_one
thf(fact_5853_floor__diff__one,axiom,
    ! [X4: real] :
      ( ( archim6058952711729229775r_real @ ( minus_minus_real @ X4 @ one_one_real ) )
      = ( minus_minus_int @ ( archim6058952711729229775r_real @ X4 ) @ one_one_int ) ) ).

% floor_diff_one
thf(fact_5854_floor__diff__one,axiom,
    ! [X4: rat] :
      ( ( archim3151403230148437115or_rat @ ( minus_minus_rat @ X4 @ one_one_rat ) )
      = ( minus_minus_int @ ( archim3151403230148437115or_rat @ X4 ) @ one_one_int ) ) ).

% floor_diff_one
thf(fact_5855_ceiling__diff__numeral,axiom,
    ! [X4: real,V: num] :
      ( ( archim7802044766580827645g_real @ ( minus_minus_real @ X4 @ ( numeral_numeral_real @ V ) ) )
      = ( minus_minus_int @ ( archim7802044766580827645g_real @ X4 ) @ ( numeral_numeral_int @ V ) ) ) ).

% ceiling_diff_numeral
thf(fact_5856_ceiling__diff__numeral,axiom,
    ! [X4: rat,V: num] :
      ( ( archim2889992004027027881ng_rat @ ( minus_minus_rat @ X4 @ ( numeral_numeral_rat @ V ) ) )
      = ( minus_minus_int @ ( archim2889992004027027881ng_rat @ X4 ) @ ( numeral_numeral_int @ V ) ) ) ).

% ceiling_diff_numeral
thf(fact_5857_ceiling__diff__one,axiom,
    ! [X4: rat] :
      ( ( archim2889992004027027881ng_rat @ ( minus_minus_rat @ X4 @ one_one_rat ) )
      = ( minus_minus_int @ ( archim2889992004027027881ng_rat @ X4 ) @ one_one_int ) ) ).

% ceiling_diff_one
thf(fact_5858_ceiling__diff__one,axiom,
    ! [X4: real] :
      ( ( archim7802044766580827645g_real @ ( minus_minus_real @ X4 @ one_one_real ) )
      = ( minus_minus_int @ ( archim7802044766580827645g_real @ X4 ) @ one_one_int ) ) ).

% ceiling_diff_one
thf(fact_5859_floor__numeral__power,axiom,
    ! [X4: num,N: nat] :
      ( ( archim6058952711729229775r_real @ ( power_power_real @ ( numeral_numeral_real @ X4 ) @ N ) )
      = ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N ) ) ).

% floor_numeral_power
thf(fact_5860_floor__numeral__power,axiom,
    ! [X4: num,N: nat] :
      ( ( archim3151403230148437115or_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X4 ) @ N ) )
      = ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N ) ) ).

% floor_numeral_power
thf(fact_5861_ceiling__numeral__power,axiom,
    ! [X4: num,N: nat] :
      ( ( archim7802044766580827645g_real @ ( power_power_real @ ( numeral_numeral_real @ X4 ) @ N ) )
      = ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N ) ) ).

% ceiling_numeral_power
thf(fact_5862_ceiling__numeral__power,axiom,
    ! [X4: num,N: nat] :
      ( ( archim2889992004027027881ng_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X4 ) @ N ) )
      = ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N ) ) ).

% ceiling_numeral_power
thf(fact_5863_floor__divide__eq__div__numeral,axiom,
    ! [A: num,B2: num] :
      ( ( archim6058952711729229775r_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B2 ) ) )
      = ( divide_divide_int @ ( numeral_numeral_int @ A ) @ ( numeral_numeral_int @ B2 ) ) ) ).

% floor_divide_eq_div_numeral
thf(fact_5864_ceiling__less__zero,axiom,
    ! [X4: real] :
      ( ( ord_less_int @ ( archim7802044766580827645g_real @ X4 ) @ zero_zero_int )
      = ( ord_less_eq_real @ X4 @ ( uminus_uminus_real @ one_one_real ) ) ) ).

% ceiling_less_zero
thf(fact_5865_ceiling__less__zero,axiom,
    ! [X4: rat] :
      ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X4 ) @ zero_zero_int )
      = ( ord_less_eq_rat @ X4 @ ( uminus_uminus_rat @ one_one_rat ) ) ) ).

% ceiling_less_zero
thf(fact_5866_zero__le__ceiling,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( archim7802044766580827645g_real @ X4 ) )
      = ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X4 ) ) ).

% zero_le_ceiling
thf(fact_5867_zero__le__ceiling,axiom,
    ! [X4: rat] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( archim2889992004027027881ng_rat @ X4 ) )
      = ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ X4 ) ) ).

% zero_le_ceiling
thf(fact_5868_ceiling__divide__eq__div__numeral,axiom,
    ! [A: num,B2: num] :
      ( ( archim7802044766580827645g_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B2 ) ) )
      = ( uminus_uminus_int @ ( divide_divide_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ A ) ) @ ( numeral_numeral_int @ B2 ) ) ) ) ).

% ceiling_divide_eq_div_numeral
thf(fact_5869_numeral__less__floor,axiom,
    ! [V: num,X4: real] :
      ( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim6058952711729229775r_real @ X4 ) )
      = ( ord_less_eq_real @ ( plus_plus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) @ X4 ) ) ).

% numeral_less_floor
thf(fact_5870_numeral__less__floor,axiom,
    ! [V: num,X4: rat] :
      ( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim3151403230148437115or_rat @ X4 ) )
      = ( ord_less_eq_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) @ X4 ) ) ).

% numeral_less_floor
thf(fact_5871_floor__le__numeral,axiom,
    ! [X4: real,V: num] :
      ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X4 ) @ ( numeral_numeral_int @ V ) )
      = ( ord_less_real @ X4 @ ( plus_plus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) ) ) ).

% floor_le_numeral
thf(fact_5872_floor__le__numeral,axiom,
    ! [X4: rat,V: num] :
      ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X4 ) @ ( numeral_numeral_int @ V ) )
      = ( ord_less_rat @ X4 @ ( plus_plus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) ) ) ).

% floor_le_numeral
thf(fact_5873_one__less__floor,axiom,
    ! [X4: real] :
      ( ( ord_less_int @ one_one_int @ ( archim6058952711729229775r_real @ X4 ) )
      = ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X4 ) ) ).

% one_less_floor
thf(fact_5874_one__less__floor,axiom,
    ! [X4: rat] :
      ( ( ord_less_int @ one_one_int @ ( archim3151403230148437115or_rat @ X4 ) )
      = ( ord_less_eq_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X4 ) ) ).

% one_less_floor
thf(fact_5875_floor__le__one,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X4 ) @ one_one_int )
      = ( ord_less_real @ X4 @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% floor_le_one
thf(fact_5876_floor__le__one,axiom,
    ! [X4: rat] :
      ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X4 ) @ one_one_int )
      = ( ord_less_rat @ X4 @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).

% floor_le_one
thf(fact_5877_ceiling__less__numeral,axiom,
    ! [X4: real,V: num] :
      ( ( ord_less_int @ ( archim7802044766580827645g_real @ X4 ) @ ( numeral_numeral_int @ V ) )
      = ( ord_less_eq_real @ X4 @ ( minus_minus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) ) ) ).

% ceiling_less_numeral
thf(fact_5878_ceiling__less__numeral,axiom,
    ! [X4: rat,V: num] :
      ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X4 ) @ ( numeral_numeral_int @ V ) )
      = ( ord_less_eq_rat @ X4 @ ( minus_minus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) ) ) ).

% ceiling_less_numeral
thf(fact_5879_numeral__le__ceiling,axiom,
    ! [V: num,X4: real] :
      ( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim7802044766580827645g_real @ X4 ) )
      = ( ord_less_real @ ( minus_minus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) @ X4 ) ) ).

% numeral_le_ceiling
thf(fact_5880_numeral__le__ceiling,axiom,
    ! [V: num,X4: rat] :
      ( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim2889992004027027881ng_rat @ X4 ) )
      = ( ord_less_rat @ ( minus_minus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) @ X4 ) ) ).

% numeral_le_ceiling
thf(fact_5881_neg__numeral__le__floor,axiom,
    ! [V: num,X4: real] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim6058952711729229775r_real @ X4 ) )
      = ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ X4 ) ) ).

% neg_numeral_le_floor
thf(fact_5882_neg__numeral__le__floor,axiom,
    ! [V: num,X4: rat] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim3151403230148437115or_rat @ X4 ) )
      = ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ X4 ) ) ).

% neg_numeral_le_floor
thf(fact_5883_floor__less__neg__numeral,axiom,
    ! [X4: real,V: num] :
      ( ( ord_less_int @ ( archim6058952711729229775r_real @ X4 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
      = ( ord_less_real @ X4 @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) ) ).

% floor_less_neg_numeral
thf(fact_5884_floor__less__neg__numeral,axiom,
    ! [X4: rat,V: num] :
      ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X4 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
      = ( ord_less_rat @ X4 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) ) ).

% floor_less_neg_numeral
thf(fact_5885_ceiling__le__neg__numeral,axiom,
    ! [X4: real,V: num] :
      ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X4 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
      = ( ord_less_eq_real @ X4 @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) ) ).

% ceiling_le_neg_numeral
thf(fact_5886_ceiling__le__neg__numeral,axiom,
    ! [X4: rat,V: num] :
      ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X4 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
      = ( ord_less_eq_rat @ X4 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) ) ).

% ceiling_le_neg_numeral
thf(fact_5887_neg__numeral__less__ceiling,axiom,
    ! [V: num,X4: real] :
      ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim7802044766580827645g_real @ X4 ) )
      = ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ X4 ) ) ).

% neg_numeral_less_ceiling
thf(fact_5888_neg__numeral__less__ceiling,axiom,
    ! [V: num,X4: rat] :
      ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim2889992004027027881ng_rat @ X4 ) )
      = ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ X4 ) ) ).

% neg_numeral_less_ceiling
thf(fact_5889_floor__one__divide__eq__div__numeral,axiom,
    ! [B2: num] :
      ( ( archim6058952711729229775r_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ B2 ) ) )
      = ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ B2 ) ) ) ).

% floor_one_divide_eq_div_numeral
thf(fact_5890_floor__minus__divide__eq__div__numeral,axiom,
    ! [A: num,B2: num] :
      ( ( archim6058952711729229775r_real @ ( uminus_uminus_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B2 ) ) ) )
      = ( divide_divide_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ A ) ) @ ( numeral_numeral_int @ B2 ) ) ) ).

% floor_minus_divide_eq_div_numeral
thf(fact_5891_ceiling__minus__divide__eq__div__numeral,axiom,
    ! [A: num,B2: num] :
      ( ( archim7802044766580827645g_real @ ( uminus_uminus_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B2 ) ) ) )
      = ( uminus_uminus_int @ ( divide_divide_int @ ( numeral_numeral_int @ A ) @ ( numeral_numeral_int @ B2 ) ) ) ) ).

% ceiling_minus_divide_eq_div_numeral
thf(fact_5892_neg__numeral__less__floor,axiom,
    ! [V: num,X4: real] :
      ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim6058952711729229775r_real @ X4 ) )
      = ( ord_less_eq_real @ ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) @ X4 ) ) ).

% neg_numeral_less_floor
thf(fact_5893_neg__numeral__less__floor,axiom,
    ! [V: num,X4: rat] :
      ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim3151403230148437115or_rat @ X4 ) )
      = ( ord_less_eq_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) @ X4 ) ) ).

% neg_numeral_less_floor
thf(fact_5894_floor__le__neg__numeral,axiom,
    ! [X4: real,V: num] :
      ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X4 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
      = ( ord_less_real @ X4 @ ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) ) ) ).

% floor_le_neg_numeral
thf(fact_5895_floor__le__neg__numeral,axiom,
    ! [X4: rat,V: num] :
      ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X4 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
      = ( ord_less_rat @ X4 @ ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) ) ) ).

% floor_le_neg_numeral
thf(fact_5896_ceiling__less__neg__numeral,axiom,
    ! [X4: real,V: num] :
      ( ( ord_less_int @ ( archim7802044766580827645g_real @ X4 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
      = ( ord_less_eq_real @ X4 @ ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) ) ) ).

% ceiling_less_neg_numeral
thf(fact_5897_ceiling__less__neg__numeral,axiom,
    ! [X4: rat,V: num] :
      ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X4 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
      = ( ord_less_eq_rat @ X4 @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) ) ) ).

% ceiling_less_neg_numeral
thf(fact_5898_neg__numeral__le__ceiling,axiom,
    ! [V: num,X4: real] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim7802044766580827645g_real @ X4 ) )
      = ( ord_less_real @ ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) @ X4 ) ) ).

% neg_numeral_le_ceiling
thf(fact_5899_neg__numeral__le__ceiling,axiom,
    ! [V: num,X4: rat] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim2889992004027027881ng_rat @ X4 ) )
      = ( ord_less_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) @ X4 ) ) ).

% neg_numeral_le_ceiling
thf(fact_5900_floor__minus__one__divide__eq__div__numeral,axiom,
    ! [B2: num] :
      ( ( archim6058952711729229775r_real @ ( uminus_uminus_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ B2 ) ) ) )
      = ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ B2 ) ) ) ).

% floor_minus_one_divide_eq_div_numeral
thf(fact_5901_int__if,axiom,
    ! [P: $o,A: nat,B2: nat] :
      ( ( P
       => ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B2 ) )
          = ( semiri1314217659103216013at_int @ A ) ) )
      & ( ~ P
       => ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B2 ) )
          = ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).

% int_if
thf(fact_5902_nat__int__comparison_I1_J,axiom,
    ( ( ^ [Y6: nat,Z4: nat] : Y6 = Z4 )
    = ( ^ [A4: nat,B3: nat] :
          ( ( semiri1314217659103216013at_int @ A4 )
          = ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).

% nat_int_comparison(1)
thf(fact_5903_int__diff__cases,axiom,
    ! [Z: int] :
      ~ ! [M4: nat,N4: nat] :
          ( Z
         != ( minus_minus_int @ ( semiri1314217659103216013at_int @ M4 ) @ ( semiri1314217659103216013at_int @ N4 ) ) ) ).

% int_diff_cases
thf(fact_5904_ceiling__altdef,axiom,
    ( archim7802044766580827645g_real
    = ( ^ [X: real] :
          ( if_int
          @ ( X
            = ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X ) ) )
          @ ( archim6058952711729229775r_real @ X )
          @ ( plus_plus_int @ ( archim6058952711729229775r_real @ X ) @ one_one_int ) ) ) ) ).

% ceiling_altdef
thf(fact_5905_ceiling__altdef,axiom,
    ( archim2889992004027027881ng_rat
    = ( ^ [X: rat] :
          ( if_int
          @ ( X
            = ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X ) ) )
          @ ( archim3151403230148437115or_rat @ X )
          @ ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ one_one_int ) ) ) ) ).

% ceiling_altdef
thf(fact_5906_ceiling__diff__floor__le__1,axiom,
    ! [X4: real] : ( ord_less_eq_int @ ( minus_minus_int @ ( archim7802044766580827645g_real @ X4 ) @ ( archim6058952711729229775r_real @ X4 ) ) @ one_one_int ) ).

% ceiling_diff_floor_le_1
thf(fact_5907_ceiling__diff__floor__le__1,axiom,
    ! [X4: rat] : ( ord_less_eq_int @ ( minus_minus_int @ ( archim2889992004027027881ng_rat @ X4 ) @ ( archim3151403230148437115or_rat @ X4 ) ) @ one_one_int ) ).

% ceiling_diff_floor_le_1
thf(fact_5908_floor__mono,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_eq_real @ X4 @ Y3 )
     => ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X4 ) @ ( archim6058952711729229775r_real @ Y3 ) ) ) ).

% floor_mono
thf(fact_5909_floor__mono,axiom,
    ! [X4: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ X4 @ Y3 )
     => ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X4 ) @ ( archim3151403230148437115or_rat @ Y3 ) ) ) ).

% floor_mono
thf(fact_5910_of__int__floor__le,axiom,
    ! [X4: real] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X4 ) ) @ X4 ) ).

% of_int_floor_le
thf(fact_5911_of__int__floor__le,axiom,
    ! [X4: rat] : ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X4 ) ) @ X4 ) ).

% of_int_floor_le
thf(fact_5912_floor__less__cancel,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_int @ ( archim6058952711729229775r_real @ X4 ) @ ( archim6058952711729229775r_real @ Y3 ) )
     => ( ord_less_real @ X4 @ Y3 ) ) ).

% floor_less_cancel
thf(fact_5913_floor__less__cancel,axiom,
    ! [X4: rat,Y3: rat] :
      ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X4 ) @ ( archim3151403230148437115or_rat @ Y3 ) )
     => ( ord_less_rat @ X4 @ Y3 ) ) ).

% floor_less_cancel
thf(fact_5914_not__bit__Suc__0__Suc,axiom,
    ! [N: nat] :
      ~ ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ ( suc @ N ) ) ).

% not_bit_Suc_0_Suc
thf(fact_5915_bit__Suc__0__iff,axiom,
    ! [N: nat] :
      ( ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ N )
      = ( N = zero_zero_nat ) ) ).

% bit_Suc_0_iff
thf(fact_5916_ceiling__mono,axiom,
    ! [Y3: real,X4: real] :
      ( ( ord_less_eq_real @ Y3 @ X4 )
     => ( ord_less_eq_int @ ( archim7802044766580827645g_real @ Y3 ) @ ( archim7802044766580827645g_real @ X4 ) ) ) ).

% ceiling_mono
thf(fact_5917_ceiling__mono,axiom,
    ! [Y3: rat,X4: rat] :
      ( ( ord_less_eq_rat @ Y3 @ X4 )
     => ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ Y3 ) @ ( archim2889992004027027881ng_rat @ X4 ) ) ) ).

% ceiling_mono
thf(fact_5918_le__of__int__ceiling,axiom,
    ! [X4: real] : ( ord_less_eq_real @ X4 @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X4 ) ) ) ).

% le_of_int_ceiling
thf(fact_5919_le__of__int__ceiling,axiom,
    ! [X4: rat] : ( ord_less_eq_rat @ X4 @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X4 ) ) ) ).

% le_of_int_ceiling
thf(fact_5920_ceiling__less__cancel,axiom,
    ! [X4: rat,Y3: rat] :
      ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X4 ) @ ( archim2889992004027027881ng_rat @ Y3 ) )
     => ( ord_less_rat @ X4 @ Y3 ) ) ).

% ceiling_less_cancel
thf(fact_5921_ceiling__less__cancel,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_int @ ( archim7802044766580827645g_real @ X4 ) @ ( archim7802044766580827645g_real @ Y3 ) )
     => ( ord_less_real @ X4 @ Y3 ) ) ).

% ceiling_less_cancel
thf(fact_5922_int__ops_I1_J,axiom,
    ( ( semiri1314217659103216013at_int @ zero_zero_nat )
    = zero_zero_int ) ).

% int_ops(1)
thf(fact_5923_complex__mod__minus__le__complex__mod,axiom,
    ! [X4: complex] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( real_V1022390504157884413omplex @ X4 ) ) @ ( real_V1022390504157884413omplex @ X4 ) ) ).

% complex_mod_minus_le_complex_mod
thf(fact_5924_int__ops_I3_J,axiom,
    ! [N: num] :
      ( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N ) )
      = ( numeral_numeral_int @ N ) ) ).

% int_ops(3)
thf(fact_5925_pochhammer__pos,axiom,
    ! [X4: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ord_less_real @ zero_zero_real @ ( comm_s7457072308508201937r_real @ X4 @ N ) ) ) ).

% pochhammer_pos
thf(fact_5926_pochhammer__pos,axiom,
    ! [X4: rat,N: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ X4 )
     => ( ord_less_rat @ zero_zero_rat @ ( comm_s4028243227959126397er_rat @ X4 @ N ) ) ) ).

% pochhammer_pos
thf(fact_5927_pochhammer__pos,axiom,
    ! [X4: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ X4 )
     => ( ord_less_nat @ zero_zero_nat @ ( comm_s4663373288045622133er_nat @ X4 @ N ) ) ) ).

% pochhammer_pos
thf(fact_5928_pochhammer__pos,axiom,
    ! [X4: int,N: nat] :
      ( ( ord_less_int @ zero_zero_int @ X4 )
     => ( ord_less_int @ zero_zero_int @ ( comm_s4660882817536571857er_int @ X4 @ N ) ) ) ).

% pochhammer_pos
thf(fact_5929_nat__int__comparison_I2_J,axiom,
    ( ord_less_nat
    = ( ^ [A4: nat,B3: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).

% nat_int_comparison(2)
thf(fact_5930_int__of__nat__induct,axiom,
    ! [P: int > $o,Z: int] :
      ( ! [N4: nat] : ( P @ ( semiri1314217659103216013at_int @ N4 ) )
     => ( ! [N4: nat] : ( P @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N4 ) ) ) )
       => ( P @ Z ) ) ) ).

% int_of_nat_induct
thf(fact_5931_int__cases,axiom,
    ! [Z: int] :
      ( ! [N4: nat] :
          ( Z
         != ( semiri1314217659103216013at_int @ N4 ) )
     => ~ ! [N4: nat] :
            ( Z
           != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N4 ) ) ) ) ) ).

% int_cases
thf(fact_5932_pi__not__less__zero,axiom,
    ~ ( ord_less_real @ pi @ zero_zero_real ) ).

% pi_not_less_zero
thf(fact_5933_pi__gt__zero,axiom,
    ord_less_real @ zero_zero_real @ pi ).

% pi_gt_zero
thf(fact_5934_pi__ge__zero,axiom,
    ord_less_eq_real @ zero_zero_real @ pi ).

% pi_ge_zero
thf(fact_5935_nat__int__comparison_I3_J,axiom,
    ( ord_less_eq_nat
    = ( ^ [A4: nat,B3: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).

% nat_int_comparison(3)
thf(fact_5936_zle__int,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% zle_int
thf(fact_5937_complex__mod__triangle__ineq2,axiom,
    ! [B2: complex,A: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ B2 @ A ) ) @ ( real_V1022390504157884413omplex @ B2 ) ) @ ( real_V1022390504157884413omplex @ A ) ) ).

% complex_mod_triangle_ineq2
thf(fact_5938_pochhammer__eq__0__mono,axiom,
    ! [A: real,N: nat,M: nat] :
      ( ( ( comm_s7457072308508201937r_real @ A @ N )
        = zero_zero_real )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( ( comm_s7457072308508201937r_real @ A @ M )
          = zero_zero_real ) ) ) ).

% pochhammer_eq_0_mono
thf(fact_5939_pochhammer__eq__0__mono,axiom,
    ! [A: rat,N: nat,M: nat] :
      ( ( ( comm_s4028243227959126397er_rat @ A @ N )
        = zero_zero_rat )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( ( comm_s4028243227959126397er_rat @ A @ M )
          = zero_zero_rat ) ) ) ).

% pochhammer_eq_0_mono
thf(fact_5940_pochhammer__neq__0__mono,axiom,
    ! [A: real,M: nat,N: nat] :
      ( ( ( comm_s7457072308508201937r_real @ A @ M )
       != zero_zero_real )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( ( comm_s7457072308508201937r_real @ A @ N )
         != zero_zero_real ) ) ) ).

% pochhammer_neq_0_mono
thf(fact_5941_pochhammer__neq__0__mono,axiom,
    ! [A: rat,M: nat,N: nat] :
      ( ( ( comm_s4028243227959126397er_rat @ A @ M )
       != zero_zero_rat )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( ( comm_s4028243227959126397er_rat @ A @ N )
         != zero_zero_rat ) ) ) ).

% pochhammer_neq_0_mono
thf(fact_5942_int__ops_I5_J,axiom,
    ! [A: nat,B2: nat] :
      ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ A @ B2 ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ).

% int_ops(5)
thf(fact_5943_int__plus,axiom,
    ! [N: nat,M: nat] :
      ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ N @ M ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M ) ) ) ).

% int_plus
thf(fact_5944_zadd__int__left,axiom,
    ! [M: nat,N: nat,Z: int] :
      ( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ Z ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) ) @ Z ) ) ).

% zadd_int_left
thf(fact_5945_int__ops_I2_J,axiom,
    ( ( semiri1314217659103216013at_int @ one_one_nat )
    = one_one_int ) ).

% int_ops(2)
thf(fact_5946_int__ops_I7_J,axiom,
    ! [A: nat,B2: nat] :
      ( ( semiri1314217659103216013at_int @ ( times_times_nat @ A @ B2 ) )
      = ( times_times_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ).

% int_ops(7)
thf(fact_5947_zle__iff__zadd,axiom,
    ( ord_less_eq_int
    = ( ^ [W2: int,Z5: int] :
        ? [N2: nat] :
          ( Z5
          = ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).

% zle_iff_zadd
thf(fact_5948_not__int__zless__negative,axiom,
    ! [N: nat,M: nat] :
      ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).

% not_int_zless_negative
thf(fact_5949_zdiv__int,axiom,
    ! [A: nat,B2: nat] :
      ( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ A @ B2 ) )
      = ( divide_divide_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ).

% zdiv_int
thf(fact_5950_zmod__int,axiom,
    ! [A: nat,B2: nat] :
      ( ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ A @ B2 ) )
      = ( modulo_modulo_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ).

% zmod_int
thf(fact_5951_not__bit__Suc__0__numeral,axiom,
    ! [N: num] :
      ~ ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ N ) ) ).

% not_bit_Suc_0_numeral
thf(fact_5952_floor__divide__of__nat__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( archim6058952711729229775r_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) )
      = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N ) ) ) ).

% floor_divide_of_nat_eq
thf(fact_5953_floor__divide__of__nat__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( archim3151403230148437115or_rat @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) )
      = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N ) ) ) ).

% floor_divide_of_nat_eq
thf(fact_5954_le__floor__iff,axiom,
    ! [Z: int,X4: real] :
      ( ( ord_less_eq_int @ Z @ ( archim6058952711729229775r_real @ X4 ) )
      = ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ X4 ) ) ).

% le_floor_iff
thf(fact_5955_le__floor__iff,axiom,
    ! [Z: int,X4: rat] :
      ( ( ord_less_eq_int @ Z @ ( archim3151403230148437115or_rat @ X4 ) )
      = ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ X4 ) ) ).

% le_floor_iff
thf(fact_5956_floor__less__iff,axiom,
    ! [X4: real,Z: int] :
      ( ( ord_less_int @ ( archim6058952711729229775r_real @ X4 ) @ Z )
      = ( ord_less_real @ X4 @ ( ring_1_of_int_real @ Z ) ) ) ).

% floor_less_iff
thf(fact_5957_floor__less__iff,axiom,
    ! [X4: rat,Z: int] :
      ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X4 ) @ Z )
      = ( ord_less_rat @ X4 @ ( ring_1_of_int_rat @ Z ) ) ) ).

% floor_less_iff
thf(fact_5958_le__floor__add,axiom,
    ! [X4: real,Y3: real] : ( ord_less_eq_int @ ( plus_plus_int @ ( archim6058952711729229775r_real @ X4 ) @ ( archim6058952711729229775r_real @ Y3 ) ) @ ( archim6058952711729229775r_real @ ( plus_plus_real @ X4 @ Y3 ) ) ) ).

% le_floor_add
thf(fact_5959_le__floor__add,axiom,
    ! [X4: rat,Y3: rat] : ( ord_less_eq_int @ ( plus_plus_int @ ( archim3151403230148437115or_rat @ X4 ) @ ( archim3151403230148437115or_rat @ Y3 ) ) @ ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X4 @ Y3 ) ) ) ).

% le_floor_add
thf(fact_5960_int__add__floor,axiom,
    ! [Z: int,X4: real] :
      ( ( plus_plus_int @ Z @ ( archim6058952711729229775r_real @ X4 ) )
      = ( archim6058952711729229775r_real @ ( plus_plus_real @ ( ring_1_of_int_real @ Z ) @ X4 ) ) ) ).

% int_add_floor
thf(fact_5961_int__add__floor,axiom,
    ! [Z: int,X4: rat] :
      ( ( plus_plus_int @ Z @ ( archim3151403230148437115or_rat @ X4 ) )
      = ( archim3151403230148437115or_rat @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z ) @ X4 ) ) ) ).

% int_add_floor
thf(fact_5962_floor__add__int,axiom,
    ! [X4: real,Z: int] :
      ( ( plus_plus_int @ ( archim6058952711729229775r_real @ X4 ) @ Z )
      = ( archim6058952711729229775r_real @ ( plus_plus_real @ X4 @ ( ring_1_of_int_real @ Z ) ) ) ) ).

% floor_add_int
thf(fact_5963_floor__add__int,axiom,
    ! [X4: rat,Z: int] :
      ( ( plus_plus_int @ ( archim3151403230148437115or_rat @ X4 ) @ Z )
      = ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X4 @ ( ring_1_of_int_rat @ Z ) ) ) ) ).

% floor_add_int
thf(fact_5964_floor__divide__of__int__eq,axiom,
    ! [K: int,L2: int] :
      ( ( archim6058952711729229775r_real @ ( divide_divide_real @ ( ring_1_of_int_real @ K ) @ ( ring_1_of_int_real @ L2 ) ) )
      = ( divide_divide_int @ K @ L2 ) ) ).

% floor_divide_of_int_eq
thf(fact_5965_floor__divide__of__int__eq,axiom,
    ! [K: int,L2: int] :
      ( ( archim3151403230148437115or_rat @ ( divide_divide_rat @ ( ring_1_of_int_rat @ K ) @ ( ring_1_of_int_rat @ L2 ) ) )
      = ( divide_divide_int @ K @ L2 ) ) ).

% floor_divide_of_int_eq
thf(fact_5966_floor__power,axiom,
    ! [X4: real,N: nat] :
      ( ( X4
        = ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X4 ) ) )
     => ( ( archim6058952711729229775r_real @ ( power_power_real @ X4 @ N ) )
        = ( power_power_int @ ( archim6058952711729229775r_real @ X4 ) @ N ) ) ) ).

% floor_power
thf(fact_5967_floor__power,axiom,
    ! [X4: rat,N: nat] :
      ( ( X4
        = ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X4 ) ) )
     => ( ( archim3151403230148437115or_rat @ ( power_power_rat @ X4 @ N ) )
        = ( power_power_int @ ( archim3151403230148437115or_rat @ X4 ) @ N ) ) ) ).

% floor_power
thf(fact_5968_ceiling__le,axiom,
    ! [X4: real,A: int] :
      ( ( ord_less_eq_real @ X4 @ ( ring_1_of_int_real @ A ) )
     => ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X4 ) @ A ) ) ).

% ceiling_le
thf(fact_5969_ceiling__le,axiom,
    ! [X4: rat,A: int] :
      ( ( ord_less_eq_rat @ X4 @ ( ring_1_of_int_rat @ A ) )
     => ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X4 ) @ A ) ) ).

% ceiling_le
thf(fact_5970_ceiling__le__iff,axiom,
    ! [X4: real,Z: int] :
      ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X4 ) @ Z )
      = ( ord_less_eq_real @ X4 @ ( ring_1_of_int_real @ Z ) ) ) ).

% ceiling_le_iff
thf(fact_5971_ceiling__le__iff,axiom,
    ! [X4: rat,Z: int] :
      ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X4 ) @ Z )
      = ( ord_less_eq_rat @ X4 @ ( ring_1_of_int_rat @ Z ) ) ) ).

% ceiling_le_iff
thf(fact_5972_less__ceiling__iff,axiom,
    ! [Z: int,X4: rat] :
      ( ( ord_less_int @ Z @ ( archim2889992004027027881ng_rat @ X4 ) )
      = ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ X4 ) ) ).

% less_ceiling_iff
thf(fact_5973_less__ceiling__iff,axiom,
    ! [Z: int,X4: real] :
      ( ( ord_less_int @ Z @ ( archim7802044766580827645g_real @ X4 ) )
      = ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ X4 ) ) ).

% less_ceiling_iff
thf(fact_5974_ceiling__add__le,axiom,
    ! [X4: rat,Y3: rat] : ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X4 @ Y3 ) ) @ ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X4 ) @ ( archim2889992004027027881ng_rat @ Y3 ) ) ) ).

% ceiling_add_le
thf(fact_5975_ceiling__add__le,axiom,
    ! [X4: real,Y3: real] : ( ord_less_eq_int @ ( archim7802044766580827645g_real @ ( plus_plus_real @ X4 @ Y3 ) ) @ ( plus_plus_int @ ( archim7802044766580827645g_real @ X4 ) @ ( archim7802044766580827645g_real @ Y3 ) ) ) ).

% ceiling_add_le
thf(fact_5976_pochhammer__nonneg,axiom,
    ! [X4: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ord_less_eq_real @ zero_zero_real @ ( comm_s7457072308508201937r_real @ X4 @ N ) ) ) ).

% pochhammer_nonneg
thf(fact_5977_pochhammer__nonneg,axiom,
    ! [X4: rat,N: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ X4 )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( comm_s4028243227959126397er_rat @ X4 @ N ) ) ) ).

% pochhammer_nonneg
thf(fact_5978_pochhammer__nonneg,axiom,
    ! [X4: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ X4 )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( comm_s4663373288045622133er_nat @ X4 @ N ) ) ) ).

% pochhammer_nonneg
thf(fact_5979_pochhammer__nonneg,axiom,
    ! [X4: int,N: nat] :
      ( ( ord_less_int @ zero_zero_int @ X4 )
     => ( ord_less_eq_int @ zero_zero_int @ ( comm_s4660882817536571857er_int @ X4 @ N ) ) ) ).

% pochhammer_nonneg
thf(fact_5980_int__cases4,axiom,
    ! [M: int] :
      ( ! [N4: nat] :
          ( M
         != ( semiri1314217659103216013at_int @ N4 ) )
     => ~ ! [N4: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N4 )
           => ( M
             != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N4 ) ) ) ) ) ).

% int_cases4
thf(fact_5981_pochhammer__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( comm_s2602460028002588243omplex @ zero_zero_complex @ N )
          = one_one_complex ) )
      & ( ( N != zero_zero_nat )
       => ( ( comm_s2602460028002588243omplex @ zero_zero_complex @ N )
          = zero_zero_complex ) ) ) ).

% pochhammer_0_left
thf(fact_5982_pochhammer__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( comm_s7457072308508201937r_real @ zero_zero_real @ N )
          = one_one_real ) )
      & ( ( N != zero_zero_nat )
       => ( ( comm_s7457072308508201937r_real @ zero_zero_real @ N )
          = zero_zero_real ) ) ) ).

% pochhammer_0_left
thf(fact_5983_pochhammer__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( comm_s4028243227959126397er_rat @ zero_zero_rat @ N )
          = one_one_rat ) )
      & ( ( N != zero_zero_nat )
       => ( ( comm_s4028243227959126397er_rat @ zero_zero_rat @ N )
          = zero_zero_rat ) ) ) ).

% pochhammer_0_left
thf(fact_5984_pochhammer__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( comm_s4663373288045622133er_nat @ zero_zero_nat @ N )
          = one_one_nat ) )
      & ( ( N != zero_zero_nat )
       => ( ( comm_s4663373288045622133er_nat @ zero_zero_nat @ N )
          = zero_zero_nat ) ) ) ).

% pochhammer_0_left
thf(fact_5985_pochhammer__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( comm_s4660882817536571857er_int @ zero_zero_int @ N )
          = one_one_int ) )
      & ( ( N != zero_zero_nat )
       => ( ( comm_s4660882817536571857er_int @ zero_zero_int @ N )
          = zero_zero_int ) ) ) ).

% pochhammer_0_left
thf(fact_5986_int__Suc,axiom,
    ! [N: nat] :
      ( ( semiri1314217659103216013at_int @ ( suc @ N ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ).

% int_Suc
thf(fact_5987_int__ops_I4_J,axiom,
    ! [A: nat] :
      ( ( semiri1314217659103216013at_int @ ( suc @ A ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ one_one_int ) ) ).

% int_ops(4)
thf(fact_5988_zless__iff__Suc__zadd,axiom,
    ( ord_less_int
    = ( ^ [W2: int,Z5: int] :
        ? [N2: nat] :
          ( Z5
          = ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ) ) ).

% zless_iff_Suc_zadd
thf(fact_5989_norm__exp,axiom,
    ! [X4: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( exp_real @ X4 ) ) @ ( exp_real @ ( real_V7735802525324610683m_real @ X4 ) ) ) ).

% norm_exp
thf(fact_5990_norm__exp,axiom,
    ! [X4: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( exp_complex @ X4 ) ) @ ( exp_real @ ( real_V1022390504157884413omplex @ X4 ) ) ) ).

% norm_exp
thf(fact_5991_one__add__floor,axiom,
    ! [X4: real] :
      ( ( plus_plus_int @ ( archim6058952711729229775r_real @ X4 ) @ one_one_int )
      = ( archim6058952711729229775r_real @ ( plus_plus_real @ X4 @ one_one_real ) ) ) ).

% one_add_floor
thf(fact_5992_one__add__floor,axiom,
    ! [X4: rat] :
      ( ( plus_plus_int @ ( archim3151403230148437115or_rat @ X4 ) @ one_one_int )
      = ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X4 @ one_one_rat ) ) ) ).

% one_add_floor
thf(fact_5993_of__int__ceiling__le__add__one,axiom,
    ! [R2: real] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ R2 ) ) @ ( plus_plus_real @ R2 @ one_one_real ) ) ).

% of_int_ceiling_le_add_one
thf(fact_5994_of__int__ceiling__le__add__one,axiom,
    ! [R2: rat] : ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ R2 ) ) @ ( plus_plus_rat @ R2 @ one_one_rat ) ) ).

% of_int_ceiling_le_add_one
thf(fact_5995_of__int__ceiling__diff__one__le,axiom,
    ! [R2: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ R2 ) ) @ one_one_real ) @ R2 ) ).

% of_int_ceiling_diff_one_le
thf(fact_5996_of__int__ceiling__diff__one__le,axiom,
    ! [R2: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ R2 ) ) @ one_one_rat ) @ R2 ) ).

% of_int_ceiling_diff_one_le
thf(fact_5997_floor__eq,axiom,
    ! [N: int,X4: real] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ N ) @ X4 )
     => ( ( ord_less_real @ X4 @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) )
       => ( ( archim6058952711729229775r_real @ X4 )
          = N ) ) ) ).

% floor_eq
thf(fact_5998_real__of__int__floor__add__one__gt,axiom,
    ! [R2: real] : ( ord_less_real @ R2 @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) @ one_one_real ) ) ).

% real_of_int_floor_add_one_gt
thf(fact_5999_real__of__int__floor__add__one__ge,axiom,
    ! [R2: real] : ( ord_less_eq_real @ R2 @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) @ one_one_real ) ) ).

% real_of_int_floor_add_one_ge
thf(fact_6000_real__of__int__floor__gt__diff__one,axiom,
    ! [R2: real] : ( ord_less_real @ ( minus_minus_real @ R2 @ one_one_real ) @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) ) ).

% real_of_int_floor_gt_diff_one
thf(fact_6001_real__of__int__floor__ge__diff__one,axiom,
    ! [R2: real] : ( ord_less_eq_real @ ( minus_minus_real @ R2 @ one_one_real ) @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) ) ).

% real_of_int_floor_ge_diff_one
thf(fact_6002_ceiling__divide__eq__div,axiom,
    ! [A: int,B2: int] :
      ( ( archim7802044766580827645g_real @ ( divide_divide_real @ ( ring_1_of_int_real @ A ) @ ( ring_1_of_int_real @ B2 ) ) )
      = ( uminus_uminus_int @ ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B2 ) ) ) ).

% ceiling_divide_eq_div
thf(fact_6003_ceiling__divide__eq__div,axiom,
    ! [A: int,B2: int] :
      ( ( archim2889992004027027881ng_rat @ ( divide_divide_rat @ ( ring_1_of_int_rat @ A ) @ ( ring_1_of_int_rat @ B2 ) ) )
      = ( uminus_uminus_int @ ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B2 ) ) ) ).

% ceiling_divide_eq_div
thf(fact_6004_zero__less__imp__eq__int,axiom,
    ! [K: int] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ? [N4: nat] :
          ( ( ord_less_nat @ zero_zero_nat @ N4 )
          & ( K
            = ( semiri1314217659103216013at_int @ N4 ) ) ) ) ).

% zero_less_imp_eq_int
thf(fact_6005_pos__int__cases,axiom,
    ! [K: int] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ~ ! [N4: nat] :
            ( ( K
              = ( semiri1314217659103216013at_int @ N4 ) )
           => ~ ( ord_less_nat @ zero_zero_nat @ N4 ) ) ) ).

% pos_int_cases
thf(fact_6006_int__cases3,axiom,
    ! [K: int] :
      ( ( K != zero_zero_int )
     => ( ! [N4: nat] :
            ( ( K
              = ( semiri1314217659103216013at_int @ N4 ) )
           => ~ ( ord_less_nat @ zero_zero_nat @ N4 ) )
       => ~ ! [N4: nat] :
              ( ( K
                = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N4 ) ) )
             => ~ ( ord_less_nat @ zero_zero_nat @ N4 ) ) ) ) ).

% int_cases3
thf(fact_6007_pi__less__4,axiom,
    ord_less_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ).

% pi_less_4
thf(fact_6008_pi__ge__two,axiom,
    ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ).

% pi_ge_two
thf(fact_6009_zmult__zless__mono2__lemma,axiom,
    ! [I: int,J: int,K: nat] :
      ( ( ord_less_int @ I @ J )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ord_less_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ I ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ J ) ) ) ) ).

% zmult_zless_mono2_lemma
thf(fact_6010_not__zle__0__negative,axiom,
    ! [N: nat] :
      ~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) ) ).

% not_zle_0_negative
thf(fact_6011_negative__zless__0,axiom,
    ! [N: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ zero_zero_int ) ).

% negative_zless_0
thf(fact_6012_negD,axiom,
    ! [X4: int] :
      ( ( ord_less_int @ X4 @ zero_zero_int )
     => ? [N4: nat] :
          ( X4
          = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N4 ) ) ) ) ) ).

% negD
thf(fact_6013_pi__half__neq__two,axiom,
    ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
   != ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).

% pi_half_neq_two
thf(fact_6014_pochhammer__rec,axiom,
    ! [A: complex,N: nat] :
      ( ( comm_s2602460028002588243omplex @ A @ ( suc @ N ) )
      = ( times_times_complex @ A @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ A @ one_one_complex ) @ N ) ) ) ).

% pochhammer_rec
thf(fact_6015_pochhammer__rec,axiom,
    ! [A: real,N: nat] :
      ( ( comm_s7457072308508201937r_real @ A @ ( suc @ N ) )
      = ( times_times_real @ A @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ A @ one_one_real ) @ N ) ) ) ).

% pochhammer_rec
thf(fact_6016_pochhammer__rec,axiom,
    ! [A: rat,N: nat] :
      ( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N ) )
      = ( times_times_rat @ A @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ N ) ) ) ).

% pochhammer_rec
thf(fact_6017_pochhammer__rec,axiom,
    ! [A: nat,N: nat] :
      ( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N ) )
      = ( times_times_nat @ A @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ N ) ) ) ).

% pochhammer_rec
thf(fact_6018_pochhammer__rec,axiom,
    ! [A: int,N: nat] :
      ( ( comm_s4660882817536571857er_int @ A @ ( suc @ N ) )
      = ( times_times_int @ A @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ A @ one_one_int ) @ N ) ) ) ).

% pochhammer_rec
thf(fact_6019_pochhammer__Suc,axiom,
    ! [A: complex,N: nat] :
      ( ( comm_s2602460028002588243omplex @ A @ ( suc @ N ) )
      = ( times_times_complex @ ( comm_s2602460028002588243omplex @ A @ N ) @ ( plus_plus_complex @ A @ ( semiri8010041392384452111omplex @ N ) ) ) ) ).

% pochhammer_Suc
thf(fact_6020_pochhammer__Suc,axiom,
    ! [A: real,N: nat] :
      ( ( comm_s7457072308508201937r_real @ A @ ( suc @ N ) )
      = ( times_times_real @ ( comm_s7457072308508201937r_real @ A @ N ) @ ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).

% pochhammer_Suc
thf(fact_6021_pochhammer__Suc,axiom,
    ! [A: rat,N: nat] :
      ( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N ) )
      = ( times_times_rat @ ( comm_s4028243227959126397er_rat @ A @ N ) @ ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ N ) ) ) ) ).

% pochhammer_Suc
thf(fact_6022_pochhammer__Suc,axiom,
    ! [A: nat,N: nat] :
      ( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N ) )
      = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ A @ N ) @ ( plus_plus_nat @ A @ ( semiri1316708129612266289at_nat @ N ) ) ) ) ).

% pochhammer_Suc
thf(fact_6023_pochhammer__Suc,axiom,
    ! [A: int,N: nat] :
      ( ( comm_s4660882817536571857er_int @ A @ ( suc @ N ) )
      = ( times_times_int @ ( comm_s4660882817536571857er_int @ A @ N ) @ ( plus_plus_int @ A @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).

% pochhammer_Suc
thf(fact_6024_pochhammer__rec_H,axiom,
    ! [Z: complex,N: nat] :
      ( ( comm_s2602460028002588243omplex @ Z @ ( suc @ N ) )
      = ( times_times_complex @ ( plus_plus_complex @ Z @ ( semiri8010041392384452111omplex @ N ) ) @ ( comm_s2602460028002588243omplex @ Z @ N ) ) ) ).

% pochhammer_rec'
thf(fact_6025_pochhammer__rec_H,axiom,
    ! [Z: real,N: nat] :
      ( ( comm_s7457072308508201937r_real @ Z @ ( suc @ N ) )
      = ( times_times_real @ ( plus_plus_real @ Z @ ( semiri5074537144036343181t_real @ N ) ) @ ( comm_s7457072308508201937r_real @ Z @ N ) ) ) ).

% pochhammer_rec'
thf(fact_6026_pochhammer__rec_H,axiom,
    ! [Z: rat,N: nat] :
      ( ( comm_s4028243227959126397er_rat @ Z @ ( suc @ N ) )
      = ( times_times_rat @ ( plus_plus_rat @ Z @ ( semiri681578069525770553at_rat @ N ) ) @ ( comm_s4028243227959126397er_rat @ Z @ N ) ) ) ).

% pochhammer_rec'
thf(fact_6027_pochhammer__rec_H,axiom,
    ! [Z: nat,N: nat] :
      ( ( comm_s4663373288045622133er_nat @ Z @ ( suc @ N ) )
      = ( times_times_nat @ ( plus_plus_nat @ Z @ ( semiri1316708129612266289at_nat @ N ) ) @ ( comm_s4663373288045622133er_nat @ Z @ N ) ) ) ).

% pochhammer_rec'
thf(fact_6028_pochhammer__rec_H,axiom,
    ! [Z: int,N: nat] :
      ( ( comm_s4660882817536571857er_int @ Z @ ( suc @ N ) )
      = ( times_times_int @ ( plus_plus_int @ Z @ ( semiri1314217659103216013at_int @ N ) ) @ ( comm_s4660882817536571857er_int @ Z @ N ) ) ) ).

% pochhammer_rec'
thf(fact_6029_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N: nat,K: nat] :
      ( ( ord_less_nat @ N @ K )
     => ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N ) ) @ K )
        = zero_z3403309356797280102nteger ) ) ).

% pochhammer_of_nat_eq_0_lemma
thf(fact_6030_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N: nat,K: nat] :
      ( ( ord_less_nat @ N @ K )
     => ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ K )
        = zero_zero_complex ) ) ).

% pochhammer_of_nat_eq_0_lemma
thf(fact_6031_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N: nat,K: nat] :
      ( ( ord_less_nat @ N @ K )
     => ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ K )
        = zero_zero_real ) ) ).

% pochhammer_of_nat_eq_0_lemma
thf(fact_6032_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N: nat,K: nat] :
      ( ( ord_less_nat @ N @ K )
     => ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ K )
        = zero_zero_rat ) ) ).

% pochhammer_of_nat_eq_0_lemma
thf(fact_6033_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N: nat,K: nat] :
      ( ( ord_less_nat @ N @ K )
     => ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ K )
        = zero_zero_int ) ) ).

% pochhammer_of_nat_eq_0_lemma
thf(fact_6034_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N: nat,K: nat] :
      ( ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N ) ) @ K )
        = zero_z3403309356797280102nteger )
      = ( ord_less_nat @ N @ K ) ) ).

% pochhammer_of_nat_eq_0_iff
thf(fact_6035_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N: nat,K: nat] :
      ( ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ K )
        = zero_zero_complex )
      = ( ord_less_nat @ N @ K ) ) ).

% pochhammer_of_nat_eq_0_iff
thf(fact_6036_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N: nat,K: nat] :
      ( ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ K )
        = zero_zero_real )
      = ( ord_less_nat @ N @ K ) ) ).

% pochhammer_of_nat_eq_0_iff
thf(fact_6037_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N: nat,K: nat] :
      ( ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ K )
        = zero_zero_rat )
      = ( ord_less_nat @ N @ K ) ) ).

% pochhammer_of_nat_eq_0_iff
thf(fact_6038_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N: nat,K: nat] :
      ( ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ K )
        = zero_zero_int )
      = ( ord_less_nat @ N @ K ) ) ).

% pochhammer_of_nat_eq_0_iff
thf(fact_6039_pochhammer__eq__0__iff,axiom,
    ! [A: complex,N: nat] :
      ( ( ( comm_s2602460028002588243omplex @ A @ N )
        = zero_zero_complex )
      = ( ? [K3: nat] :
            ( ( ord_less_nat @ K3 @ N )
            & ( A
              = ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ K3 ) ) ) ) ) ) ).

% pochhammer_eq_0_iff
thf(fact_6040_pochhammer__eq__0__iff,axiom,
    ! [A: real,N: nat] :
      ( ( ( comm_s7457072308508201937r_real @ A @ N )
        = zero_zero_real )
      = ( ? [K3: nat] :
            ( ( ord_less_nat @ K3 @ N )
            & ( A
              = ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ K3 ) ) ) ) ) ) ).

% pochhammer_eq_0_iff
thf(fact_6041_pochhammer__eq__0__iff,axiom,
    ! [A: rat,N: nat] :
      ( ( ( comm_s4028243227959126397er_rat @ A @ N )
        = zero_zero_rat )
      = ( ? [K3: nat] :
            ( ( ord_less_nat @ K3 @ N )
            & ( A
              = ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ K3 ) ) ) ) ) ) ).

% pochhammer_eq_0_iff
thf(fact_6042_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N ) ) @ K )
       != zero_z3403309356797280102nteger ) ) ).

% pochhammer_of_nat_eq_0_lemma'
thf(fact_6043_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ K )
       != zero_zero_complex ) ) ).

% pochhammer_of_nat_eq_0_lemma'
thf(fact_6044_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ K )
       != zero_zero_real ) ) ).

% pochhammer_of_nat_eq_0_lemma'
thf(fact_6045_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ K )
       != zero_zero_rat ) ) ).

% pochhammer_of_nat_eq_0_lemma'
thf(fact_6046_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ K )
       != zero_zero_int ) ) ).

% pochhammer_of_nat_eq_0_lemma'
thf(fact_6047_int__ops_I6_J,axiom,
    ! [A: nat,B2: nat] :
      ( ( ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B2 ) )
       => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B2 ) )
          = zero_zero_int ) )
      & ( ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B2 ) )
       => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B2 ) )
          = ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ) ).

% int_ops(6)
thf(fact_6048_pochhammer__product_H,axiom,
    ! [Z: complex,N: nat,M: nat] :
      ( ( comm_s2602460028002588243omplex @ Z @ ( plus_plus_nat @ N @ M ) )
      = ( times_times_complex @ ( comm_s2602460028002588243omplex @ Z @ N ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z @ ( semiri8010041392384452111omplex @ N ) ) @ M ) ) ) ).

% pochhammer_product'
thf(fact_6049_pochhammer__product_H,axiom,
    ! [Z: real,N: nat,M: nat] :
      ( ( comm_s7457072308508201937r_real @ Z @ ( plus_plus_nat @ N @ M ) )
      = ( times_times_real @ ( comm_s7457072308508201937r_real @ Z @ N ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z @ ( semiri5074537144036343181t_real @ N ) ) @ M ) ) ) ).

% pochhammer_product'
thf(fact_6050_pochhammer__product_H,axiom,
    ! [Z: rat,N: nat,M: nat] :
      ( ( comm_s4028243227959126397er_rat @ Z @ ( plus_plus_nat @ N @ M ) )
      = ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z @ N ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z @ ( semiri681578069525770553at_rat @ N ) ) @ M ) ) ) ).

% pochhammer_product'
thf(fact_6051_pochhammer__product_H,axiom,
    ! [Z: nat,N: nat,M: nat] :
      ( ( comm_s4663373288045622133er_nat @ Z @ ( plus_plus_nat @ N @ M ) )
      = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ Z @ N ) @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ Z @ ( semiri1316708129612266289at_nat @ N ) ) @ M ) ) ) ).

% pochhammer_product'
thf(fact_6052_pochhammer__product_H,axiom,
    ! [Z: int,N: nat,M: nat] :
      ( ( comm_s4660882817536571857er_int @ Z @ ( plus_plus_nat @ N @ M ) )
      = ( times_times_int @ ( comm_s4660882817536571857er_int @ Z @ N ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ Z @ ( semiri1314217659103216013at_int @ N ) ) @ M ) ) ) ).

% pochhammer_product'
thf(fact_6053_floor__split,axiom,
    ! [P: int > $o,T: real] :
      ( ( P @ ( archim6058952711729229775r_real @ T ) )
      = ( ! [I2: int] :
            ( ( ( ord_less_eq_real @ ( ring_1_of_int_real @ I2 ) @ T )
              & ( ord_less_real @ T @ ( plus_plus_real @ ( ring_1_of_int_real @ I2 ) @ one_one_real ) ) )
           => ( P @ I2 ) ) ) ) ).

% floor_split
thf(fact_6054_floor__split,axiom,
    ! [P: int > $o,T: rat] :
      ( ( P @ ( archim3151403230148437115or_rat @ T ) )
      = ( ! [I2: int] :
            ( ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ I2 ) @ T )
              & ( ord_less_rat @ T @ ( plus_plus_rat @ ( ring_1_of_int_rat @ I2 ) @ one_one_rat ) ) )
           => ( P @ I2 ) ) ) ) ).

% floor_split
thf(fact_6055_floor__eq__iff,axiom,
    ! [X4: real,A: int] :
      ( ( ( archim6058952711729229775r_real @ X4 )
        = A )
      = ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ X4 )
        & ( ord_less_real @ X4 @ ( plus_plus_real @ ( ring_1_of_int_real @ A ) @ one_one_real ) ) ) ) ).

% floor_eq_iff
thf(fact_6056_floor__eq__iff,axiom,
    ! [X4: rat,A: int] :
      ( ( ( archim3151403230148437115or_rat @ X4 )
        = A )
      = ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A ) @ X4 )
        & ( ord_less_rat @ X4 @ ( plus_plus_rat @ ( ring_1_of_int_rat @ A ) @ one_one_rat ) ) ) ) ).

% floor_eq_iff
thf(fact_6057_floor__unique,axiom,
    ! [Z: int,X4: real] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ X4 )
     => ( ( ord_less_real @ X4 @ ( plus_plus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) )
       => ( ( archim6058952711729229775r_real @ X4 )
          = Z ) ) ) ).

% floor_unique
thf(fact_6058_floor__unique,axiom,
    ! [Z: int,X4: rat] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ X4 )
     => ( ( ord_less_rat @ X4 @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) )
       => ( ( archim3151403230148437115or_rat @ X4 )
          = Z ) ) ) ).

% floor_unique
thf(fact_6059_le__mult__floor,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
       => ( ord_less_eq_int @ ( times_times_int @ ( archim6058952711729229775r_real @ A ) @ ( archim6058952711729229775r_real @ B2 ) ) @ ( archim6058952711729229775r_real @ ( times_times_real @ A @ B2 ) ) ) ) ) ).

% le_mult_floor
thf(fact_6060_le__mult__floor,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
       => ( ord_less_eq_int @ ( times_times_int @ ( archim3151403230148437115or_rat @ A ) @ ( archim3151403230148437115or_rat @ B2 ) ) @ ( archim3151403230148437115or_rat @ ( times_times_rat @ A @ B2 ) ) ) ) ) ).

% le_mult_floor
thf(fact_6061_less__floor__iff,axiom,
    ! [Z: int,X4: real] :
      ( ( ord_less_int @ Z @ ( archim6058952711729229775r_real @ X4 ) )
      = ( ord_less_eq_real @ ( plus_plus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) @ X4 ) ) ).

% less_floor_iff
thf(fact_6062_less__floor__iff,axiom,
    ! [Z: int,X4: rat] :
      ( ( ord_less_int @ Z @ ( archim3151403230148437115or_rat @ X4 ) )
      = ( ord_less_eq_rat @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) @ X4 ) ) ).

% less_floor_iff
thf(fact_6063_floor__le__iff,axiom,
    ! [X4: real,Z: int] :
      ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X4 ) @ Z )
      = ( ord_less_real @ X4 @ ( plus_plus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) ) ) ).

% floor_le_iff
thf(fact_6064_floor__le__iff,axiom,
    ! [X4: rat,Z: int] :
      ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X4 ) @ Z )
      = ( ord_less_rat @ X4 @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) ) ) ).

% floor_le_iff
thf(fact_6065_binomial__mono,axiom,
    ! [K: nat,K6: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ K6 )
     => ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K6 ) @ N )
       => ( ord_less_eq_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ K6 ) ) ) ) ).

% binomial_mono
thf(fact_6066_binomial__maximum_H,axiom,
    ! [N: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ K ) @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ N ) ) ).

% binomial_maximum'
thf(fact_6067_binomial__maximum,axiom,
    ! [N: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% binomial_maximum
thf(fact_6068_binomial__antimono,axiom,
    ! [K: nat,K6: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ K6 )
     => ( ( ord_less_eq_nat @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ K )
       => ( ( ord_less_eq_nat @ K6 @ N )
         => ( ord_less_eq_nat @ ( binomial @ N @ K6 ) @ ( binomial @ N @ K ) ) ) ) ) ).

% binomial_antimono
thf(fact_6069_floor__correct,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X4 ) ) @ X4 )
      & ( ord_less_real @ X4 @ ( ring_1_of_int_real @ ( plus_plus_int @ ( archim6058952711729229775r_real @ X4 ) @ one_one_int ) ) ) ) ).

% floor_correct
thf(fact_6070_floor__correct,axiom,
    ! [X4: rat] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X4 ) ) @ X4 )
      & ( ord_less_rat @ X4 @ ( ring_1_of_int_rat @ ( plus_plus_int @ ( archim3151403230148437115or_rat @ X4 ) @ one_one_int ) ) ) ) ).

% floor_correct
thf(fact_6071_ceiling__split,axiom,
    ! [P: int > $o,T: real] :
      ( ( P @ ( archim7802044766580827645g_real @ T ) )
      = ( ! [I2: int] :
            ( ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ I2 ) @ one_one_real ) @ T )
              & ( ord_less_eq_real @ T @ ( ring_1_of_int_real @ I2 ) ) )
           => ( P @ I2 ) ) ) ) ).

% ceiling_split
thf(fact_6072_ceiling__split,axiom,
    ! [P: int > $o,T: rat] :
      ( ( P @ ( archim2889992004027027881ng_rat @ T ) )
      = ( ! [I2: int] :
            ( ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ I2 ) @ one_one_rat ) @ T )
              & ( ord_less_eq_rat @ T @ ( ring_1_of_int_rat @ I2 ) ) )
           => ( P @ I2 ) ) ) ) ).

% ceiling_split
thf(fact_6073_ceiling__eq__iff,axiom,
    ! [X4: real,A: int] :
      ( ( ( archim7802044766580827645g_real @ X4 )
        = A )
      = ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ A ) @ one_one_real ) @ X4 )
        & ( ord_less_eq_real @ X4 @ ( ring_1_of_int_real @ A ) ) ) ) ).

% ceiling_eq_iff
thf(fact_6074_ceiling__eq__iff,axiom,
    ! [X4: rat,A: int] :
      ( ( ( archim2889992004027027881ng_rat @ X4 )
        = A )
      = ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ A ) @ one_one_rat ) @ X4 )
        & ( ord_less_eq_rat @ X4 @ ( ring_1_of_int_rat @ A ) ) ) ) ).

% ceiling_eq_iff
thf(fact_6075_ceiling__unique,axiom,
    ! [Z: int,X4: real] :
      ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) @ X4 )
     => ( ( ord_less_eq_real @ X4 @ ( ring_1_of_int_real @ Z ) )
       => ( ( archim7802044766580827645g_real @ X4 )
          = Z ) ) ) ).

% ceiling_unique
thf(fact_6076_ceiling__unique,axiom,
    ! [Z: int,X4: rat] :
      ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) @ X4 )
     => ( ( ord_less_eq_rat @ X4 @ ( ring_1_of_int_rat @ Z ) )
       => ( ( archim2889992004027027881ng_rat @ X4 )
          = Z ) ) ) ).

% ceiling_unique
thf(fact_6077_ceiling__correct,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X4 ) ) @ one_one_real ) @ X4 )
      & ( ord_less_eq_real @ X4 @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X4 ) ) ) ) ).

% ceiling_correct
thf(fact_6078_ceiling__correct,axiom,
    ! [X4: rat] :
      ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X4 ) ) @ one_one_rat ) @ X4 )
      & ( ord_less_eq_rat @ X4 @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X4 ) ) ) ) ).

% ceiling_correct
thf(fact_6079_mult__ceiling__le,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
       => ( ord_less_eq_int @ ( archim7802044766580827645g_real @ ( times_times_real @ A @ B2 ) ) @ ( times_times_int @ ( archim7802044766580827645g_real @ A ) @ ( archim7802044766580827645g_real @ B2 ) ) ) ) ) ).

% mult_ceiling_le
thf(fact_6080_mult__ceiling__le,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
       => ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ ( times_times_rat @ A @ B2 ) ) @ ( times_times_int @ ( archim2889992004027027881ng_rat @ A ) @ ( archim2889992004027027881ng_rat @ B2 ) ) ) ) ) ).

% mult_ceiling_le
thf(fact_6081_floor__log__nat__eq__if,axiom,
    ! [B2: nat,N: nat,K: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ B2 @ N ) @ K )
     => ( ( ord_less_nat @ K @ ( power_power_nat @ B2 @ ( plus_plus_nat @ N @ one_one_nat ) ) )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 )
         => ( ( archim6058952711729229775r_real @ ( log @ ( semiri5074537144036343181t_real @ B2 ) @ ( semiri5074537144036343181t_real @ K ) ) )
            = ( semiri1314217659103216013at_int @ N ) ) ) ) ) ).

% floor_log_nat_eq_if
thf(fact_6082_ceiling__less__iff,axiom,
    ! [X4: real,Z: int] :
      ( ( ord_less_int @ ( archim7802044766580827645g_real @ X4 ) @ Z )
      = ( ord_less_eq_real @ X4 @ ( minus_minus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) ) ) ).

% ceiling_less_iff
thf(fact_6083_ceiling__less__iff,axiom,
    ! [X4: rat,Z: int] :
      ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X4 ) @ Z )
      = ( ord_less_eq_rat @ X4 @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) ) ) ).

% ceiling_less_iff
thf(fact_6084_le__ceiling__iff,axiom,
    ! [Z: int,X4: rat] :
      ( ( ord_less_eq_int @ Z @ ( archim2889992004027027881ng_rat @ X4 ) )
      = ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) @ X4 ) ) ).

% le_ceiling_iff
thf(fact_6085_le__ceiling__iff,axiom,
    ! [Z: int,X4: real] :
      ( ( ord_less_eq_int @ Z @ ( archim7802044766580827645g_real @ X4 ) )
      = ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) @ X4 ) ) ).

% le_ceiling_iff
thf(fact_6086_floor__eq2,axiom,
    ! [N: int,X4: real] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ N ) @ X4 )
     => ( ( ord_less_real @ X4 @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) )
       => ( ( archim6058952711729229775r_real @ X4 )
          = N ) ) ) ).

% floor_eq2
thf(fact_6087_floor__divide__real__eq__div,axiom,
    ! [B2: int,A: real] :
      ( ( ord_less_eq_int @ zero_zero_int @ B2 )
     => ( ( archim6058952711729229775r_real @ ( divide_divide_real @ A @ ( ring_1_of_int_real @ B2 ) ) )
        = ( divide_divide_int @ ( archim6058952711729229775r_real @ A ) @ B2 ) ) ) ).

% floor_divide_real_eq_div
thf(fact_6088_neg__int__cases,axiom,
    ! [K: int] :
      ( ( ord_less_int @ K @ zero_zero_int )
     => ~ ! [N4: nat] :
            ( ( K
              = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N4 ) ) )
           => ~ ( ord_less_nat @ zero_zero_nat @ N4 ) ) ) ).

% neg_int_cases
thf(fact_6089_pi__half__neq__zero,axiom,
    ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
   != zero_zero_real ) ).

% pi_half_neq_zero
thf(fact_6090_pi__half__less__two,axiom,
    ord_less_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).

% pi_half_less_two
thf(fact_6091_pi__half__le__two,axiom,
    ord_less_eq_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).

% pi_half_le_two
thf(fact_6092_zdiff__int__split,axiom,
    ! [P: int > $o,X4: nat,Y3: nat] :
      ( ( P @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X4 @ Y3 ) ) )
      = ( ( ( ord_less_eq_nat @ Y3 @ X4 )
         => ( P @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X4 ) @ ( semiri1314217659103216013at_int @ Y3 ) ) ) )
        & ( ( ord_less_nat @ X4 @ Y3 )
         => ( P @ zero_zero_int ) ) ) ) ).

% zdiff_int_split
thf(fact_6093_floor__log__nat__eq__powr__iff,axiom,
    ! [B2: nat,K: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ( ( archim6058952711729229775r_real @ ( log @ ( semiri5074537144036343181t_real @ B2 ) @ ( semiri5074537144036343181t_real @ K ) ) )
            = ( semiri1314217659103216013at_int @ N ) )
          = ( ( ord_less_eq_nat @ ( power_power_nat @ B2 @ N ) @ K )
            & ( ord_less_nat @ K @ ( power_power_nat @ B2 @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) ) ) ) ).

% floor_log_nat_eq_powr_iff
thf(fact_6094_pochhammer__product,axiom,
    ! [M: nat,N: nat,Z: complex] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( comm_s2602460028002588243omplex @ Z @ N )
        = ( times_times_complex @ ( comm_s2602460028002588243omplex @ Z @ M ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z @ ( semiri8010041392384452111omplex @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).

% pochhammer_product
thf(fact_6095_pochhammer__product,axiom,
    ! [M: nat,N: nat,Z: real] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( comm_s7457072308508201937r_real @ Z @ N )
        = ( times_times_real @ ( comm_s7457072308508201937r_real @ Z @ M ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z @ ( semiri5074537144036343181t_real @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).

% pochhammer_product
thf(fact_6096_pochhammer__product,axiom,
    ! [M: nat,N: nat,Z: rat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( comm_s4028243227959126397er_rat @ Z @ N )
        = ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z @ M ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z @ ( semiri681578069525770553at_rat @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).

% pochhammer_product
thf(fact_6097_pochhammer__product,axiom,
    ! [M: nat,N: nat,Z: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( comm_s4663373288045622133er_nat @ Z @ N )
        = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ Z @ M ) @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ Z @ ( semiri1316708129612266289at_nat @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).

% pochhammer_product
thf(fact_6098_pochhammer__product,axiom,
    ! [M: nat,N: nat,Z: int] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( comm_s4660882817536571857er_int @ Z @ N )
        = ( times_times_int @ ( comm_s4660882817536571857er_int @ Z @ M ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ Z @ ( semiri1314217659103216013at_int @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).

% pochhammer_product
thf(fact_6099_floor__divide__lower,axiom,
    ! [Q2: real,P2: real] :
      ( ( ord_less_real @ zero_zero_real @ Q2 )
     => ( ord_less_eq_real @ ( times_times_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ ( divide_divide_real @ P2 @ Q2 ) ) ) @ Q2 ) @ P2 ) ) ).

% floor_divide_lower
thf(fact_6100_floor__divide__lower,axiom,
    ! [Q2: rat,P2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Q2 )
     => ( ord_less_eq_rat @ ( times_times_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ ( divide_divide_rat @ P2 @ Q2 ) ) ) @ Q2 ) @ P2 ) ) ).

% floor_divide_lower
thf(fact_6101_binomial__less__binomial__Suc,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_nat @ K @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ord_less_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( suc @ K ) ) ) ) ).

% binomial_less_binomial_Suc
thf(fact_6102_binomial__strict__mono,axiom,
    ! [K: nat,K6: nat,N: nat] :
      ( ( ord_less_nat @ K @ K6 )
     => ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K6 ) @ N )
       => ( ord_less_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ K6 ) ) ) ) ).

% binomial_strict_mono
thf(fact_6103_binomial__strict__antimono,axiom,
    ! [K: nat,K6: nat,N: nat] :
      ( ( ord_less_nat @ K @ K6 )
     => ( ( ord_less_eq_nat @ N @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K ) )
       => ( ( ord_less_eq_nat @ K6 @ N )
         => ( ord_less_nat @ ( binomial @ N @ K6 ) @ ( binomial @ N @ K ) ) ) ) ) ).

% binomial_strict_antimono
thf(fact_6104_central__binomial__odd,axiom,
    ! [N: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( binomial @ N @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
        = ( binomial @ N @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% central_binomial_odd
thf(fact_6105_ceiling__divide__upper,axiom,
    ! [Q2: real,P2: real] :
      ( ( ord_less_real @ zero_zero_real @ Q2 )
     => ( ord_less_eq_real @ P2 @ ( times_times_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ ( divide_divide_real @ P2 @ Q2 ) ) ) @ Q2 ) ) ) ).

% ceiling_divide_upper
thf(fact_6106_ceiling__divide__upper,axiom,
    ! [Q2: rat,P2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Q2 )
     => ( ord_less_eq_rat @ P2 @ ( times_times_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ ( divide_divide_rat @ P2 @ Q2 ) ) ) @ Q2 ) ) ) ).

% ceiling_divide_upper
thf(fact_6107_bit__nat__def,axiom,
    ( bit_se1148574629649215175it_nat
    = ( ^ [M6: nat,N2: nat] :
          ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ M6 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).

% bit_nat_def
thf(fact_6108_ceiling__log__nat__eq__if,axiom,
    ! [B2: nat,N: nat,K: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ B2 @ N ) @ K )
     => ( ( ord_less_eq_nat @ K @ ( power_power_nat @ B2 @ ( plus_plus_nat @ N @ one_one_nat ) ) )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 )
         => ( ( archim7802044766580827645g_real @ ( log @ ( semiri5074537144036343181t_real @ B2 ) @ ( semiri5074537144036343181t_real @ K ) ) )
            = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ) ) ) ).

% ceiling_log_nat_eq_if
thf(fact_6109_pi__half__gt__zero,axiom,
    ord_less_real @ zero_zero_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).

% pi_half_gt_zero
thf(fact_6110_pi__half__ge__zero,axiom,
    ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).

% pi_half_ge_zero
thf(fact_6111_pochhammer__absorb__comp,axiom,
    ! [R2: code_integer,K: nat] :
      ( ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ R2 @ ( semiri4939895301339042750nteger @ K ) ) @ ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ R2 ) @ K ) )
      = ( times_3573771949741848930nteger @ R2 @ ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ R2 ) @ one_one_Code_integer ) @ K ) ) ) ).

% pochhammer_absorb_comp
thf(fact_6112_pochhammer__absorb__comp,axiom,
    ! [R2: complex,K: nat] :
      ( ( times_times_complex @ ( minus_minus_complex @ R2 @ ( semiri8010041392384452111omplex @ K ) ) @ ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ R2 ) @ K ) )
      = ( times_times_complex @ R2 @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ R2 ) @ one_one_complex ) @ K ) ) ) ).

% pochhammer_absorb_comp
thf(fact_6113_pochhammer__absorb__comp,axiom,
    ! [R2: real,K: nat] :
      ( ( times_times_real @ ( minus_minus_real @ R2 @ ( semiri5074537144036343181t_real @ K ) ) @ ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ R2 ) @ K ) )
      = ( times_times_real @ R2 @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( uminus_uminus_real @ R2 ) @ one_one_real ) @ K ) ) ) ).

% pochhammer_absorb_comp
thf(fact_6114_pochhammer__absorb__comp,axiom,
    ! [R2: rat,K: nat] :
      ( ( times_times_rat @ ( minus_minus_rat @ R2 @ ( semiri681578069525770553at_rat @ K ) ) @ ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ R2 ) @ K ) )
      = ( times_times_rat @ R2 @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ R2 ) @ one_one_rat ) @ K ) ) ) ).

% pochhammer_absorb_comp
thf(fact_6115_pochhammer__absorb__comp,axiom,
    ! [R2: int,K: nat] :
      ( ( times_times_int @ ( minus_minus_int @ R2 @ ( semiri1314217659103216013at_int @ K ) ) @ ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ R2 ) @ K ) )
      = ( times_times_int @ R2 @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( uminus_uminus_int @ R2 ) @ one_one_int ) @ K ) ) ) ).

% pochhammer_absorb_comp
thf(fact_6116_m2pi__less__pi,axiom,
    ord_less_real @ ( uminus_uminus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) @ pi ).

% m2pi_less_pi
thf(fact_6117_ceiling__log__nat__eq__powr__iff,axiom,
    ! [B2: nat,K: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ( ( archim7802044766580827645g_real @ ( log @ ( semiri5074537144036343181t_real @ B2 ) @ ( semiri5074537144036343181t_real @ K ) ) )
            = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) )
          = ( ( ord_less_nat @ ( power_power_nat @ B2 @ N ) @ K )
            & ( ord_less_eq_nat @ K @ ( power_power_nat @ B2 @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) ) ) ) ).

% ceiling_log_nat_eq_powr_iff
thf(fact_6118_floor__divide__upper,axiom,
    ! [Q2: real,P2: real] :
      ( ( ord_less_real @ zero_zero_real @ Q2 )
     => ( ord_less_real @ P2 @ ( times_times_real @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ ( divide_divide_real @ P2 @ Q2 ) ) ) @ one_one_real ) @ Q2 ) ) ) ).

% floor_divide_upper
thf(fact_6119_floor__divide__upper,axiom,
    ! [Q2: rat,P2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Q2 )
     => ( ord_less_rat @ P2 @ ( times_times_rat @ ( plus_plus_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ ( divide_divide_rat @ P2 @ Q2 ) ) ) @ one_one_rat ) @ Q2 ) ) ) ).

% floor_divide_upper
thf(fact_6120_arctan__ubound,axiom,
    ! [Y3: real] : ( ord_less_real @ ( arctan @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% arctan_ubound
thf(fact_6121_arctan__one,axiom,
    ( ( arctan @ one_one_real )
    = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ).

% arctan_one
thf(fact_6122_round__def,axiom,
    ( archim8280529875227126926d_real
    = ( ^ [X: real] : ( archim6058952711729229775r_real @ ( plus_plus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).

% round_def
thf(fact_6123_round__def,axiom,
    ( archim7778729529865785530nd_rat
    = ( ^ [X: rat] : ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% round_def
thf(fact_6124_ceiling__divide__lower,axiom,
    ! [Q2: real,P2: real] :
      ( ( ord_less_real @ zero_zero_real @ Q2 )
     => ( ord_less_real @ ( times_times_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ ( divide_divide_real @ P2 @ Q2 ) ) ) @ one_one_real ) @ Q2 ) @ P2 ) ) ).

% ceiling_divide_lower
thf(fact_6125_ceiling__divide__lower,axiom,
    ! [Q2: rat,P2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Q2 )
     => ( ord_less_rat @ ( times_times_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ ( divide_divide_rat @ P2 @ Q2 ) ) ) @ one_one_rat ) @ Q2 ) @ P2 ) ) ).

% ceiling_divide_lower
thf(fact_6126_ceiling__eq,axiom,
    ! [N: int,X4: real] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ N ) @ X4 )
     => ( ( ord_less_eq_real @ X4 @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) )
       => ( ( archim7802044766580827645g_real @ X4 )
          = ( plus_plus_int @ N @ one_one_int ) ) ) ) ).

% ceiling_eq
thf(fact_6127_ceiling__eq,axiom,
    ! [N: int,X4: rat] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ N ) @ X4 )
     => ( ( ord_less_eq_rat @ X4 @ ( plus_plus_rat @ ( ring_1_of_int_rat @ N ) @ one_one_rat ) )
       => ( ( archim2889992004027027881ng_rat @ X4 )
          = ( plus_plus_int @ N @ one_one_int ) ) ) ) ).

% ceiling_eq
thf(fact_6128_exp__bound__half,axiom,
    ! [Z: real] :
      ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
     => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( exp_real @ Z ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% exp_bound_half
thf(fact_6129_exp__bound__half,axiom,
    ! [Z: complex] :
      ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( exp_complex @ Z ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% exp_bound_half
thf(fact_6130_minus__pi__half__less__zero,axiom,
    ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ zero_zero_real ).

% minus_pi_half_less_zero
thf(fact_6131_pochhammer__minus,axiom,
    ! [B2: code_integer,K: nat] :
      ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ B2 ) @ K )
      = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ K ) @ ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ B2 @ ( semiri4939895301339042750nteger @ K ) ) @ one_one_Code_integer ) @ K ) ) ) ).

% pochhammer_minus
thf(fact_6132_pochhammer__minus,axiom,
    ! [B2: complex,K: nat] :
      ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ B2 ) @ K )
      = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( minus_minus_complex @ B2 @ ( semiri8010041392384452111omplex @ K ) ) @ one_one_complex ) @ K ) ) ) ).

% pochhammer_minus
thf(fact_6133_pochhammer__minus,axiom,
    ! [B2: real,K: nat] :
      ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ B2 ) @ K )
      = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( minus_minus_real @ B2 @ ( semiri5074537144036343181t_real @ K ) ) @ one_one_real ) @ K ) ) ) ).

% pochhammer_minus
thf(fact_6134_pochhammer__minus,axiom,
    ! [B2: rat,K: nat] :
      ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ B2 ) @ K )
      = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ B2 @ ( semiri681578069525770553at_rat @ K ) ) @ one_one_rat ) @ K ) ) ) ).

% pochhammer_minus
thf(fact_6135_pochhammer__minus,axiom,
    ! [B2: int,K: nat] :
      ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ B2 ) @ K )
      = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ K ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( minus_minus_int @ B2 @ ( semiri1314217659103216013at_int @ K ) ) @ one_one_int ) @ K ) ) ) ).

% pochhammer_minus
thf(fact_6136_pochhammer__minus_H,axiom,
    ! [B2: code_integer,K: nat] :
      ( ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ B2 @ ( semiri4939895301339042750nteger @ K ) ) @ one_one_Code_integer ) @ K )
      = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ K ) @ ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ B2 ) @ K ) ) ) ).

% pochhammer_minus'
thf(fact_6137_pochhammer__minus_H,axiom,
    ! [B2: complex,K: nat] :
      ( ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( minus_minus_complex @ B2 @ ( semiri8010041392384452111omplex @ K ) ) @ one_one_complex ) @ K )
      = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K ) @ ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ B2 ) @ K ) ) ) ).

% pochhammer_minus'
thf(fact_6138_pochhammer__minus_H,axiom,
    ! [B2: real,K: nat] :
      ( ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( minus_minus_real @ B2 @ ( semiri5074537144036343181t_real @ K ) ) @ one_one_real ) @ K )
      = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K ) @ ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ B2 ) @ K ) ) ) ).

% pochhammer_minus'
thf(fact_6139_pochhammer__minus_H,axiom,
    ! [B2: rat,K: nat] :
      ( ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ B2 @ ( semiri681578069525770553at_rat @ K ) ) @ one_one_rat ) @ K )
      = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K ) @ ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ B2 ) @ K ) ) ) ).

% pochhammer_minus'
thf(fact_6140_pochhammer__minus_H,axiom,
    ! [B2: int,K: nat] :
      ( ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( minus_minus_int @ B2 @ ( semiri1314217659103216013at_int @ K ) ) @ one_one_int ) @ K )
      = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ K ) @ ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ B2 ) @ K ) ) ) ).

% pochhammer_minus'
thf(fact_6141_arctan__lbound,axiom,
    ! [Y3: real] : ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y3 ) ) ).

% arctan_lbound
thf(fact_6142_arctan__bounded,axiom,
    ! [Y3: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y3 ) )
      & ( ord_less_real @ ( arctan @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% arctan_bounded
thf(fact_6143_exp__bound__lemma,axiom,
    ! [Z: real] :
      ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
     => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( exp_real @ Z ) ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( real_V7735802525324610683m_real @ Z ) ) ) ) ) ).

% exp_bound_lemma
thf(fact_6144_exp__bound__lemma,axiom,
    ! [Z: complex] :
      ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( exp_complex @ Z ) ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( real_V1022390504157884413omplex @ Z ) ) ) ) ) ).

% exp_bound_lemma
thf(fact_6145_machin__Euler,axiom,
    ( ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit1 @ ( bit0 @ one ) ) ) @ ( arctan @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( arctan @ ( divide_divide_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ ( numeral_numeral_real @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
    = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ).

% machin_Euler
thf(fact_6146_norm__divide__numeral,axiom,
    ! [A: real,W: num] :
      ( ( real_V7735802525324610683m_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ W ) ) )
      = ( divide_divide_real @ ( real_V7735802525324610683m_real @ A ) @ ( numeral_numeral_real @ W ) ) ) ).

% norm_divide_numeral
thf(fact_6147_norm__divide__numeral,axiom,
    ! [A: complex,W: num] :
      ( ( real_V1022390504157884413omplex @ ( divide1717551699836669952omplex @ A @ ( numera6690914467698888265omplex @ W ) ) )
      = ( divide_divide_real @ ( real_V1022390504157884413omplex @ A ) @ ( numeral_numeral_real @ W ) ) ) ).

% norm_divide_numeral
thf(fact_6148_norm__mult__numeral1,axiom,
    ! [W: num,A: real] :
      ( ( real_V7735802525324610683m_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ A ) )
      = ( times_times_real @ ( numeral_numeral_real @ W ) @ ( real_V7735802525324610683m_real @ A ) ) ) ).

% norm_mult_numeral1
thf(fact_6149_norm__mult__numeral1,axiom,
    ! [W: num,A: complex] :
      ( ( real_V1022390504157884413omplex @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ A ) )
      = ( times_times_real @ ( numeral_numeral_real @ W ) @ ( real_V1022390504157884413omplex @ A ) ) ) ).

% norm_mult_numeral1
thf(fact_6150_norm__mult__numeral2,axiom,
    ! [A: real,W: num] :
      ( ( real_V7735802525324610683m_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) )
      = ( times_times_real @ ( real_V7735802525324610683m_real @ A ) @ ( numeral_numeral_real @ W ) ) ) ).

% norm_mult_numeral2
thf(fact_6151_norm__mult__numeral2,axiom,
    ! [A: complex,W: num] :
      ( ( real_V1022390504157884413omplex @ ( times_times_complex @ A @ ( numera6690914467698888265omplex @ W ) ) )
      = ( times_times_real @ ( real_V1022390504157884413omplex @ A ) @ ( numeral_numeral_real @ W ) ) ) ).

% norm_mult_numeral2
thf(fact_6152_zero__less__binomial__iff,axiom,
    ! [N: nat,K: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( binomial @ N @ K ) )
      = ( ord_less_eq_nat @ K @ N ) ) ).

% zero_less_binomial_iff
thf(fact_6153_norm__neg__numeral,axiom,
    ! [W: num] :
      ( ( real_V7735802525324610683m_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
      = ( numeral_numeral_real @ W ) ) ).

% norm_neg_numeral
thf(fact_6154_norm__neg__numeral,axiom,
    ! [W: num] :
      ( ( real_V1022390504157884413omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
      = ( numeral_numeral_real @ W ) ) ).

% norm_neg_numeral
thf(fact_6155_norm__le__zero__iff,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ X4 ) @ zero_zero_real )
      = ( X4 = zero_zero_real ) ) ).

% norm_le_zero_iff
thf(fact_6156_norm__le__zero__iff,axiom,
    ! [X4: complex] :
      ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X4 ) @ zero_zero_real )
      = ( X4 = zero_zero_complex ) ) ).

% norm_le_zero_iff
thf(fact_6157_binomial__Suc__n,axiom,
    ! [N: nat] :
      ( ( binomial @ ( suc @ N ) @ N )
      = ( suc @ N ) ) ).

% binomial_Suc_n
thf(fact_6158_binomial__n__n,axiom,
    ! [N: nat] :
      ( ( binomial @ N @ N )
      = one_one_nat ) ).

% binomial_n_n
thf(fact_6159_norm__one,axiom,
    ( ( real_V7735802525324610683m_real @ one_one_real )
    = one_one_real ) ).

% norm_one
thf(fact_6160_norm__one,axiom,
    ( ( real_V1022390504157884413omplex @ one_one_complex )
    = one_one_real ) ).

% norm_one
thf(fact_6161_norm__numeral,axiom,
    ! [W: num] :
      ( ( real_V7735802525324610683m_real @ ( numeral_numeral_real @ W ) )
      = ( numeral_numeral_real @ W ) ) ).

% norm_numeral
thf(fact_6162_norm__numeral,axiom,
    ! [W: num] :
      ( ( real_V1022390504157884413omplex @ ( numera6690914467698888265omplex @ W ) )
      = ( numeral_numeral_real @ W ) ) ).

% norm_numeral
thf(fact_6163_binomial__0__Suc,axiom,
    ! [K: nat] :
      ( ( binomial @ zero_zero_nat @ ( suc @ K ) )
      = zero_zero_nat ) ).

% binomial_0_Suc
thf(fact_6164_binomial__1,axiom,
    ! [N: nat] :
      ( ( binomial @ N @ ( suc @ zero_zero_nat ) )
      = N ) ).

% binomial_1
thf(fact_6165_binomial__eq__0__iff,axiom,
    ! [N: nat,K: nat] :
      ( ( ( binomial @ N @ K )
        = zero_zero_nat )
      = ( ord_less_nat @ N @ K ) ) ).

% binomial_eq_0_iff
thf(fact_6166_binomial__Suc__Suc,axiom,
    ! [N: nat,K: nat] :
      ( ( binomial @ ( suc @ N ) @ ( suc @ K ) )
      = ( plus_plus_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( suc @ K ) ) ) ) ).

% binomial_Suc_Suc
thf(fact_6167_binomial__n__0,axiom,
    ! [N: nat] :
      ( ( binomial @ N @ zero_zero_nat )
      = one_one_nat ) ).

% binomial_n_0
thf(fact_6168_zero__less__norm__iff,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ ( real_V7735802525324610683m_real @ X4 ) )
      = ( X4 != zero_zero_real ) ) ).

% zero_less_norm_iff
thf(fact_6169_zero__less__norm__iff,axiom,
    ! [X4: complex] :
      ( ( ord_less_real @ zero_zero_real @ ( real_V1022390504157884413omplex @ X4 ) )
      = ( X4 != zero_zero_complex ) ) ).

% zero_less_norm_iff
thf(fact_6170_norm__minus__commute,axiom,
    ! [A: real,B2: real] :
      ( ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ B2 ) )
      = ( real_V7735802525324610683m_real @ ( minus_minus_real @ B2 @ A ) ) ) ).

% norm_minus_commute
thf(fact_6171_norm__minus__commute,axiom,
    ! [A: complex,B2: complex] :
      ( ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ B2 ) )
      = ( real_V1022390504157884413omplex @ ( minus_minus_complex @ B2 @ A ) ) ) ).

% norm_minus_commute
thf(fact_6172_choose__one,axiom,
    ! [N: nat] :
      ( ( binomial @ N @ one_one_nat )
      = N ) ).

% choose_one
thf(fact_6173_norm__not__less__zero,axiom,
    ! [X4: complex] :
      ~ ( ord_less_real @ ( real_V1022390504157884413omplex @ X4 ) @ zero_zero_real ) ).

% norm_not_less_zero
thf(fact_6174_norm__ge__zero,axiom,
    ! [X4: complex] : ( ord_less_eq_real @ zero_zero_real @ ( real_V1022390504157884413omplex @ X4 ) ) ).

% norm_ge_zero
thf(fact_6175_norm__mult,axiom,
    ! [X4: real,Y3: real] :
      ( ( real_V7735802525324610683m_real @ ( times_times_real @ X4 @ Y3 ) )
      = ( times_times_real @ ( real_V7735802525324610683m_real @ X4 ) @ ( real_V7735802525324610683m_real @ Y3 ) ) ) ).

% norm_mult
thf(fact_6176_norm__mult,axiom,
    ! [X4: complex,Y3: complex] :
      ( ( real_V1022390504157884413omplex @ ( times_times_complex @ X4 @ Y3 ) )
      = ( times_times_real @ ( real_V1022390504157884413omplex @ X4 ) @ ( real_V1022390504157884413omplex @ Y3 ) ) ) ).

% norm_mult
thf(fact_6177_norm__divide,axiom,
    ! [A: real,B2: real] :
      ( ( real_V7735802525324610683m_real @ ( divide_divide_real @ A @ B2 ) )
      = ( divide_divide_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B2 ) ) ) ).

% norm_divide
thf(fact_6178_norm__divide,axiom,
    ! [A: complex,B2: complex] :
      ( ( real_V1022390504157884413omplex @ ( divide1717551699836669952omplex @ A @ B2 ) )
      = ( divide_divide_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B2 ) ) ) ).

% norm_divide
thf(fact_6179_norm__power,axiom,
    ! [X4: real,N: nat] :
      ( ( real_V7735802525324610683m_real @ ( power_power_real @ X4 @ N ) )
      = ( power_power_real @ ( real_V7735802525324610683m_real @ X4 ) @ N ) ) ).

% norm_power
thf(fact_6180_norm__power,axiom,
    ! [X4: complex,N: nat] :
      ( ( real_V1022390504157884413omplex @ ( power_power_complex @ X4 @ N ) )
      = ( power_power_real @ ( real_V1022390504157884413omplex @ X4 ) @ N ) ) ).

% norm_power
thf(fact_6181_binomial__eq__0,axiom,
    ! [N: nat,K: nat] :
      ( ( ord_less_nat @ N @ K )
     => ( ( binomial @ N @ K )
        = zero_zero_nat ) ) ).

% binomial_eq_0
thf(fact_6182_Suc__times__binomial,axiom,
    ! [K: nat,N: nat] :
      ( ( times_times_nat @ ( suc @ K ) @ ( binomial @ ( suc @ N ) @ ( suc @ K ) ) )
      = ( times_times_nat @ ( suc @ N ) @ ( binomial @ N @ K ) ) ) ).

% Suc_times_binomial
thf(fact_6183_Suc__times__binomial__eq,axiom,
    ! [N: nat,K: nat] :
      ( ( times_times_nat @ ( suc @ N ) @ ( binomial @ N @ K ) )
      = ( times_times_nat @ ( binomial @ ( suc @ N ) @ ( suc @ K ) ) @ ( suc @ K ) ) ) ).

% Suc_times_binomial_eq
thf(fact_6184_binomial__symmetric,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( binomial @ N @ K )
        = ( binomial @ N @ ( minus_minus_nat @ N @ K ) ) ) ) ).

% binomial_symmetric
thf(fact_6185_choose__mult__lemma,axiom,
    ! [M: nat,R2: nat,K: nat] :
      ( ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M @ R2 ) @ K ) @ ( plus_plus_nat @ M @ K ) ) @ ( binomial @ ( plus_plus_nat @ M @ K ) @ K ) )
      = ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M @ R2 ) @ K ) @ K ) @ ( binomial @ ( plus_plus_nat @ M @ R2 ) @ M ) ) ) ).

% choose_mult_lemma
thf(fact_6186_binomial__le__pow,axiom,
    ! [R2: nat,N: nat] :
      ( ( ord_less_eq_nat @ R2 @ N )
     => ( ord_less_eq_nat @ ( binomial @ N @ R2 ) @ ( power_power_nat @ N @ R2 ) ) ) ).

% binomial_le_pow
thf(fact_6187_norm__uminus__minus,axiom,
    ! [X4: real,Y3: real] :
      ( ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( uminus_uminus_real @ X4 ) @ Y3 ) )
      = ( real_V7735802525324610683m_real @ ( plus_plus_real @ X4 @ Y3 ) ) ) ).

% norm_uminus_minus
thf(fact_6188_norm__uminus__minus,axiom,
    ! [X4: complex,Y3: complex] :
      ( ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ X4 ) @ Y3 ) )
      = ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X4 @ Y3 ) ) ) ).

% norm_uminus_minus
thf(fact_6189_nonzero__norm__divide,axiom,
    ! [B2: real,A: real] :
      ( ( B2 != zero_zero_real )
     => ( ( real_V7735802525324610683m_real @ ( divide_divide_real @ A @ B2 ) )
        = ( divide_divide_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B2 ) ) ) ) ).

% nonzero_norm_divide
thf(fact_6190_nonzero__norm__divide,axiom,
    ! [B2: complex,A: complex] :
      ( ( B2 != zero_zero_complex )
     => ( ( real_V1022390504157884413omplex @ ( divide1717551699836669952omplex @ A @ B2 ) )
        = ( divide_divide_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B2 ) ) ) ) ).

% nonzero_norm_divide
thf(fact_6191_power__eq__imp__eq__norm,axiom,
    ! [W: real,N: nat,Z: real] :
      ( ( ( power_power_real @ W @ N )
        = ( power_power_real @ Z @ N ) )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( real_V7735802525324610683m_real @ W )
          = ( real_V7735802525324610683m_real @ Z ) ) ) ) ).

% power_eq_imp_eq_norm
thf(fact_6192_power__eq__imp__eq__norm,axiom,
    ! [W: complex,N: nat,Z: complex] :
      ( ( ( power_power_complex @ W @ N )
        = ( power_power_complex @ Z @ N ) )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( real_V1022390504157884413omplex @ W )
          = ( real_V1022390504157884413omplex @ Z ) ) ) ) ).

% power_eq_imp_eq_norm
thf(fact_6193_norm__mult__less,axiom,
    ! [X4: real,R2: real,Y3: real,S: real] :
      ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X4 ) @ R2 )
     => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Y3 ) @ S )
       => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( times_times_real @ X4 @ Y3 ) ) @ ( times_times_real @ R2 @ S ) ) ) ) ).

% norm_mult_less
thf(fact_6194_norm__mult__less,axiom,
    ! [X4: complex,R2: real,Y3: complex,S: real] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X4 ) @ R2 )
     => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Y3 ) @ S )
       => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( times_times_complex @ X4 @ Y3 ) ) @ ( times_times_real @ R2 @ S ) ) ) ) ).

% norm_mult_less
thf(fact_6195_norm__mult__ineq,axiom,
    ! [X4: real,Y3: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( times_times_real @ X4 @ Y3 ) ) @ ( times_times_real @ ( real_V7735802525324610683m_real @ X4 ) @ ( real_V7735802525324610683m_real @ Y3 ) ) ) ).

% norm_mult_ineq
thf(fact_6196_norm__mult__ineq,axiom,
    ! [X4: complex,Y3: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( times_times_complex @ X4 @ Y3 ) ) @ ( times_times_real @ ( real_V1022390504157884413omplex @ X4 ) @ ( real_V1022390504157884413omplex @ Y3 ) ) ) ).

% norm_mult_ineq
thf(fact_6197_norm__triangle__lt,axiom,
    ! [X4: real,Y3: real,E: real] :
      ( ( ord_less_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X4 ) @ ( real_V7735802525324610683m_real @ Y3 ) ) @ E )
     => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X4 @ Y3 ) ) @ E ) ) ).

% norm_triangle_lt
thf(fact_6198_norm__triangle__lt,axiom,
    ! [X4: complex,Y3: complex,E: real] :
      ( ( ord_less_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X4 ) @ ( real_V1022390504157884413omplex @ Y3 ) ) @ E )
     => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X4 @ Y3 ) ) @ E ) ) ).

% norm_triangle_lt
thf(fact_6199_norm__add__less,axiom,
    ! [X4: real,R2: real,Y3: real,S: real] :
      ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X4 ) @ R2 )
     => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Y3 ) @ S )
       => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X4 @ Y3 ) ) @ ( plus_plus_real @ R2 @ S ) ) ) ) ).

% norm_add_less
thf(fact_6200_norm__add__less,axiom,
    ! [X4: complex,R2: real,Y3: complex,S: real] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X4 ) @ R2 )
     => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Y3 ) @ S )
       => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X4 @ Y3 ) ) @ ( plus_plus_real @ R2 @ S ) ) ) ) ).

% norm_add_less
thf(fact_6201_norm__power__ineq,axiom,
    ! [X4: real,N: nat] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( power_power_real @ X4 @ N ) ) @ ( power_power_real @ ( real_V7735802525324610683m_real @ X4 ) @ N ) ) ).

% norm_power_ineq
thf(fact_6202_norm__power__ineq,axiom,
    ! [X4: complex,N: nat] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( power_power_complex @ X4 @ N ) ) @ ( power_power_real @ ( real_V1022390504157884413omplex @ X4 ) @ N ) ) ).

% norm_power_ineq
thf(fact_6203_norm__triangle__mono,axiom,
    ! [A: real,R2: real,B2: real,S: real] :
      ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ A ) @ R2 )
     => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ B2 ) @ S )
       => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B2 ) ) @ ( plus_plus_real @ R2 @ S ) ) ) ) ).

% norm_triangle_mono
thf(fact_6204_norm__triangle__mono,axiom,
    ! [A: complex,R2: real,B2: complex,S: real] :
      ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ A ) @ R2 )
     => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ B2 ) @ S )
       => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B2 ) ) @ ( plus_plus_real @ R2 @ S ) ) ) ) ).

% norm_triangle_mono
thf(fact_6205_norm__triangle__ineq,axiom,
    ! [X4: real,Y3: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X4 @ Y3 ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X4 ) @ ( real_V7735802525324610683m_real @ Y3 ) ) ) ).

% norm_triangle_ineq
thf(fact_6206_norm__triangle__ineq,axiom,
    ! [X4: complex,Y3: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X4 @ Y3 ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X4 ) @ ( real_V1022390504157884413omplex @ Y3 ) ) ) ).

% norm_triangle_ineq
thf(fact_6207_norm__triangle__le,axiom,
    ! [X4: real,Y3: real,E: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X4 ) @ ( real_V7735802525324610683m_real @ Y3 ) ) @ E )
     => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X4 @ Y3 ) ) @ E ) ) ).

% norm_triangle_le
thf(fact_6208_norm__triangle__le,axiom,
    ! [X4: complex,Y3: complex,E: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X4 ) @ ( real_V1022390504157884413omplex @ Y3 ) ) @ E )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X4 @ Y3 ) ) @ E ) ) ).

% norm_triangle_le
thf(fact_6209_norm__add__leD,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B2 ) ) @ C )
     => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ B2 ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ A ) @ C ) ) ) ).

% norm_add_leD
thf(fact_6210_norm__add__leD,axiom,
    ! [A: complex,B2: complex,C: real] :
      ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B2 ) ) @ C )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ B2 ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ A ) @ C ) ) ) ).

% norm_add_leD
thf(fact_6211_norm__diff__triangle__less,axiom,
    ! [X4: real,Y3: real,E1: real,Z: real,E22: real] :
      ( ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X4 @ Y3 ) ) @ E1 )
     => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ Y3 @ Z ) ) @ E22 )
       => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X4 @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).

% norm_diff_triangle_less
thf(fact_6212_norm__diff__triangle__less,axiom,
    ! [X4: complex,Y3: complex,E1: real,Z: complex,E22: real] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X4 @ Y3 ) ) @ E1 )
     => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Y3 @ Z ) ) @ E22 )
       => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X4 @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).

% norm_diff_triangle_less
thf(fact_6213_norm__triangle__sub,axiom,
    ! [X4: real,Y3: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ X4 ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ Y3 ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X4 @ Y3 ) ) ) ) ).

% norm_triangle_sub
thf(fact_6214_norm__triangle__sub,axiom,
    ! [X4: complex,Y3: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X4 ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Y3 ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X4 @ Y3 ) ) ) ) ).

% norm_triangle_sub
thf(fact_6215_norm__triangle__ineq4,axiom,
    ! [A: real,B2: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ B2 ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B2 ) ) ) ).

% norm_triangle_ineq4
thf(fact_6216_norm__triangle__ineq4,axiom,
    ! [A: complex,B2: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ B2 ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B2 ) ) ) ).

% norm_triangle_ineq4
thf(fact_6217_norm__diff__triangle__le,axiom,
    ! [X4: real,Y3: real,E1: real,Z: real,E22: real] :
      ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X4 @ Y3 ) ) @ E1 )
     => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ Y3 @ Z ) ) @ E22 )
       => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X4 @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).

% norm_diff_triangle_le
thf(fact_6218_norm__diff__triangle__le,axiom,
    ! [X4: complex,Y3: complex,E1: real,Z: complex,E22: real] :
      ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X4 @ Y3 ) ) @ E1 )
     => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Y3 @ Z ) ) @ E22 )
       => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X4 @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).

% norm_diff_triangle_le
thf(fact_6219_norm__triangle__le__diff,axiom,
    ! [X4: real,Y3: real,E: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X4 ) @ ( real_V7735802525324610683m_real @ Y3 ) ) @ E )
     => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X4 @ Y3 ) ) @ E ) ) ).

% norm_triangle_le_diff
thf(fact_6220_norm__triangle__le__diff,axiom,
    ! [X4: complex,Y3: complex,E: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X4 ) @ ( real_V1022390504157884413omplex @ Y3 ) ) @ E )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X4 @ Y3 ) ) @ E ) ) ).

% norm_triangle_le_diff
thf(fact_6221_norm__diff__ineq,axiom,
    ! [A: real,B2: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B2 ) ) @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B2 ) ) ) ).

% norm_diff_ineq
thf(fact_6222_norm__diff__ineq,axiom,
    ! [A: complex,B2: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B2 ) ) @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B2 ) ) ) ).

% norm_diff_ineq
thf(fact_6223_norm__triangle__ineq2,axiom,
    ! [A: real,B2: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B2 ) ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ B2 ) ) ) ).

% norm_triangle_ineq2
thf(fact_6224_norm__triangle__ineq2,axiom,
    ! [A: complex,B2: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B2 ) ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ B2 ) ) ) ).

% norm_triangle_ineq2
thf(fact_6225_zero__less__binomial,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ord_less_nat @ zero_zero_nat @ ( binomial @ N @ K ) ) ) ).

% zero_less_binomial
thf(fact_6226_Suc__times__binomial__add,axiom,
    ! [A: nat,B2: nat] :
      ( ( times_times_nat @ ( suc @ A ) @ ( binomial @ ( suc @ ( plus_plus_nat @ A @ B2 ) ) @ ( suc @ A ) ) )
      = ( times_times_nat @ ( suc @ B2 ) @ ( binomial @ ( suc @ ( plus_plus_nat @ A @ B2 ) ) @ A ) ) ) ).

% Suc_times_binomial_add
thf(fact_6227_binomial__Suc__Suc__eq__times,axiom,
    ! [N: nat,K: nat] :
      ( ( binomial @ ( suc @ N ) @ ( suc @ K ) )
      = ( divide_divide_nat @ ( times_times_nat @ ( suc @ N ) @ ( binomial @ N @ K ) ) @ ( suc @ K ) ) ) ).

% binomial_Suc_Suc_eq_times
thf(fact_6228_choose__mult,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ M @ N )
       => ( ( times_times_nat @ ( binomial @ N @ M ) @ ( binomial @ M @ K ) )
          = ( times_times_nat @ ( binomial @ N @ K ) @ ( binomial @ ( minus_minus_nat @ N @ K ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ) ).

% choose_mult
thf(fact_6229_binomial__absorb__comp,axiom,
    ! [N: nat,K: nat] :
      ( ( times_times_nat @ ( minus_minus_nat @ N @ K ) @ ( binomial @ N @ K ) )
      = ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ).

% binomial_absorb_comp
thf(fact_6230_power__eq__1__iff,axiom,
    ! [W: real,N: nat] :
      ( ( ( power_power_real @ W @ N )
        = one_one_real )
     => ( ( ( real_V7735802525324610683m_real @ W )
          = one_one_real )
        | ( N = zero_zero_nat ) ) ) ).

% power_eq_1_iff
thf(fact_6231_power__eq__1__iff,axiom,
    ! [W: complex,N: nat] :
      ( ( ( power_power_complex @ W @ N )
        = one_one_complex )
     => ( ( ( real_V1022390504157884413omplex @ W )
          = one_one_real )
        | ( N = zero_zero_nat ) ) ) ).

% power_eq_1_iff
thf(fact_6232_norm__diff__triangle__ineq,axiom,
    ! [A: real,B2: real,C: real,D: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( plus_plus_real @ A @ B2 ) @ ( plus_plus_real @ C @ D ) ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ C ) ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ B2 @ D ) ) ) ) ).

% norm_diff_triangle_ineq
thf(fact_6233_norm__diff__triangle__ineq,axiom,
    ! [A: complex,B2: complex,C: complex,D: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( plus_plus_complex @ A @ B2 ) @ ( plus_plus_complex @ C @ D ) ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ C ) ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ B2 @ D ) ) ) ) ).

% norm_diff_triangle_ineq
thf(fact_6234_norm__triangle__ineq3,axiom,
    ! [A: real,B2: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B2 ) ) ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ B2 ) ) ) ).

% norm_triangle_ineq3
thf(fact_6235_norm__triangle__ineq3,axiom,
    ! [A: complex,B2: complex] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B2 ) ) ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ B2 ) ) ) ).

% norm_triangle_ineq3
thf(fact_6236_binomial__absorption,axiom,
    ! [K: nat,N: nat] :
      ( ( times_times_nat @ ( suc @ K ) @ ( binomial @ N @ ( suc @ K ) ) )
      = ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ).

% binomial_absorption
thf(fact_6237_binomial__ge__n__over__k__pow__k,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ord_less_eq_real @ ( power_power_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ K ) ) @ K ) @ ( semiri5074537144036343181t_real @ ( binomial @ N @ K ) ) ) ) ).

% binomial_ge_n_over_k_pow_k
thf(fact_6238_binomial__ge__n__over__k__pow__k,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ord_less_eq_rat @ ( power_power_rat @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ N ) @ ( semiri681578069525770553at_rat @ K ) ) @ K ) @ ( semiri681578069525770553at_rat @ ( binomial @ N @ K ) ) ) ) ).

% binomial_ge_n_over_k_pow_k
thf(fact_6239_binomial__le__pow2,axiom,
    ! [N: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ N @ K ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).

% binomial_le_pow2
thf(fact_6240_choose__reduce__nat,axiom,
    ! [N: nat,K: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ( binomial @ N @ K )
          = ( plus_plus_nat @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ) ) ).

% choose_reduce_nat
thf(fact_6241_times__binomial__minus1__eq,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( times_times_nat @ K @ ( binomial @ N @ K ) )
        = ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).

% times_binomial_minus1_eq
thf(fact_6242_square__norm__one,axiom,
    ! [X4: real] :
      ( ( ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_real )
     => ( ( real_V7735802525324610683m_real @ X4 )
        = one_one_real ) ) ).

% square_norm_one
thf(fact_6243_square__norm__one,axiom,
    ! [X4: complex] :
      ( ( ( power_power_complex @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_complex )
     => ( ( real_V1022390504157884413omplex @ X4 )
        = one_one_real ) ) ).

% square_norm_one
thf(fact_6244_norm__power__diff,axiom,
    ! [Z: real,W: real,M: nat] :
      ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ one_one_real )
     => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ W ) @ one_one_real )
       => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( power_power_real @ Z @ M ) @ ( power_power_real @ W @ M ) ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ Z @ W ) ) ) ) ) ) ).

% norm_power_diff
thf(fact_6245_norm__power__diff,axiom,
    ! [Z: complex,W: complex,M: nat] :
      ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ one_one_real )
     => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ W ) @ one_one_real )
       => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( power_power_complex @ Z @ M ) @ ( power_power_complex @ W @ M ) ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Z @ W ) ) ) ) ) ) ).

% norm_power_diff
thf(fact_6246_binomial__addition__formula,axiom,
    ! [N: nat,K: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( binomial @ N @ ( suc @ K ) )
        = ( plus_plus_nat @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( suc @ K ) ) @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ) ).

% binomial_addition_formula
thf(fact_6247_choose__two,axiom,
    ! [N: nat] :
      ( ( binomial @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( divide_divide_nat @ ( times_times_nat @ N @ ( minus_minus_nat @ N @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% choose_two
thf(fact_6248_sin__cos__npi,axiom,
    ! [N: nat] :
      ( ( sin_real @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
      = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) ) ).

% sin_cos_npi
thf(fact_6249_cos__pi__eq__zero,axiom,
    ! [M: nat] :
      ( ( cos_real @ ( divide_divide_real @ ( times_times_real @ pi @ ( semiri5074537144036343181t_real @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
      = zero_zero_real ) ).

% cos_pi_eq_zero
thf(fact_6250_round__altdef,axiom,
    ( archim8280529875227126926d_real
    = ( ^ [X: real] : ( if_int @ ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( archim2898591450579166408c_real @ X ) ) @ ( archim7802044766580827645g_real @ X ) @ ( archim6058952711729229775r_real @ X ) ) ) ) ).

% round_altdef
thf(fact_6251_round__altdef,axiom,
    ( archim7778729529865785530nd_rat
    = ( ^ [X: rat] : ( if_int @ ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( archimedean_frac_rat @ X ) ) @ ( archim2889992004027027881ng_rat @ X ) @ ( archim3151403230148437115or_rat @ X ) ) ) ) ).

% round_altdef
thf(fact_6252_ceiling__log__eq__powr__iff,axiom,
    ! [X4: real,B2: real,K: nat] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( ord_less_real @ one_one_real @ B2 )
       => ( ( ( archim7802044766580827645g_real @ ( log @ B2 @ X4 ) )
            = ( plus_plus_int @ ( semiri1314217659103216013at_int @ K ) @ one_one_int ) )
          = ( ( ord_less_real @ ( powr_real @ B2 @ ( semiri5074537144036343181t_real @ K ) ) @ X4 )
            & ( ord_less_eq_real @ X4 @ ( powr_real @ B2 @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ) ) ) ) ) ).

% ceiling_log_eq_powr_iff
thf(fact_6253_cos__zero__iff,axiom,
    ! [X4: real] :
      ( ( ( cos_real @ X4 )
        = zero_zero_real )
      = ( ? [N2: nat] :
            ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
            & ( X4
              = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) )
        | ? [N2: nat] :
            ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
            & ( X4
              = ( uminus_uminus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% cos_zero_iff
thf(fact_6254_cot__less__zero,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X4 )
     => ( ( ord_less_real @ X4 @ zero_zero_real )
       => ( ord_less_real @ ( cot_real @ X4 ) @ zero_zero_real ) ) ) ).

% cot_less_zero
thf(fact_6255_powr__one__eq__one,axiom,
    ! [A: real] :
      ( ( powr_real @ one_one_real @ A )
      = one_one_real ) ).

% powr_one_eq_one
thf(fact_6256_cos__zero,axiom,
    ( ( cos_complex @ zero_zero_complex )
    = one_one_complex ) ).

% cos_zero
thf(fact_6257_cos__zero,axiom,
    ( ( cos_real @ zero_zero_real )
    = one_one_real ) ).

% cos_zero
thf(fact_6258_powr__zero__eq__one,axiom,
    ! [X4: real] :
      ( ( ( X4 = zero_zero_real )
       => ( ( powr_real @ X4 @ zero_zero_real )
          = zero_zero_real ) )
      & ( ( X4 != zero_zero_real )
       => ( ( powr_real @ X4 @ zero_zero_real )
          = one_one_real ) ) ) ).

% powr_zero_eq_one
thf(fact_6259_powr__gt__zero,axiom,
    ! [X4: real,A: real] :
      ( ( ord_less_real @ zero_zero_real @ ( powr_real @ X4 @ A ) )
      = ( X4 != zero_zero_real ) ) ).

% powr_gt_zero
thf(fact_6260_powr__nonneg__iff,axiom,
    ! [A: real,X4: real] :
      ( ( ord_less_eq_real @ ( powr_real @ A @ X4 ) @ zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% powr_nonneg_iff
thf(fact_6261_powr__less__cancel__iff,axiom,
    ! [X4: real,A: real,B2: real] :
      ( ( ord_less_real @ one_one_real @ X4 )
     => ( ( ord_less_real @ ( powr_real @ X4 @ A ) @ ( powr_real @ X4 @ B2 ) )
        = ( ord_less_real @ A @ B2 ) ) ) ).

% powr_less_cancel_iff
thf(fact_6262_sin__pi__minus,axiom,
    ! [X4: real] :
      ( ( sin_real @ ( minus_minus_real @ pi @ X4 ) )
      = ( sin_real @ X4 ) ) ).

% sin_pi_minus
thf(fact_6263_powr__eq__one__iff,axiom,
    ! [A: real,X4: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ( powr_real @ A @ X4 )
          = one_one_real )
        = ( X4 = zero_zero_real ) ) ) ).

% powr_eq_one_iff
thf(fact_6264_powr__one,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( powr_real @ X4 @ one_one_real )
        = X4 ) ) ).

% powr_one
thf(fact_6265_powr__one__gt__zero__iff,axiom,
    ! [X4: real] :
      ( ( ( powr_real @ X4 @ one_one_real )
        = X4 )
      = ( ord_less_eq_real @ zero_zero_real @ X4 ) ) ).

% powr_one_gt_zero_iff
thf(fact_6266_powr__le__cancel__iff,axiom,
    ! [X4: real,A: real,B2: real] :
      ( ( ord_less_real @ one_one_real @ X4 )
     => ( ( ord_less_eq_real @ ( powr_real @ X4 @ A ) @ ( powr_real @ X4 @ B2 ) )
        = ( ord_less_eq_real @ A @ B2 ) ) ) ).

% powr_le_cancel_iff
thf(fact_6267_numeral__powr__numeral__real,axiom,
    ! [M: num,N: num] :
      ( ( powr_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
      = ( power_power_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_nat @ N ) ) ) ).

% numeral_powr_numeral_real
thf(fact_6268_cos__pi,axiom,
    ( ( cos_real @ pi )
    = ( uminus_uminus_real @ one_one_real ) ) ).

% cos_pi
thf(fact_6269_cos__periodic__pi2,axiom,
    ! [X4: real] :
      ( ( cos_real @ ( plus_plus_real @ pi @ X4 ) )
      = ( uminus_uminus_real @ ( cos_real @ X4 ) ) ) ).

% cos_periodic_pi2
thf(fact_6270_cos__periodic__pi,axiom,
    ! [X4: real] :
      ( ( cos_real @ ( plus_plus_real @ X4 @ pi ) )
      = ( uminus_uminus_real @ ( cos_real @ X4 ) ) ) ).

% cos_periodic_pi
thf(fact_6271_sin__periodic__pi2,axiom,
    ! [X4: real] :
      ( ( sin_real @ ( plus_plus_real @ pi @ X4 ) )
      = ( uminus_uminus_real @ ( sin_real @ X4 ) ) ) ).

% sin_periodic_pi2
thf(fact_6272_sin__periodic__pi,axiom,
    ! [X4: real] :
      ( ( sin_real @ ( plus_plus_real @ X4 @ pi ) )
      = ( uminus_uminus_real @ ( sin_real @ X4 ) ) ) ).

% sin_periodic_pi
thf(fact_6273_cos__pi__minus,axiom,
    ! [X4: real] :
      ( ( cos_real @ ( minus_minus_real @ pi @ X4 ) )
      = ( uminus_uminus_real @ ( cos_real @ X4 ) ) ) ).

% cos_pi_minus
thf(fact_6274_cos__minus__pi,axiom,
    ! [X4: real] :
      ( ( cos_real @ ( minus_minus_real @ X4 @ pi ) )
      = ( uminus_uminus_real @ ( cos_real @ X4 ) ) ) ).

% cos_minus_pi
thf(fact_6275_sin__minus__pi,axiom,
    ! [X4: real] :
      ( ( sin_real @ ( minus_minus_real @ X4 @ pi ) )
      = ( uminus_uminus_real @ ( sin_real @ X4 ) ) ) ).

% sin_minus_pi
thf(fact_6276_sin__cos__squared__add3,axiom,
    ! [X4: complex] :
      ( ( plus_plus_complex @ ( times_times_complex @ ( cos_complex @ X4 ) @ ( cos_complex @ X4 ) ) @ ( times_times_complex @ ( sin_complex @ X4 ) @ ( sin_complex @ X4 ) ) )
      = one_one_complex ) ).

% sin_cos_squared_add3
thf(fact_6277_sin__cos__squared__add3,axiom,
    ! [X4: real] :
      ( ( plus_plus_real @ ( times_times_real @ ( cos_real @ X4 ) @ ( cos_real @ X4 ) ) @ ( times_times_real @ ( sin_real @ X4 ) @ ( sin_real @ X4 ) ) )
      = one_one_real ) ).

% sin_cos_squared_add3
thf(fact_6278_log__powr__cancel,axiom,
    ! [A: real,Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( log @ A @ ( powr_real @ A @ Y3 ) )
          = Y3 ) ) ) ).

% log_powr_cancel
thf(fact_6279_powr__log__cancel,axiom,
    ! [A: real,X4: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X4 )
         => ( ( powr_real @ A @ ( log @ A @ X4 ) )
            = X4 ) ) ) ) ).

% powr_log_cancel
thf(fact_6280_sin__npi2,axiom,
    ! [N: nat] :
      ( ( sin_real @ ( times_times_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) )
      = zero_zero_real ) ).

% sin_npi2
thf(fact_6281_sin__npi,axiom,
    ! [N: nat] :
      ( ( sin_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) )
      = zero_zero_real ) ).

% sin_npi
thf(fact_6282_sin__npi__int,axiom,
    ! [N: int] :
      ( ( sin_real @ ( times_times_real @ pi @ ( ring_1_of_int_real @ N ) ) )
      = zero_zero_real ) ).

% sin_npi_int
thf(fact_6283_cot__npi,axiom,
    ! [N: nat] :
      ( ( cot_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) )
      = zero_zero_real ) ).

% cot_npi
thf(fact_6284_powr__numeral,axiom,
    ! [X4: real,N: num] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( powr_real @ X4 @ ( numeral_numeral_real @ N ) )
        = ( power_power_real @ X4 @ ( numeral_numeral_nat @ N ) ) ) ) ).

% powr_numeral
thf(fact_6285_cos__pi__half,axiom,
    ( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
    = zero_zero_real ) ).

% cos_pi_half
thf(fact_6286_sin__two__pi,axiom,
    ( ( sin_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
    = zero_zero_real ) ).

% sin_two_pi
thf(fact_6287_sin__pi__half,axiom,
    ( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
    = one_one_real ) ).

% sin_pi_half
thf(fact_6288_cos__two__pi,axiom,
    ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
    = one_one_real ) ).

% cos_two_pi
thf(fact_6289_cos__periodic,axiom,
    ! [X4: real] :
      ( ( cos_real @ ( plus_plus_real @ X4 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
      = ( cos_real @ X4 ) ) ).

% cos_periodic
thf(fact_6290_sin__periodic,axiom,
    ! [X4: real] :
      ( ( sin_real @ ( plus_plus_real @ X4 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
      = ( sin_real @ X4 ) ) ).

% sin_periodic
thf(fact_6291_cos__2pi__minus,axiom,
    ! [X4: real] :
      ( ( cos_real @ ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ X4 ) )
      = ( cos_real @ X4 ) ) ).

% cos_2pi_minus
thf(fact_6292_cos__npi,axiom,
    ! [N: nat] :
      ( ( cos_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) )
      = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) ) ).

% cos_npi
thf(fact_6293_cos__npi2,axiom,
    ! [N: nat] :
      ( ( cos_real @ ( times_times_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) )
      = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) ) ).

% cos_npi2
thf(fact_6294_cot__periodic,axiom,
    ! [X4: real] :
      ( ( cot_real @ ( plus_plus_real @ X4 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
      = ( cot_real @ X4 ) ) ).

% cot_periodic
thf(fact_6295_sin__cos__squared__add2,axiom,
    ! [X4: real] :
      ( ( plus_plus_real @ ( power_power_real @ ( cos_real @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( sin_real @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = one_one_real ) ).

% sin_cos_squared_add2
thf(fact_6296_sin__cos__squared__add2,axiom,
    ! [X4: complex] :
      ( ( plus_plus_complex @ ( power_power_complex @ ( cos_complex @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( sin_complex @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = one_one_complex ) ).

% sin_cos_squared_add2
thf(fact_6297_sin__cos__squared__add,axiom,
    ! [X4: real] :
      ( ( plus_plus_real @ ( power_power_real @ ( sin_real @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( cos_real @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = one_one_real ) ).

% sin_cos_squared_add
thf(fact_6298_sin__cos__squared__add,axiom,
    ! [X4: complex] :
      ( ( plus_plus_complex @ ( power_power_complex @ ( sin_complex @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( cos_complex @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = one_one_complex ) ).

% sin_cos_squared_add
thf(fact_6299_sin__2npi,axiom,
    ! [N: nat] :
      ( ( sin_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) )
      = zero_zero_real ) ).

% sin_2npi
thf(fact_6300_cos__2npi,axiom,
    ! [N: nat] :
      ( ( cos_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) )
      = one_one_real ) ).

% cos_2npi
thf(fact_6301_sin__2pi__minus,axiom,
    ! [X4: real] :
      ( ( sin_real @ ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ X4 ) )
      = ( uminus_uminus_real @ ( sin_real @ X4 ) ) ) ).

% sin_2pi_minus
thf(fact_6302_sin__int__2pin,axiom,
    ! [N: int] :
      ( ( sin_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( ring_1_of_int_real @ N ) ) )
      = zero_zero_real ) ).

% sin_int_2pin
thf(fact_6303_cos__int__2pin,axiom,
    ! [N: int] :
      ( ( cos_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( ring_1_of_int_real @ N ) ) )
      = one_one_real ) ).

% cos_int_2pin
thf(fact_6304_cos__3over2__pi,axiom,
    ( ( cos_real @ ( times_times_real @ ( divide_divide_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) )
    = zero_zero_real ) ).

% cos_3over2_pi
thf(fact_6305_square__powr__half,axiom,
    ! [X4: real] :
      ( ( powr_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
      = ( abs_abs_real @ X4 ) ) ).

% square_powr_half
thf(fact_6306_sin__3over2__pi,axiom,
    ( ( sin_real @ ( times_times_real @ ( divide_divide_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) )
    = ( uminus_uminus_real @ one_one_real ) ) ).

% sin_3over2_pi
thf(fact_6307_cos__npi__int,axiom,
    ! [N: int] :
      ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
       => ( ( cos_real @ ( times_times_real @ pi @ ( ring_1_of_int_real @ N ) ) )
          = one_one_real ) )
      & ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
       => ( ( cos_real @ ( times_times_real @ pi @ ( ring_1_of_int_real @ N ) ) )
          = ( uminus_uminus_real @ one_one_real ) ) ) ) ).

% cos_npi_int
thf(fact_6308_sin__add,axiom,
    ! [X4: real,Y3: real] :
      ( ( sin_real @ ( plus_plus_real @ X4 @ Y3 ) )
      = ( plus_plus_real @ ( times_times_real @ ( sin_real @ X4 ) @ ( cos_real @ Y3 ) ) @ ( times_times_real @ ( cos_real @ X4 ) @ ( sin_real @ Y3 ) ) ) ) ).

% sin_add
thf(fact_6309_sin__diff,axiom,
    ! [X4: real,Y3: real] :
      ( ( sin_real @ ( minus_minus_real @ X4 @ Y3 ) )
      = ( minus_minus_real @ ( times_times_real @ ( sin_real @ X4 ) @ ( cos_real @ Y3 ) ) @ ( times_times_real @ ( cos_real @ X4 ) @ ( sin_real @ Y3 ) ) ) ) ).

% sin_diff
thf(fact_6310_cot__def,axiom,
    ( cot_complex
    = ( ^ [X: complex] : ( divide1717551699836669952omplex @ ( cos_complex @ X ) @ ( sin_complex @ X ) ) ) ) ).

% cot_def
thf(fact_6311_cot__def,axiom,
    ( cot_real
    = ( ^ [X: real] : ( divide_divide_real @ ( cos_real @ X ) @ ( sin_real @ X ) ) ) ) ).

% cot_def
thf(fact_6312_polar__Ex,axiom,
    ! [X4: real,Y3: real] :
    ? [R3: real,A5: real] :
      ( ( X4
        = ( times_times_real @ R3 @ ( cos_real @ A5 ) ) )
      & ( Y3
        = ( times_times_real @ R3 @ ( sin_real @ A5 ) ) ) ) ).

% polar_Ex
thf(fact_6313_cos__one__sin__zero,axiom,
    ! [X4: complex] :
      ( ( ( cos_complex @ X4 )
        = one_one_complex )
     => ( ( sin_complex @ X4 )
        = zero_zero_complex ) ) ).

% cos_one_sin_zero
thf(fact_6314_cos__one__sin__zero,axiom,
    ! [X4: real] :
      ( ( ( cos_real @ X4 )
        = one_one_real )
     => ( ( sin_real @ X4 )
        = zero_zero_real ) ) ).

% cos_one_sin_zero
thf(fact_6315_powr__powr,axiom,
    ! [X4: real,A: real,B2: real] :
      ( ( powr_real @ ( powr_real @ X4 @ A ) @ B2 )
      = ( powr_real @ X4 @ ( times_times_real @ A @ B2 ) ) ) ).

% powr_powr
thf(fact_6316_cos__add,axiom,
    ! [X4: real,Y3: real] :
      ( ( cos_real @ ( plus_plus_real @ X4 @ Y3 ) )
      = ( minus_minus_real @ ( times_times_real @ ( cos_real @ X4 ) @ ( cos_real @ Y3 ) ) @ ( times_times_real @ ( sin_real @ X4 ) @ ( sin_real @ Y3 ) ) ) ) ).

% cos_add
thf(fact_6317_cos__diff,axiom,
    ! [X4: real,Y3: real] :
      ( ( cos_real @ ( minus_minus_real @ X4 @ Y3 ) )
      = ( plus_plus_real @ ( times_times_real @ ( cos_real @ X4 ) @ ( cos_real @ Y3 ) ) @ ( times_times_real @ ( sin_real @ X4 ) @ ( sin_real @ Y3 ) ) ) ) ).

% cos_diff
thf(fact_6318_sin__zero__norm__cos__one,axiom,
    ! [X4: real] :
      ( ( ( sin_real @ X4 )
        = zero_zero_real )
     => ( ( real_V7735802525324610683m_real @ ( cos_real @ X4 ) )
        = one_one_real ) ) ).

% sin_zero_norm_cos_one
thf(fact_6319_sin__zero__norm__cos__one,axiom,
    ! [X4: complex] :
      ( ( ( sin_complex @ X4 )
        = zero_zero_complex )
     => ( ( real_V1022390504157884413omplex @ ( cos_complex @ X4 ) )
        = one_one_real ) ) ).

% sin_zero_norm_cos_one
thf(fact_6320_sin__zero__abs__cos__one,axiom,
    ! [X4: real] :
      ( ( ( sin_real @ X4 )
        = zero_zero_real )
     => ( ( abs_abs_real @ ( cos_real @ X4 ) )
        = one_one_real ) ) ).

% sin_zero_abs_cos_one
thf(fact_6321_sin__double,axiom,
    ! [X4: complex] :
      ( ( sin_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X4 ) )
      = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sin_complex @ X4 ) ) @ ( cos_complex @ X4 ) ) ) ).

% sin_double
thf(fact_6322_sin__double,axiom,
    ! [X4: real] :
      ( ( sin_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X4 ) )
      = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sin_real @ X4 ) ) @ ( cos_real @ X4 ) ) ) ).

% sin_double
thf(fact_6323_sincos__principal__value,axiom,
    ! [X4: real] :
    ? [Y4: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ Y4 )
      & ( ord_less_eq_real @ Y4 @ pi )
      & ( ( sin_real @ Y4 )
        = ( sin_real @ X4 ) )
      & ( ( cos_real @ Y4 )
        = ( cos_real @ X4 ) ) ) ).

% sincos_principal_value
thf(fact_6324_powr__less__mono2__neg,axiom,
    ! [A: real,X4: real,Y3: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( ord_less_real @ zero_zero_real @ X4 )
       => ( ( ord_less_real @ X4 @ Y3 )
         => ( ord_less_real @ ( powr_real @ Y3 @ A ) @ ( powr_real @ X4 @ A ) ) ) ) ) ).

% powr_less_mono2_neg
thf(fact_6325_powr__non__neg,axiom,
    ! [A: real,X4: real] :
      ~ ( ord_less_real @ ( powr_real @ A @ X4 ) @ zero_zero_real ) ).

% powr_non_neg
thf(fact_6326_powr__mono2,axiom,
    ! [A: real,X4: real,Y3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ X4 )
       => ( ( ord_less_eq_real @ X4 @ Y3 )
         => ( ord_less_eq_real @ ( powr_real @ X4 @ A ) @ ( powr_real @ Y3 @ A ) ) ) ) ) ).

% powr_mono2
thf(fact_6327_powr__ge__pzero,axiom,
    ! [X4: real,Y3: real] : ( ord_less_eq_real @ zero_zero_real @ ( powr_real @ X4 @ Y3 ) ) ).

% powr_ge_pzero
thf(fact_6328_powr__less__mono,axiom,
    ! [A: real,B2: real,X4: real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ( ord_less_real @ one_one_real @ X4 )
       => ( ord_less_real @ ( powr_real @ X4 @ A ) @ ( powr_real @ X4 @ B2 ) ) ) ) ).

% powr_less_mono
thf(fact_6329_powr__less__cancel,axiom,
    ! [X4: real,A: real,B2: real] :
      ( ( ord_less_real @ ( powr_real @ X4 @ A ) @ ( powr_real @ X4 @ B2 ) )
     => ( ( ord_less_real @ one_one_real @ X4 )
       => ( ord_less_real @ A @ B2 ) ) ) ).

% powr_less_cancel
thf(fact_6330_powr__mono,axiom,
    ! [A: real,B2: real,X4: real] :
      ( ( ord_less_eq_real @ A @ B2 )
     => ( ( ord_less_eq_real @ one_one_real @ X4 )
       => ( ord_less_eq_real @ ( powr_real @ X4 @ A ) @ ( powr_real @ X4 @ B2 ) ) ) ) ).

% powr_mono
thf(fact_6331_sin__x__le__x,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ord_less_eq_real @ ( sin_real @ X4 ) @ X4 ) ) ).

% sin_x_le_x
thf(fact_6332_sin__le__one,axiom,
    ! [X4: real] : ( ord_less_eq_real @ ( sin_real @ X4 ) @ one_one_real ) ).

% sin_le_one
thf(fact_6333_cos__le__one,axiom,
    ! [X4: real] : ( ord_less_eq_real @ ( cos_real @ X4 ) @ one_one_real ) ).

% cos_le_one
thf(fact_6334_abs__sin__x__le__abs__x,axiom,
    ! [X4: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( sin_real @ X4 ) ) @ ( abs_abs_real @ X4 ) ) ).

% abs_sin_x_le_abs_x
thf(fact_6335_sin__cos__le1,axiom,
    ! [X4: real,Y3: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( plus_plus_real @ ( times_times_real @ ( sin_real @ X4 ) @ ( sin_real @ Y3 ) ) @ ( times_times_real @ ( cos_real @ X4 ) @ ( cos_real @ Y3 ) ) ) ) @ one_one_real ) ).

% sin_cos_le1
thf(fact_6336_frac__ge__0,axiom,
    ! [X4: real] : ( ord_less_eq_real @ zero_zero_real @ ( archim2898591450579166408c_real @ X4 ) ) ).

% frac_ge_0
thf(fact_6337_frac__ge__0,axiom,
    ! [X4: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( archimedean_frac_rat @ X4 ) ) ).

% frac_ge_0
thf(fact_6338_sin__squared__eq,axiom,
    ! [X4: complex] :
      ( ( power_power_complex @ ( sin_complex @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ ( cos_complex @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sin_squared_eq
thf(fact_6339_sin__squared__eq,axiom,
    ! [X4: real] :
      ( ( power_power_real @ ( sin_real @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_real @ one_one_real @ ( power_power_real @ ( cos_real @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sin_squared_eq
thf(fact_6340_cos__squared__eq,axiom,
    ! [X4: complex] :
      ( ( power_power_complex @ ( cos_complex @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ ( sin_complex @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% cos_squared_eq
thf(fact_6341_cos__squared__eq,axiom,
    ! [X4: real] :
      ( ( power_power_real @ ( cos_real @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_real @ one_one_real @ ( power_power_real @ ( sin_real @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% cos_squared_eq
thf(fact_6342_frac__lt__1,axiom,
    ! [X4: real] : ( ord_less_real @ ( archim2898591450579166408c_real @ X4 ) @ one_one_real ) ).

% frac_lt_1
thf(fact_6343_frac__lt__1,axiom,
    ! [X4: rat] : ( ord_less_rat @ ( archimedean_frac_rat @ X4 ) @ one_one_rat ) ).

% frac_lt_1
thf(fact_6344_frac__1__eq,axiom,
    ! [X4: real] :
      ( ( archim2898591450579166408c_real @ ( plus_plus_real @ X4 @ one_one_real ) )
      = ( archim2898591450579166408c_real @ X4 ) ) ).

% frac_1_eq
thf(fact_6345_frac__1__eq,axiom,
    ! [X4: rat] :
      ( ( archimedean_frac_rat @ ( plus_plus_rat @ X4 @ one_one_rat ) )
      = ( archimedean_frac_rat @ X4 ) ) ).

% frac_1_eq
thf(fact_6346_powr__mono2_H,axiom,
    ! [A: real,X4: real,Y3: real] :
      ( ( ord_less_eq_real @ A @ zero_zero_real )
     => ( ( ord_less_real @ zero_zero_real @ X4 )
       => ( ( ord_less_eq_real @ X4 @ Y3 )
         => ( ord_less_eq_real @ ( powr_real @ Y3 @ A ) @ ( powr_real @ X4 @ A ) ) ) ) ) ).

% powr_mono2'
thf(fact_6347_powr__less__mono2,axiom,
    ! [A: real,X4: real,Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ X4 )
       => ( ( ord_less_real @ X4 @ Y3 )
         => ( ord_less_real @ ( powr_real @ X4 @ A ) @ ( powr_real @ Y3 @ A ) ) ) ) ) ).

% powr_less_mono2
thf(fact_6348_gr__one__powr,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_real @ one_one_real @ X4 )
     => ( ( ord_less_real @ zero_zero_real @ Y3 )
       => ( ord_less_real @ one_one_real @ ( powr_real @ X4 @ Y3 ) ) ) ) ).

% gr_one_powr
thf(fact_6349_powr__inj,axiom,
    ! [A: real,X4: real,Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( ( powr_real @ A @ X4 )
            = ( powr_real @ A @ Y3 ) )
          = ( X4 = Y3 ) ) ) ) ).

% powr_inj
thf(fact_6350_powr__le1,axiom,
    ! [A: real,X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ X4 )
       => ( ( ord_less_eq_real @ X4 @ one_one_real )
         => ( ord_less_eq_real @ ( powr_real @ X4 @ A ) @ one_one_real ) ) ) ) ).

% powr_le1
thf(fact_6351_powr__mono__both,axiom,
    ! [A: real,B2: real,X4: real,Y3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ A @ B2 )
       => ( ( ord_less_eq_real @ one_one_real @ X4 )
         => ( ( ord_less_eq_real @ X4 @ Y3 )
           => ( ord_less_eq_real @ ( powr_real @ X4 @ A ) @ ( powr_real @ Y3 @ B2 ) ) ) ) ) ) ).

% powr_mono_both
thf(fact_6352_ge__one__powr__ge__zero,axiom,
    ! [X4: real,A: real] :
      ( ( ord_less_eq_real @ one_one_real @ X4 )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ord_less_eq_real @ one_one_real @ ( powr_real @ X4 @ A ) ) ) ) ).

% ge_one_powr_ge_zero
thf(fact_6353_powr__divide,axiom,
    ! [X4: real,Y3: real,A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
       => ( ( powr_real @ ( divide_divide_real @ X4 @ Y3 ) @ A )
          = ( divide_divide_real @ ( powr_real @ X4 @ A ) @ ( powr_real @ Y3 @ A ) ) ) ) ) ).

% powr_divide
thf(fact_6354_powr__mult,axiom,
    ! [X4: real,Y3: real,A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
       => ( ( powr_real @ ( times_times_real @ X4 @ Y3 ) @ A )
          = ( times_times_real @ ( powr_real @ X4 @ A ) @ ( powr_real @ Y3 @ A ) ) ) ) ) ).

% powr_mult
thf(fact_6355_divide__powr__uminus,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( divide_divide_real @ A @ ( powr_real @ B2 @ C ) )
      = ( times_times_real @ A @ ( powr_real @ B2 @ ( uminus_uminus_real @ C ) ) ) ) ).

% divide_powr_uminus
thf(fact_6356_sin__gt__zero,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( ord_less_real @ X4 @ pi )
       => ( ord_less_real @ zero_zero_real @ ( sin_real @ X4 ) ) ) ) ).

% sin_gt_zero
thf(fact_6357_sin__x__ge__neg__x,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ord_less_eq_real @ ( uminus_uminus_real @ X4 ) @ ( sin_real @ X4 ) ) ) ).

% sin_x_ge_neg_x
thf(fact_6358_sin__ge__zero,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_eq_real @ X4 @ pi )
       => ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ X4 ) ) ) ) ).

% sin_ge_zero
thf(fact_6359_log__base__powr,axiom,
    ! [A: real,B2: real,X4: real] :
      ( ( A != zero_zero_real )
     => ( ( log @ ( powr_real @ A @ B2 ) @ X4 )
        = ( divide_divide_real @ ( log @ A @ X4 ) @ B2 ) ) ) ).

% log_base_powr
thf(fact_6360_ln__powr,axiom,
    ! [X4: real,Y3: real] :
      ( ( X4 != zero_zero_real )
     => ( ( ln_ln_real @ ( powr_real @ X4 @ Y3 ) )
        = ( times_times_real @ Y3 @ ( ln_ln_real @ X4 ) ) ) ) ).

% ln_powr
thf(fact_6361_log__powr,axiom,
    ! [X4: real,B2: real,Y3: real] :
      ( ( X4 != zero_zero_real )
     => ( ( log @ B2 @ ( powr_real @ X4 @ Y3 ) )
        = ( times_times_real @ Y3 @ ( log @ B2 @ X4 ) ) ) ) ).

% log_powr
thf(fact_6362_sin__ge__minus__one,axiom,
    ! [X4: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( sin_real @ X4 ) ) ).

% sin_ge_minus_one
thf(fact_6363_cos__monotone__0__pi__le,axiom,
    ! [Y3: real,X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
     => ( ( ord_less_eq_real @ Y3 @ X4 )
       => ( ( ord_less_eq_real @ X4 @ pi )
         => ( ord_less_eq_real @ ( cos_real @ X4 ) @ ( cos_real @ Y3 ) ) ) ) ) ).

% cos_monotone_0_pi_le
thf(fact_6364_cos__mono__le__eq,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_eq_real @ X4 @ pi )
       => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
         => ( ( ord_less_eq_real @ Y3 @ pi )
           => ( ( ord_less_eq_real @ ( cos_real @ X4 ) @ ( cos_real @ Y3 ) )
              = ( ord_less_eq_real @ Y3 @ X4 ) ) ) ) ) ) ).

% cos_mono_le_eq
thf(fact_6365_cos__inj__pi,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_eq_real @ X4 @ pi )
       => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
         => ( ( ord_less_eq_real @ Y3 @ pi )
           => ( ( ( cos_real @ X4 )
                = ( cos_real @ Y3 ) )
             => ( X4 = Y3 ) ) ) ) ) ) ).

% cos_inj_pi
thf(fact_6366_cos__ge__minus__one,axiom,
    ! [X4: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( cos_real @ X4 ) ) ).

% cos_ge_minus_one
thf(fact_6367_abs__sin__le__one,axiom,
    ! [X4: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( sin_real @ X4 ) ) @ one_one_real ) ).

% abs_sin_le_one
thf(fact_6368_abs__cos__le__one,axiom,
    ! [X4: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( cos_real @ X4 ) ) @ one_one_real ) ).

% abs_cos_le_one
thf(fact_6369_cos__diff__cos,axiom,
    ! [W: complex,Z: complex] :
      ( ( minus_minus_complex @ ( cos_complex @ W ) @ ( cos_complex @ Z ) )
      = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sin_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ W @ Z ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) @ ( sin_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ Z @ W ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).

% cos_diff_cos
thf(fact_6370_cos__diff__cos,axiom,
    ! [W: real,Z: real] :
      ( ( minus_minus_real @ ( cos_real @ W ) @ ( cos_real @ Z ) )
      = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sin_real @ ( divide_divide_real @ ( plus_plus_real @ W @ Z ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ ( sin_real @ ( divide_divide_real @ ( minus_minus_real @ Z @ W ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% cos_diff_cos
thf(fact_6371_sin__diff__sin,axiom,
    ! [W: complex,Z: complex] :
      ( ( minus_minus_complex @ ( sin_complex @ W ) @ ( sin_complex @ Z ) )
      = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sin_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ W @ Z ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) @ ( cos_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ W @ Z ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).

% sin_diff_sin
thf(fact_6372_sin__diff__sin,axiom,
    ! [W: real,Z: real] :
      ( ( minus_minus_real @ ( sin_real @ W ) @ ( sin_real @ Z ) )
      = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sin_real @ ( divide_divide_real @ ( minus_minus_real @ W @ Z ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ ( cos_real @ ( divide_divide_real @ ( plus_plus_real @ W @ Z ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% sin_diff_sin
thf(fact_6373_sin__plus__sin,axiom,
    ! [W: complex,Z: complex] :
      ( ( plus_plus_complex @ ( sin_complex @ W ) @ ( sin_complex @ Z ) )
      = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sin_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ W @ Z ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) @ ( cos_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ W @ Z ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).

% sin_plus_sin
thf(fact_6374_sin__plus__sin,axiom,
    ! [W: real,Z: real] :
      ( ( plus_plus_real @ ( sin_real @ W ) @ ( sin_real @ Z ) )
      = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sin_real @ ( divide_divide_real @ ( plus_plus_real @ W @ Z ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ ( cos_real @ ( divide_divide_real @ ( minus_minus_real @ W @ Z ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% sin_plus_sin
thf(fact_6375_cos__times__sin,axiom,
    ! [W: complex,Z: complex] :
      ( ( times_times_complex @ ( cos_complex @ W ) @ ( sin_complex @ Z ) )
      = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( sin_complex @ ( plus_plus_complex @ W @ Z ) ) @ ( sin_complex @ ( minus_minus_complex @ W @ Z ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).

% cos_times_sin
thf(fact_6376_cos__times__sin,axiom,
    ! [W: real,Z: real] :
      ( ( times_times_real @ ( cos_real @ W ) @ ( sin_real @ Z ) )
      = ( divide_divide_real @ ( minus_minus_real @ ( sin_real @ ( plus_plus_real @ W @ Z ) ) @ ( sin_real @ ( minus_minus_real @ W @ Z ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% cos_times_sin
thf(fact_6377_sin__times__cos,axiom,
    ! [W: complex,Z: complex] :
      ( ( times_times_complex @ ( sin_complex @ W ) @ ( cos_complex @ Z ) )
      = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( sin_complex @ ( plus_plus_complex @ W @ Z ) ) @ ( sin_complex @ ( minus_minus_complex @ W @ Z ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).

% sin_times_cos
thf(fact_6378_sin__times__cos,axiom,
    ! [W: real,Z: real] :
      ( ( times_times_real @ ( sin_real @ W ) @ ( cos_real @ Z ) )
      = ( divide_divide_real @ ( plus_plus_real @ ( sin_real @ ( plus_plus_real @ W @ Z ) ) @ ( sin_real @ ( minus_minus_real @ W @ Z ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% sin_times_cos
thf(fact_6379_sin__times__sin,axiom,
    ! [W: complex,Z: complex] :
      ( ( times_times_complex @ ( sin_complex @ W ) @ ( sin_complex @ Z ) )
      = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( cos_complex @ ( minus_minus_complex @ W @ Z ) ) @ ( cos_complex @ ( plus_plus_complex @ W @ Z ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).

% sin_times_sin
thf(fact_6380_sin__times__sin,axiom,
    ! [W: real,Z: real] :
      ( ( times_times_real @ ( sin_real @ W ) @ ( sin_real @ Z ) )
      = ( divide_divide_real @ ( minus_minus_real @ ( cos_real @ ( minus_minus_real @ W @ Z ) ) @ ( cos_real @ ( plus_plus_real @ W @ Z ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% sin_times_sin
thf(fact_6381_cos__double,axiom,
    ! [X4: complex] :
      ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X4 ) )
      = ( minus_minus_complex @ ( power_power_complex @ ( cos_complex @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( sin_complex @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% cos_double
thf(fact_6382_cos__double,axiom,
    ! [X4: real] :
      ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X4 ) )
      = ( minus_minus_real @ ( power_power_real @ ( cos_real @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( sin_real @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% cos_double
thf(fact_6383_powr__add,axiom,
    ! [X4: real,A: real,B2: real] :
      ( ( powr_real @ X4 @ ( plus_plus_real @ A @ B2 ) )
      = ( times_times_real @ ( powr_real @ X4 @ A ) @ ( powr_real @ X4 @ B2 ) ) ) ).

% powr_add
thf(fact_6384_powr__diff,axiom,
    ! [W: real,Z1: real,Z22: real] :
      ( ( powr_real @ W @ ( minus_minus_real @ Z1 @ Z22 ) )
      = ( divide_divide_real @ ( powr_real @ W @ Z1 ) @ ( powr_real @ W @ Z22 ) ) ) ).

% powr_diff
thf(fact_6385_cos__double__sin,axiom,
    ! [W: complex] :
      ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ W ) )
      = ( minus_minus_complex @ one_one_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( power_power_complex @ ( sin_complex @ W ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% cos_double_sin
thf(fact_6386_cos__double__sin,axiom,
    ! [W: real] :
      ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ W ) )
      = ( minus_minus_real @ one_one_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ ( sin_real @ W ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% cos_double_sin
thf(fact_6387_powr__realpow,axiom,
    ! [X4: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( powr_real @ X4 @ ( semiri5074537144036343181t_real @ N ) )
        = ( power_power_real @ X4 @ N ) ) ) ).

% powr_realpow
thf(fact_6388_cos__two__neq__zero,axiom,
    ( ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
   != zero_zero_real ) ).

% cos_two_neq_zero
thf(fact_6389_less__log__iff,axiom,
    ! [B2: real,X4: real,Y3: real] :
      ( ( ord_less_real @ one_one_real @ B2 )
     => ( ( ord_less_real @ zero_zero_real @ X4 )
       => ( ( ord_less_real @ Y3 @ ( log @ B2 @ X4 ) )
          = ( ord_less_real @ ( powr_real @ B2 @ Y3 ) @ X4 ) ) ) ) ).

% less_log_iff
thf(fact_6390_log__less__iff,axiom,
    ! [B2: real,X4: real,Y3: real] :
      ( ( ord_less_real @ one_one_real @ B2 )
     => ( ( ord_less_real @ zero_zero_real @ X4 )
       => ( ( ord_less_real @ ( log @ B2 @ X4 ) @ Y3 )
          = ( ord_less_real @ X4 @ ( powr_real @ B2 @ Y3 ) ) ) ) ) ).

% log_less_iff
thf(fact_6391_less__powr__iff,axiom,
    ! [B2: real,X4: real,Y3: real] :
      ( ( ord_less_real @ one_one_real @ B2 )
     => ( ( ord_less_real @ zero_zero_real @ X4 )
       => ( ( ord_less_real @ X4 @ ( powr_real @ B2 @ Y3 ) )
          = ( ord_less_real @ ( log @ B2 @ X4 ) @ Y3 ) ) ) ) ).

% less_powr_iff
thf(fact_6392_powr__less__iff,axiom,
    ! [B2: real,X4: real,Y3: real] :
      ( ( ord_less_real @ one_one_real @ B2 )
     => ( ( ord_less_real @ zero_zero_real @ X4 )
       => ( ( ord_less_real @ ( powr_real @ B2 @ Y3 ) @ X4 )
          = ( ord_less_real @ Y3 @ ( log @ B2 @ X4 ) ) ) ) ) ).

% powr_less_iff
thf(fact_6393_cos__monotone__0__pi,axiom,
    ! [Y3: real,X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
     => ( ( ord_less_real @ Y3 @ X4 )
       => ( ( ord_less_eq_real @ X4 @ pi )
         => ( ord_less_real @ ( cos_real @ X4 ) @ ( cos_real @ Y3 ) ) ) ) ) ).

% cos_monotone_0_pi
thf(fact_6394_cos__mono__less__eq,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_eq_real @ X4 @ pi )
       => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
         => ( ( ord_less_eq_real @ Y3 @ pi )
           => ( ( ord_less_real @ ( cos_real @ X4 ) @ ( cos_real @ Y3 ) )
              = ( ord_less_real @ Y3 @ X4 ) ) ) ) ) ) ).

% cos_mono_less_eq
thf(fact_6395_sin__eq__0__pi,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ X4 )
     => ( ( ord_less_real @ X4 @ pi )
       => ( ( ( sin_real @ X4 )
            = zero_zero_real )
         => ( X4 = zero_zero_real ) ) ) ) ).

% sin_eq_0_pi
thf(fact_6396_sin__zero__pi__iff,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X4 ) @ pi )
     => ( ( ( sin_real @ X4 )
          = zero_zero_real )
        = ( X4 = zero_zero_real ) ) ) ).

% sin_zero_pi_iff
thf(fact_6397_cos__monotone__minus__pi__0_H,axiom,
    ! [Y3: real,X4: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ Y3 )
     => ( ( ord_less_eq_real @ Y3 @ X4 )
       => ( ( ord_less_eq_real @ X4 @ zero_zero_real )
         => ( ord_less_eq_real @ ( cos_real @ Y3 ) @ ( cos_real @ X4 ) ) ) ) ) ).

% cos_monotone_minus_pi_0'
thf(fact_6398_sin__zero__iff__int2,axiom,
    ! [X4: real] :
      ( ( ( sin_real @ X4 )
        = zero_zero_real )
      = ( ? [I2: int] :
            ( X4
            = ( times_times_real @ ( ring_1_of_int_real @ I2 ) @ pi ) ) ) ) ).

% sin_zero_iff_int2
thf(fact_6399_frac__def,axiom,
    ( archim2898591450579166408c_real
    = ( ^ [X: real] : ( minus_minus_real @ X @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X ) ) ) ) ) ).

% frac_def
thf(fact_6400_frac__def,axiom,
    ( archimedean_frac_rat
    = ( ^ [X: rat] : ( minus_minus_rat @ X @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X ) ) ) ) ) ).

% frac_def
thf(fact_6401_sincos__total__pi,axiom,
    ! [Y3: real,X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
     => ( ( ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          = one_one_real )
       => ? [T2: real] :
            ( ( ord_less_eq_real @ zero_zero_real @ T2 )
            & ( ord_less_eq_real @ T2 @ pi )
            & ( X4
              = ( cos_real @ T2 ) )
            & ( Y3
              = ( sin_real @ T2 ) ) ) ) ) ).

% sincos_total_pi
thf(fact_6402_sin__cos__sqrt,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ X4 ) )
     => ( ( sin_real @ X4 )
        = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ ( cos_real @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% sin_cos_sqrt
thf(fact_6403_sin__expansion__lemma,axiom,
    ! [X4: real,M: nat] :
      ( ( sin_real @ ( plus_plus_real @ X4 @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ M ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
      = ( cos_real @ ( plus_plus_real @ X4 @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% sin_expansion_lemma
thf(fact_6404_powr__minus__divide,axiom,
    ! [X4: real,A: real] :
      ( ( powr_real @ X4 @ ( uminus_uminus_real @ A ) )
      = ( divide_divide_real @ one_one_real @ ( powr_real @ X4 @ A ) ) ) ).

% powr_minus_divide
thf(fact_6405_cos__expansion__lemma,axiom,
    ! [X4: real,M: nat] :
      ( ( cos_real @ ( plus_plus_real @ X4 @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ M ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
      = ( uminus_uminus_real @ ( sin_real @ ( plus_plus_real @ X4 @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).

% cos_expansion_lemma
thf(fact_6406_powr__neg__one,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( powr_real @ X4 @ ( uminus_uminus_real @ one_one_real ) )
        = ( divide_divide_real @ one_one_real @ X4 ) ) ) ).

% powr_neg_one
thf(fact_6407_sin__gt__zero__02,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( ord_less_real @ X4 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
       => ( ord_less_real @ zero_zero_real @ ( sin_real @ X4 ) ) ) ) ).

% sin_gt_zero_02
thf(fact_6408_powr__mult__base,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( times_times_real @ X4 @ ( powr_real @ X4 @ Y3 ) )
        = ( powr_real @ X4 @ ( plus_plus_real @ one_one_real @ Y3 ) ) ) ) ).

% powr_mult_base
thf(fact_6409_powr__le__iff,axiom,
    ! [B2: real,X4: real,Y3: real] :
      ( ( ord_less_real @ one_one_real @ B2 )
     => ( ( ord_less_real @ zero_zero_real @ X4 )
       => ( ( ord_less_eq_real @ ( powr_real @ B2 @ Y3 ) @ X4 )
          = ( ord_less_eq_real @ Y3 @ ( log @ B2 @ X4 ) ) ) ) ) ).

% powr_le_iff
thf(fact_6410_le__powr__iff,axiom,
    ! [B2: real,X4: real,Y3: real] :
      ( ( ord_less_real @ one_one_real @ B2 )
     => ( ( ord_less_real @ zero_zero_real @ X4 )
       => ( ( ord_less_eq_real @ X4 @ ( powr_real @ B2 @ Y3 ) )
          = ( ord_less_eq_real @ ( log @ B2 @ X4 ) @ Y3 ) ) ) ) ).

% le_powr_iff
thf(fact_6411_log__le__iff,axiom,
    ! [B2: real,X4: real,Y3: real] :
      ( ( ord_less_real @ one_one_real @ B2 )
     => ( ( ord_less_real @ zero_zero_real @ X4 )
       => ( ( ord_less_eq_real @ ( log @ B2 @ X4 ) @ Y3 )
          = ( ord_less_eq_real @ X4 @ ( powr_real @ B2 @ Y3 ) ) ) ) ) ).

% log_le_iff
thf(fact_6412_le__log__iff,axiom,
    ! [B2: real,X4: real,Y3: real] :
      ( ( ord_less_real @ one_one_real @ B2 )
     => ( ( ord_less_real @ zero_zero_real @ X4 )
       => ( ( ord_less_eq_real @ Y3 @ ( log @ B2 @ X4 ) )
          = ( ord_less_eq_real @ ( powr_real @ B2 @ Y3 ) @ X4 ) ) ) ) ).

% le_log_iff
thf(fact_6413_cos__two__less__zero,axiom,
    ord_less_real @ ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ zero_zero_real ).

% cos_two_less_zero
thf(fact_6414_cos__two__le__zero,axiom,
    ord_less_eq_real @ ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ zero_zero_real ).

% cos_two_le_zero
thf(fact_6415_cos__is__zero,axiom,
    ? [X3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X3 )
      & ( ord_less_eq_real @ X3 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
      & ( ( cos_real @ X3 )
        = zero_zero_real )
      & ! [Y5: real] :
          ( ( ( ord_less_eq_real @ zero_zero_real @ Y5 )
            & ( ord_less_eq_real @ Y5 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
            & ( ( cos_real @ Y5 )
              = zero_zero_real ) )
         => ( Y5 = X3 ) ) ) ).

% cos_is_zero
thf(fact_6416_cos__monotone__minus__pi__0,axiom,
    ! [Y3: real,X4: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ Y3 )
     => ( ( ord_less_real @ Y3 @ X4 )
       => ( ( ord_less_eq_real @ X4 @ zero_zero_real )
         => ( ord_less_real @ ( cos_real @ Y3 ) @ ( cos_real @ X4 ) ) ) ) ) ).

% cos_monotone_minus_pi_0
thf(fact_6417_cos__total,axiom,
    ! [Y3: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
     => ( ( ord_less_eq_real @ Y3 @ one_one_real )
       => ? [X3: real] :
            ( ( ord_less_eq_real @ zero_zero_real @ X3 )
            & ( ord_less_eq_real @ X3 @ pi )
            & ( ( cos_real @ X3 )
              = Y3 )
            & ! [Y5: real] :
                ( ( ( ord_less_eq_real @ zero_zero_real @ Y5 )
                  & ( ord_less_eq_real @ Y5 @ pi )
                  & ( ( cos_real @ Y5 )
                    = Y3 ) )
               => ( Y5 = X3 ) ) ) ) ) ).

% cos_total
thf(fact_6418_frac__eq,axiom,
    ! [X4: real] :
      ( ( ( archim2898591450579166408c_real @ X4 )
        = X4 )
      = ( ( ord_less_eq_real @ zero_zero_real @ X4 )
        & ( ord_less_real @ X4 @ one_one_real ) ) ) ).

% frac_eq
thf(fact_6419_frac__eq,axiom,
    ! [X4: rat] :
      ( ( ( archimedean_frac_rat @ X4 )
        = X4 )
      = ( ( ord_less_eq_rat @ zero_zero_rat @ X4 )
        & ( ord_less_rat @ X4 @ one_one_rat ) ) ) ).

% frac_eq
thf(fact_6420_frac__add,axiom,
    ! [X4: real,Y3: real] :
      ( ( ( ord_less_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X4 ) @ ( archim2898591450579166408c_real @ Y3 ) ) @ one_one_real )
       => ( ( archim2898591450579166408c_real @ ( plus_plus_real @ X4 @ Y3 ) )
          = ( plus_plus_real @ ( archim2898591450579166408c_real @ X4 ) @ ( archim2898591450579166408c_real @ Y3 ) ) ) )
      & ( ~ ( ord_less_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X4 ) @ ( archim2898591450579166408c_real @ Y3 ) ) @ one_one_real )
       => ( ( archim2898591450579166408c_real @ ( plus_plus_real @ X4 @ Y3 ) )
          = ( minus_minus_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X4 ) @ ( archim2898591450579166408c_real @ Y3 ) ) @ one_one_real ) ) ) ) ).

% frac_add
thf(fact_6421_frac__add,axiom,
    ! [X4: rat,Y3: rat] :
      ( ( ( ord_less_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X4 ) @ ( archimedean_frac_rat @ Y3 ) ) @ one_one_rat )
       => ( ( archimedean_frac_rat @ ( plus_plus_rat @ X4 @ Y3 ) )
          = ( plus_plus_rat @ ( archimedean_frac_rat @ X4 ) @ ( archimedean_frac_rat @ Y3 ) ) ) )
      & ( ~ ( ord_less_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X4 ) @ ( archimedean_frac_rat @ Y3 ) ) @ one_one_rat )
       => ( ( archimedean_frac_rat @ ( plus_plus_rat @ X4 @ Y3 ) )
          = ( minus_minus_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X4 ) @ ( archimedean_frac_rat @ Y3 ) ) @ one_one_rat ) ) ) ) ).

% frac_add
thf(fact_6422_sincos__total__pi__half,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
       => ( ( ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
            = one_one_real )
         => ? [T2: real] :
              ( ( ord_less_eq_real @ zero_zero_real @ T2 )
              & ( ord_less_eq_real @ T2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
              & ( X4
                = ( cos_real @ T2 ) )
              & ( Y3
                = ( sin_real @ T2 ) ) ) ) ) ) ).

% sincos_total_pi_half
thf(fact_6423_sincos__total__2pi__le,axiom,
    ! [X4: real,Y3: real] :
      ( ( ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = one_one_real )
     => ? [T2: real] :
          ( ( ord_less_eq_real @ zero_zero_real @ T2 )
          & ( ord_less_eq_real @ T2 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
          & ( X4
            = ( cos_real @ T2 ) )
          & ( Y3
            = ( sin_real @ T2 ) ) ) ) ).

% sincos_total_2pi_le
thf(fact_6424_sincos__total__2pi,axiom,
    ! [X4: real,Y3: real] :
      ( ( ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = one_one_real )
     => ~ ! [T2: real] :
            ( ( ord_less_eq_real @ zero_zero_real @ T2 )
           => ( ( ord_less_real @ T2 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
             => ( ( X4
                  = ( cos_real @ T2 ) )
               => ( Y3
                 != ( sin_real @ T2 ) ) ) ) ) ) ).

% sincos_total_2pi
thf(fact_6425_ln__powr__bound,axiom,
    ! [X4: real,A: real] :
      ( ( ord_less_eq_real @ one_one_real @ X4 )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ( ord_less_eq_real @ ( ln_ln_real @ X4 ) @ ( divide_divide_real @ ( powr_real @ X4 @ A ) @ A ) ) ) ) ).

% ln_powr_bound
thf(fact_6426_ln__powr__bound2,axiom,
    ! [X4: real,A: real] :
      ( ( ord_less_real @ one_one_real @ X4 )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ( ord_less_eq_real @ ( powr_real @ ( ln_ln_real @ X4 ) @ A ) @ ( times_times_real @ ( powr_real @ A @ A ) @ X4 ) ) ) ) ).

% ln_powr_bound2
thf(fact_6427_log__add__eq__powr,axiom,
    ! [B2: real,X4: real,Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ B2 )
     => ( ( B2 != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X4 )
         => ( ( plus_plus_real @ ( log @ B2 @ X4 ) @ Y3 )
            = ( log @ B2 @ ( times_times_real @ X4 @ ( powr_real @ B2 @ Y3 ) ) ) ) ) ) ) ).

% log_add_eq_powr
thf(fact_6428_add__log__eq__powr,axiom,
    ! [B2: real,X4: real,Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ B2 )
     => ( ( B2 != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X4 )
         => ( ( plus_plus_real @ Y3 @ ( log @ B2 @ X4 ) )
            = ( log @ B2 @ ( times_times_real @ ( powr_real @ B2 @ Y3 ) @ X4 ) ) ) ) ) ) ).

% add_log_eq_powr
thf(fact_6429_minus__log__eq__powr,axiom,
    ! [B2: real,X4: real,Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ B2 )
     => ( ( B2 != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X4 )
         => ( ( minus_minus_real @ Y3 @ ( log @ B2 @ X4 ) )
            = ( log @ B2 @ ( divide_divide_real @ ( powr_real @ B2 @ Y3 ) @ X4 ) ) ) ) ) ) ).

% minus_log_eq_powr
thf(fact_6430_sin__pi__divide__n__ge__0,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ ( divide_divide_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).

% sin_pi_divide_n_ge_0
thf(fact_6431_sin__45,axiom,
    ( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
    = ( divide_divide_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% sin_45
thf(fact_6432_cos__45,axiom,
    ( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
    = ( divide_divide_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% cos_45
thf(fact_6433_powr__def,axiom,
    ( powr_real
    = ( ^ [X: real,A4: real] : ( if_real @ ( X = zero_zero_real ) @ zero_zero_real @ ( exp_real @ ( times_times_real @ A4 @ ( ln_ln_real @ X ) ) ) ) ) ) ).

% powr_def
thf(fact_6434_cos__plus__cos,axiom,
    ! [W: complex,Z: complex] :
      ( ( plus_plus_complex @ ( cos_complex @ W ) @ ( cos_complex @ Z ) )
      = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( cos_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ W @ Z ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) @ ( cos_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ W @ Z ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).

% cos_plus_cos
thf(fact_6435_cos__plus__cos,axiom,
    ! [W: real,Z: real] :
      ( ( plus_plus_real @ ( cos_real @ W ) @ ( cos_real @ Z ) )
      = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( cos_real @ ( divide_divide_real @ ( plus_plus_real @ W @ Z ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ ( cos_real @ ( divide_divide_real @ ( minus_minus_real @ W @ Z ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% cos_plus_cos
thf(fact_6436_cos__times__cos,axiom,
    ! [W: complex,Z: complex] :
      ( ( times_times_complex @ ( cos_complex @ W ) @ ( cos_complex @ Z ) )
      = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( cos_complex @ ( minus_minus_complex @ W @ Z ) ) @ ( cos_complex @ ( plus_plus_complex @ W @ Z ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).

% cos_times_cos
thf(fact_6437_cos__times__cos,axiom,
    ! [W: real,Z: real] :
      ( ( times_times_real @ ( cos_real @ W ) @ ( cos_real @ Z ) )
      = ( divide_divide_real @ ( plus_plus_real @ ( cos_real @ ( minus_minus_real @ W @ Z ) ) @ ( cos_real @ ( plus_plus_real @ W @ Z ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% cos_times_cos
thf(fact_6438_sin__gt__zero2,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_real @ zero_zero_real @ ( sin_real @ X4 ) ) ) ) ).

% sin_gt_zero2
thf(fact_6439_sin__lt__zero,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ pi @ X4 )
     => ( ( ord_less_real @ X4 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
       => ( ord_less_real @ ( sin_real @ X4 ) @ zero_zero_real ) ) ) ).

% sin_lt_zero
thf(fact_6440_cos__double__less__one,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( ord_less_real @ X4 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
       => ( ord_less_real @ ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X4 ) ) @ one_one_real ) ) ) ).

% cos_double_less_one
thf(fact_6441_sin__30,axiom,
    ( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ one ) ) ) ) )
    = ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% sin_30
thf(fact_6442_cos__gt__zero,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_real @ zero_zero_real @ ( cos_real @ X4 ) ) ) ) ).

% cos_gt_zero
thf(fact_6443_sin__monotone__2pi__le,axiom,
    ! [Y3: real,X4: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
     => ( ( ord_less_eq_real @ Y3 @ X4 )
       => ( ( ord_less_eq_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
         => ( ord_less_eq_real @ ( sin_real @ Y3 ) @ ( sin_real @ X4 ) ) ) ) ) ).

% sin_monotone_2pi_le
thf(fact_6444_sin__mono__le__eq,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X4 )
     => ( ( ord_less_eq_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
         => ( ( ord_less_eq_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ord_less_eq_real @ ( sin_real @ X4 ) @ ( sin_real @ Y3 ) )
              = ( ord_less_eq_real @ X4 @ Y3 ) ) ) ) ) ) ).

% sin_mono_le_eq
thf(fact_6445_sin__inj__pi,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X4 )
     => ( ( ord_less_eq_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
         => ( ( ord_less_eq_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ( sin_real @ X4 )
                = ( sin_real @ Y3 ) )
             => ( X4 = Y3 ) ) ) ) ) ) ).

% sin_inj_pi
thf(fact_6446_log__minus__eq__powr,axiom,
    ! [B2: real,X4: real,Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ B2 )
     => ( ( B2 != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X4 )
         => ( ( minus_minus_real @ ( log @ B2 @ X4 ) @ Y3 )
            = ( log @ B2 @ ( times_times_real @ X4 @ ( powr_real @ B2 @ ( uminus_uminus_real @ Y3 ) ) ) ) ) ) ) ) ).

% log_minus_eq_powr
thf(fact_6447_cos__60,axiom,
    ( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) )
    = ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% cos_60
thf(fact_6448_sin__60,axiom,
    ( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) )
    = ( divide_divide_real @ ( sqrt @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% sin_60
thf(fact_6449_cos__30,axiom,
    ( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ one ) ) ) ) )
    = ( divide_divide_real @ ( sqrt @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% cos_30
thf(fact_6450_cos__one__2pi__int,axiom,
    ! [X4: real] :
      ( ( ( cos_real @ X4 )
        = one_one_real )
      = ( ? [X: int] :
            ( X4
            = ( times_times_real @ ( times_times_real @ ( ring_1_of_int_real @ X ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) ) ) ) ).

% cos_one_2pi_int
thf(fact_6451_cos__double__cos,axiom,
    ! [W: complex] :
      ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ W ) )
      = ( minus_minus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( power_power_complex @ ( cos_complex @ W ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_complex ) ) ).

% cos_double_cos
thf(fact_6452_cos__double__cos,axiom,
    ! [W: real] :
      ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ W ) )
      = ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ ( cos_real @ W ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_real ) ) ).

% cos_double_cos
thf(fact_6453_cos__treble__cos,axiom,
    ! [X4: complex] :
      ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit1 @ one ) ) @ X4 ) )
      = ( minus_minus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( cos_complex @ X4 ) @ ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit1 @ one ) ) @ ( cos_complex @ X4 ) ) ) ) ).

% cos_treble_cos
thf(fact_6454_cos__treble__cos,axiom,
    ! [X4: real] :
      ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ X4 ) )
      = ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( cos_real @ X4 ) @ ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ ( cos_real @ X4 ) ) ) ) ).

% cos_treble_cos
thf(fact_6455_powr__half__sqrt,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( powr_real @ X4 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
        = ( sqrt @ X4 ) ) ) ).

% powr_half_sqrt
thf(fact_6456_powr__neg__numeral,axiom,
    ! [X4: real,N: num] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( powr_real @ X4 @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
        = ( divide_divide_real @ one_one_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ N ) ) ) ) ) ).

% powr_neg_numeral
thf(fact_6457_sin__le__zero,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ pi @ X4 )
     => ( ( ord_less_real @ X4 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
       => ( ord_less_eq_real @ ( sin_real @ X4 ) @ zero_zero_real ) ) ) ).

% sin_le_zero
thf(fact_6458_sin__less__zero,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X4 )
     => ( ( ord_less_real @ X4 @ zero_zero_real )
       => ( ord_less_real @ ( sin_real @ X4 ) @ zero_zero_real ) ) ) ).

% sin_less_zero
thf(fact_6459_sin__monotone__2pi,axiom,
    ! [Y3: real,X4: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
     => ( ( ord_less_real @ Y3 @ X4 )
       => ( ( ord_less_eq_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
         => ( ord_less_real @ ( sin_real @ Y3 ) @ ( sin_real @ X4 ) ) ) ) ) ).

% sin_monotone_2pi
thf(fact_6460_sin__mono__less__eq,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X4 )
     => ( ( ord_less_eq_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
         => ( ( ord_less_eq_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ord_less_real @ ( sin_real @ X4 ) @ ( sin_real @ Y3 ) )
              = ( ord_less_real @ X4 @ Y3 ) ) ) ) ) ) ).

% sin_mono_less_eq
thf(fact_6461_sin__total,axiom,
    ! [Y3: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
     => ( ( ord_less_eq_real @ Y3 @ one_one_real )
       => ? [X3: real] :
            ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
            & ( ord_less_eq_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
            & ( ( sin_real @ X3 )
              = Y3 )
            & ! [Y5: real] :
                ( ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y5 )
                  & ( ord_less_eq_real @ Y5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
                  & ( ( sin_real @ Y5 )
                    = Y3 ) )
               => ( Y5 = X3 ) ) ) ) ) ).

% sin_total
thf(fact_6462_cos__gt__zero__pi,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X4 )
     => ( ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_real @ zero_zero_real @ ( cos_real @ X4 ) ) ) ) ).

% cos_gt_zero_pi
thf(fact_6463_cos__ge__zero,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X4 )
     => ( ( ord_less_eq_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_eq_real @ zero_zero_real @ ( cos_real @ X4 ) ) ) ) ).

% cos_ge_zero
thf(fact_6464_cos__one__2pi,axiom,
    ! [X4: real] :
      ( ( ( cos_real @ X4 )
        = one_one_real )
      = ( ? [X: nat] :
            ( X4
            = ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ X ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) )
        | ? [X: nat] :
            ( X4
            = ( uminus_uminus_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ X ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) ) ) ) ) ).

% cos_one_2pi
thf(fact_6465_floor__add,axiom,
    ! [X4: real,Y3: real] :
      ( ( ( ord_less_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X4 ) @ ( archim2898591450579166408c_real @ Y3 ) ) @ one_one_real )
       => ( ( archim6058952711729229775r_real @ ( plus_plus_real @ X4 @ Y3 ) )
          = ( plus_plus_int @ ( archim6058952711729229775r_real @ X4 ) @ ( archim6058952711729229775r_real @ Y3 ) ) ) )
      & ( ~ ( ord_less_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X4 ) @ ( archim2898591450579166408c_real @ Y3 ) ) @ one_one_real )
       => ( ( archim6058952711729229775r_real @ ( plus_plus_real @ X4 @ Y3 ) )
          = ( plus_plus_int @ ( plus_plus_int @ ( archim6058952711729229775r_real @ X4 ) @ ( archim6058952711729229775r_real @ Y3 ) ) @ one_one_int ) ) ) ) ).

% floor_add
thf(fact_6466_floor__add,axiom,
    ! [X4: rat,Y3: rat] :
      ( ( ( ord_less_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X4 ) @ ( archimedean_frac_rat @ Y3 ) ) @ one_one_rat )
       => ( ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X4 @ Y3 ) )
          = ( plus_plus_int @ ( archim3151403230148437115or_rat @ X4 ) @ ( archim3151403230148437115or_rat @ Y3 ) ) ) )
      & ( ~ ( ord_less_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X4 ) @ ( archimedean_frac_rat @ Y3 ) ) @ one_one_rat )
       => ( ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X4 @ Y3 ) )
          = ( plus_plus_int @ ( plus_plus_int @ ( archim3151403230148437115or_rat @ X4 ) @ ( archim3151403230148437115or_rat @ Y3 ) ) @ one_one_int ) ) ) ) ).

% floor_add
thf(fact_6467_sin__pi__divide__n__gt__0,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ord_less_real @ zero_zero_real @ ( sin_real @ ( divide_divide_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).

% sin_pi_divide_n_gt_0
thf(fact_6468_sin__arctan,axiom,
    ! [X4: real] :
      ( ( sin_real @ ( arctan @ X4 ) )
      = ( divide_divide_real @ X4 @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% sin_arctan
thf(fact_6469_floor__log__eq__powr__iff,axiom,
    ! [X4: real,B2: real,K: int] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( ord_less_real @ one_one_real @ B2 )
       => ( ( ( archim6058952711729229775r_real @ ( log @ B2 @ X4 ) )
            = K )
          = ( ( ord_less_eq_real @ ( powr_real @ B2 @ ( ring_1_of_int_real @ K ) ) @ X4 )
            & ( ord_less_real @ X4 @ ( powr_real @ B2 @ ( ring_1_of_int_real @ ( plus_plus_int @ K @ one_one_int ) ) ) ) ) ) ) ) ).

% floor_log_eq_powr_iff
thf(fact_6470_cos__arctan,axiom,
    ! [X4: real] :
      ( ( cos_real @ ( arctan @ X4 ) )
      = ( divide_divide_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% cos_arctan
thf(fact_6471_cot__gt__zero,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_real @ zero_zero_real @ ( cot_real @ X4 ) ) ) ) ).

% cot_gt_zero
thf(fact_6472_sin__zero__iff__int,axiom,
    ! [X4: real] :
      ( ( ( sin_real @ X4 )
        = zero_zero_real )
      = ( ? [I2: int] :
            ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ I2 )
            & ( X4
              = ( times_times_real @ ( ring_1_of_int_real @ I2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% sin_zero_iff_int
thf(fact_6473_cos__zero__iff__int,axiom,
    ! [X4: real] :
      ( ( ( cos_real @ X4 )
        = zero_zero_real )
      = ( ? [I2: int] :
            ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ I2 )
            & ( X4
              = ( times_times_real @ ( ring_1_of_int_real @ I2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% cos_zero_iff_int
thf(fact_6474_sin__zero__lemma,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ( sin_real @ X4 )
          = zero_zero_real )
       => ? [N4: nat] :
            ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 )
            & ( X4
              = ( times_times_real @ ( semiri5074537144036343181t_real @ N4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% sin_zero_lemma
thf(fact_6475_sin__zero__iff,axiom,
    ! [X4: real] :
      ( ( ( sin_real @ X4 )
        = zero_zero_real )
      = ( ? [N2: nat] :
            ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
            & ( X4
              = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) )
        | ? [N2: nat] :
            ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
            & ( X4
              = ( uminus_uminus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% sin_zero_iff
thf(fact_6476_cos__zero__lemma,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ( cos_real @ X4 )
          = zero_zero_real )
       => ? [N4: nat] :
            ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 )
            & ( X4
              = ( times_times_real @ ( semiri5074537144036343181t_real @ N4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% cos_zero_lemma
thf(fact_6477_tan__double,axiom,
    ! [X4: complex] :
      ( ( ( cos_complex @ X4 )
       != zero_zero_complex )
     => ( ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X4 ) )
         != zero_zero_complex )
       => ( ( tan_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X4 ) )
          = ( divide1717551699836669952omplex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( tan_complex @ X4 ) ) @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ ( tan_complex @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% tan_double
thf(fact_6478_tan__double,axiom,
    ! [X4: real] :
      ( ( ( cos_real @ X4 )
       != zero_zero_real )
     => ( ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X4 ) )
         != zero_zero_real )
       => ( ( tan_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X4 ) )
          = ( divide_divide_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( tan_real @ X4 ) ) @ ( minus_minus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% tan_double
thf(fact_6479_sin__tan,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
     => ( ( sin_real @ X4 )
        = ( divide_divide_real @ ( tan_real @ X4 ) @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% sin_tan
thf(fact_6480_cos__tan,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
     => ( ( cos_real @ X4 )
        = ( divide_divide_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% cos_tan
thf(fact_6481_arcosh__def,axiom,
    ( arcosh_real
    = ( ^ [X: real] : ( ln_ln_real @ ( plus_plus_real @ X @ ( powr_real @ ( minus_minus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) @ ( real_V1803761363581548252l_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% arcosh_def
thf(fact_6482_complex__unimodular__polar,axiom,
    ! [Z: complex] :
      ( ( ( real_V1022390504157884413omplex @ Z )
        = one_one_real )
     => ~ ! [T2: real] :
            ( ( ord_less_eq_real @ zero_zero_real @ T2 )
           => ( ( ord_less_real @ T2 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
             => ( Z
               != ( complex2 @ ( cos_real @ T2 ) @ ( sin_real @ T2 ) ) ) ) ) ) ).

% complex_unimodular_polar
thf(fact_6483_cos__arcsin,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X4 )
     => ( ( ord_less_eq_real @ X4 @ one_one_real )
       => ( ( cos_real @ ( arcsin @ X4 ) )
          = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% cos_arcsin
thf(fact_6484_of__real__1,axiom,
    ( ( real_V1803761363581548252l_real @ one_one_real )
    = one_one_real ) ).

% of_real_1
thf(fact_6485_of__real__1,axiom,
    ( ( real_V4546457046886955230omplex @ one_one_real )
    = one_one_complex ) ).

% of_real_1
thf(fact_6486_of__real__eq__1__iff,axiom,
    ! [X4: real] :
      ( ( ( real_V1803761363581548252l_real @ X4 )
        = one_one_real )
      = ( X4 = one_one_real ) ) ).

% of_real_eq_1_iff
thf(fact_6487_of__real__eq__1__iff,axiom,
    ! [X4: real] :
      ( ( ( real_V4546457046886955230omplex @ X4 )
        = one_one_complex )
      = ( X4 = one_one_real ) ) ).

% of_real_eq_1_iff
thf(fact_6488_of__real__mult,axiom,
    ! [X4: real,Y3: real] :
      ( ( real_V1803761363581548252l_real @ ( times_times_real @ X4 @ Y3 ) )
      = ( times_times_real @ ( real_V1803761363581548252l_real @ X4 ) @ ( real_V1803761363581548252l_real @ Y3 ) ) ) ).

% of_real_mult
thf(fact_6489_of__real__mult,axiom,
    ! [X4: real,Y3: real] :
      ( ( real_V4546457046886955230omplex @ ( times_times_real @ X4 @ Y3 ) )
      = ( times_times_complex @ ( real_V4546457046886955230omplex @ X4 ) @ ( real_V4546457046886955230omplex @ Y3 ) ) ) ).

% of_real_mult
thf(fact_6490_of__real__numeral,axiom,
    ! [W: num] :
      ( ( real_V1803761363581548252l_real @ ( numeral_numeral_real @ W ) )
      = ( numeral_numeral_real @ W ) ) ).

% of_real_numeral
thf(fact_6491_of__real__numeral,axiom,
    ! [W: num] :
      ( ( real_V4546457046886955230omplex @ ( numeral_numeral_real @ W ) )
      = ( numera6690914467698888265omplex @ W ) ) ).

% of_real_numeral
thf(fact_6492_of__real__divide,axiom,
    ! [X4: real,Y3: real] :
      ( ( real_V1803761363581548252l_real @ ( divide_divide_real @ X4 @ Y3 ) )
      = ( divide_divide_real @ ( real_V1803761363581548252l_real @ X4 ) @ ( real_V1803761363581548252l_real @ Y3 ) ) ) ).

% of_real_divide
thf(fact_6493_of__real__divide,axiom,
    ! [X4: real,Y3: real] :
      ( ( real_V4546457046886955230omplex @ ( divide_divide_real @ X4 @ Y3 ) )
      = ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ X4 ) @ ( real_V4546457046886955230omplex @ Y3 ) ) ) ).

% of_real_divide
thf(fact_6494_of__real__power,axiom,
    ! [X4: real,N: nat] :
      ( ( real_V1803761363581548252l_real @ ( power_power_real @ X4 @ N ) )
      = ( power_power_real @ ( real_V1803761363581548252l_real @ X4 ) @ N ) ) ).

% of_real_power
thf(fact_6495_of__real__power,axiom,
    ! [X4: real,N: nat] :
      ( ( real_V4546457046886955230omplex @ ( power_power_real @ X4 @ N ) )
      = ( power_power_complex @ ( real_V4546457046886955230omplex @ X4 ) @ N ) ) ).

% of_real_power
thf(fact_6496_of__real__add,axiom,
    ! [X4: real,Y3: real] :
      ( ( real_V1803761363581548252l_real @ ( plus_plus_real @ X4 @ Y3 ) )
      = ( plus_plus_real @ ( real_V1803761363581548252l_real @ X4 ) @ ( real_V1803761363581548252l_real @ Y3 ) ) ) ).

% of_real_add
thf(fact_6497_of__real__add,axiom,
    ! [X4: real,Y3: real] :
      ( ( real_V4546457046886955230omplex @ ( plus_plus_real @ X4 @ Y3 ) )
      = ( plus_plus_complex @ ( real_V4546457046886955230omplex @ X4 ) @ ( real_V4546457046886955230omplex @ Y3 ) ) ) ).

% of_real_add
thf(fact_6498_of__real__diff,axiom,
    ! [X4: real,Y3: real] :
      ( ( real_V1803761363581548252l_real @ ( minus_minus_real @ X4 @ Y3 ) )
      = ( minus_minus_real @ ( real_V1803761363581548252l_real @ X4 ) @ ( real_V1803761363581548252l_real @ Y3 ) ) ) ).

% of_real_diff
thf(fact_6499_of__real__diff,axiom,
    ! [X4: real,Y3: real] :
      ( ( real_V4546457046886955230omplex @ ( minus_minus_real @ X4 @ Y3 ) )
      = ( minus_minus_complex @ ( real_V4546457046886955230omplex @ X4 ) @ ( real_V4546457046886955230omplex @ Y3 ) ) ) ).

% of_real_diff
thf(fact_6500_tan__periodic__pi,axiom,
    ! [X4: real] :
      ( ( tan_real @ ( plus_plus_real @ X4 @ pi ) )
      = ( tan_real @ X4 ) ) ).

% tan_periodic_pi
thf(fact_6501_of__real__neg__numeral,axiom,
    ! [W: num] :
      ( ( real_V1803761363581548252l_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
      = ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ).

% of_real_neg_numeral
thf(fact_6502_of__real__neg__numeral,axiom,
    ! [W: num] :
      ( ( real_V4546457046886955230omplex @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ).

% of_real_neg_numeral
thf(fact_6503_cos__of__real__pi,axiom,
    ( ( cos_real @ ( real_V1803761363581548252l_real @ pi ) )
    = ( uminus_uminus_real @ one_one_real ) ) ).

% cos_of_real_pi
thf(fact_6504_cos__of__real__pi,axiom,
    ( ( cos_complex @ ( real_V4546457046886955230omplex @ pi ) )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% cos_of_real_pi
thf(fact_6505_tan__npi,axiom,
    ! [N: nat] :
      ( ( tan_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) )
      = zero_zero_real ) ).

% tan_npi
thf(fact_6506_tan__periodic__n,axiom,
    ! [X4: real,N: num] :
      ( ( tan_real @ ( plus_plus_real @ X4 @ ( times_times_real @ ( numeral_numeral_real @ N ) @ pi ) ) )
      = ( tan_real @ X4 ) ) ).

% tan_periodic_n
thf(fact_6507_sin__arcsin,axiom,
    ! [Y3: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
     => ( ( ord_less_eq_real @ Y3 @ one_one_real )
       => ( ( sin_real @ ( arcsin @ Y3 ) )
          = Y3 ) ) ) ).

% sin_arcsin
thf(fact_6508_tan__periodic__nat,axiom,
    ! [X4: real,N: nat] :
      ( ( tan_real @ ( plus_plus_real @ X4 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) ) )
      = ( tan_real @ X4 ) ) ).

% tan_periodic_nat
thf(fact_6509_tan__periodic__int,axiom,
    ! [X4: real,I: int] :
      ( ( tan_real @ ( plus_plus_real @ X4 @ ( times_times_real @ ( ring_1_of_int_real @ I ) @ pi ) ) )
      = ( tan_real @ X4 ) ) ).

% tan_periodic_int
thf(fact_6510_norm__cos__sin,axiom,
    ! [T: real] :
      ( ( real_V1022390504157884413omplex @ ( complex2 @ ( cos_real @ T ) @ ( sin_real @ T ) ) )
      = one_one_real ) ).

% norm_cos_sin
thf(fact_6511_norm__of__real__add1,axiom,
    ! [X4: real] :
      ( ( real_V7735802525324610683m_real @ ( plus_plus_real @ ( real_V1803761363581548252l_real @ X4 ) @ one_one_real ) )
      = ( abs_abs_real @ ( plus_plus_real @ X4 @ one_one_real ) ) ) ).

% norm_of_real_add1
thf(fact_6512_norm__of__real__add1,axiom,
    ! [X4: real] :
      ( ( real_V1022390504157884413omplex @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ X4 ) @ one_one_complex ) )
      = ( abs_abs_real @ ( plus_plus_real @ X4 @ one_one_real ) ) ) ).

% norm_of_real_add1
thf(fact_6513_norm__of__real__addn,axiom,
    ! [X4: real,B2: num] :
      ( ( real_V7735802525324610683m_real @ ( plus_plus_real @ ( real_V1803761363581548252l_real @ X4 ) @ ( numeral_numeral_real @ B2 ) ) )
      = ( abs_abs_real @ ( plus_plus_real @ X4 @ ( numeral_numeral_real @ B2 ) ) ) ) ).

% norm_of_real_addn
thf(fact_6514_norm__of__real__addn,axiom,
    ! [X4: real,B2: num] :
      ( ( real_V1022390504157884413omplex @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ X4 ) @ ( numera6690914467698888265omplex @ B2 ) ) )
      = ( abs_abs_real @ ( plus_plus_real @ X4 @ ( numeral_numeral_real @ B2 ) ) ) ) ).

% norm_of_real_addn
thf(fact_6515_arcsin__1,axiom,
    ( ( arcsin @ one_one_real )
    = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% arcsin_1
thf(fact_6516_tan__periodic,axiom,
    ! [X4: real] :
      ( ( tan_real @ ( plus_plus_real @ X4 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
      = ( tan_real @ X4 ) ) ).

% tan_periodic
thf(fact_6517_cos__of__real__pi__half,axiom,
    ( ( cos_real @ ( divide_divide_real @ ( real_V1803761363581548252l_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
    = zero_zero_real ) ).

% cos_of_real_pi_half
thf(fact_6518_cos__of__real__pi__half,axiom,
    ( ( cos_complex @ ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) )
    = zero_zero_complex ) ).

% cos_of_real_pi_half
thf(fact_6519_sin__of__real__pi__half,axiom,
    ( ( sin_real @ ( divide_divide_real @ ( real_V1803761363581548252l_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
    = one_one_real ) ).

% sin_of_real_pi_half
thf(fact_6520_sin__of__real__pi__half,axiom,
    ( ( sin_complex @ ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) )
    = one_one_complex ) ).

% sin_of_real_pi_half
thf(fact_6521_arcsin__minus__1,axiom,
    ( ( arcsin @ ( uminus_uminus_real @ one_one_real ) )
    = ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% arcsin_minus_1
thf(fact_6522_complex__of__real__mult__Complex,axiom,
    ! [R2: real,X4: real,Y3: real] :
      ( ( times_times_complex @ ( real_V4546457046886955230omplex @ R2 ) @ ( complex2 @ X4 @ Y3 ) )
      = ( complex2 @ ( times_times_real @ R2 @ X4 ) @ ( times_times_real @ R2 @ Y3 ) ) ) ).

% complex_of_real_mult_Complex
thf(fact_6523_Complex__mult__complex__of__real,axiom,
    ! [X4: real,Y3: real,R2: real] :
      ( ( times_times_complex @ ( complex2 @ X4 @ Y3 ) @ ( real_V4546457046886955230omplex @ R2 ) )
      = ( complex2 @ ( times_times_real @ X4 @ R2 ) @ ( times_times_real @ Y3 @ R2 ) ) ) ).

% Complex_mult_complex_of_real
thf(fact_6524_complex__of__real__add__Complex,axiom,
    ! [R2: real,X4: real,Y3: real] :
      ( ( plus_plus_complex @ ( real_V4546457046886955230omplex @ R2 ) @ ( complex2 @ X4 @ Y3 ) )
      = ( complex2 @ ( plus_plus_real @ R2 @ X4 ) @ Y3 ) ) ).

% complex_of_real_add_Complex
thf(fact_6525_Complex__add__complex__of__real,axiom,
    ! [X4: real,Y3: real,R2: real] :
      ( ( plus_plus_complex @ ( complex2 @ X4 @ Y3 ) @ ( real_V4546457046886955230omplex @ R2 ) )
      = ( complex2 @ ( plus_plus_real @ X4 @ R2 ) @ Y3 ) ) ).

% Complex_add_complex_of_real
thf(fact_6526_complex__diff,axiom,
    ! [A: real,B2: real,C: real,D: real] :
      ( ( minus_minus_complex @ ( complex2 @ A @ B2 ) @ ( complex2 @ C @ D ) )
      = ( complex2 @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B2 @ D ) ) ) ).

% complex_diff
thf(fact_6527_Complex__eq__1,axiom,
    ! [A: real,B2: real] :
      ( ( ( complex2 @ A @ B2 )
        = one_one_complex )
      = ( ( A = one_one_real )
        & ( B2 = zero_zero_real ) ) ) ).

% Complex_eq_1
thf(fact_6528_one__complex_Ocode,axiom,
    ( one_one_complex
    = ( complex2 @ one_one_real @ zero_zero_real ) ) ).

% one_complex.code
thf(fact_6529_Complex__eq__numeral,axiom,
    ! [A: real,B2: real,W: num] :
      ( ( ( complex2 @ A @ B2 )
        = ( numera6690914467698888265omplex @ W ) )
      = ( ( A
          = ( numeral_numeral_real @ W ) )
        & ( B2 = zero_zero_real ) ) ) ).

% Complex_eq_numeral
thf(fact_6530_complex__add,axiom,
    ! [A: real,B2: real,C: real,D: real] :
      ( ( plus_plus_complex @ ( complex2 @ A @ B2 ) @ ( complex2 @ C @ D ) )
      = ( complex2 @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ D ) ) ) ).

% complex_add
thf(fact_6531_nonzero__of__real__divide,axiom,
    ! [Y3: real,X4: real] :
      ( ( Y3 != zero_zero_real )
     => ( ( real_V1803761363581548252l_real @ ( divide_divide_real @ X4 @ Y3 ) )
        = ( divide_divide_real @ ( real_V1803761363581548252l_real @ X4 ) @ ( real_V1803761363581548252l_real @ Y3 ) ) ) ) ).

% nonzero_of_real_divide
thf(fact_6532_nonzero__of__real__divide,axiom,
    ! [Y3: real,X4: real] :
      ( ( Y3 != zero_zero_real )
     => ( ( real_V4546457046886955230omplex @ ( divide_divide_real @ X4 @ Y3 ) )
        = ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ X4 ) @ ( real_V4546457046886955230omplex @ Y3 ) ) ) ) ).

% nonzero_of_real_divide
thf(fact_6533_Complex__eq__neg__1,axiom,
    ! [A: real,B2: real] :
      ( ( ( complex2 @ A @ B2 )
        = ( uminus1482373934393186551omplex @ one_one_complex ) )
      = ( ( A
          = ( uminus_uminus_real @ one_one_real ) )
        & ( B2 = zero_zero_real ) ) ) ).

% Complex_eq_neg_1
thf(fact_6534_Complex__eq__neg__numeral,axiom,
    ! [A: real,B2: real,W: num] :
      ( ( ( complex2 @ A @ B2 )
        = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
      = ( ( A
          = ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
        & ( B2 = zero_zero_real ) ) ) ).

% Complex_eq_neg_numeral
thf(fact_6535_complex__mult,axiom,
    ! [A: real,B2: real,C: real,D: real] :
      ( ( times_times_complex @ ( complex2 @ A @ B2 ) @ ( complex2 @ C @ D ) )
      = ( complex2 @ ( minus_minus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ D ) ) @ ( plus_plus_real @ ( times_times_real @ A @ D ) @ ( times_times_real @ B2 @ C ) ) ) ) ).

% complex_mult
thf(fact_6536_arcsin__le__arcsin,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X4 )
     => ( ( ord_less_eq_real @ X4 @ Y3 )
       => ( ( ord_less_eq_real @ Y3 @ one_one_real )
         => ( ord_less_eq_real @ ( arcsin @ X4 ) @ ( arcsin @ Y3 ) ) ) ) ) ).

% arcsin_le_arcsin
thf(fact_6537_arcsin__minus,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X4 )
     => ( ( ord_less_eq_real @ X4 @ one_one_real )
       => ( ( arcsin @ ( uminus_uminus_real @ X4 ) )
          = ( uminus_uminus_real @ ( arcsin @ X4 ) ) ) ) ) ).

% arcsin_minus
thf(fact_6538_arcsin__le__mono,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X4 ) @ one_one_real )
     => ( ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
       => ( ( ord_less_eq_real @ ( arcsin @ X4 ) @ ( arcsin @ Y3 ) )
          = ( ord_less_eq_real @ X4 @ Y3 ) ) ) ) ).

% arcsin_le_mono
thf(fact_6539_arcsin__eq__iff,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X4 ) @ one_one_real )
     => ( ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
       => ( ( ( arcsin @ X4 )
            = ( arcsin @ Y3 ) )
          = ( X4 = Y3 ) ) ) ) ).

% arcsin_eq_iff
thf(fact_6540_norm__less__p1,axiom,
    ! [X4: real] : ( ord_less_real @ ( real_V7735802525324610683m_real @ X4 ) @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ ( real_V1803761363581548252l_real @ ( real_V7735802525324610683m_real @ X4 ) ) @ one_one_real ) ) ) ).

% norm_less_p1
thf(fact_6541_norm__less__p1,axiom,
    ! [X4: complex] : ( ord_less_real @ ( real_V1022390504157884413omplex @ X4 ) @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( real_V1022390504157884413omplex @ X4 ) ) @ one_one_complex ) ) ) ).

% norm_less_p1
thf(fact_6542_tan__def,axiom,
    ( tan_complex
    = ( ^ [X: complex] : ( divide1717551699836669952omplex @ ( sin_complex @ X ) @ ( cos_complex @ X ) ) ) ) ).

% tan_def
thf(fact_6543_tan__def,axiom,
    ( tan_real
    = ( ^ [X: real] : ( divide_divide_real @ ( sin_real @ X ) @ ( cos_real @ X ) ) ) ) ).

% tan_def
thf(fact_6544_arcsin__less__arcsin,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X4 )
     => ( ( ord_less_real @ X4 @ Y3 )
       => ( ( ord_less_eq_real @ Y3 @ one_one_real )
         => ( ord_less_real @ ( arcsin @ X4 ) @ ( arcsin @ Y3 ) ) ) ) ) ).

% arcsin_less_arcsin
thf(fact_6545_arcsin__less__mono,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X4 ) @ one_one_real )
     => ( ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
       => ( ( ord_less_real @ ( arcsin @ X4 ) @ ( arcsin @ Y3 ) )
          = ( ord_less_real @ X4 @ Y3 ) ) ) ) ).

% arcsin_less_mono
thf(fact_6546_norm__of__real__diff,axiom,
    ! [B2: real,A: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( real_V1803761363581548252l_real @ B2 ) @ ( real_V1803761363581548252l_real @ A ) ) ) @ ( abs_abs_real @ ( minus_minus_real @ B2 @ A ) ) ) ).

% norm_of_real_diff
thf(fact_6547_norm__of__real__diff,axiom,
    ! [B2: real,A: real] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( real_V4546457046886955230omplex @ B2 ) @ ( real_V4546457046886955230omplex @ A ) ) ) @ ( abs_abs_real @ ( minus_minus_real @ B2 @ A ) ) ) ).

% norm_of_real_diff
thf(fact_6548_tan__45,axiom,
    ( ( tan_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
    = one_one_real ) ).

% tan_45
thf(fact_6549_tan__60,axiom,
    ( ( tan_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) )
    = ( sqrt @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) ) ).

% tan_60
thf(fact_6550_cos__int__times__real,axiom,
    ! [M: int,X4: real] :
      ( ( cos_real @ ( times_times_real @ ( ring_1_of_int_real @ M ) @ ( real_V1803761363581548252l_real @ X4 ) ) )
      = ( real_V1803761363581548252l_real @ ( cos_real @ ( times_times_real @ ( ring_1_of_int_real @ M ) @ X4 ) ) ) ) ).

% cos_int_times_real
thf(fact_6551_cos__int__times__real,axiom,
    ! [M: int,X4: real] :
      ( ( cos_complex @ ( times_times_complex @ ( ring_17405671764205052669omplex @ M ) @ ( real_V4546457046886955230omplex @ X4 ) ) )
      = ( real_V4546457046886955230omplex @ ( cos_real @ ( times_times_real @ ( ring_1_of_int_real @ M ) @ X4 ) ) ) ) ).

% cos_int_times_real
thf(fact_6552_sin__int__times__real,axiom,
    ! [M: int,X4: real] :
      ( ( sin_real @ ( times_times_real @ ( ring_1_of_int_real @ M ) @ ( real_V1803761363581548252l_real @ X4 ) ) )
      = ( real_V1803761363581548252l_real @ ( sin_real @ ( times_times_real @ ( ring_1_of_int_real @ M ) @ X4 ) ) ) ) ).

% sin_int_times_real
thf(fact_6553_sin__int__times__real,axiom,
    ! [M: int,X4: real] :
      ( ( sin_complex @ ( times_times_complex @ ( ring_17405671764205052669omplex @ M ) @ ( real_V4546457046886955230omplex @ X4 ) ) )
      = ( real_V4546457046886955230omplex @ ( sin_real @ ( times_times_real @ ( ring_1_of_int_real @ M ) @ X4 ) ) ) ) ).

% sin_int_times_real
thf(fact_6554_cos__arcsin__nonzero,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X4 )
     => ( ( ord_less_real @ X4 @ one_one_real )
       => ( ( cos_real @ ( arcsin @ X4 ) )
         != zero_zero_real ) ) ) ).

% cos_arcsin_nonzero
thf(fact_6555_lemma__tan__total,axiom,
    ! [Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ Y3 )
     => ? [X3: real] :
          ( ( ord_less_real @ zero_zero_real @ X3 )
          & ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
          & ( ord_less_real @ Y3 @ ( tan_real @ X3 ) ) ) ) ).

% lemma_tan_total
thf(fact_6556_tan__gt__zero,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_real @ zero_zero_real @ ( tan_real @ X4 ) ) ) ) ).

% tan_gt_zero
thf(fact_6557_tan__total,axiom,
    ! [Y3: real] :
    ? [X3: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
      & ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
      & ( ( tan_real @ X3 )
        = Y3 )
      & ! [Y5: real] :
          ( ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y5 )
            & ( ord_less_real @ Y5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
            & ( ( tan_real @ Y5 )
              = Y3 ) )
         => ( Y5 = X3 ) ) ) ).

% tan_total
thf(fact_6558_tan__monotone,axiom,
    ! [Y3: real,X4: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
     => ( ( ord_less_real @ Y3 @ X4 )
       => ( ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
         => ( ord_less_real @ ( tan_real @ Y3 ) @ ( tan_real @ X4 ) ) ) ) ) ).

% tan_monotone
thf(fact_6559_tan__monotone_H,axiom,
    ! [Y3: real,X4: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
     => ( ( ord_less_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X4 )
         => ( ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ord_less_real @ Y3 @ X4 )
              = ( ord_less_real @ ( tan_real @ Y3 ) @ ( tan_real @ X4 ) ) ) ) ) ) ) ).

% tan_monotone'
thf(fact_6560_tan__mono__lt__eq,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X4 )
     => ( ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
         => ( ( ord_less_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ord_less_real @ ( tan_real @ X4 ) @ ( tan_real @ Y3 ) )
              = ( ord_less_real @ X4 @ Y3 ) ) ) ) ) ) ).

% tan_mono_lt_eq
thf(fact_6561_lemma__tan__total1,axiom,
    ! [Y3: real] :
    ? [X3: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
      & ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
      & ( ( tan_real @ X3 )
        = Y3 ) ) ).

% lemma_tan_total1
thf(fact_6562_tan__minus__45,axiom,
    ( ( tan_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) )
    = ( uminus_uminus_real @ one_one_real ) ) ).

% tan_minus_45
thf(fact_6563_tan__inverse,axiom,
    ! [Y3: real] :
      ( ( divide_divide_real @ one_one_real @ ( tan_real @ Y3 ) )
      = ( tan_real @ ( minus_minus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ Y3 ) ) ) ).

% tan_inverse
thf(fact_6564_complex__norm,axiom,
    ! [X4: real,Y3: real] :
      ( ( real_V1022390504157884413omplex @ ( complex2 @ X4 @ Y3 ) )
      = ( sqrt @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% complex_norm
thf(fact_6565_add__tan__eq,axiom,
    ! [X4: complex,Y3: complex] :
      ( ( ( cos_complex @ X4 )
       != zero_zero_complex )
     => ( ( ( cos_complex @ Y3 )
         != zero_zero_complex )
       => ( ( plus_plus_complex @ ( tan_complex @ X4 ) @ ( tan_complex @ Y3 ) )
          = ( divide1717551699836669952omplex @ ( sin_complex @ ( plus_plus_complex @ X4 @ Y3 ) ) @ ( times_times_complex @ ( cos_complex @ X4 ) @ ( cos_complex @ Y3 ) ) ) ) ) ) ).

% add_tan_eq
thf(fact_6566_add__tan__eq,axiom,
    ! [X4: real,Y3: real] :
      ( ( ( cos_real @ X4 )
       != zero_zero_real )
     => ( ( ( cos_real @ Y3 )
         != zero_zero_real )
       => ( ( plus_plus_real @ ( tan_real @ X4 ) @ ( tan_real @ Y3 ) )
          = ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ X4 @ Y3 ) ) @ ( times_times_real @ ( cos_real @ X4 ) @ ( cos_real @ Y3 ) ) ) ) ) ) ).

% add_tan_eq
thf(fact_6567_tan__cot_H,axiom,
    ! [X4: real] :
      ( ( tan_real @ ( minus_minus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X4 ) )
      = ( cot_real @ X4 ) ) ).

% tan_cot'
thf(fact_6568_sin__cos__eq,axiom,
    ( sin_real
    = ( ^ [X: real] : ( cos_real @ ( minus_minus_real @ ( divide_divide_real @ ( real_V1803761363581548252l_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X ) ) ) ) ).

% sin_cos_eq
thf(fact_6569_sin__cos__eq,axiom,
    ( sin_complex
    = ( ^ [X: complex] : ( cos_complex @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ X ) ) ) ) ).

% sin_cos_eq
thf(fact_6570_cos__sin__eq,axiom,
    ( cos_real
    = ( ^ [X: real] : ( sin_real @ ( minus_minus_real @ ( divide_divide_real @ ( real_V1803761363581548252l_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X ) ) ) ) ).

% cos_sin_eq
thf(fact_6571_cos__sin__eq,axiom,
    ( cos_complex
    = ( ^ [X: complex] : ( sin_complex @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ X ) ) ) ) ).

% cos_sin_eq
thf(fact_6572_tan__pos__pi2__le,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_eq_real @ zero_zero_real @ ( tan_real @ X4 ) ) ) ) ).

% tan_pos_pi2_le
thf(fact_6573_tan__total__pos,axiom,
    ! [Y3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
     => ? [X3: real] :
          ( ( ord_less_eq_real @ zero_zero_real @ X3 )
          & ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
          & ( ( tan_real @ X3 )
            = Y3 ) ) ) ).

% tan_total_pos
thf(fact_6574_tan__less__zero,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X4 )
     => ( ( ord_less_real @ X4 @ zero_zero_real )
       => ( ord_less_real @ ( tan_real @ X4 ) @ zero_zero_real ) ) ) ).

% tan_less_zero
thf(fact_6575_tan__mono__le,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X4 )
     => ( ( ord_less_eq_real @ X4 @ Y3 )
       => ( ( ord_less_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
         => ( ord_less_eq_real @ ( tan_real @ X4 ) @ ( tan_real @ Y3 ) ) ) ) ) ).

% tan_mono_le
thf(fact_6576_tan__mono__le__eq,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X4 )
     => ( ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
         => ( ( ord_less_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ord_less_eq_real @ ( tan_real @ X4 ) @ ( tan_real @ Y3 ) )
              = ( ord_less_eq_real @ X4 @ Y3 ) ) ) ) ) ) ).

% tan_mono_le_eq
thf(fact_6577_tan__bound__pi2,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
     => ( ord_less_real @ ( abs_abs_real @ ( tan_real @ X4 ) ) @ one_one_real ) ) ).

% tan_bound_pi2
thf(fact_6578_tan__30,axiom,
    ( ( tan_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ one ) ) ) ) )
    = ( divide_divide_real @ one_one_real @ ( sqrt @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) ) ) ).

% tan_30
thf(fact_6579_arctan,axiom,
    ! [Y3: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y3 ) )
      & ( ord_less_real @ ( arctan @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
      & ( ( tan_real @ ( arctan @ Y3 ) )
        = Y3 ) ) ).

% arctan
thf(fact_6580_arctan__tan,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X4 )
     => ( ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( arctan @ ( tan_real @ X4 ) )
          = X4 ) ) ) ).

% arctan_tan
thf(fact_6581_arctan__unique,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X4 )
     => ( ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( ( tan_real @ X4 )
            = Y3 )
         => ( ( arctan @ Y3 )
            = X4 ) ) ) ) ).

% arctan_unique
thf(fact_6582_lemma__tan__add1,axiom,
    ! [X4: complex,Y3: complex] :
      ( ( ( cos_complex @ X4 )
       != zero_zero_complex )
     => ( ( ( cos_complex @ Y3 )
         != zero_zero_complex )
       => ( ( minus_minus_complex @ one_one_complex @ ( times_times_complex @ ( tan_complex @ X4 ) @ ( tan_complex @ Y3 ) ) )
          = ( divide1717551699836669952omplex @ ( cos_complex @ ( plus_plus_complex @ X4 @ Y3 ) ) @ ( times_times_complex @ ( cos_complex @ X4 ) @ ( cos_complex @ Y3 ) ) ) ) ) ) ).

% lemma_tan_add1
thf(fact_6583_lemma__tan__add1,axiom,
    ! [X4: real,Y3: real] :
      ( ( ( cos_real @ X4 )
       != zero_zero_real )
     => ( ( ( cos_real @ Y3 )
         != zero_zero_real )
       => ( ( minus_minus_real @ one_one_real @ ( times_times_real @ ( tan_real @ X4 ) @ ( tan_real @ Y3 ) ) )
          = ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ X4 @ Y3 ) ) @ ( times_times_real @ ( cos_real @ X4 ) @ ( cos_real @ Y3 ) ) ) ) ) ) ).

% lemma_tan_add1
thf(fact_6584_tan__diff,axiom,
    ! [X4: complex,Y3: complex] :
      ( ( ( cos_complex @ X4 )
       != zero_zero_complex )
     => ( ( ( cos_complex @ Y3 )
         != zero_zero_complex )
       => ( ( ( cos_complex @ ( minus_minus_complex @ X4 @ Y3 ) )
           != zero_zero_complex )
         => ( ( tan_complex @ ( minus_minus_complex @ X4 @ Y3 ) )
            = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( tan_complex @ X4 ) @ ( tan_complex @ Y3 ) ) @ ( plus_plus_complex @ one_one_complex @ ( times_times_complex @ ( tan_complex @ X4 ) @ ( tan_complex @ Y3 ) ) ) ) ) ) ) ) ).

% tan_diff
thf(fact_6585_tan__diff,axiom,
    ! [X4: real,Y3: real] :
      ( ( ( cos_real @ X4 )
       != zero_zero_real )
     => ( ( ( cos_real @ Y3 )
         != zero_zero_real )
       => ( ( ( cos_real @ ( minus_minus_real @ X4 @ Y3 ) )
           != zero_zero_real )
         => ( ( tan_real @ ( minus_minus_real @ X4 @ Y3 ) )
            = ( divide_divide_real @ ( minus_minus_real @ ( tan_real @ X4 ) @ ( tan_real @ Y3 ) ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( tan_real @ X4 ) @ ( tan_real @ Y3 ) ) ) ) ) ) ) ) ).

% tan_diff
thf(fact_6586_tan__add,axiom,
    ! [X4: complex,Y3: complex] :
      ( ( ( cos_complex @ X4 )
       != zero_zero_complex )
     => ( ( ( cos_complex @ Y3 )
         != zero_zero_complex )
       => ( ( ( cos_complex @ ( plus_plus_complex @ X4 @ Y3 ) )
           != zero_zero_complex )
         => ( ( tan_complex @ ( plus_plus_complex @ X4 @ Y3 ) )
            = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( tan_complex @ X4 ) @ ( tan_complex @ Y3 ) ) @ ( minus_minus_complex @ one_one_complex @ ( times_times_complex @ ( tan_complex @ X4 ) @ ( tan_complex @ Y3 ) ) ) ) ) ) ) ) ).

% tan_add
thf(fact_6587_tan__add,axiom,
    ! [X4: real,Y3: real] :
      ( ( ( cos_real @ X4 )
       != zero_zero_real )
     => ( ( ( cos_real @ Y3 )
         != zero_zero_real )
       => ( ( ( cos_real @ ( plus_plus_real @ X4 @ Y3 ) )
           != zero_zero_real )
         => ( ( tan_real @ ( plus_plus_real @ X4 @ Y3 ) )
            = ( divide_divide_real @ ( plus_plus_real @ ( tan_real @ X4 ) @ ( tan_real @ Y3 ) ) @ ( minus_minus_real @ one_one_real @ ( times_times_real @ ( tan_real @ X4 ) @ ( tan_real @ Y3 ) ) ) ) ) ) ) ) ).

% tan_add
thf(fact_6588_minus__sin__cos__eq,axiom,
    ! [X4: real] :
      ( ( uminus_uminus_real @ ( sin_real @ X4 ) )
      = ( cos_real @ ( plus_plus_real @ X4 @ ( divide_divide_real @ ( real_V1803761363581548252l_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% minus_sin_cos_eq
thf(fact_6589_minus__sin__cos__eq,axiom,
    ! [X4: complex] :
      ( ( uminus1482373934393186551omplex @ ( sin_complex @ X4 ) )
      = ( cos_complex @ ( plus_plus_complex @ X4 @ ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).

% minus_sin_cos_eq
thf(fact_6590_tan__total__pi4,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X4 ) @ one_one_real )
     => ? [Z2: real] :
          ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) @ Z2 )
          & ( ord_less_real @ Z2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
          & ( ( tan_real @ Z2 )
            = X4 ) ) ) ).

% tan_total_pi4
thf(fact_6591_arcsin__lt__bounded,axiom,
    ! [Y3: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
     => ( ( ord_less_real @ Y3 @ one_one_real )
       => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y3 ) )
          & ( ord_less_real @ ( arcsin @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).

% arcsin_lt_bounded
thf(fact_6592_arcsin__lbound,axiom,
    ! [Y3: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
     => ( ( ord_less_eq_real @ Y3 @ one_one_real )
       => ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y3 ) ) ) ) ).

% arcsin_lbound
thf(fact_6593_arcsin__ubound,axiom,
    ! [Y3: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
     => ( ( ord_less_eq_real @ Y3 @ one_one_real )
       => ( ord_less_eq_real @ ( arcsin @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% arcsin_ubound
thf(fact_6594_arcsin__bounded,axiom,
    ! [Y3: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
     => ( ( ord_less_eq_real @ Y3 @ one_one_real )
       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y3 ) )
          & ( ord_less_eq_real @ ( arcsin @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).

% arcsin_bounded
thf(fact_6595_arcsin__sin,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X4 )
     => ( ( ord_less_eq_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( arcsin @ ( sin_real @ X4 ) )
          = X4 ) ) ) ).

% arcsin_sin
thf(fact_6596_tan__half,axiom,
    ( tan_complex
    = ( ^ [X: complex] : ( divide1717551699836669952omplex @ ( sin_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) ) @ ( plus_plus_complex @ ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) ) @ one_one_complex ) ) ) ) ).

% tan_half
thf(fact_6597_tan__half,axiom,
    ( tan_real
    = ( ^ [X: real] : ( divide_divide_real @ ( sin_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) ) @ ( plus_plus_real @ ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) ) @ one_one_real ) ) ) ) ).

% tan_half
thf(fact_6598_le__arcsin__iff,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X4 )
     => ( ( ord_less_eq_real @ X4 @ one_one_real )
       => ( ( ord_less_eq_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ Y3 )
         => ( ( ord_less_eq_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ord_less_eq_real @ Y3 @ ( arcsin @ X4 ) )
              = ( ord_less_eq_real @ ( sin_real @ Y3 ) @ X4 ) ) ) ) ) ) ).

% le_arcsin_iff
thf(fact_6599_arcsin__le__iff,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X4 )
     => ( ( ord_less_eq_real @ X4 @ one_one_real )
       => ( ( ord_less_eq_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ Y3 )
         => ( ( ord_less_eq_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ord_less_eq_real @ ( arcsin @ X4 ) @ Y3 )
              = ( ord_less_eq_real @ X4 @ ( sin_real @ Y3 ) ) ) ) ) ) ) ).

% arcsin_le_iff
thf(fact_6600_arcsin__pi,axiom,
    ! [Y3: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
     => ( ( ord_less_eq_real @ Y3 @ one_one_real )
       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y3 ) )
          & ( ord_less_eq_real @ ( arcsin @ Y3 ) @ pi )
          & ( ( sin_real @ ( arcsin @ Y3 ) )
            = Y3 ) ) ) ) ).

% arcsin_pi
thf(fact_6601_arcsin,axiom,
    ! [Y3: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
     => ( ( ord_less_eq_real @ Y3 @ one_one_real )
       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y3 ) )
          & ( ord_less_eq_real @ ( arcsin @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
          & ( ( sin_real @ ( arcsin @ Y3 ) )
            = Y3 ) ) ) ) ).

% arcsin
thf(fact_6602_arsinh__def,axiom,
    ( arsinh_real
    = ( ^ [X: real] : ( ln_ln_real @ ( plus_plus_real @ X @ ( powr_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) @ ( real_V1803761363581548252l_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% arsinh_def
thf(fact_6603_sin__arccos__abs,axiom,
    ! [Y3: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
     => ( ( sin_real @ ( arccos @ Y3 ) )
        = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% sin_arccos_abs
thf(fact_6604_sin__arccos,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X4 )
     => ( ( ord_less_eq_real @ X4 @ one_one_real )
       => ( ( sin_real @ ( arccos @ X4 ) )
          = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% sin_arccos
thf(fact_6605_arccos__cos__eq__abs__2pi,axiom,
    ! [Theta: real] :
      ~ ! [K2: int] :
          ( ( arccos @ ( cos_real @ Theta ) )
         != ( abs_abs_real @ ( minus_minus_real @ Theta @ ( times_times_real @ ( ring_1_of_int_real @ K2 ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) ) ) ) ).

% arccos_cos_eq_abs_2pi
thf(fact_6606_fact__double,axiom,
    ! [N: nat] :
      ( ( semiri5044797733671781792omplex @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( times_times_complex @ ( times_times_complex @ ( power_power_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( comm_s2602460028002588243omplex @ ( divide1717551699836669952omplex @ one_one_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ N ) ) @ ( semiri5044797733671781792omplex @ N ) ) ) ).

% fact_double
thf(fact_6607_fact__double,axiom,
    ! [N: nat] :
      ( ( semiri773545260158071498ct_rat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( times_times_rat @ ( times_times_rat @ ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( comm_s4028243227959126397er_rat @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ N ) ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).

% fact_double
thf(fact_6608_fact__double,axiom,
    ! [N: nat] :
      ( ( semiri2265585572941072030t_real @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( times_times_real @ ( times_times_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( comm_s7457072308508201937r_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ N ) ) @ ( semiri2265585572941072030t_real @ N ) ) ) ).

% fact_double
thf(fact_6609_modulo__int__unfold,axiom,
    ! [L2: int,K: int,N: nat,M: nat] :
      ( ( ( ( ( sgn_sgn_int @ L2 )
            = zero_zero_int )
          | ( ( sgn_sgn_int @ K )
            = zero_zero_int )
          | ( N = zero_zero_nat ) )
       => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ N ) ) )
          = ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) ) )
      & ( ~ ( ( ( sgn_sgn_int @ L2 )
              = zero_zero_int )
            | ( ( sgn_sgn_int @ K )
              = zero_zero_int )
            | ( N = zero_zero_nat ) )
       => ( ( ( ( sgn_sgn_int @ K )
              = ( sgn_sgn_int @ L2 ) )
           => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ N ) ) )
              = ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N ) ) ) ) )
          & ( ( ( sgn_sgn_int @ K )
             != ( sgn_sgn_int @ L2 ) )
           => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ N ) ) )
              = ( times_times_int @ ( sgn_sgn_int @ L2 )
                @ ( minus_minus_int
                  @ ( semiri1314217659103216013at_int
                    @ ( times_times_nat @ N
                      @ ( zero_n2687167440665602831ol_nat
                        @ ~ ( dvd_dvd_nat @ N @ M ) ) ) )
                  @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N ) ) ) ) ) ) ) ) ) ).

% modulo_int_unfold
thf(fact_6610_sgn__sgn,axiom,
    ! [A: int] :
      ( ( sgn_sgn_int @ ( sgn_sgn_int @ A ) )
      = ( sgn_sgn_int @ A ) ) ).

% sgn_sgn
thf(fact_6611_sgn__sgn,axiom,
    ! [A: real] :
      ( ( sgn_sgn_real @ ( sgn_sgn_real @ A ) )
      = ( sgn_sgn_real @ A ) ) ).

% sgn_sgn
thf(fact_6612_sgn__sgn,axiom,
    ! [A: complex] :
      ( ( sgn_sgn_complex @ ( sgn_sgn_complex @ A ) )
      = ( sgn_sgn_complex @ A ) ) ).

% sgn_sgn
thf(fact_6613_sgn__sgn,axiom,
    ! [A: code_integer] :
      ( ( sgn_sgn_Code_integer @ ( sgn_sgn_Code_integer @ A ) )
      = ( sgn_sgn_Code_integer @ A ) ) ).

% sgn_sgn
thf(fact_6614_sgn__sgn,axiom,
    ! [A: rat] :
      ( ( sgn_sgn_rat @ ( sgn_sgn_rat @ A ) )
      = ( sgn_sgn_rat @ A ) ) ).

% sgn_sgn
thf(fact_6615_sgn__0,axiom,
    ( ( sgn_sgn_complex @ zero_zero_complex )
    = zero_zero_complex ) ).

% sgn_0
thf(fact_6616_sgn__0,axiom,
    ( ( sgn_sgn_Code_integer @ zero_z3403309356797280102nteger )
    = zero_z3403309356797280102nteger ) ).

% sgn_0
thf(fact_6617_sgn__0,axiom,
    ( ( sgn_sgn_real @ zero_zero_real )
    = zero_zero_real ) ).

% sgn_0
thf(fact_6618_sgn__0,axiom,
    ( ( sgn_sgn_rat @ zero_zero_rat )
    = zero_zero_rat ) ).

% sgn_0
thf(fact_6619_sgn__0,axiom,
    ( ( sgn_sgn_int @ zero_zero_int )
    = zero_zero_int ) ).

% sgn_0
thf(fact_6620_sgn__1,axiom,
    ( ( sgn_sgn_int @ one_one_int )
    = one_one_int ) ).

% sgn_1
thf(fact_6621_sgn__1,axiom,
    ( ( sgn_sgn_real @ one_one_real )
    = one_one_real ) ).

% sgn_1
thf(fact_6622_sgn__1,axiom,
    ( ( sgn_sgn_complex @ one_one_complex )
    = one_one_complex ) ).

% sgn_1
thf(fact_6623_sgn__1,axiom,
    ( ( sgn_sgn_Code_integer @ one_one_Code_integer )
    = one_one_Code_integer ) ).

% sgn_1
thf(fact_6624_sgn__1,axiom,
    ( ( sgn_sgn_rat @ one_one_rat )
    = one_one_rat ) ).

% sgn_1
thf(fact_6625_sgn__one,axiom,
    ( ( sgn_sgn_real @ one_one_real )
    = one_one_real ) ).

% sgn_one
thf(fact_6626_sgn__one,axiom,
    ( ( sgn_sgn_complex @ one_one_complex )
    = one_one_complex ) ).

% sgn_one
thf(fact_6627_sgn__divide,axiom,
    ! [A: complex,B2: complex] :
      ( ( sgn_sgn_complex @ ( divide1717551699836669952omplex @ A @ B2 ) )
      = ( divide1717551699836669952omplex @ ( sgn_sgn_complex @ A ) @ ( sgn_sgn_complex @ B2 ) ) ) ).

% sgn_divide
thf(fact_6628_sgn__divide,axiom,
    ! [A: real,B2: real] :
      ( ( sgn_sgn_real @ ( divide_divide_real @ A @ B2 ) )
      = ( divide_divide_real @ ( sgn_sgn_real @ A ) @ ( sgn_sgn_real @ B2 ) ) ) ).

% sgn_divide
thf(fact_6629_sgn__divide,axiom,
    ! [A: rat,B2: rat] :
      ( ( sgn_sgn_rat @ ( divide_divide_rat @ A @ B2 ) )
      = ( divide_divide_rat @ ( sgn_sgn_rat @ A ) @ ( sgn_sgn_rat @ B2 ) ) ) ).

% sgn_divide
thf(fact_6630_idom__abs__sgn__class_Osgn__minus,axiom,
    ! [A: real] :
      ( ( sgn_sgn_real @ ( uminus_uminus_real @ A ) )
      = ( uminus_uminus_real @ ( sgn_sgn_real @ A ) ) ) ).

% idom_abs_sgn_class.sgn_minus
thf(fact_6631_idom__abs__sgn__class_Osgn__minus,axiom,
    ! [A: int] :
      ( ( sgn_sgn_int @ ( uminus_uminus_int @ A ) )
      = ( uminus_uminus_int @ ( sgn_sgn_int @ A ) ) ) ).

% idom_abs_sgn_class.sgn_minus
thf(fact_6632_idom__abs__sgn__class_Osgn__minus,axiom,
    ! [A: complex] :
      ( ( sgn_sgn_complex @ ( uminus1482373934393186551omplex @ A ) )
      = ( uminus1482373934393186551omplex @ ( sgn_sgn_complex @ A ) ) ) ).

% idom_abs_sgn_class.sgn_minus
thf(fact_6633_idom__abs__sgn__class_Osgn__minus,axiom,
    ! [A: code_integer] :
      ( ( sgn_sgn_Code_integer @ ( uminus1351360451143612070nteger @ A ) )
      = ( uminus1351360451143612070nteger @ ( sgn_sgn_Code_integer @ A ) ) ) ).

% idom_abs_sgn_class.sgn_minus
thf(fact_6634_idom__abs__sgn__class_Osgn__minus,axiom,
    ! [A: rat] :
      ( ( sgn_sgn_rat @ ( uminus_uminus_rat @ A ) )
      = ( uminus_uminus_rat @ ( sgn_sgn_rat @ A ) ) ) ).

% idom_abs_sgn_class.sgn_minus
thf(fact_6635_power__sgn,axiom,
    ! [A: code_integer,N: nat] :
      ( ( sgn_sgn_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) )
      = ( power_8256067586552552935nteger @ ( sgn_sgn_Code_integer @ A ) @ N ) ) ).

% power_sgn
thf(fact_6636_power__sgn,axiom,
    ! [A: rat,N: nat] :
      ( ( sgn_sgn_rat @ ( power_power_rat @ A @ N ) )
      = ( power_power_rat @ ( sgn_sgn_rat @ A ) @ N ) ) ).

% power_sgn
thf(fact_6637_power__sgn,axiom,
    ! [A: real,N: nat] :
      ( ( sgn_sgn_real @ ( power_power_real @ A @ N ) )
      = ( power_power_real @ ( sgn_sgn_real @ A ) @ N ) ) ).

% power_sgn
thf(fact_6638_power__sgn,axiom,
    ! [A: int,N: nat] :
      ( ( sgn_sgn_int @ ( power_power_int @ A @ N ) )
      = ( power_power_int @ ( sgn_sgn_int @ A ) @ N ) ) ).

% power_sgn
thf(fact_6639_sgn__greater,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( sgn_sgn_Code_integer @ A ) )
      = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).

% sgn_greater
thf(fact_6640_sgn__greater,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ ( sgn_sgn_real @ A ) )
      = ( ord_less_real @ zero_zero_real @ A ) ) ).

% sgn_greater
thf(fact_6641_sgn__greater,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( sgn_sgn_rat @ A ) )
      = ( ord_less_rat @ zero_zero_rat @ A ) ) ).

% sgn_greater
thf(fact_6642_sgn__greater,axiom,
    ! [A: int] :
      ( ( ord_less_int @ zero_zero_int @ ( sgn_sgn_int @ A ) )
      = ( ord_less_int @ zero_zero_int @ A ) ) ).

% sgn_greater
thf(fact_6643_sgn__less,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( sgn_sgn_Code_integer @ A ) @ zero_z3403309356797280102nteger )
      = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).

% sgn_less
thf(fact_6644_sgn__less,axiom,
    ! [A: real] :
      ( ( ord_less_real @ ( sgn_sgn_real @ A ) @ zero_zero_real )
      = ( ord_less_real @ A @ zero_zero_real ) ) ).

% sgn_less
thf(fact_6645_sgn__less,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ ( sgn_sgn_rat @ A ) @ zero_zero_rat )
      = ( ord_less_rat @ A @ zero_zero_rat ) ) ).

% sgn_less
thf(fact_6646_sgn__less,axiom,
    ! [A: int] :
      ( ( ord_less_int @ ( sgn_sgn_int @ A ) @ zero_zero_int )
      = ( ord_less_int @ A @ zero_zero_int ) ) ).

% sgn_less
thf(fact_6647_divide__sgn,axiom,
    ! [A: real,B2: real] :
      ( ( divide_divide_real @ A @ ( sgn_sgn_real @ B2 ) )
      = ( times_times_real @ A @ ( sgn_sgn_real @ B2 ) ) ) ).

% divide_sgn
thf(fact_6648_divide__sgn,axiom,
    ! [A: rat,B2: rat] :
      ( ( divide_divide_rat @ A @ ( sgn_sgn_rat @ B2 ) )
      = ( times_times_rat @ A @ ( sgn_sgn_rat @ B2 ) ) ) ).

% divide_sgn
thf(fact_6649_fact__0,axiom,
    ( ( semiri5044797733671781792omplex @ zero_zero_nat )
    = one_one_complex ) ).

% fact_0
thf(fact_6650_fact__0,axiom,
    ( ( semiri773545260158071498ct_rat @ zero_zero_nat )
    = one_one_rat ) ).

% fact_0
thf(fact_6651_fact__0,axiom,
    ( ( semiri1406184849735516958ct_int @ zero_zero_nat )
    = one_one_int ) ).

% fact_0
thf(fact_6652_fact__0,axiom,
    ( ( semiri1408675320244567234ct_nat @ zero_zero_nat )
    = one_one_nat ) ).

% fact_0
thf(fact_6653_fact__0,axiom,
    ( ( semiri2265585572941072030t_real @ zero_zero_nat )
    = one_one_real ) ).

% fact_0
thf(fact_6654_fact__1,axiom,
    ( ( semiri5044797733671781792omplex @ one_one_nat )
    = one_one_complex ) ).

% fact_1
thf(fact_6655_fact__1,axiom,
    ( ( semiri773545260158071498ct_rat @ one_one_nat )
    = one_one_rat ) ).

% fact_1
thf(fact_6656_fact__1,axiom,
    ( ( semiri1406184849735516958ct_int @ one_one_nat )
    = one_one_int ) ).

% fact_1
thf(fact_6657_fact__1,axiom,
    ( ( semiri1408675320244567234ct_nat @ one_one_nat )
    = one_one_nat ) ).

% fact_1
thf(fact_6658_fact__1,axiom,
    ( ( semiri2265585572941072030t_real @ one_one_nat )
    = one_one_real ) ).

% fact_1
thf(fact_6659_arccos__1,axiom,
    ( ( arccos @ one_one_real )
    = zero_zero_real ) ).

% arccos_1
thf(fact_6660_sgn__pos,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( sgn_sgn_Code_integer @ A )
        = one_one_Code_integer ) ) ).

% sgn_pos
thf(fact_6661_sgn__pos,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( sgn_sgn_real @ A )
        = one_one_real ) ) ).

% sgn_pos
thf(fact_6662_sgn__pos,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( sgn_sgn_rat @ A )
        = one_one_rat ) ) ).

% sgn_pos
thf(fact_6663_sgn__pos,axiom,
    ! [A: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( sgn_sgn_int @ A )
        = one_one_int ) ) ).

% sgn_pos
thf(fact_6664_abs__sgn__eq__1,axiom,
    ! [A: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( abs_abs_Code_integer @ ( sgn_sgn_Code_integer @ A ) )
        = one_one_Code_integer ) ) ).

% abs_sgn_eq_1
thf(fact_6665_abs__sgn__eq__1,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( ( abs_abs_real @ ( sgn_sgn_real @ A ) )
        = one_one_real ) ) ).

% abs_sgn_eq_1
thf(fact_6666_abs__sgn__eq__1,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( ( abs_abs_rat @ ( sgn_sgn_rat @ A ) )
        = one_one_rat ) ) ).

% abs_sgn_eq_1
thf(fact_6667_abs__sgn__eq__1,axiom,
    ! [A: int] :
      ( ( A != zero_zero_int )
     => ( ( abs_abs_int @ ( sgn_sgn_int @ A ) )
        = one_one_int ) ) ).

% abs_sgn_eq_1
thf(fact_6668_fact__Suc__0,axiom,
    ( ( semiri5044797733671781792omplex @ ( suc @ zero_zero_nat ) )
    = one_one_complex ) ).

% fact_Suc_0
thf(fact_6669_fact__Suc__0,axiom,
    ( ( semiri773545260158071498ct_rat @ ( suc @ zero_zero_nat ) )
    = one_one_rat ) ).

% fact_Suc_0
thf(fact_6670_fact__Suc__0,axiom,
    ( ( semiri1406184849735516958ct_int @ ( suc @ zero_zero_nat ) )
    = one_one_int ) ).

% fact_Suc_0
thf(fact_6671_fact__Suc__0,axiom,
    ( ( semiri1408675320244567234ct_nat @ ( suc @ zero_zero_nat ) )
    = one_one_nat ) ).

% fact_Suc_0
thf(fact_6672_fact__Suc__0,axiom,
    ( ( semiri2265585572941072030t_real @ ( suc @ zero_zero_nat ) )
    = one_one_real ) ).

% fact_Suc_0
thf(fact_6673_fact__Suc,axiom,
    ! [N: nat] :
      ( ( semiri5044797733671781792omplex @ ( suc @ N ) )
      = ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ N ) ) @ ( semiri5044797733671781792omplex @ N ) ) ) ).

% fact_Suc
thf(fact_6674_fact__Suc,axiom,
    ! [N: nat] :
      ( ( semiri773545260158071498ct_rat @ ( suc @ N ) )
      = ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ N ) ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).

% fact_Suc
thf(fact_6675_fact__Suc,axiom,
    ! [N: nat] :
      ( ( semiri1406184849735516958ct_int @ ( suc @ N ) )
      = ( times_times_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ).

% fact_Suc
thf(fact_6676_fact__Suc,axiom,
    ! [N: nat] :
      ( ( semiri1408675320244567234ct_nat @ ( suc @ N ) )
      = ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( suc @ N ) ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).

% fact_Suc
thf(fact_6677_fact__Suc,axiom,
    ! [N: nat] :
      ( ( semiri2265585572941072030t_real @ ( suc @ N ) )
      = ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) @ ( semiri2265585572941072030t_real @ N ) ) ) ).

% fact_Suc
thf(fact_6678_sgn__mult__self__eq,axiom,
    ! [A: real] :
      ( ( times_times_real @ ( sgn_sgn_real @ A ) @ ( sgn_sgn_real @ A ) )
      = ( zero_n3304061248610475627l_real @ ( A != zero_zero_real ) ) ) ).

% sgn_mult_self_eq
thf(fact_6679_sgn__mult__self__eq,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ ( sgn_sgn_rat @ A ) @ ( sgn_sgn_rat @ A ) )
      = ( zero_n2052037380579107095ol_rat @ ( A != zero_zero_rat ) ) ) ).

% sgn_mult_self_eq
thf(fact_6680_sgn__mult__self__eq,axiom,
    ! [A: int] :
      ( ( times_times_int @ ( sgn_sgn_int @ A ) @ ( sgn_sgn_int @ A ) )
      = ( zero_n2684676970156552555ol_int @ ( A != zero_zero_int ) ) ) ).

% sgn_mult_self_eq
thf(fact_6681_sgn__mult__self__eq,axiom,
    ! [A: code_integer] :
      ( ( times_3573771949741848930nteger @ ( sgn_sgn_Code_integer @ A ) @ ( sgn_sgn_Code_integer @ A ) )
      = ( zero_n356916108424825756nteger @ ( A != zero_z3403309356797280102nteger ) ) ) ).

% sgn_mult_self_eq
thf(fact_6682_idom__abs__sgn__class_Oabs__sgn,axiom,
    ! [A: complex] :
      ( ( sgn_sgn_complex @ ( abs_abs_complex @ A ) )
      = ( zero_n1201886186963655149omplex @ ( A != zero_zero_complex ) ) ) ).

% idom_abs_sgn_class.abs_sgn
thf(fact_6683_idom__abs__sgn__class_Oabs__sgn,axiom,
    ! [A: real] :
      ( ( sgn_sgn_real @ ( abs_abs_real @ A ) )
      = ( zero_n3304061248610475627l_real @ ( A != zero_zero_real ) ) ) ).

% idom_abs_sgn_class.abs_sgn
thf(fact_6684_idom__abs__sgn__class_Oabs__sgn,axiom,
    ! [A: rat] :
      ( ( sgn_sgn_rat @ ( abs_abs_rat @ A ) )
      = ( zero_n2052037380579107095ol_rat @ ( A != zero_zero_rat ) ) ) ).

% idom_abs_sgn_class.abs_sgn
thf(fact_6685_idom__abs__sgn__class_Oabs__sgn,axiom,
    ! [A: int] :
      ( ( sgn_sgn_int @ ( abs_abs_int @ A ) )
      = ( zero_n2684676970156552555ol_int @ ( A != zero_zero_int ) ) ) ).

% idom_abs_sgn_class.abs_sgn
thf(fact_6686_idom__abs__sgn__class_Oabs__sgn,axiom,
    ! [A: code_integer] :
      ( ( sgn_sgn_Code_integer @ ( abs_abs_Code_integer @ A ) )
      = ( zero_n356916108424825756nteger @ ( A != zero_z3403309356797280102nteger ) ) ) ).

% idom_abs_sgn_class.abs_sgn
thf(fact_6687_sgn__abs,axiom,
    ! [A: complex] :
      ( ( abs_abs_complex @ ( sgn_sgn_complex @ A ) )
      = ( zero_n1201886186963655149omplex @ ( A != zero_zero_complex ) ) ) ).

% sgn_abs
thf(fact_6688_sgn__abs,axiom,
    ! [A: real] :
      ( ( abs_abs_real @ ( sgn_sgn_real @ A ) )
      = ( zero_n3304061248610475627l_real @ ( A != zero_zero_real ) ) ) ).

% sgn_abs
thf(fact_6689_sgn__abs,axiom,
    ! [A: rat] :
      ( ( abs_abs_rat @ ( sgn_sgn_rat @ A ) )
      = ( zero_n2052037380579107095ol_rat @ ( A != zero_zero_rat ) ) ) ).

% sgn_abs
thf(fact_6690_sgn__abs,axiom,
    ! [A: int] :
      ( ( abs_abs_int @ ( sgn_sgn_int @ A ) )
      = ( zero_n2684676970156552555ol_int @ ( A != zero_zero_int ) ) ) ).

% sgn_abs
thf(fact_6691_sgn__abs,axiom,
    ! [A: code_integer] :
      ( ( abs_abs_Code_integer @ ( sgn_sgn_Code_integer @ A ) )
      = ( zero_n356916108424825756nteger @ ( A != zero_z3403309356797280102nteger ) ) ) ).

% sgn_abs
thf(fact_6692_arccos__minus__1,axiom,
    ( ( arccos @ ( uminus_uminus_real @ one_one_real ) )
    = pi ) ).

% arccos_minus_1
thf(fact_6693_sgn__mult__dvd__iff,axiom,
    ! [R2: int,L2: int,K: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ ( sgn_sgn_int @ R2 ) @ L2 ) @ K )
      = ( ( dvd_dvd_int @ L2 @ K )
        & ( ( R2 = zero_zero_int )
         => ( K = zero_zero_int ) ) ) ) ).

% sgn_mult_dvd_iff
thf(fact_6694_mult__sgn__dvd__iff,axiom,
    ! [L2: int,R2: int,K: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ L2 @ ( sgn_sgn_int @ R2 ) ) @ K )
      = ( ( dvd_dvd_int @ L2 @ K )
        & ( ( R2 = zero_zero_int )
         => ( K = zero_zero_int ) ) ) ) ).

% mult_sgn_dvd_iff
thf(fact_6695_dvd__sgn__mult__iff,axiom,
    ! [L2: int,R2: int,K: int] :
      ( ( dvd_dvd_int @ L2 @ ( times_times_int @ ( sgn_sgn_int @ R2 ) @ K ) )
      = ( ( dvd_dvd_int @ L2 @ K )
        | ( R2 = zero_zero_int ) ) ) ).

% dvd_sgn_mult_iff
thf(fact_6696_dvd__mult__sgn__iff,axiom,
    ! [L2: int,K: int,R2: int] :
      ( ( dvd_dvd_int @ L2 @ ( times_times_int @ K @ ( sgn_sgn_int @ R2 ) ) )
      = ( ( dvd_dvd_int @ L2 @ K )
        | ( R2 = zero_zero_int ) ) ) ).

% dvd_mult_sgn_iff
thf(fact_6697_sgn__neg,axiom,
    ! [A: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( sgn_sgn_real @ A )
        = ( uminus_uminus_real @ one_one_real ) ) ) ).

% sgn_neg
thf(fact_6698_sgn__neg,axiom,
    ! [A: int] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ( sgn_sgn_int @ A )
        = ( uminus_uminus_int @ one_one_int ) ) ) ).

% sgn_neg
thf(fact_6699_sgn__neg,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger )
     => ( ( sgn_sgn_Code_integer @ A )
        = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ).

% sgn_neg
thf(fact_6700_sgn__neg,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( sgn_sgn_rat @ A )
        = ( uminus_uminus_rat @ one_one_rat ) ) ) ).

% sgn_neg
thf(fact_6701_fact__2,axiom,
    ( ( semiri5044797733671781792omplex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).

% fact_2
thf(fact_6702_fact__2,axiom,
    ( ( semiri773545260158071498ct_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).

% fact_2
thf(fact_6703_fact__2,axiom,
    ( ( semiri1406184849735516958ct_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).

% fact_2
thf(fact_6704_fact__2,axiom,
    ( ( semiri1408675320244567234ct_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% fact_2
thf(fact_6705_fact__2,axiom,
    ( ( semiri2265585572941072030t_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).

% fact_2
thf(fact_6706_sgn__of__nat,axiom,
    ! [N: nat] :
      ( ( sgn_sgn_real @ ( semiri5074537144036343181t_real @ N ) )
      = ( zero_n3304061248610475627l_real @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% sgn_of_nat
thf(fact_6707_sgn__of__nat,axiom,
    ! [N: nat] :
      ( ( sgn_sgn_rat @ ( semiri681578069525770553at_rat @ N ) )
      = ( zero_n2052037380579107095ol_rat @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% sgn_of_nat
thf(fact_6708_sgn__of__nat,axiom,
    ! [N: nat] :
      ( ( sgn_sgn_int @ ( semiri1314217659103216013at_int @ N ) )
      = ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% sgn_of_nat
thf(fact_6709_sgn__of__nat,axiom,
    ! [N: nat] :
      ( ( sgn_sgn_Code_integer @ ( semiri4939895301339042750nteger @ N ) )
      = ( zero_n356916108424825756nteger @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% sgn_of_nat
thf(fact_6710_cos__arccos,axiom,
    ! [Y3: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
     => ( ( ord_less_eq_real @ Y3 @ one_one_real )
       => ( ( cos_real @ ( arccos @ Y3 ) )
          = Y3 ) ) ) ).

% cos_arccos
thf(fact_6711_arccos__0,axiom,
    ( ( arccos @ zero_zero_real )
    = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% arccos_0
thf(fact_6712_complex__exp__exists,axiom,
    ! [Z: complex] :
    ? [A5: complex,R3: real] :
      ( Z
      = ( times_times_complex @ ( real_V4546457046886955230omplex @ R3 ) @ ( exp_complex @ A5 ) ) ) ).

% complex_exp_exists
thf(fact_6713_fact__mono__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).

% fact_mono_nat
thf(fact_6714_fact__ge__self,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ N @ ( semiri1408675320244567234ct_nat @ N ) ) ).

% fact_ge_self
thf(fact_6715_sgn__eq__0__iff,axiom,
    ! [A: complex] :
      ( ( ( sgn_sgn_complex @ A )
        = zero_zero_complex )
      = ( A = zero_zero_complex ) ) ).

% sgn_eq_0_iff
thf(fact_6716_sgn__eq__0__iff,axiom,
    ! [A: code_integer] :
      ( ( ( sgn_sgn_Code_integer @ A )
        = zero_z3403309356797280102nteger )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% sgn_eq_0_iff
thf(fact_6717_sgn__eq__0__iff,axiom,
    ! [A: real] :
      ( ( ( sgn_sgn_real @ A )
        = zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% sgn_eq_0_iff
thf(fact_6718_sgn__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( ( sgn_sgn_rat @ A )
        = zero_zero_rat )
      = ( A = zero_zero_rat ) ) ).

% sgn_eq_0_iff
thf(fact_6719_sgn__eq__0__iff,axiom,
    ! [A: int] :
      ( ( ( sgn_sgn_int @ A )
        = zero_zero_int )
      = ( A = zero_zero_int ) ) ).

% sgn_eq_0_iff
thf(fact_6720_sgn__0__0,axiom,
    ! [A: code_integer] :
      ( ( ( sgn_sgn_Code_integer @ A )
        = zero_z3403309356797280102nteger )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% sgn_0_0
thf(fact_6721_sgn__0__0,axiom,
    ! [A: real] :
      ( ( ( sgn_sgn_real @ A )
        = zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% sgn_0_0
thf(fact_6722_sgn__0__0,axiom,
    ! [A: rat] :
      ( ( ( sgn_sgn_rat @ A )
        = zero_zero_rat )
      = ( A = zero_zero_rat ) ) ).

% sgn_0_0
thf(fact_6723_sgn__0__0,axiom,
    ! [A: int] :
      ( ( ( sgn_sgn_int @ A )
        = zero_zero_int )
      = ( A = zero_zero_int ) ) ).

% sgn_0_0
thf(fact_6724_Real__Vector__Spaces_Osgn__mult,axiom,
    ! [X4: complex,Y3: complex] :
      ( ( sgn_sgn_complex @ ( times_times_complex @ X4 @ Y3 ) )
      = ( times_times_complex @ ( sgn_sgn_complex @ X4 ) @ ( sgn_sgn_complex @ Y3 ) ) ) ).

% Real_Vector_Spaces.sgn_mult
thf(fact_6725_Real__Vector__Spaces_Osgn__mult,axiom,
    ! [X4: real,Y3: real] :
      ( ( sgn_sgn_real @ ( times_times_real @ X4 @ Y3 ) )
      = ( times_times_real @ ( sgn_sgn_real @ X4 ) @ ( sgn_sgn_real @ Y3 ) ) ) ).

% Real_Vector_Spaces.sgn_mult
thf(fact_6726_sgn__mult,axiom,
    ! [A: complex,B2: complex] :
      ( ( sgn_sgn_complex @ ( times_times_complex @ A @ B2 ) )
      = ( times_times_complex @ ( sgn_sgn_complex @ A ) @ ( sgn_sgn_complex @ B2 ) ) ) ).

% sgn_mult
thf(fact_6727_sgn__mult,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( sgn_sgn_Code_integer @ ( times_3573771949741848930nteger @ A @ B2 ) )
      = ( times_3573771949741848930nteger @ ( sgn_sgn_Code_integer @ A ) @ ( sgn_sgn_Code_integer @ B2 ) ) ) ).

% sgn_mult
thf(fact_6728_sgn__mult,axiom,
    ! [A: real,B2: real] :
      ( ( sgn_sgn_real @ ( times_times_real @ A @ B2 ) )
      = ( times_times_real @ ( sgn_sgn_real @ A ) @ ( sgn_sgn_real @ B2 ) ) ) ).

% sgn_mult
thf(fact_6729_sgn__mult,axiom,
    ! [A: rat,B2: rat] :
      ( ( sgn_sgn_rat @ ( times_times_rat @ A @ B2 ) )
      = ( times_times_rat @ ( sgn_sgn_rat @ A ) @ ( sgn_sgn_rat @ B2 ) ) ) ).

% sgn_mult
thf(fact_6730_sgn__mult,axiom,
    ! [A: int,B2: int] :
      ( ( sgn_sgn_int @ ( times_times_int @ A @ B2 ) )
      = ( times_times_int @ ( sgn_sgn_int @ A ) @ ( sgn_sgn_int @ B2 ) ) ) ).

% sgn_mult
thf(fact_6731_same__sgn__sgn__add,axiom,
    ! [B2: code_integer,A: code_integer] :
      ( ( ( sgn_sgn_Code_integer @ B2 )
        = ( sgn_sgn_Code_integer @ A ) )
     => ( ( sgn_sgn_Code_integer @ ( plus_p5714425477246183910nteger @ A @ B2 ) )
        = ( sgn_sgn_Code_integer @ A ) ) ) ).

% same_sgn_sgn_add
thf(fact_6732_same__sgn__sgn__add,axiom,
    ! [B2: real,A: real] :
      ( ( ( sgn_sgn_real @ B2 )
        = ( sgn_sgn_real @ A ) )
     => ( ( sgn_sgn_real @ ( plus_plus_real @ A @ B2 ) )
        = ( sgn_sgn_real @ A ) ) ) ).

% same_sgn_sgn_add
thf(fact_6733_same__sgn__sgn__add,axiom,
    ! [B2: rat,A: rat] :
      ( ( ( sgn_sgn_rat @ B2 )
        = ( sgn_sgn_rat @ A ) )
     => ( ( sgn_sgn_rat @ ( plus_plus_rat @ A @ B2 ) )
        = ( sgn_sgn_rat @ A ) ) ) ).

% same_sgn_sgn_add
thf(fact_6734_same__sgn__sgn__add,axiom,
    ! [B2: int,A: int] :
      ( ( ( sgn_sgn_int @ B2 )
        = ( sgn_sgn_int @ A ) )
     => ( ( sgn_sgn_int @ ( plus_plus_int @ A @ B2 ) )
        = ( sgn_sgn_int @ A ) ) ) ).

% same_sgn_sgn_add
thf(fact_6735_fact__less__mono__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_nat @ M @ N )
       => ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ) ).

% fact_less_mono_nat
thf(fact_6736_sgn__not__eq__imp,axiom,
    ! [B2: real,A: real] :
      ( ( ( sgn_sgn_real @ B2 )
       != ( sgn_sgn_real @ A ) )
     => ( ( ( sgn_sgn_real @ A )
         != zero_zero_real )
       => ( ( ( sgn_sgn_real @ B2 )
           != zero_zero_real )
         => ( ( sgn_sgn_real @ A )
            = ( uminus_uminus_real @ ( sgn_sgn_real @ B2 ) ) ) ) ) ) ).

% sgn_not_eq_imp
thf(fact_6737_sgn__not__eq__imp,axiom,
    ! [B2: int,A: int] :
      ( ( ( sgn_sgn_int @ B2 )
       != ( sgn_sgn_int @ A ) )
     => ( ( ( sgn_sgn_int @ A )
         != zero_zero_int )
       => ( ( ( sgn_sgn_int @ B2 )
           != zero_zero_int )
         => ( ( sgn_sgn_int @ A )
            = ( uminus_uminus_int @ ( sgn_sgn_int @ B2 ) ) ) ) ) ) ).

% sgn_not_eq_imp
thf(fact_6738_sgn__not__eq__imp,axiom,
    ! [B2: code_integer,A: code_integer] :
      ( ( ( sgn_sgn_Code_integer @ B2 )
       != ( sgn_sgn_Code_integer @ A ) )
     => ( ( ( sgn_sgn_Code_integer @ A )
         != zero_z3403309356797280102nteger )
       => ( ( ( sgn_sgn_Code_integer @ B2 )
           != zero_z3403309356797280102nteger )
         => ( ( sgn_sgn_Code_integer @ A )
            = ( uminus1351360451143612070nteger @ ( sgn_sgn_Code_integer @ B2 ) ) ) ) ) ) ).

% sgn_not_eq_imp
thf(fact_6739_sgn__not__eq__imp,axiom,
    ! [B2: rat,A: rat] :
      ( ( ( sgn_sgn_rat @ B2 )
       != ( sgn_sgn_rat @ A ) )
     => ( ( ( sgn_sgn_rat @ A )
         != zero_zero_rat )
       => ( ( ( sgn_sgn_rat @ B2 )
           != zero_zero_rat )
         => ( ( sgn_sgn_rat @ A )
            = ( uminus_uminus_rat @ ( sgn_sgn_rat @ B2 ) ) ) ) ) ) ).

% sgn_not_eq_imp
thf(fact_6740_sgn__minus__1,axiom,
    ( ( sgn_sgn_real @ ( uminus_uminus_real @ one_one_real ) )
    = ( uminus_uminus_real @ one_one_real ) ) ).

% sgn_minus_1
thf(fact_6741_sgn__minus__1,axiom,
    ( ( sgn_sgn_int @ ( uminus_uminus_int @ one_one_int ) )
    = ( uminus_uminus_int @ one_one_int ) ) ).

% sgn_minus_1
thf(fact_6742_sgn__minus__1,axiom,
    ( ( sgn_sgn_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% sgn_minus_1
thf(fact_6743_sgn__minus__1,axiom,
    ( ( sgn_sgn_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% sgn_minus_1
thf(fact_6744_sgn__minus__1,axiom,
    ( ( sgn_sgn_rat @ ( uminus_uminus_rat @ one_one_rat ) )
    = ( uminus_uminus_rat @ one_one_rat ) ) ).

% sgn_minus_1
thf(fact_6745_fact__ge__zero,axiom,
    ! [N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( semiri773545260158071498ct_rat @ N ) ) ).

% fact_ge_zero
thf(fact_6746_fact__ge__zero,axiom,
    ! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1406184849735516958ct_int @ N ) ) ).

% fact_ge_zero
thf(fact_6747_fact__ge__zero,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1408675320244567234ct_nat @ N ) ) ).

% fact_ge_zero
thf(fact_6748_fact__ge__zero,axiom,
    ! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri2265585572941072030t_real @ N ) ) ).

% fact_ge_zero
thf(fact_6749_linordered__idom__class_Oabs__sgn,axiom,
    ( abs_abs_Code_integer
    = ( ^ [K3: code_integer] : ( times_3573771949741848930nteger @ K3 @ ( sgn_sgn_Code_integer @ K3 ) ) ) ) ).

% linordered_idom_class.abs_sgn
thf(fact_6750_linordered__idom__class_Oabs__sgn,axiom,
    ( abs_abs_real
    = ( ^ [K3: real] : ( times_times_real @ K3 @ ( sgn_sgn_real @ K3 ) ) ) ) ).

% linordered_idom_class.abs_sgn
thf(fact_6751_linordered__idom__class_Oabs__sgn,axiom,
    ( abs_abs_rat
    = ( ^ [K3: rat] : ( times_times_rat @ K3 @ ( sgn_sgn_rat @ K3 ) ) ) ) ).

% linordered_idom_class.abs_sgn
thf(fact_6752_linordered__idom__class_Oabs__sgn,axiom,
    ( abs_abs_int
    = ( ^ [K3: int] : ( times_times_int @ K3 @ ( sgn_sgn_int @ K3 ) ) ) ) ).

% linordered_idom_class.abs_sgn
thf(fact_6753_abs__mult__sgn,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ ( abs_abs_complex @ A ) @ ( sgn_sgn_complex @ A ) )
      = A ) ).

% abs_mult_sgn
thf(fact_6754_abs__mult__sgn,axiom,
    ! [A: code_integer] :
      ( ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( sgn_sgn_Code_integer @ A ) )
      = A ) ).

% abs_mult_sgn
thf(fact_6755_abs__mult__sgn,axiom,
    ! [A: real] :
      ( ( times_times_real @ ( abs_abs_real @ A ) @ ( sgn_sgn_real @ A ) )
      = A ) ).

% abs_mult_sgn
thf(fact_6756_abs__mult__sgn,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ ( abs_abs_rat @ A ) @ ( sgn_sgn_rat @ A ) )
      = A ) ).

% abs_mult_sgn
thf(fact_6757_abs__mult__sgn,axiom,
    ! [A: int] :
      ( ( times_times_int @ ( abs_abs_int @ A ) @ ( sgn_sgn_int @ A ) )
      = A ) ).

% abs_mult_sgn
thf(fact_6758_sgn__mult__abs,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ ( sgn_sgn_complex @ A ) @ ( abs_abs_complex @ A ) )
      = A ) ).

% sgn_mult_abs
thf(fact_6759_sgn__mult__abs,axiom,
    ! [A: code_integer] :
      ( ( times_3573771949741848930nteger @ ( sgn_sgn_Code_integer @ A ) @ ( abs_abs_Code_integer @ A ) )
      = A ) ).

% sgn_mult_abs
thf(fact_6760_sgn__mult__abs,axiom,
    ! [A: real] :
      ( ( times_times_real @ ( sgn_sgn_real @ A ) @ ( abs_abs_real @ A ) )
      = A ) ).

% sgn_mult_abs
thf(fact_6761_sgn__mult__abs,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ ( sgn_sgn_rat @ A ) @ ( abs_abs_rat @ A ) )
      = A ) ).

% sgn_mult_abs
thf(fact_6762_sgn__mult__abs,axiom,
    ! [A: int] :
      ( ( times_times_int @ ( sgn_sgn_int @ A ) @ ( abs_abs_int @ A ) )
      = A ) ).

% sgn_mult_abs
thf(fact_6763_mult__sgn__abs,axiom,
    ! [X4: code_integer] :
      ( ( times_3573771949741848930nteger @ ( sgn_sgn_Code_integer @ X4 ) @ ( abs_abs_Code_integer @ X4 ) )
      = X4 ) ).

% mult_sgn_abs
thf(fact_6764_mult__sgn__abs,axiom,
    ! [X4: real] :
      ( ( times_times_real @ ( sgn_sgn_real @ X4 ) @ ( abs_abs_real @ X4 ) )
      = X4 ) ).

% mult_sgn_abs
thf(fact_6765_mult__sgn__abs,axiom,
    ! [X4: rat] :
      ( ( times_times_rat @ ( sgn_sgn_rat @ X4 ) @ ( abs_abs_rat @ X4 ) )
      = X4 ) ).

% mult_sgn_abs
thf(fact_6766_mult__sgn__abs,axiom,
    ! [X4: int] :
      ( ( times_times_int @ ( sgn_sgn_int @ X4 ) @ ( abs_abs_int @ X4 ) )
      = X4 ) ).

% mult_sgn_abs
thf(fact_6767_fact__not__neg,axiom,
    ! [N: nat] :
      ~ ( ord_less_rat @ ( semiri773545260158071498ct_rat @ N ) @ zero_zero_rat ) ).

% fact_not_neg
thf(fact_6768_fact__not__neg,axiom,
    ! [N: nat] :
      ~ ( ord_less_int @ ( semiri1406184849735516958ct_int @ N ) @ zero_zero_int ) ).

% fact_not_neg
thf(fact_6769_fact__not__neg,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ N ) @ zero_zero_nat ) ).

% fact_not_neg
thf(fact_6770_fact__not__neg,axiom,
    ! [N: nat] :
      ~ ( ord_less_real @ ( semiri2265585572941072030t_real @ N ) @ zero_zero_real ) ).

% fact_not_neg
thf(fact_6771_fact__gt__zero,axiom,
    ! [N: nat] : ( ord_less_rat @ zero_zero_rat @ ( semiri773545260158071498ct_rat @ N ) ) ).

% fact_gt_zero
thf(fact_6772_fact__gt__zero,axiom,
    ! [N: nat] : ( ord_less_int @ zero_zero_int @ ( semiri1406184849735516958ct_int @ N ) ) ).

% fact_gt_zero
thf(fact_6773_fact__gt__zero,axiom,
    ! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( semiri1408675320244567234ct_nat @ N ) ) ).

% fact_gt_zero
thf(fact_6774_fact__gt__zero,axiom,
    ! [N: nat] : ( ord_less_real @ zero_zero_real @ ( semiri2265585572941072030t_real @ N ) ) ).

% fact_gt_zero
thf(fact_6775_int__sgnE,axiom,
    ! [K: int] :
      ~ ! [N4: nat,L3: int] :
          ( K
         != ( times_times_int @ ( sgn_sgn_int @ L3 ) @ ( semiri1314217659103216013at_int @ N4 ) ) ) ).

% int_sgnE
thf(fact_6776_same__sgn__abs__add,axiom,
    ! [B2: code_integer,A: code_integer] :
      ( ( ( sgn_sgn_Code_integer @ B2 )
        = ( sgn_sgn_Code_integer @ A ) )
     => ( ( abs_abs_Code_integer @ ( plus_p5714425477246183910nteger @ A @ B2 ) )
        = ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B2 ) ) ) ) ).

% same_sgn_abs_add
thf(fact_6777_same__sgn__abs__add,axiom,
    ! [B2: real,A: real] :
      ( ( ( sgn_sgn_real @ B2 )
        = ( sgn_sgn_real @ A ) )
     => ( ( abs_abs_real @ ( plus_plus_real @ A @ B2 ) )
        = ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B2 ) ) ) ) ).

% same_sgn_abs_add
thf(fact_6778_same__sgn__abs__add,axiom,
    ! [B2: rat,A: rat] :
      ( ( ( sgn_sgn_rat @ B2 )
        = ( sgn_sgn_rat @ A ) )
     => ( ( abs_abs_rat @ ( plus_plus_rat @ A @ B2 ) )
        = ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B2 ) ) ) ) ).

% same_sgn_abs_add
thf(fact_6779_same__sgn__abs__add,axiom,
    ! [B2: int,A: int] :
      ( ( ( sgn_sgn_int @ B2 )
        = ( sgn_sgn_int @ A ) )
     => ( ( abs_abs_int @ ( plus_plus_int @ A @ B2 ) )
        = ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B2 ) ) ) ) ).

% same_sgn_abs_add
thf(fact_6780_fact__ge__1,axiom,
    ! [N: nat] : ( ord_less_eq_rat @ one_one_rat @ ( semiri773545260158071498ct_rat @ N ) ) ).

% fact_ge_1
thf(fact_6781_fact__ge__1,axiom,
    ! [N: nat] : ( ord_less_eq_int @ one_one_int @ ( semiri1406184849735516958ct_int @ N ) ) ).

% fact_ge_1
thf(fact_6782_fact__ge__1,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ one_one_nat @ ( semiri1408675320244567234ct_nat @ N ) ) ).

% fact_ge_1
thf(fact_6783_fact__ge__1,axiom,
    ! [N: nat] : ( ord_less_eq_real @ one_one_real @ ( semiri2265585572941072030t_real @ N ) ) ).

% fact_ge_1
thf(fact_6784_fact__mono,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_rat @ ( semiri773545260158071498ct_rat @ M ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).

% fact_mono
thf(fact_6785_fact__mono,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_int @ ( semiri1406184849735516958ct_int @ M ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ).

% fact_mono
thf(fact_6786_fact__mono,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).

% fact_mono
thf(fact_6787_fact__mono,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_real @ ( semiri2265585572941072030t_real @ M ) @ ( semiri2265585572941072030t_real @ N ) ) ) ).

% fact_mono
thf(fact_6788_div__eq__sgn__abs,axiom,
    ! [K: int,L2: int] :
      ( ( ( sgn_sgn_int @ K )
        = ( sgn_sgn_int @ L2 ) )
     => ( ( divide_divide_int @ K @ L2 )
        = ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L2 ) ) ) ) ).

% div_eq_sgn_abs
thf(fact_6789_fact__dvd,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( dvd_dvd_int @ ( semiri1406184849735516958ct_int @ N ) @ ( semiri1406184849735516958ct_int @ M ) ) ) ).

% fact_dvd
thf(fact_6790_fact__dvd,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( dvd_dvd_Code_integer @ ( semiri3624122377584611663nteger @ N ) @ ( semiri3624122377584611663nteger @ M ) ) ) ).

% fact_dvd
thf(fact_6791_fact__dvd,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( dvd_dvd_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1408675320244567234ct_nat @ M ) ) ) ).

% fact_dvd
thf(fact_6792_fact__dvd,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( dvd_dvd_real @ ( semiri2265585572941072030t_real @ N ) @ ( semiri2265585572941072030t_real @ M ) ) ) ).

% fact_dvd
thf(fact_6793_pochhammer__fact,axiom,
    ( semiri5044797733671781792omplex
    = ( comm_s2602460028002588243omplex @ one_one_complex ) ) ).

% pochhammer_fact
thf(fact_6794_pochhammer__fact,axiom,
    ( semiri773545260158071498ct_rat
    = ( comm_s4028243227959126397er_rat @ one_one_rat ) ) ).

% pochhammer_fact
thf(fact_6795_pochhammer__fact,axiom,
    ( semiri1406184849735516958ct_int
    = ( comm_s4660882817536571857er_int @ one_one_int ) ) ).

% pochhammer_fact
thf(fact_6796_pochhammer__fact,axiom,
    ( semiri1408675320244567234ct_nat
    = ( comm_s4663373288045622133er_nat @ one_one_nat ) ) ).

% pochhammer_fact
thf(fact_6797_pochhammer__fact,axiom,
    ( semiri2265585572941072030t_real
    = ( comm_s7457072308508201937r_real @ one_one_real ) ) ).

% pochhammer_fact
thf(fact_6798_fact__ge__Suc__0__nat,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( semiri1408675320244567234ct_nat @ N ) ) ).

% fact_ge_Suc_0_nat
thf(fact_6799_sgn__1__pos,axiom,
    ! [A: code_integer] :
      ( ( ( sgn_sgn_Code_integer @ A )
        = one_one_Code_integer )
      = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).

% sgn_1_pos
thf(fact_6800_sgn__1__pos,axiom,
    ! [A: real] :
      ( ( ( sgn_sgn_real @ A )
        = one_one_real )
      = ( ord_less_real @ zero_zero_real @ A ) ) ).

% sgn_1_pos
thf(fact_6801_sgn__1__pos,axiom,
    ! [A: rat] :
      ( ( ( sgn_sgn_rat @ A )
        = one_one_rat )
      = ( ord_less_rat @ zero_zero_rat @ A ) ) ).

% sgn_1_pos
thf(fact_6802_sgn__1__pos,axiom,
    ! [A: int] :
      ( ( ( sgn_sgn_int @ A )
        = one_one_int )
      = ( ord_less_int @ zero_zero_int @ A ) ) ).

% sgn_1_pos
thf(fact_6803_abs__sgn__eq,axiom,
    ! [A: code_integer] :
      ( ( ( A = zero_z3403309356797280102nteger )
       => ( ( abs_abs_Code_integer @ ( sgn_sgn_Code_integer @ A ) )
          = zero_z3403309356797280102nteger ) )
      & ( ( A != zero_z3403309356797280102nteger )
       => ( ( abs_abs_Code_integer @ ( sgn_sgn_Code_integer @ A ) )
          = one_one_Code_integer ) ) ) ).

% abs_sgn_eq
thf(fact_6804_abs__sgn__eq,axiom,
    ! [A: real] :
      ( ( ( A = zero_zero_real )
       => ( ( abs_abs_real @ ( sgn_sgn_real @ A ) )
          = zero_zero_real ) )
      & ( ( A != zero_zero_real )
       => ( ( abs_abs_real @ ( sgn_sgn_real @ A ) )
          = one_one_real ) ) ) ).

% abs_sgn_eq
thf(fact_6805_abs__sgn__eq,axiom,
    ! [A: rat] :
      ( ( ( A = zero_zero_rat )
       => ( ( abs_abs_rat @ ( sgn_sgn_rat @ A ) )
          = zero_zero_rat ) )
      & ( ( A != zero_zero_rat )
       => ( ( abs_abs_rat @ ( sgn_sgn_rat @ A ) )
          = one_one_rat ) ) ) ).

% abs_sgn_eq
thf(fact_6806_abs__sgn__eq,axiom,
    ! [A: int] :
      ( ( ( A = zero_zero_int )
       => ( ( abs_abs_int @ ( sgn_sgn_int @ A ) )
          = zero_zero_int ) )
      & ( ( A != zero_zero_int )
       => ( ( abs_abs_int @ ( sgn_sgn_int @ A ) )
          = one_one_int ) ) ) ).

% abs_sgn_eq
thf(fact_6807_dvd__fact,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ M )
     => ( ( ord_less_eq_nat @ M @ N )
       => ( dvd_dvd_nat @ M @ ( semiri1408675320244567234ct_nat @ N ) ) ) ) ).

% dvd_fact
thf(fact_6808_fact__less__mono,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_nat @ M @ N )
       => ( ord_less_rat @ ( semiri773545260158071498ct_rat @ M ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ) ).

% fact_less_mono
thf(fact_6809_fact__less__mono,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_nat @ M @ N )
       => ( ord_less_int @ ( semiri1406184849735516958ct_int @ M ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ) ).

% fact_less_mono
thf(fact_6810_fact__less__mono,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_nat @ M @ N )
       => ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ) ).

% fact_less_mono
thf(fact_6811_fact__less__mono,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_nat @ M @ N )
       => ( ord_less_real @ ( semiri2265585572941072030t_real @ M ) @ ( semiri2265585572941072030t_real @ N ) ) ) ) ).

% fact_less_mono
thf(fact_6812_sgn__mod,axiom,
    ! [L2: int,K: int] :
      ( ( L2 != zero_zero_int )
     => ( ~ ( dvd_dvd_int @ L2 @ K )
       => ( ( sgn_sgn_int @ ( modulo_modulo_int @ K @ L2 ) )
          = ( sgn_sgn_int @ L2 ) ) ) ) ).

% sgn_mod
thf(fact_6813_fact__mod,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( modulo_modulo_int @ ( semiri1406184849735516958ct_int @ N ) @ ( semiri1406184849735516958ct_int @ M ) )
        = zero_zero_int ) ) ).

% fact_mod
thf(fact_6814_fact__mod,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( modulo364778990260209775nteger @ ( semiri3624122377584611663nteger @ N ) @ ( semiri3624122377584611663nteger @ M ) )
        = zero_z3403309356797280102nteger ) ) ).

% fact_mod
thf(fact_6815_fact__mod,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( modulo_modulo_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1408675320244567234ct_nat @ M ) )
        = zero_zero_nat ) ) ).

% fact_mod
thf(fact_6816_fact__fact__dvd__fact,axiom,
    ! [K: nat,N: nat] : ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ ( semiri3624122377584611663nteger @ K ) @ ( semiri3624122377584611663nteger @ N ) ) @ ( semiri3624122377584611663nteger @ ( plus_plus_nat @ K @ N ) ) ) ).

% fact_fact_dvd_fact
thf(fact_6817_fact__fact__dvd__fact,axiom,
    ! [K: nat,N: nat] : ( dvd_dvd_rat @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri773545260158071498ct_rat @ N ) ) @ ( semiri773545260158071498ct_rat @ ( plus_plus_nat @ K @ N ) ) ) ).

% fact_fact_dvd_fact
thf(fact_6818_fact__fact__dvd__fact,axiom,
    ! [K: nat,N: nat] : ( dvd_dvd_int @ ( times_times_int @ ( semiri1406184849735516958ct_int @ K ) @ ( semiri1406184849735516958ct_int @ N ) ) @ ( semiri1406184849735516958ct_int @ ( plus_plus_nat @ K @ N ) ) ) ).

% fact_fact_dvd_fact
thf(fact_6819_fact__fact__dvd__fact,axiom,
    ! [K: nat,N: nat] : ( dvd_dvd_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ N ) ) @ ( semiri1408675320244567234ct_nat @ ( plus_plus_nat @ K @ N ) ) ) ).

% fact_fact_dvd_fact
thf(fact_6820_fact__fact__dvd__fact,axiom,
    ! [K: nat,N: nat] : ( dvd_dvd_real @ ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( semiri2265585572941072030t_real @ ( plus_plus_nat @ K @ N ) ) ) ).

% fact_fact_dvd_fact
thf(fact_6821_fact__le__power,axiom,
    ! [N: nat] : ( ord_less_eq_rat @ ( semiri773545260158071498ct_rat @ N ) @ ( semiri681578069525770553at_rat @ ( power_power_nat @ N @ N ) ) ) ).

% fact_le_power
thf(fact_6822_fact__le__power,axiom,
    ! [N: nat] : ( ord_less_eq_int @ ( semiri1406184849735516958ct_int @ N ) @ ( semiri1314217659103216013at_int @ ( power_power_nat @ N @ N ) ) ) ).

% fact_le_power
thf(fact_6823_fact__le__power,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1316708129612266289at_nat @ ( power_power_nat @ N @ N ) ) ) ).

% fact_le_power
thf(fact_6824_fact__le__power,axiom,
    ! [N: nat] : ( ord_less_eq_real @ ( semiri2265585572941072030t_real @ N ) @ ( semiri5074537144036343181t_real @ ( power_power_nat @ N @ N ) ) ) ).

% fact_le_power
thf(fact_6825_arccos__le__arccos,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X4 )
     => ( ( ord_less_eq_real @ X4 @ Y3 )
       => ( ( ord_less_eq_real @ Y3 @ one_one_real )
         => ( ord_less_eq_real @ ( arccos @ Y3 ) @ ( arccos @ X4 ) ) ) ) ) ).

% arccos_le_arccos
thf(fact_6826_arccos__eq__iff,axiom,
    ! [X4: real,Y3: real] :
      ( ( ( ord_less_eq_real @ ( abs_abs_real @ X4 ) @ one_one_real )
        & ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real ) )
     => ( ( ( arccos @ X4 )
          = ( arccos @ Y3 ) )
        = ( X4 = Y3 ) ) ) ).

% arccos_eq_iff
thf(fact_6827_arccos__le__mono,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X4 ) @ one_one_real )
     => ( ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
       => ( ( ord_less_eq_real @ ( arccos @ X4 ) @ ( arccos @ Y3 ) )
          = ( ord_less_eq_real @ Y3 @ X4 ) ) ) ) ).

% arccos_le_mono
thf(fact_6828_fact__diff__Suc,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ N @ ( suc @ M ) )
     => ( ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) )
        = ( times_times_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ M @ N ) ) ) ) ) ).

% fact_diff_Suc
thf(fact_6829_sgn__if,axiom,
    ( sgn_sgn_real
    = ( ^ [X: real] : ( if_real @ ( X = zero_zero_real ) @ zero_zero_real @ ( if_real @ ( ord_less_real @ zero_zero_real @ X ) @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ) ) ) ).

% sgn_if
thf(fact_6830_sgn__if,axiom,
    ( sgn_sgn_int
    = ( ^ [X: int] : ( if_int @ ( X = zero_zero_int ) @ zero_zero_int @ ( if_int @ ( ord_less_int @ zero_zero_int @ X ) @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).

% sgn_if
thf(fact_6831_sgn__if,axiom,
    ( sgn_sgn_Code_integer
    = ( ^ [X: code_integer] : ( if_Code_integer @ ( X = zero_z3403309356797280102nteger ) @ zero_z3403309356797280102nteger @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ X ) @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ) ).

% sgn_if
thf(fact_6832_sgn__if,axiom,
    ( sgn_sgn_rat
    = ( ^ [X: rat] : ( if_rat @ ( X = zero_zero_rat ) @ zero_zero_rat @ ( if_rat @ ( ord_less_rat @ zero_zero_rat @ X ) @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ) ) ) ).

% sgn_if
thf(fact_6833_sgn__1__neg,axiom,
    ! [A: real] :
      ( ( ( sgn_sgn_real @ A )
        = ( uminus_uminus_real @ one_one_real ) )
      = ( ord_less_real @ A @ zero_zero_real ) ) ).

% sgn_1_neg
thf(fact_6834_sgn__1__neg,axiom,
    ! [A: int] :
      ( ( ( sgn_sgn_int @ A )
        = ( uminus_uminus_int @ one_one_int ) )
      = ( ord_less_int @ A @ zero_zero_int ) ) ).

% sgn_1_neg
thf(fact_6835_sgn__1__neg,axiom,
    ! [A: code_integer] :
      ( ( ( sgn_sgn_Code_integer @ A )
        = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).

% sgn_1_neg
thf(fact_6836_sgn__1__neg,axiom,
    ! [A: rat] :
      ( ( ( sgn_sgn_rat @ A )
        = ( uminus_uminus_rat @ one_one_rat ) )
      = ( ord_less_rat @ A @ zero_zero_rat ) ) ).

% sgn_1_neg
thf(fact_6837_zsgn__def,axiom,
    ( sgn_sgn_int
    = ( ^ [I2: int] : ( if_int @ ( I2 = zero_zero_int ) @ zero_zero_int @ ( if_int @ ( ord_less_int @ zero_zero_int @ I2 ) @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).

% zsgn_def
thf(fact_6838_fact__div__fact__le__pow,axiom,
    ! [R2: nat,N: nat] :
      ( ( ord_less_eq_nat @ R2 @ N )
     => ( ord_less_eq_nat @ ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ R2 ) ) ) @ ( power_power_nat @ N @ R2 ) ) ) ).

% fact_div_fact_le_pow
thf(fact_6839_binomial__fact__lemma,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( times_times_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( binomial @ N @ K ) )
        = ( semiri1408675320244567234ct_nat @ N ) ) ) ).

% binomial_fact_lemma
thf(fact_6840_norm__sgn,axiom,
    ! [X4: real] :
      ( ( ( X4 = zero_zero_real )
       => ( ( real_V7735802525324610683m_real @ ( sgn_sgn_real @ X4 ) )
          = zero_zero_real ) )
      & ( ( X4 != zero_zero_real )
       => ( ( real_V7735802525324610683m_real @ ( sgn_sgn_real @ X4 ) )
          = one_one_real ) ) ) ).

% norm_sgn
thf(fact_6841_norm__sgn,axiom,
    ! [X4: complex] :
      ( ( ( X4 = zero_zero_complex )
       => ( ( real_V1022390504157884413omplex @ ( sgn_sgn_complex @ X4 ) )
          = zero_zero_real ) )
      & ( ( X4 != zero_zero_complex )
       => ( ( real_V1022390504157884413omplex @ ( sgn_sgn_complex @ X4 ) )
          = one_one_real ) ) ) ).

% norm_sgn
thf(fact_6842_div__sgn__abs__cancel,axiom,
    ! [V: int,K: int,L2: int] :
      ( ( V != zero_zero_int )
     => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ V ) @ ( abs_abs_int @ K ) ) @ ( times_times_int @ ( sgn_sgn_int @ V ) @ ( abs_abs_int @ L2 ) ) )
        = ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L2 ) ) ) ) ).

% div_sgn_abs_cancel
thf(fact_6843_div__dvd__sgn__abs,axiom,
    ! [L2: int,K: int] :
      ( ( dvd_dvd_int @ L2 @ K )
     => ( ( divide_divide_int @ K @ L2 )
        = ( times_times_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( sgn_sgn_int @ L2 ) ) @ ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L2 ) ) ) ) ) ).

% div_dvd_sgn_abs
thf(fact_6844_choose__dvd,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ ( semiri3624122377584611663nteger @ K ) @ ( semiri3624122377584611663nteger @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri3624122377584611663nteger @ N ) ) ) ).

% choose_dvd
thf(fact_6845_choose__dvd,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( dvd_dvd_rat @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).

% choose_dvd
thf(fact_6846_choose__dvd,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( dvd_dvd_int @ ( times_times_int @ ( semiri1406184849735516958ct_int @ K ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ).

% choose_dvd
thf(fact_6847_choose__dvd,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( dvd_dvd_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).

% choose_dvd
thf(fact_6848_choose__dvd,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( dvd_dvd_real @ ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) ) ).

% choose_dvd
thf(fact_6849_fact__numeral,axiom,
    ! [K: num] :
      ( ( semiri5044797733671781792omplex @ ( numeral_numeral_nat @ K ) )
      = ( times_times_complex @ ( numera6690914467698888265omplex @ K ) @ ( semiri5044797733671781792omplex @ ( pred_numeral @ K ) ) ) ) ).

% fact_numeral
thf(fact_6850_fact__numeral,axiom,
    ! [K: num] :
      ( ( semiri773545260158071498ct_rat @ ( numeral_numeral_nat @ K ) )
      = ( times_times_rat @ ( numeral_numeral_rat @ K ) @ ( semiri773545260158071498ct_rat @ ( pred_numeral @ K ) ) ) ) ).

% fact_numeral
thf(fact_6851_fact__numeral,axiom,
    ! [K: num] :
      ( ( semiri1406184849735516958ct_int @ ( numeral_numeral_nat @ K ) )
      = ( times_times_int @ ( numeral_numeral_int @ K ) @ ( semiri1406184849735516958ct_int @ ( pred_numeral @ K ) ) ) ) ).

% fact_numeral
thf(fact_6852_fact__numeral,axiom,
    ! [K: num] :
      ( ( semiri1408675320244567234ct_nat @ ( numeral_numeral_nat @ K ) )
      = ( times_times_nat @ ( numeral_numeral_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( pred_numeral @ K ) ) ) ) ).

% fact_numeral
thf(fact_6853_fact__numeral,axiom,
    ! [K: num] :
      ( ( semiri2265585572941072030t_real @ ( numeral_numeral_nat @ K ) )
      = ( times_times_real @ ( numeral_numeral_real @ K ) @ ( semiri2265585572941072030t_real @ ( pred_numeral @ K ) ) ) ) ).

% fact_numeral
thf(fact_6854_arccos__lbound,axiom,
    ! [Y3: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
     => ( ( ord_less_eq_real @ Y3 @ one_one_real )
       => ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y3 ) ) ) ) ).

% arccos_lbound
thf(fact_6855_arccos__less__arccos,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X4 )
     => ( ( ord_less_real @ X4 @ Y3 )
       => ( ( ord_less_eq_real @ Y3 @ one_one_real )
         => ( ord_less_real @ ( arccos @ Y3 ) @ ( arccos @ X4 ) ) ) ) ) ).

% arccos_less_arccos
thf(fact_6856_arccos__less__mono,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X4 ) @ one_one_real )
     => ( ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
       => ( ( ord_less_real @ ( arccos @ X4 ) @ ( arccos @ Y3 ) )
          = ( ord_less_real @ Y3 @ X4 ) ) ) ) ).

% arccos_less_mono
thf(fact_6857_arccos__ubound,axiom,
    ! [Y3: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
     => ( ( ord_less_eq_real @ Y3 @ one_one_real )
       => ( ord_less_eq_real @ ( arccos @ Y3 ) @ pi ) ) ) ).

% arccos_ubound
thf(fact_6858_arccos__cos,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_eq_real @ X4 @ pi )
       => ( ( arccos @ ( cos_real @ X4 ) )
          = X4 ) ) ) ).

% arccos_cos
thf(fact_6859_cos__arccos__abs,axiom,
    ! [Y3: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
     => ( ( cos_real @ ( arccos @ Y3 ) )
        = Y3 ) ) ).

% cos_arccos_abs
thf(fact_6860_arccos__cos__eq__abs,axiom,
    ! [Theta: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ Theta ) @ pi )
     => ( ( arccos @ ( cos_real @ Theta ) )
        = ( abs_abs_real @ Theta ) ) ) ).

% arccos_cos_eq_abs
thf(fact_6861_binomial__altdef__nat,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( binomial @ N @ K )
        = ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ K ) ) ) ) ) ) ).

% binomial_altdef_nat
thf(fact_6862_square__fact__le__2__fact,axiom,
    ! [N: nat] : ( ord_less_eq_real @ ( times_times_real @ ( semiri2265585572941072030t_real @ N ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( semiri2265585572941072030t_real @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% square_fact_le_2_fact
thf(fact_6863_arccos__lt__bounded,axiom,
    ! [Y3: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
     => ( ( ord_less_real @ Y3 @ one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ ( arccos @ Y3 ) )
          & ( ord_less_real @ ( arccos @ Y3 ) @ pi ) ) ) ) ).

% arccos_lt_bounded
thf(fact_6864_arccos__bounded,axiom,
    ! [Y3: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
     => ( ( ord_less_eq_real @ Y3 @ one_one_real )
       => ( ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y3 ) )
          & ( ord_less_eq_real @ ( arccos @ Y3 ) @ pi ) ) ) ) ).

% arccos_bounded
thf(fact_6865_sin__arccos__nonzero,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X4 )
     => ( ( ord_less_real @ X4 @ one_one_real )
       => ( ( sin_real @ ( arccos @ X4 ) )
         != zero_zero_real ) ) ) ).

% sin_arccos_nonzero
thf(fact_6866_arccos__cos2,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ X4 @ zero_zero_real )
     => ( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ X4 )
       => ( ( arccos @ ( cos_real @ X4 ) )
          = ( uminus_uminus_real @ X4 ) ) ) ) ).

% arccos_cos2
thf(fact_6867_arccos__minus,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X4 )
     => ( ( ord_less_eq_real @ X4 @ one_one_real )
       => ( ( arccos @ ( uminus_uminus_real @ X4 ) )
          = ( minus_minus_real @ pi @ ( arccos @ X4 ) ) ) ) ) ).

% arccos_minus
thf(fact_6868_fact__num__eq__if,axiom,
    ( semiri5044797733671781792omplex
    = ( ^ [M6: nat] : ( if_complex @ ( M6 = zero_zero_nat ) @ one_one_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ M6 ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).

% fact_num_eq_if
thf(fact_6869_fact__num__eq__if,axiom,
    ( semiri773545260158071498ct_rat
    = ( ^ [M6: nat] : ( if_rat @ ( M6 = zero_zero_nat ) @ one_one_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ M6 ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).

% fact_num_eq_if
thf(fact_6870_fact__num__eq__if,axiom,
    ( semiri1406184849735516958ct_int
    = ( ^ [M6: nat] : ( if_int @ ( M6 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ M6 ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).

% fact_num_eq_if
thf(fact_6871_fact__num__eq__if,axiom,
    ( semiri1408675320244567234ct_nat
    = ( ^ [M6: nat] : ( if_nat @ ( M6 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M6 ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).

% fact_num_eq_if
thf(fact_6872_fact__num__eq__if,axiom,
    ( semiri2265585572941072030t_real
    = ( ^ [M6: nat] : ( if_real @ ( M6 = zero_zero_nat ) @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M6 ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).

% fact_num_eq_if
thf(fact_6873_fact__reduce,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( semiri5044797733671781792omplex @ N )
        = ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).

% fact_reduce
thf(fact_6874_fact__reduce,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( semiri773545260158071498ct_rat @ N )
        = ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).

% fact_reduce
thf(fact_6875_fact__reduce,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( semiri1406184849735516958ct_int @ N )
        = ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).

% fact_reduce
thf(fact_6876_fact__reduce,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( semiri1408675320244567234ct_nat @ N )
        = ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).

% fact_reduce
thf(fact_6877_fact__reduce,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( semiri2265585572941072030t_real @ N )
        = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).

% fact_reduce
thf(fact_6878_pochhammer__same,axiom,
    ! [N: nat] :
      ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N ) ) @ N )
      = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ ( semiri3624122377584611663nteger @ N ) ) ) ).

% pochhammer_same
thf(fact_6879_pochhammer__same,axiom,
    ! [N: nat] :
      ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ N )
      = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( semiri5044797733671781792omplex @ N ) ) ) ).

% pochhammer_same
thf(fact_6880_pochhammer__same,axiom,
    ! [N: nat] :
      ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ N )
      = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).

% pochhammer_same
thf(fact_6881_pochhammer__same,axiom,
    ! [N: nat] :
      ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ N )
      = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ).

% pochhammer_same
thf(fact_6882_pochhammer__same,axiom,
    ! [N: nat] :
      ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ N )
      = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( semiri2265585572941072030t_real @ N ) ) ) ).

% pochhammer_same
thf(fact_6883_binomial__fact,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( semiri8010041392384452111omplex @ ( binomial @ N @ K ) )
        = ( divide1717551699836669952omplex @ ( semiri5044797733671781792omplex @ N ) @ ( times_times_complex @ ( semiri5044797733671781792omplex @ K ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ N @ K ) ) ) ) ) ) ).

% binomial_fact
thf(fact_6884_binomial__fact,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( semiri681578069525770553at_rat @ ( binomial @ N @ K ) )
        = ( divide_divide_rat @ ( semiri773545260158071498ct_rat @ N ) @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N @ K ) ) ) ) ) ) ).

% binomial_fact
thf(fact_6885_binomial__fact,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( semiri5074537144036343181t_real @ ( binomial @ N @ K ) )
        = ( divide_divide_real @ ( semiri2265585572941072030t_real @ N ) @ ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ K ) ) ) ) ) ) ).

% binomial_fact
thf(fact_6886_fact__binomial,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( times_times_complex @ ( semiri5044797733671781792omplex @ K ) @ ( semiri8010041392384452111omplex @ ( binomial @ N @ K ) ) )
        = ( divide1717551699836669952omplex @ ( semiri5044797733671781792omplex @ N ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ N @ K ) ) ) ) ) ).

% fact_binomial
thf(fact_6887_fact__binomial,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri681578069525770553at_rat @ ( binomial @ N @ K ) ) )
        = ( divide_divide_rat @ ( semiri773545260158071498ct_rat @ N ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N @ K ) ) ) ) ) ).

% fact_binomial
thf(fact_6888_fact__binomial,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri5074537144036343181t_real @ ( binomial @ N @ K ) ) )
        = ( divide_divide_real @ ( semiri2265585572941072030t_real @ N ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ K ) ) ) ) ) ).

% fact_binomial
thf(fact_6889_arccos,axiom,
    ! [Y3: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
     => ( ( ord_less_eq_real @ Y3 @ one_one_real )
       => ( ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y3 ) )
          & ( ord_less_eq_real @ ( arccos @ Y3 ) @ pi )
          & ( ( cos_real @ ( arccos @ Y3 ) )
            = Y3 ) ) ) ) ).

% arccos
thf(fact_6890_arccos__minus__abs,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X4 ) @ one_one_real )
     => ( ( arccos @ ( uminus_uminus_real @ X4 ) )
        = ( minus_minus_real @ pi @ ( arccos @ X4 ) ) ) ) ).

% arccos_minus_abs
thf(fact_6891_div__noneq__sgn__abs,axiom,
    ! [L2: int,K: int] :
      ( ( L2 != zero_zero_int )
     => ( ( ( sgn_sgn_int @ K )
         != ( sgn_sgn_int @ L2 ) )
       => ( ( divide_divide_int @ K @ L2 )
          = ( minus_minus_int @ ( uminus_uminus_int @ ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L2 ) ) )
            @ ( zero_n2684676970156552555ol_int
              @ ~ ( dvd_dvd_int @ L2 @ K ) ) ) ) ) ) ).

% div_noneq_sgn_abs
thf(fact_6892_arccos__le__pi2,axiom,
    ! [Y3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
     => ( ( ord_less_eq_real @ Y3 @ one_one_real )
       => ( ord_less_eq_real @ ( arccos @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% arccos_le_pi2
thf(fact_6893_divide__int__unfold,axiom,
    ! [L2: int,K: int,N: nat,M: nat] :
      ( ( ( ( ( sgn_sgn_int @ L2 )
            = zero_zero_int )
          | ( ( sgn_sgn_int @ K )
            = zero_zero_int )
          | ( N = zero_zero_nat ) )
       => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ N ) ) )
          = zero_zero_int ) )
      & ( ~ ( ( ( sgn_sgn_int @ L2 )
              = zero_zero_int )
            | ( ( sgn_sgn_int @ K )
              = zero_zero_int )
            | ( N = zero_zero_nat ) )
       => ( ( ( ( sgn_sgn_int @ K )
              = ( sgn_sgn_int @ L2 ) )
           => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ N ) ) )
              = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N ) ) ) )
          & ( ( ( sgn_sgn_int @ K )
             != ( sgn_sgn_int @ L2 ) )
           => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ N ) ) )
              = ( uminus_uminus_int
                @ ( semiri1314217659103216013at_int
                  @ ( plus_plus_nat @ ( divide_divide_nat @ M @ N )
                    @ ( zero_n2687167440665602831ol_nat
                      @ ~ ( dvd_dvd_nat @ N @ M ) ) ) ) ) ) ) ) ) ) ).

% divide_int_unfold
thf(fact_6894_sin__coeff__def,axiom,
    ( sin_coeff
    = ( ^ [N2: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ zero_zero_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( divide_divide_nat @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) ) ) ) ).

% sin_coeff_def
thf(fact_6895_binomial__code,axiom,
    ( binomial
    = ( ^ [N2: nat,K3: nat] : ( if_nat @ ( ord_less_nat @ N2 @ K3 ) @ zero_zero_nat @ ( if_nat @ ( ord_less_nat @ N2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K3 ) ) @ ( binomial @ N2 @ ( minus_minus_nat @ N2 @ K3 ) ) @ ( divide_divide_nat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( plus_plus_nat @ ( minus_minus_nat @ N2 @ K3 ) @ one_one_nat ) @ N2 @ one_one_nat ) @ ( semiri1408675320244567234ct_nat @ K3 ) ) ) ) ) ) ).

% binomial_code
thf(fact_6896_cos__coeff__def,axiom,
    ( cos_coeff
    = ( ^ [N2: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ zero_zero_real ) ) ) ).

% cos_coeff_def
thf(fact_6897_exp__two__pi__i,axiom,
    ( ( exp_complex @ ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( real_V4546457046886955230omplex @ pi ) ) @ imaginary_unit ) )
    = one_one_complex ) ).

% exp_two_pi_i
thf(fact_6898_exp__two__pi__i_H,axiom,
    ( ( exp_complex @ ( times_times_complex @ imaginary_unit @ ( times_times_complex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) )
    = one_one_complex ) ).

% exp_two_pi_i'
thf(fact_6899_zero__le__sgn__iff,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( sgn_sgn_real @ X4 ) )
      = ( ord_less_eq_real @ zero_zero_real @ X4 ) ) ).

% zero_le_sgn_iff
thf(fact_6900_sgn__le__0__iff,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ ( sgn_sgn_real @ X4 ) @ zero_zero_real )
      = ( ord_less_eq_real @ X4 @ zero_zero_real ) ) ).

% sgn_le_0_iff
thf(fact_6901_norm__ii,axiom,
    ( ( real_V1022390504157884413omplex @ imaginary_unit )
    = one_one_real ) ).

% norm_ii
thf(fact_6902_cos__coeff__0,axiom,
    ( ( cos_coeff @ zero_zero_nat )
    = one_one_real ) ).

% cos_coeff_0
thf(fact_6903_divide__i,axiom,
    ! [X4: complex] :
      ( ( divide1717551699836669952omplex @ X4 @ imaginary_unit )
      = ( times_times_complex @ ( uminus1482373934393186551omplex @ imaginary_unit ) @ X4 ) ) ).

% divide_i
thf(fact_6904_complex__i__mult__minus,axiom,
    ! [X4: complex] :
      ( ( times_times_complex @ imaginary_unit @ ( times_times_complex @ imaginary_unit @ X4 ) )
      = ( uminus1482373934393186551omplex @ X4 ) ) ).

% complex_i_mult_minus
thf(fact_6905_i__squared,axiom,
    ( ( times_times_complex @ imaginary_unit @ imaginary_unit )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% i_squared
thf(fact_6906_divide__numeral__i,axiom,
    ! [Z: complex,N: num] :
      ( ( divide1717551699836669952omplex @ Z @ ( times_times_complex @ ( numera6690914467698888265omplex @ N ) @ imaginary_unit ) )
      = ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ ( times_times_complex @ imaginary_unit @ Z ) ) @ ( numera6690914467698888265omplex @ N ) ) ) ).

% divide_numeral_i
thf(fact_6907_power2__i,axiom,
    ( ( power_power_complex @ imaginary_unit @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% power2_i
thf(fact_6908_i__even__power,axiom,
    ! [N: nat] :
      ( ( power_power_complex @ imaginary_unit @ ( times_times_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) ) ).

% i_even_power
thf(fact_6909_exp__pi__i_H,axiom,
    ( ( exp_complex @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ pi ) ) )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% exp_pi_i'
thf(fact_6910_exp__pi__i,axiom,
    ( ( exp_complex @ ( times_times_complex @ ( real_V4546457046886955230omplex @ pi ) @ imaginary_unit ) )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% exp_pi_i
thf(fact_6911_real__sgn__eq,axiom,
    ( sgn_sgn_real
    = ( ^ [X: real] : ( divide_divide_real @ X @ ( abs_abs_real @ X ) ) ) ) ).

% real_sgn_eq
thf(fact_6912_sgn__eq,axiom,
    ( sgn_sgn_complex
    = ( ^ [Z5: complex] : ( divide1717551699836669952omplex @ Z5 @ ( real_V4546457046886955230omplex @ ( real_V1022390504157884413omplex @ Z5 ) ) ) ) ) ).

% sgn_eq
thf(fact_6913_sin__coeff__Suc,axiom,
    ! [N: nat] :
      ( ( sin_coeff @ ( suc @ N ) )
      = ( divide_divide_real @ ( cos_coeff @ N ) @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) ) ).

% sin_coeff_Suc
thf(fact_6914_i__times__eq__iff,axiom,
    ! [W: complex,Z: complex] :
      ( ( ( times_times_complex @ imaginary_unit @ W )
        = Z )
      = ( W
        = ( uminus1482373934393186551omplex @ ( times_times_complex @ imaginary_unit @ Z ) ) ) ) ).

% i_times_eq_iff
thf(fact_6915_cos__coeff__Suc,axiom,
    ! [N: nat] :
      ( ( cos_coeff @ ( suc @ N ) )
      = ( divide_divide_real @ ( uminus_uminus_real @ ( sin_coeff @ N ) ) @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) ) ).

% cos_coeff_Suc
thf(fact_6916_imaginary__unit_Ocode,axiom,
    ( imaginary_unit
    = ( complex2 @ zero_zero_real @ one_one_real ) ) ).

% imaginary_unit.code
thf(fact_6917_Complex__eq__i,axiom,
    ! [X4: real,Y3: real] :
      ( ( ( complex2 @ X4 @ Y3 )
        = imaginary_unit )
      = ( ( X4 = zero_zero_real )
        & ( Y3 = one_one_real ) ) ) ).

% Complex_eq_i
thf(fact_6918_i__mult__Complex,axiom,
    ! [A: real,B2: real] :
      ( ( times_times_complex @ imaginary_unit @ ( complex2 @ A @ B2 ) )
      = ( complex2 @ ( uminus_uminus_real @ B2 ) @ A ) ) ).

% i_mult_Complex
thf(fact_6919_Complex__mult__i,axiom,
    ! [A: real,B2: real] :
      ( ( times_times_complex @ ( complex2 @ A @ B2 ) @ imaginary_unit )
      = ( complex2 @ ( uminus_uminus_real @ B2 ) @ A ) ) ).

% Complex_mult_i
thf(fact_6920_sgn__real__def,axiom,
    ( sgn_sgn_real
    = ( ^ [A4: real] : ( if_real @ ( A4 = zero_zero_real ) @ zero_zero_real @ ( if_real @ ( ord_less_real @ zero_zero_real @ A4 ) @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ) ) ) ).

% sgn_real_def
thf(fact_6921_sgn__power__injE,axiom,
    ! [A: real,N: nat,X4: real,B2: real] :
      ( ( ( times_times_real @ ( sgn_sgn_real @ A ) @ ( power_power_real @ ( abs_abs_real @ A ) @ N ) )
        = X4 )
     => ( ( X4
          = ( times_times_real @ ( sgn_sgn_real @ B2 ) @ ( power_power_real @ ( abs_abs_real @ B2 ) @ N ) ) )
       => ( ( ord_less_nat @ zero_zero_nat @ N )
         => ( A = B2 ) ) ) ) ).

% sgn_power_injE
thf(fact_6922_fold__atLeastAtMost__nat_Osimps,axiom,
    ( set_fo2584398358068434914at_nat
    = ( ^ [F2: nat > nat > nat,A4: nat,B3: nat,Acc: nat] : ( if_nat @ ( ord_less_nat @ B3 @ A4 ) @ Acc @ ( set_fo2584398358068434914at_nat @ F2 @ ( plus_plus_nat @ A4 @ one_one_nat ) @ B3 @ ( F2 @ A4 @ Acc ) ) ) ) ) ).

% fold_atLeastAtMost_nat.simps
thf(fact_6923_fold__atLeastAtMost__nat_Oelims,axiom,
    ! [X4: nat > nat > nat,Xa: nat,Xb: nat,Xc: nat,Y3: nat] :
      ( ( ( set_fo2584398358068434914at_nat @ X4 @ Xa @ Xb @ Xc )
        = Y3 )
     => ( ( ( ord_less_nat @ Xb @ Xa )
         => ( Y3 = Xc ) )
        & ( ~ ( ord_less_nat @ Xb @ Xa )
         => ( Y3
            = ( set_fo2584398358068434914at_nat @ X4 @ ( plus_plus_nat @ Xa @ one_one_nat ) @ Xb @ ( X4 @ Xa @ Xc ) ) ) ) ) ) ).

% fold_atLeastAtMost_nat.elims
thf(fact_6924_complex__of__real__i,axiom,
    ! [R2: real] :
      ( ( times_times_complex @ ( real_V4546457046886955230omplex @ R2 ) @ imaginary_unit )
      = ( complex2 @ zero_zero_real @ R2 ) ) ).

% complex_of_real_i
thf(fact_6925_i__complex__of__real,axiom,
    ! [R2: real] :
      ( ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ R2 ) )
      = ( complex2 @ zero_zero_real @ R2 ) ) ).

% i_complex_of_real
thf(fact_6926_Complex__eq,axiom,
    ( complex2
    = ( ^ [A4: real,B3: real] : ( plus_plus_complex @ ( real_V4546457046886955230omplex @ A4 ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ B3 ) ) ) ) ) ).

% Complex_eq
thf(fact_6927_complex__split__polar,axiom,
    ! [Z: complex] :
    ? [R3: real,A5: real] :
      ( Z
      = ( times_times_complex @ ( real_V4546457046886955230omplex @ R3 ) @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( cos_real @ A5 ) ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sin_real @ A5 ) ) ) ) ) ) ).

% complex_split_polar
thf(fact_6928_cmod__unit__one,axiom,
    ! [A: real] :
      ( ( real_V1022390504157884413omplex @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( cos_real @ A ) ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sin_real @ A ) ) ) ) )
      = one_one_real ) ).

% cmod_unit_one
thf(fact_6929_cmod__complex__polar,axiom,
    ! [R2: real,A: real] :
      ( ( real_V1022390504157884413omplex @ ( times_times_complex @ ( real_V4546457046886955230omplex @ R2 ) @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( cos_real @ A ) ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sin_real @ A ) ) ) ) ) )
      = ( abs_abs_real @ R2 ) ) ).

% cmod_complex_polar
thf(fact_6930_arctan__inverse,axiom,
    ! [X4: real] :
      ( ( X4 != zero_zero_real )
     => ( ( arctan @ ( divide_divide_real @ one_one_real @ X4 ) )
        = ( minus_minus_real @ ( divide_divide_real @ ( times_times_real @ ( sgn_sgn_real @ X4 ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( arctan @ X4 ) ) ) ) ).

% arctan_inverse
thf(fact_6931_fact__code,axiom,
    ( semiri5044797733671781792omplex
    = ( ^ [N2: nat] : ( semiri8010041392384452111omplex @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 @ one_one_nat ) ) ) ) ).

% fact_code
thf(fact_6932_fact__code,axiom,
    ( semiri773545260158071498ct_rat
    = ( ^ [N2: nat] : ( semiri681578069525770553at_rat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 @ one_one_nat ) ) ) ) ).

% fact_code
thf(fact_6933_fact__code,axiom,
    ( semiri1406184849735516958ct_int
    = ( ^ [N2: nat] : ( semiri1314217659103216013at_int @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 @ one_one_nat ) ) ) ) ).

% fact_code
thf(fact_6934_fact__code,axiom,
    ( semiri1408675320244567234ct_nat
    = ( ^ [N2: nat] : ( semiri1316708129612266289at_nat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 @ one_one_nat ) ) ) ) ).

% fact_code
thf(fact_6935_fact__code,axiom,
    ( semiri2265585572941072030t_real
    = ( ^ [N2: nat] : ( semiri5074537144036343181t_real @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 @ one_one_nat ) ) ) ) ).

% fact_code
thf(fact_6936_Arg__minus__ii,axiom,
    ( ( arg @ ( uminus1482373934393186551omplex @ imaginary_unit ) )
    = ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% Arg_minus_ii
thf(fact_6937_csqrt__ii,axiom,
    ( ( csqrt @ imaginary_unit )
    = ( divide1717551699836669952omplex @ ( plus_plus_complex @ one_one_complex @ imaginary_unit ) @ ( real_V4546457046886955230omplex @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% csqrt_ii
thf(fact_6938_Arg__ii,axiom,
    ( ( arg @ imaginary_unit )
    = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% Arg_ii
thf(fact_6939_cis__minus__pi__half,axiom,
    ( ( cis @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
    = ( uminus1482373934393186551omplex @ imaginary_unit ) ) ).

% cis_minus_pi_half
thf(fact_6940_modulo__int__def,axiom,
    ( modulo_modulo_int
    = ( ^ [K3: int,L: int] :
          ( if_int @ ( L = zero_zero_int ) @ K3
          @ ( if_int
            @ ( ( sgn_sgn_int @ K3 )
              = ( sgn_sgn_int @ L ) )
            @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K3 ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) ) )
            @ ( times_times_int @ ( sgn_sgn_int @ L )
              @ ( minus_minus_int
                @ ( times_times_int @ ( abs_abs_int @ L )
                  @ ( zero_n2684676970156552555ol_int
                    @ ~ ( dvd_dvd_int @ L @ K3 ) ) )
                @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K3 ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) ) ) ) ) ) ) ) ).

% modulo_int_def
thf(fact_6941_nat__numeral,axiom,
    ! [K: num] :
      ( ( nat2 @ ( numeral_numeral_int @ K ) )
      = ( numeral_numeral_nat @ K ) ) ).

% nat_numeral
thf(fact_6942_norm__cis,axiom,
    ! [A: real] :
      ( ( real_V1022390504157884413omplex @ ( cis @ A ) )
      = one_one_real ) ).

% norm_cis
thf(fact_6943_nat__1,axiom,
    ( ( nat2 @ one_one_int )
    = ( suc @ zero_zero_nat ) ) ).

% nat_1
thf(fact_6944_zless__nat__conj,axiom,
    ! [W: int,Z: int] :
      ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
      = ( ( ord_less_int @ zero_zero_int @ Z )
        & ( ord_less_int @ W @ Z ) ) ) ).

% zless_nat_conj
thf(fact_6945_of__nat__nat__take__bit__eq,axiom,
    ! [N: nat,K: int] :
      ( ( semiri8010041392384452111omplex @ ( nat2 @ ( bit_se2923211474154528505it_int @ N @ K ) ) )
      = ( ring_17405671764205052669omplex @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).

% of_nat_nat_take_bit_eq
thf(fact_6946_of__nat__nat__take__bit__eq,axiom,
    ! [N: nat,K: int] :
      ( ( semiri5074537144036343181t_real @ ( nat2 @ ( bit_se2923211474154528505it_int @ N @ K ) ) )
      = ( ring_1_of_int_real @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).

% of_nat_nat_take_bit_eq
thf(fact_6947_of__nat__nat__take__bit__eq,axiom,
    ! [N: nat,K: int] :
      ( ( semiri681578069525770553at_rat @ ( nat2 @ ( bit_se2923211474154528505it_int @ N @ K ) ) )
      = ( ring_1_of_int_rat @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).

% of_nat_nat_take_bit_eq
thf(fact_6948_of__nat__nat__take__bit__eq,axiom,
    ! [N: nat,K: int] :
      ( ( semiri1314217659103216013at_int @ ( nat2 @ ( bit_se2923211474154528505it_int @ N @ K ) ) )
      = ( ring_1_of_int_int @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).

% of_nat_nat_take_bit_eq
thf(fact_6949_zero__less__nat__eq,axiom,
    ! [Z: int] :
      ( ( ord_less_nat @ zero_zero_nat @ ( nat2 @ Z ) )
      = ( ord_less_int @ zero_zero_int @ Z ) ) ).

% zero_less_nat_eq
thf(fact_6950_diff__nat__numeral,axiom,
    ! [V: num,V2: num] :
      ( ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ ( numeral_numeral_nat @ V2 ) )
      = ( nat2 @ ( minus_minus_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ V2 ) ) ) ) ).

% diff_nat_numeral
thf(fact_6951_numeral__power__eq__nat__cancel__iff,axiom,
    ! [X4: num,N: nat,Y3: int] :
      ( ( ( power_power_nat @ ( numeral_numeral_nat @ X4 ) @ N )
        = ( nat2 @ Y3 ) )
      = ( ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N )
        = Y3 ) ) ).

% numeral_power_eq_nat_cancel_iff
thf(fact_6952_nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y3: int,X4: num,N: nat] :
      ( ( ( nat2 @ Y3 )
        = ( power_power_nat @ ( numeral_numeral_nat @ X4 ) @ N ) )
      = ( Y3
        = ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N ) ) ) ).

% nat_eq_numeral_power_cancel_iff
thf(fact_6953_power2__csqrt,axiom,
    ! [Z: complex] :
      ( ( power_power_complex @ ( csqrt @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = Z ) ).

% power2_csqrt
thf(fact_6954_nat__ceiling__le__eq,axiom,
    ! [X4: real,A: nat] :
      ( ( ord_less_eq_nat @ ( nat2 @ ( archim7802044766580827645g_real @ X4 ) ) @ A )
      = ( ord_less_eq_real @ X4 @ ( semiri5074537144036343181t_real @ A ) ) ) ).

% nat_ceiling_le_eq
thf(fact_6955_one__less__nat__eq,axiom,
    ! [Z: int] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( nat2 @ Z ) )
      = ( ord_less_int @ one_one_int @ Z ) ) ).

% one_less_nat_eq
thf(fact_6956_nat__numeral__diff__1,axiom,
    ! [V: num] :
      ( ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ one_one_nat )
      = ( nat2 @ ( minus_minus_int @ ( numeral_numeral_int @ V ) @ one_one_int ) ) ) ).

% nat_numeral_diff_1
thf(fact_6957_numeral__power__less__nat__cancel__iff,axiom,
    ! [X4: num,N: nat,A: int] :
      ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X4 ) @ N ) @ ( nat2 @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N ) @ A ) ) ).

% numeral_power_less_nat_cancel_iff
thf(fact_6958_nat__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X4: num,N: nat] :
      ( ( ord_less_nat @ ( nat2 @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ X4 ) @ N ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N ) ) ) ).

% nat_less_numeral_power_cancel_iff
thf(fact_6959_numeral__power__le__nat__cancel__iff,axiom,
    ! [X4: num,N: nat,A: int] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X4 ) @ N ) @ ( nat2 @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N ) @ A ) ) ).

% numeral_power_le_nat_cancel_iff
thf(fact_6960_nat__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X4: num,N: nat] :
      ( ( ord_less_eq_nat @ ( nat2 @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ X4 ) @ N ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N ) ) ) ).

% nat_le_numeral_power_cancel_iff
thf(fact_6961_cis__pi__half,axiom,
    ( ( cis @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
    = imaginary_unit ) ).

% cis_pi_half
thf(fact_6962_cis__2pi,axiom,
    ( ( cis @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
    = one_one_complex ) ).

% cis_2pi
thf(fact_6963_cis__divide,axiom,
    ! [A: real,B2: real] :
      ( ( divide1717551699836669952omplex @ ( cis @ A ) @ ( cis @ B2 ) )
      = ( cis @ ( minus_minus_real @ A @ B2 ) ) ) ).

% cis_divide
thf(fact_6964_nat__zero__as__int,axiom,
    ( zero_zero_nat
    = ( nat2 @ zero_zero_int ) ) ).

% nat_zero_as_int
thf(fact_6965_nat__numeral__as__int,axiom,
    ( numeral_numeral_nat
    = ( ^ [I2: num] : ( nat2 @ ( numeral_numeral_int @ I2 ) ) ) ) ).

% nat_numeral_as_int
thf(fact_6966_nat__mono,axiom,
    ! [X4: int,Y3: int] :
      ( ( ord_less_eq_int @ X4 @ Y3 )
     => ( ord_less_eq_nat @ ( nat2 @ X4 ) @ ( nat2 @ Y3 ) ) ) ).

% nat_mono
thf(fact_6967_nat__one__as__int,axiom,
    ( one_one_nat
    = ( nat2 @ one_one_int ) ) ).

% nat_one_as_int
thf(fact_6968_unset__bit__nat__def,axiom,
    ( bit_se4205575877204974255it_nat
    = ( ^ [M6: nat,N2: nat] : ( nat2 @ ( bit_se4203085406695923979it_int @ M6 @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).

% unset_bit_nat_def
thf(fact_6969_DeMoivre,axiom,
    ! [A: real,N: nat] :
      ( ( power_power_complex @ ( cis @ A ) @ N )
      = ( cis @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ A ) ) ) ).

% DeMoivre
thf(fact_6970_nat__mask__eq,axiom,
    ! [N: nat] :
      ( ( nat2 @ ( bit_se2000444600071755411sk_int @ N ) )
      = ( bit_se2002935070580805687sk_nat @ N ) ) ).

% nat_mask_eq
thf(fact_6971_Arg__correct,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero_complex )
     => ( ( ( sgn_sgn_complex @ Z )
          = ( cis @ ( arg @ Z ) ) )
        & ( ord_less_real @ ( uminus_uminus_real @ pi ) @ ( arg @ Z ) )
        & ( ord_less_eq_real @ ( arg @ Z ) @ pi ) ) ) ).

% Arg_correct
thf(fact_6972_cis__Arg__unique,axiom,
    ! [Z: complex,X4: real] :
      ( ( ( sgn_sgn_complex @ Z )
        = ( cis @ X4 ) )
     => ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ X4 )
       => ( ( ord_less_eq_real @ X4 @ pi )
         => ( ( arg @ Z )
            = X4 ) ) ) ) ).

% cis_Arg_unique
thf(fact_6973_cis__mult,axiom,
    ! [A: real,B2: real] :
      ( ( times_times_complex @ ( cis @ A ) @ ( cis @ B2 ) )
      = ( cis @ ( plus_plus_real @ A @ B2 ) ) ) ).

% cis_mult
thf(fact_6974_nat__mono__iff,axiom,
    ! [Z: int,W: int] :
      ( ( ord_less_int @ zero_zero_int @ Z )
     => ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
        = ( ord_less_int @ W @ Z ) ) ) ).

% nat_mono_iff
thf(fact_6975_of__nat__ceiling,axiom,
    ! [R2: real] : ( ord_less_eq_real @ R2 @ ( semiri5074537144036343181t_real @ ( nat2 @ ( archim7802044766580827645g_real @ R2 ) ) ) ) ).

% of_nat_ceiling
thf(fact_6976_of__nat__ceiling,axiom,
    ! [R2: rat] : ( ord_less_eq_rat @ R2 @ ( semiri681578069525770553at_rat @ ( nat2 @ ( archim2889992004027027881ng_rat @ R2 ) ) ) ) ).

% of_nat_ceiling
thf(fact_6977_zless__nat__eq__int__zless,axiom,
    ! [M: nat,Z: int] :
      ( ( ord_less_nat @ M @ ( nat2 @ Z ) )
      = ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ Z ) ) ).

% zless_nat_eq_int_zless
thf(fact_6978_nat__le__iff,axiom,
    ! [X4: int,N: nat] :
      ( ( ord_less_eq_nat @ ( nat2 @ X4 ) @ N )
      = ( ord_less_eq_int @ X4 @ ( semiri1314217659103216013at_int @ N ) ) ) ).

% nat_le_iff
thf(fact_6979_nat__int__add,axiom,
    ! [A: nat,B2: nat] :
      ( ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B2 ) ) )
      = ( plus_plus_nat @ A @ B2 ) ) ).

% nat_int_add
thf(fact_6980_int__minus,axiom,
    ! [N: nat,M: nat] :
      ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ N @ M ) )
      = ( semiri1314217659103216013at_int @ ( nat2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M ) ) ) ) ) ).

% int_minus
thf(fact_6981_nat__abs__mult__distrib,axiom,
    ! [W: int,Z: int] :
      ( ( nat2 @ ( abs_abs_int @ ( times_times_int @ W @ Z ) ) )
      = ( times_times_nat @ ( nat2 @ ( abs_abs_int @ W ) ) @ ( nat2 @ ( abs_abs_int @ Z ) ) ) ) ).

% nat_abs_mult_distrib
thf(fact_6982_and__nat__def,axiom,
    ( bit_se727722235901077358nd_nat
    = ( ^ [M6: nat,N2: nat] : ( nat2 @ ( bit_se725231765392027082nd_int @ ( semiri1314217659103216013at_int @ M6 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).

% and_nat_def
thf(fact_6983_real__nat__ceiling__ge,axiom,
    ! [X4: real] : ( ord_less_eq_real @ X4 @ ( semiri5074537144036343181t_real @ ( nat2 @ ( archim7802044766580827645g_real @ X4 ) ) ) ) ).

% real_nat_ceiling_ge
thf(fact_6984_of__nat__floor,axiom,
    ! [R2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ R2 )
     => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( nat2 @ ( archim6058952711729229775r_real @ R2 ) ) ) @ R2 ) ) ).

% of_nat_floor
thf(fact_6985_of__nat__floor,axiom,
    ! [R2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ R2 )
     => ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ ( nat2 @ ( archim3151403230148437115or_rat @ R2 ) ) ) @ R2 ) ) ).

% of_nat_floor
thf(fact_6986_nat__less__eq__zless,axiom,
    ! [W: int,Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ W )
     => ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
        = ( ord_less_int @ W @ Z ) ) ) ).

% nat_less_eq_zless
thf(fact_6987_nat__le__eq__zle,axiom,
    ! [W: int,Z: int] :
      ( ( ( ord_less_int @ zero_zero_int @ W )
        | ( ord_less_eq_int @ zero_zero_int @ Z ) )
     => ( ( ord_less_eq_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
        = ( ord_less_eq_int @ W @ Z ) ) ) ).

% nat_le_eq_zle
thf(fact_6988_split__nat,axiom,
    ! [P: nat > $o,I: int] :
      ( ( P @ ( nat2 @ I ) )
      = ( ! [N2: nat] :
            ( ( I
              = ( semiri1314217659103216013at_int @ N2 ) )
           => ( P @ N2 ) )
        & ( ( ord_less_int @ I @ zero_zero_int )
         => ( P @ zero_zero_nat ) ) ) ) ).

% split_nat
thf(fact_6989_le__mult__nat__floor,axiom,
    ! [A: real,B2: real] : ( ord_less_eq_nat @ ( times_times_nat @ ( nat2 @ ( archim6058952711729229775r_real @ A ) ) @ ( nat2 @ ( archim6058952711729229775r_real @ B2 ) ) ) @ ( nat2 @ ( archim6058952711729229775r_real @ ( times_times_real @ A @ B2 ) ) ) ) ).

% le_mult_nat_floor
thf(fact_6990_le__mult__nat__floor,axiom,
    ! [A: rat,B2: rat] : ( ord_less_eq_nat @ ( times_times_nat @ ( nat2 @ ( archim3151403230148437115or_rat @ A ) ) @ ( nat2 @ ( archim3151403230148437115or_rat @ B2 ) ) ) @ ( nat2 @ ( archim3151403230148437115or_rat @ ( times_times_rat @ A @ B2 ) ) ) ) ).

% le_mult_nat_floor
thf(fact_6991_le__nat__iff,axiom,
    ! [K: int,N: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ( ( ord_less_eq_nat @ N @ ( nat2 @ K ) )
        = ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ K ) ) ) ).

% le_nat_iff
thf(fact_6992_nat__add__distrib,axiom,
    ! [Z: int,Z6: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( ord_less_eq_int @ zero_zero_int @ Z6 )
       => ( ( nat2 @ ( plus_plus_int @ Z @ Z6 ) )
          = ( plus_plus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z6 ) ) ) ) ) ).

% nat_add_distrib
thf(fact_6993_nat__mult__distrib,axiom,
    ! [Z: int,Z6: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( nat2 @ ( times_times_int @ Z @ Z6 ) )
        = ( times_times_nat @ ( nat2 @ Z ) @ ( nat2 @ Z6 ) ) ) ) ).

% nat_mult_distrib
thf(fact_6994_Suc__as__int,axiom,
    ( suc
    = ( ^ [A4: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A4 ) @ one_one_int ) ) ) ) ).

% Suc_as_int
thf(fact_6995_nat__diff__distrib,axiom,
    ! [Z6: int,Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z6 )
     => ( ( ord_less_eq_int @ Z6 @ Z )
       => ( ( nat2 @ ( minus_minus_int @ Z @ Z6 ) )
          = ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z6 ) ) ) ) ) ).

% nat_diff_distrib
thf(fact_6996_nat__diff__distrib_H,axiom,
    ! [X4: int,Y3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X4 )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
       => ( ( nat2 @ ( minus_minus_int @ X4 @ Y3 ) )
          = ( minus_minus_nat @ ( nat2 @ X4 ) @ ( nat2 @ Y3 ) ) ) ) ) ).

% nat_diff_distrib'
thf(fact_6997_nat__abs__triangle__ineq,axiom,
    ! [K: int,L2: int] : ( ord_less_eq_nat @ ( nat2 @ ( abs_abs_int @ ( plus_plus_int @ K @ L2 ) ) ) @ ( plus_plus_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) ) ) ).

% nat_abs_triangle_ineq
thf(fact_6998_nat__div__distrib,axiom,
    ! [X4: int,Y3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X4 )
     => ( ( nat2 @ ( divide_divide_int @ X4 @ Y3 ) )
        = ( divide_divide_nat @ ( nat2 @ X4 ) @ ( nat2 @ Y3 ) ) ) ) ).

% nat_div_distrib
thf(fact_6999_nat__div__distrib_H,axiom,
    ! [Y3: int,X4: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
     => ( ( nat2 @ ( divide_divide_int @ X4 @ Y3 ) )
        = ( divide_divide_nat @ ( nat2 @ X4 ) @ ( nat2 @ Y3 ) ) ) ) ).

% nat_div_distrib'
thf(fact_7000_nat__power__eq,axiom,
    ! [Z: int,N: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( nat2 @ ( power_power_int @ Z @ N ) )
        = ( power_power_nat @ ( nat2 @ Z ) @ N ) ) ) ).

% nat_power_eq
thf(fact_7001_nat__floor__neg,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ X4 @ zero_zero_real )
     => ( ( nat2 @ ( archim6058952711729229775r_real @ X4 ) )
        = zero_zero_nat ) ) ).

% nat_floor_neg
thf(fact_7002_nat__mod__distrib,axiom,
    ! [X4: int,Y3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X4 )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
       => ( ( nat2 @ ( modulo_modulo_int @ X4 @ Y3 ) )
          = ( modulo_modulo_nat @ ( nat2 @ X4 ) @ ( nat2 @ Y3 ) ) ) ) ) ).

% nat_mod_distrib
thf(fact_7003_div__abs__eq__div__nat,axiom,
    ! [K: int,L2: int] :
      ( ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L2 ) )
      = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) ) ) ) ).

% div_abs_eq_div_nat
thf(fact_7004_floor__eq3,axiom,
    ! [N: nat,X4: real] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ X4 )
     => ( ( ord_less_real @ X4 @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) )
       => ( ( nat2 @ ( archim6058952711729229775r_real @ X4 ) )
          = N ) ) ) ).

% floor_eq3
thf(fact_7005_le__nat__floor,axiom,
    ! [X4: nat,A: real] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X4 ) @ A )
     => ( ord_less_eq_nat @ X4 @ ( nat2 @ ( archim6058952711729229775r_real @ A ) ) ) ) ).

% le_nat_floor
thf(fact_7006_mod__abs__eq__div__nat,axiom,
    ! [K: int,L2: int] :
      ( ( modulo_modulo_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L2 ) )
      = ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) ) ) ) ).

% mod_abs_eq_div_nat
thf(fact_7007_nat__take__bit__eq,axiom,
    ! [K: int,N: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ( ( nat2 @ ( bit_se2923211474154528505it_int @ N @ K ) )
        = ( bit_se2925701944663578781it_nat @ N @ ( nat2 @ K ) ) ) ) ).

% nat_take_bit_eq
thf(fact_7008_take__bit__nat__eq,axiom,
    ! [K: int,N: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ( ( bit_se2925701944663578781it_nat @ N @ ( nat2 @ K ) )
        = ( nat2 @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ) ).

% take_bit_nat_eq
thf(fact_7009_bit__nat__iff,axiom,
    ! [K: int,N: nat] :
      ( ( bit_se1148574629649215175it_nat @ ( nat2 @ K ) @ N )
      = ( ( ord_less_eq_int @ zero_zero_int @ K )
        & ( bit_se1146084159140164899it_int @ K @ N ) ) ) ).

% bit_nat_iff
thf(fact_7010_nat__2,axiom,
    ( ( nat2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
    = ( suc @ ( suc @ zero_zero_nat ) ) ) ).

% nat_2
thf(fact_7011_Suc__nat__eq__nat__zadd1,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( suc @ ( nat2 @ Z ) )
        = ( nat2 @ ( plus_plus_int @ one_one_int @ Z ) ) ) ) ).

% Suc_nat_eq_nat_zadd1
thf(fact_7012_nat__less__iff,axiom,
    ! [W: int,M: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ W )
     => ( ( ord_less_nat @ ( nat2 @ W ) @ M )
        = ( ord_less_int @ W @ ( semiri1314217659103216013at_int @ M ) ) ) ) ).

% nat_less_iff
thf(fact_7013_nat__mult__distrib__neg,axiom,
    ! [Z: int,Z6: int] :
      ( ( ord_less_eq_int @ Z @ zero_zero_int )
     => ( ( nat2 @ ( times_times_int @ Z @ Z6 ) )
        = ( times_times_nat @ ( nat2 @ ( uminus_uminus_int @ Z ) ) @ ( nat2 @ ( uminus_uminus_int @ Z6 ) ) ) ) ) ).

% nat_mult_distrib_neg
thf(fact_7014_nat__abs__int__diff,axiom,
    ! [A: nat,B2: nat] :
      ( ( ( ord_less_eq_nat @ A @ B2 )
       => ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) )
          = ( minus_minus_nat @ B2 @ A ) ) )
      & ( ~ ( ord_less_eq_nat @ A @ B2 )
       => ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) )
          = ( minus_minus_nat @ A @ B2 ) ) ) ) ).

% nat_abs_int_diff
thf(fact_7015_floor__eq4,axiom,
    ! [N: nat,X4: real] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ X4 )
     => ( ( ord_less_real @ X4 @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) )
       => ( ( nat2 @ ( archim6058952711729229775r_real @ X4 ) )
          = N ) ) ) ).

% floor_eq4
thf(fact_7016_cis__conv__exp,axiom,
    ( cis
    = ( ^ [B3: real] : ( exp_complex @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ B3 ) ) ) ) ) ).

% cis_conv_exp
thf(fact_7017_of__int__of__nat,axiom,
    ( ring_18347121197199848620nteger
    = ( ^ [K3: int] : ( if_Code_integer @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ ( nat2 @ ( uminus_uminus_int @ K3 ) ) ) ) @ ( semiri4939895301339042750nteger @ ( nat2 @ K3 ) ) ) ) ) ).

% of_int_of_nat
thf(fact_7018_of__int__of__nat,axiom,
    ( ring_17405671764205052669omplex
    = ( ^ [K3: int] : ( if_complex @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ ( nat2 @ ( uminus_uminus_int @ K3 ) ) ) ) @ ( semiri8010041392384452111omplex @ ( nat2 @ K3 ) ) ) ) ) ).

% of_int_of_nat
thf(fact_7019_of__int__of__nat,axiom,
    ( ring_1_of_int_real
    = ( ^ [K3: int] : ( if_real @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ ( nat2 @ ( uminus_uminus_int @ K3 ) ) ) ) @ ( semiri5074537144036343181t_real @ ( nat2 @ K3 ) ) ) ) ) ).

% of_int_of_nat
thf(fact_7020_of__int__of__nat,axiom,
    ( ring_1_of_int_rat
    = ( ^ [K3: int] : ( if_rat @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ ( nat2 @ ( uminus_uminus_int @ K3 ) ) ) ) @ ( semiri681578069525770553at_rat @ ( nat2 @ K3 ) ) ) ) ) ).

% of_int_of_nat
thf(fact_7021_of__int__of__nat,axiom,
    ( ring_1_of_int_int
    = ( ^ [K3: int] : ( if_int @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( nat2 @ ( uminus_uminus_int @ K3 ) ) ) ) @ ( semiri1314217659103216013at_int @ ( nat2 @ K3 ) ) ) ) ) ).

% of_int_of_nat
thf(fact_7022_of__real__sqrt,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( real_V4546457046886955230omplex @ ( sqrt @ X4 ) )
        = ( csqrt @ ( real_V4546457046886955230omplex @ X4 ) ) ) ) ).

% of_real_sqrt
thf(fact_7023_Arg__bounded,axiom,
    ! [Z: complex] :
      ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ ( arg @ Z ) )
      & ( ord_less_eq_real @ ( arg @ Z ) @ pi ) ) ).

% Arg_bounded
thf(fact_7024_even__nat__iff,axiom,
    ! [K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( nat2 @ K ) )
        = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ) ).

% even_nat_iff
thf(fact_7025_powr__int,axiom,
    ! [X4: real,I: int] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( ( ord_less_eq_int @ zero_zero_int @ I )
         => ( ( powr_real @ X4 @ ( ring_1_of_int_real @ I ) )
            = ( power_power_real @ X4 @ ( nat2 @ I ) ) ) )
        & ( ~ ( ord_less_eq_int @ zero_zero_int @ I )
         => ( ( powr_real @ X4 @ ( ring_1_of_int_real @ I ) )
            = ( divide_divide_real @ one_one_real @ ( power_power_real @ X4 @ ( nat2 @ ( uminus_uminus_int @ I ) ) ) ) ) ) ) ) ).

% powr_int
thf(fact_7026_divide__int__def,axiom,
    ( divide_divide_int
    = ( ^ [K3: int,L: int] :
          ( if_int @ ( L = zero_zero_int ) @ zero_zero_int
          @ ( if_int
            @ ( ( sgn_sgn_int @ K3 )
              = ( sgn_sgn_int @ L ) )
            @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K3 ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) )
            @ ( uminus_uminus_int
              @ ( semiri1314217659103216013at_int
                @ ( plus_plus_nat @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K3 ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) )
                  @ ( zero_n2687167440665602831ol_nat
                    @ ~ ( dvd_dvd_int @ L @ K3 ) ) ) ) ) ) ) ) ) ).

% divide_int_def
thf(fact_7027_cis__multiple__2pi,axiom,
    ! [N: real] :
      ( ( member_real @ N @ ring_1_Ints_real )
     => ( ( cis @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ N ) )
        = one_one_complex ) ) ).

% cis_multiple_2pi
thf(fact_7028_powr__real__of__int,axiom,
    ! [X4: real,N: int] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( ( ord_less_eq_int @ zero_zero_int @ N )
         => ( ( powr_real @ X4 @ ( ring_1_of_int_real @ N ) )
            = ( power_power_real @ X4 @ ( nat2 @ N ) ) ) )
        & ( ~ ( ord_less_eq_int @ zero_zero_int @ N )
         => ( ( powr_real @ X4 @ ( ring_1_of_int_real @ N ) )
            = ( inverse_inverse_real @ ( power_power_real @ X4 @ ( nat2 @ ( uminus_uminus_int @ N ) ) ) ) ) ) ) ) ).

% powr_real_of_int
thf(fact_7029_Suc__0__xor__eq,axiom,
    ! [N: nat] :
      ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ N )
      = ( minus_minus_nat @ ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
        @ ( zero_n2687167440665602831ol_nat
          @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% Suc_0_xor_eq
thf(fact_7030_xor__Suc__0__eq,axiom,
    ! [N: nat] :
      ( ( bit_se6528837805403552850or_nat @ N @ ( suc @ zero_zero_nat ) )
      = ( minus_minus_nat @ ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
        @ ( zero_n2687167440665602831ol_nat
          @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% xor_Suc_0_eq
thf(fact_7031_gbinomial__absorption_H,axiom,
    ! [K: nat,A: complex] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( gbinomial_complex @ A @ K )
        = ( times_times_complex @ ( divide1717551699836669952omplex @ A @ ( semiri8010041392384452111omplex @ K ) ) @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).

% gbinomial_absorption'
thf(fact_7032_gbinomial__absorption_H,axiom,
    ! [K: nat,A: real] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( gbinomial_real @ A @ K )
        = ( times_times_real @ ( divide_divide_real @ A @ ( semiri5074537144036343181t_real @ K ) ) @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).

% gbinomial_absorption'
thf(fact_7033_gbinomial__absorption_H,axiom,
    ! [K: nat,A: rat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( gbinomial_rat @ A @ K )
        = ( times_times_rat @ ( divide_divide_rat @ A @ ( semiri681578069525770553at_rat @ K ) ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).

% gbinomial_absorption'
thf(fact_7034_inverse__inverse__eq,axiom,
    ! [A: real] :
      ( ( inverse_inverse_real @ ( inverse_inverse_real @ A ) )
      = A ) ).

% inverse_inverse_eq
thf(fact_7035_inverse__inverse__eq,axiom,
    ! [A: complex] :
      ( ( invers8013647133539491842omplex @ ( invers8013647133539491842omplex @ A ) )
      = A ) ).

% inverse_inverse_eq
thf(fact_7036_inverse__inverse__eq,axiom,
    ! [A: rat] :
      ( ( inverse_inverse_rat @ ( inverse_inverse_rat @ A ) )
      = A ) ).

% inverse_inverse_eq
thf(fact_7037_inverse__eq__iff__eq,axiom,
    ! [A: real,B2: real] :
      ( ( ( inverse_inverse_real @ A )
        = ( inverse_inverse_real @ B2 ) )
      = ( A = B2 ) ) ).

% inverse_eq_iff_eq
thf(fact_7038_inverse__eq__iff__eq,axiom,
    ! [A: complex,B2: complex] :
      ( ( ( invers8013647133539491842omplex @ A )
        = ( invers8013647133539491842omplex @ B2 ) )
      = ( A = B2 ) ) ).

% inverse_eq_iff_eq
thf(fact_7039_inverse__eq__iff__eq,axiom,
    ! [A: rat,B2: rat] :
      ( ( ( inverse_inverse_rat @ A )
        = ( inverse_inverse_rat @ B2 ) )
      = ( A = B2 ) ) ).

% inverse_eq_iff_eq
thf(fact_7040_bit_Oxor__left__self,axiom,
    ! [X4: int,Y3: int] :
      ( ( bit_se6526347334894502574or_int @ X4 @ ( bit_se6526347334894502574or_int @ X4 @ Y3 ) )
      = Y3 ) ).

% bit.xor_left_self
thf(fact_7041_inverse__nonzero__iff__nonzero,axiom,
    ! [A: real] :
      ( ( ( inverse_inverse_real @ A )
        = zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% inverse_nonzero_iff_nonzero
thf(fact_7042_inverse__nonzero__iff__nonzero,axiom,
    ! [A: complex] :
      ( ( ( invers8013647133539491842omplex @ A )
        = zero_zero_complex )
      = ( A = zero_zero_complex ) ) ).

% inverse_nonzero_iff_nonzero
thf(fact_7043_inverse__nonzero__iff__nonzero,axiom,
    ! [A: rat] :
      ( ( ( inverse_inverse_rat @ A )
        = zero_zero_rat )
      = ( A = zero_zero_rat ) ) ).

% inverse_nonzero_iff_nonzero
thf(fact_7044_inverse__zero,axiom,
    ( ( inverse_inverse_real @ zero_zero_real )
    = zero_zero_real ) ).

% inverse_zero
thf(fact_7045_inverse__zero,axiom,
    ( ( invers8013647133539491842omplex @ zero_zero_complex )
    = zero_zero_complex ) ).

% inverse_zero
thf(fact_7046_inverse__zero,axiom,
    ( ( inverse_inverse_rat @ zero_zero_rat )
    = zero_zero_rat ) ).

% inverse_zero
thf(fact_7047_inverse__mult__distrib,axiom,
    ! [A: real,B2: real] :
      ( ( inverse_inverse_real @ ( times_times_real @ A @ B2 ) )
      = ( times_times_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B2 ) ) ) ).

% inverse_mult_distrib
thf(fact_7048_inverse__mult__distrib,axiom,
    ! [A: complex,B2: complex] :
      ( ( invers8013647133539491842omplex @ ( times_times_complex @ A @ B2 ) )
      = ( times_times_complex @ ( invers8013647133539491842omplex @ A ) @ ( invers8013647133539491842omplex @ B2 ) ) ) ).

% inverse_mult_distrib
thf(fact_7049_inverse__mult__distrib,axiom,
    ! [A: rat,B2: rat] :
      ( ( inverse_inverse_rat @ ( times_times_rat @ A @ B2 ) )
      = ( times_times_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B2 ) ) ) ).

% inverse_mult_distrib
thf(fact_7050_inverse__eq__1__iff,axiom,
    ! [X4: real] :
      ( ( ( inverse_inverse_real @ X4 )
        = one_one_real )
      = ( X4 = one_one_real ) ) ).

% inverse_eq_1_iff
thf(fact_7051_inverse__eq__1__iff,axiom,
    ! [X4: complex] :
      ( ( ( invers8013647133539491842omplex @ X4 )
        = one_one_complex )
      = ( X4 = one_one_complex ) ) ).

% inverse_eq_1_iff
thf(fact_7052_inverse__eq__1__iff,axiom,
    ! [X4: rat] :
      ( ( ( inverse_inverse_rat @ X4 )
        = one_one_rat )
      = ( X4 = one_one_rat ) ) ).

% inverse_eq_1_iff
thf(fact_7053_inverse__1,axiom,
    ( ( inverse_inverse_real @ one_one_real )
    = one_one_real ) ).

% inverse_1
thf(fact_7054_inverse__1,axiom,
    ( ( invers8013647133539491842omplex @ one_one_complex )
    = one_one_complex ) ).

% inverse_1
thf(fact_7055_inverse__1,axiom,
    ( ( inverse_inverse_rat @ one_one_rat )
    = one_one_rat ) ).

% inverse_1
thf(fact_7056_inverse__divide,axiom,
    ! [A: real,B2: real] :
      ( ( inverse_inverse_real @ ( divide_divide_real @ A @ B2 ) )
      = ( divide_divide_real @ B2 @ A ) ) ).

% inverse_divide
thf(fact_7057_inverse__divide,axiom,
    ! [A: complex,B2: complex] :
      ( ( invers8013647133539491842omplex @ ( divide1717551699836669952omplex @ A @ B2 ) )
      = ( divide1717551699836669952omplex @ B2 @ A ) ) ).

% inverse_divide
thf(fact_7058_inverse__divide,axiom,
    ! [A: rat,B2: rat] :
      ( ( inverse_inverse_rat @ ( divide_divide_rat @ A @ B2 ) )
      = ( divide_divide_rat @ B2 @ A ) ) ).

% inverse_divide
thf(fact_7059_inverse__minus__eq,axiom,
    ! [A: real] :
      ( ( inverse_inverse_real @ ( uminus_uminus_real @ A ) )
      = ( uminus_uminus_real @ ( inverse_inverse_real @ A ) ) ) ).

% inverse_minus_eq
thf(fact_7060_inverse__minus__eq,axiom,
    ! [A: complex] :
      ( ( invers8013647133539491842omplex @ ( uminus1482373934393186551omplex @ A ) )
      = ( uminus1482373934393186551omplex @ ( invers8013647133539491842omplex @ A ) ) ) ).

% inverse_minus_eq
thf(fact_7061_inverse__minus__eq,axiom,
    ! [A: rat] :
      ( ( inverse_inverse_rat @ ( uminus_uminus_rat @ A ) )
      = ( uminus_uminus_rat @ ( inverse_inverse_rat @ A ) ) ) ).

% inverse_minus_eq
thf(fact_7062_xor_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( bit_se6528837805403552850or_nat @ A @ zero_zero_nat )
      = A ) ).

% xor.right_neutral
thf(fact_7063_xor_Oright__neutral,axiom,
    ! [A: int] :
      ( ( bit_se6526347334894502574or_int @ A @ zero_zero_int )
      = A ) ).

% xor.right_neutral
thf(fact_7064_xor_Oleft__neutral,axiom,
    ! [A: nat] :
      ( ( bit_se6528837805403552850or_nat @ zero_zero_nat @ A )
      = A ) ).

% xor.left_neutral
thf(fact_7065_xor_Oleft__neutral,axiom,
    ! [A: int] :
      ( ( bit_se6526347334894502574or_int @ zero_zero_int @ A )
      = A ) ).

% xor.left_neutral
thf(fact_7066_xor__self__eq,axiom,
    ! [A: nat] :
      ( ( bit_se6528837805403552850or_nat @ A @ A )
      = zero_zero_nat ) ).

% xor_self_eq
thf(fact_7067_xor__self__eq,axiom,
    ! [A: int] :
      ( ( bit_se6526347334894502574or_int @ A @ A )
      = zero_zero_int ) ).

% xor_self_eq
thf(fact_7068_bit_Oxor__self,axiom,
    ! [X4: int] :
      ( ( bit_se6526347334894502574or_int @ X4 @ X4 )
      = zero_zero_int ) ).

% bit.xor_self
thf(fact_7069_abs__inverse,axiom,
    ! [A: real] :
      ( ( abs_abs_real @ ( inverse_inverse_real @ A ) )
      = ( inverse_inverse_real @ ( abs_abs_real @ A ) ) ) ).

% abs_inverse
thf(fact_7070_abs__inverse,axiom,
    ! [A: complex] :
      ( ( abs_abs_complex @ ( invers8013647133539491842omplex @ A ) )
      = ( invers8013647133539491842omplex @ ( abs_abs_complex @ A ) ) ) ).

% abs_inverse
thf(fact_7071_abs__inverse,axiom,
    ! [A: rat] :
      ( ( abs_abs_rat @ ( inverse_inverse_rat @ A ) )
      = ( inverse_inverse_rat @ ( abs_abs_rat @ A ) ) ) ).

% abs_inverse
thf(fact_7072_inverse__sgn,axiom,
    ! [A: real] :
      ( ( inverse_inverse_real @ ( sgn_sgn_real @ A ) )
      = ( sgn_sgn_real @ A ) ) ).

% inverse_sgn
thf(fact_7073_inverse__sgn,axiom,
    ! [A: rat] :
      ( ( inverse_inverse_rat @ ( sgn_sgn_rat @ A ) )
      = ( sgn_sgn_rat @ A ) ) ).

% inverse_sgn
thf(fact_7074_sgn__inverse,axiom,
    ! [A: real] :
      ( ( sgn_sgn_real @ ( inverse_inverse_real @ A ) )
      = ( inverse_inverse_real @ ( sgn_sgn_real @ A ) ) ) ).

% sgn_inverse
thf(fact_7075_sgn__inverse,axiom,
    ! [A: complex] :
      ( ( sgn_sgn_complex @ ( invers8013647133539491842omplex @ A ) )
      = ( invers8013647133539491842omplex @ ( sgn_sgn_complex @ A ) ) ) ).

% sgn_inverse
thf(fact_7076_sgn__inverse,axiom,
    ! [A: rat] :
      ( ( sgn_sgn_rat @ ( inverse_inverse_rat @ A ) )
      = ( inverse_inverse_rat @ ( sgn_sgn_rat @ A ) ) ) ).

% sgn_inverse
thf(fact_7077_take__bit__xor,axiom,
    ! [N: nat,A: int,B2: int] :
      ( ( bit_se2923211474154528505it_int @ N @ ( bit_se6526347334894502574or_int @ A @ B2 ) )
      = ( bit_se6526347334894502574or_int @ ( bit_se2923211474154528505it_int @ N @ A ) @ ( bit_se2923211474154528505it_int @ N @ B2 ) ) ) ).

% take_bit_xor
thf(fact_7078_take__bit__xor,axiom,
    ! [N: nat,A: nat,B2: nat] :
      ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se6528837805403552850or_nat @ A @ B2 ) )
      = ( bit_se6528837805403552850or_nat @ ( bit_se2925701944663578781it_nat @ N @ A ) @ ( bit_se2925701944663578781it_nat @ N @ B2 ) ) ) ).

% take_bit_xor
thf(fact_7079_inverse__nonpositive__iff__nonpositive,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ zero_zero_real )
      = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).

% inverse_nonpositive_iff_nonpositive
thf(fact_7080_inverse__nonpositive__iff__nonpositive,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ zero_zero_rat )
      = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).

% inverse_nonpositive_iff_nonpositive
thf(fact_7081_inverse__nonnegative__iff__nonnegative,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( inverse_inverse_real @ A ) )
      = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).

% inverse_nonnegative_iff_nonnegative
thf(fact_7082_inverse__nonnegative__iff__nonnegative,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( inverse_inverse_rat @ A ) )
      = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).

% inverse_nonnegative_iff_nonnegative
thf(fact_7083_inverse__positive__iff__positive,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ A ) )
      = ( ord_less_real @ zero_zero_real @ A ) ) ).

% inverse_positive_iff_positive
thf(fact_7084_inverse__positive__iff__positive,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( inverse_inverse_rat @ A ) )
      = ( ord_less_rat @ zero_zero_rat @ A ) ) ).

% inverse_positive_iff_positive
thf(fact_7085_inverse__negative__iff__negative,axiom,
    ! [A: real] :
      ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ zero_zero_real )
      = ( ord_less_real @ A @ zero_zero_real ) ) ).

% inverse_negative_iff_negative
thf(fact_7086_inverse__negative__iff__negative,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ zero_zero_rat )
      = ( ord_less_rat @ A @ zero_zero_rat ) ) ).

% inverse_negative_iff_negative
thf(fact_7087_inverse__less__iff__less__neg,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( ord_less_real @ B2 @ zero_zero_real )
       => ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B2 ) )
          = ( ord_less_real @ B2 @ A ) ) ) ) ).

% inverse_less_iff_less_neg
thf(fact_7088_inverse__less__iff__less__neg,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( ord_less_rat @ B2 @ zero_zero_rat )
       => ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B2 ) )
          = ( ord_less_rat @ B2 @ A ) ) ) ) ).

% inverse_less_iff_less_neg
thf(fact_7089_inverse__less__iff__less,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ B2 )
       => ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B2 ) )
          = ( ord_less_real @ B2 @ A ) ) ) ) ).

% inverse_less_iff_less
thf(fact_7090_inverse__less__iff__less,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ zero_zero_rat @ B2 )
       => ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B2 ) )
          = ( ord_less_rat @ B2 @ A ) ) ) ) ).

% inverse_less_iff_less
thf(fact_7091_gbinomial__0_I2_J,axiom,
    ! [K: nat] :
      ( ( gbinomial_real @ zero_zero_real @ ( suc @ K ) )
      = zero_zero_real ) ).

% gbinomial_0(2)
thf(fact_7092_gbinomial__0_I2_J,axiom,
    ! [K: nat] :
      ( ( gbinomial_rat @ zero_zero_rat @ ( suc @ K ) )
      = zero_zero_rat ) ).

% gbinomial_0(2)
thf(fact_7093_gbinomial__0_I2_J,axiom,
    ! [K: nat] :
      ( ( gbinomial_nat @ zero_zero_nat @ ( suc @ K ) )
      = zero_zero_nat ) ).

% gbinomial_0(2)
thf(fact_7094_gbinomial__0_I2_J,axiom,
    ! [K: nat] :
      ( ( gbinomial_int @ zero_zero_int @ ( suc @ K ) )
      = zero_zero_int ) ).

% gbinomial_0(2)
thf(fact_7095_gbinomial__0_I1_J,axiom,
    ! [A: complex] :
      ( ( gbinomial_complex @ A @ zero_zero_nat )
      = one_one_complex ) ).

% gbinomial_0(1)
thf(fact_7096_gbinomial__0_I1_J,axiom,
    ! [A: real] :
      ( ( gbinomial_real @ A @ zero_zero_nat )
      = one_one_real ) ).

% gbinomial_0(1)
thf(fact_7097_gbinomial__0_I1_J,axiom,
    ! [A: rat] :
      ( ( gbinomial_rat @ A @ zero_zero_nat )
      = one_one_rat ) ).

% gbinomial_0(1)
thf(fact_7098_gbinomial__0_I1_J,axiom,
    ! [A: nat] :
      ( ( gbinomial_nat @ A @ zero_zero_nat )
      = one_one_nat ) ).

% gbinomial_0(1)
thf(fact_7099_gbinomial__0_I1_J,axiom,
    ! [A: int] :
      ( ( gbinomial_int @ A @ zero_zero_nat )
      = one_one_int ) ).

% gbinomial_0(1)
thf(fact_7100_inverse__le__iff__le,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ B2 )
       => ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B2 ) )
          = ( ord_less_eq_real @ B2 @ A ) ) ) ) ).

% inverse_le_iff_le
thf(fact_7101_inverse__le__iff__le,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ zero_zero_rat @ B2 )
       => ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B2 ) )
          = ( ord_less_eq_rat @ B2 @ A ) ) ) ) ).

% inverse_le_iff_le
thf(fact_7102_inverse__le__iff__le__neg,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( ord_less_real @ B2 @ zero_zero_real )
       => ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B2 ) )
          = ( ord_less_eq_real @ B2 @ A ) ) ) ) ).

% inverse_le_iff_le_neg
thf(fact_7103_inverse__le__iff__le__neg,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( ord_less_rat @ B2 @ zero_zero_rat )
       => ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B2 ) )
          = ( ord_less_eq_rat @ B2 @ A ) ) ) ) ).

% inverse_le_iff_le_neg
thf(fact_7104_right__inverse,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( ( times_times_real @ A @ ( inverse_inverse_real @ A ) )
        = one_one_real ) ) ).

% right_inverse
thf(fact_7105_right__inverse,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( ( times_times_complex @ A @ ( invers8013647133539491842omplex @ A ) )
        = one_one_complex ) ) ).

% right_inverse
thf(fact_7106_right__inverse,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( ( times_times_rat @ A @ ( inverse_inverse_rat @ A ) )
        = one_one_rat ) ) ).

% right_inverse
thf(fact_7107_left__inverse,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( ( times_times_real @ ( inverse_inverse_real @ A ) @ A )
        = one_one_real ) ) ).

% left_inverse
thf(fact_7108_left__inverse,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( ( times_times_complex @ ( invers8013647133539491842omplex @ A ) @ A )
        = one_one_complex ) ) ).

% left_inverse
thf(fact_7109_left__inverse,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( ( times_times_rat @ ( inverse_inverse_rat @ A ) @ A )
        = one_one_rat ) ) ).

% left_inverse
thf(fact_7110_inverse__eq__divide__numeral,axiom,
    ! [W: num] :
      ( ( inverse_inverse_real @ ( numeral_numeral_real @ W ) )
      = ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ W ) ) ) ).

% inverse_eq_divide_numeral
thf(fact_7111_inverse__eq__divide__numeral,axiom,
    ! [W: num] :
      ( ( invers8013647133539491842omplex @ ( numera6690914467698888265omplex @ W ) )
      = ( divide1717551699836669952omplex @ one_one_complex @ ( numera6690914467698888265omplex @ W ) ) ) ).

% inverse_eq_divide_numeral
thf(fact_7112_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
thf(fact_7113_floor__add2,axiom,
    ! [X4: real,Y3: real] :
      ( ( ( member_real @ X4 @ ring_1_Ints_real )
        | ( member_real @ Y3 @ ring_1_Ints_real ) )
     => ( ( archim6058952711729229775r_real @ ( plus_plus_real @ X4 @ Y3 ) )
        = ( plus_plus_int @ ( archim6058952711729229775r_real @ X4 ) @ ( archim6058952711729229775r_real @ Y3 ) ) ) ) ).

% floor_add2
thf(fact_7114_floor__add2,axiom,
    ! [X4: rat,Y3: rat] :
      ( ( ( member_rat @ X4 @ ring_1_Ints_rat )
        | ( member_rat @ Y3 @ ring_1_Ints_rat ) )
     => ( ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X4 @ Y3 ) )
        = ( plus_plus_int @ ( archim3151403230148437115or_rat @ X4 ) @ ( archim3151403230148437115or_rat @ Y3 ) ) ) ) ).

% floor_add2
thf(fact_7115_frac__gt__0__iff,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ ( archim2898591450579166408c_real @ X4 ) )
      = ( ~ ( member_real @ X4 @ ring_1_Ints_real ) ) ) ).

% frac_gt_0_iff
thf(fact_7116_frac__gt__0__iff,axiom,
    ! [X4: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( archimedean_frac_rat @ X4 ) )
      = ( ~ ( member_rat @ X4 @ ring_1_Ints_rat ) ) ) ).

% frac_gt_0_iff
thf(fact_7117_inverse__eq__divide__neg__numeral,axiom,
    ! [W: num] :
      ( ( inverse_inverse_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
      = ( divide_divide_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ).

% inverse_eq_divide_neg_numeral
thf(fact_7118_inverse__eq__divide__neg__numeral,axiom,
    ! [W: num] :
      ( ( invers8013647133539491842omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
      = ( divide1717551699836669952omplex @ one_one_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ) ).

% inverse_eq_divide_neg_numeral
thf(fact_7119_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
thf(fact_7120_xor__numerals_I3_J,axiom,
    ! [X4: num,Y3: num] :
      ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit0 @ X4 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) )
      = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ X4 ) @ ( numeral_numeral_nat @ Y3 ) ) ) ) ).

% xor_numerals(3)
thf(fact_7121_xor__numerals_I3_J,axiom,
    ! [X4: num,Y3: num] :
      ( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit0 @ X4 ) ) @ ( numeral_numeral_int @ ( bit0 @ Y3 ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ X4 ) @ ( numeral_numeral_int @ Y3 ) ) ) ) ).

% xor_numerals(3)
thf(fact_7122_xor__numerals_I1_J,axiom,
    ! [Y3: num] :
      ( ( bit_se6528837805403552850or_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) )
      = ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) ) ).

% xor_numerals(1)
thf(fact_7123_xor__numerals_I1_J,axiom,
    ! [Y3: num] :
      ( ( bit_se6526347334894502574or_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ Y3 ) ) )
      = ( numeral_numeral_int @ ( bit1 @ Y3 ) ) ) ).

% xor_numerals(1)
thf(fact_7124_xor__numerals_I2_J,axiom,
    ! [Y3: num] :
      ( ( bit_se6528837805403552850or_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) )
      = ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) ) ).

% xor_numerals(2)
thf(fact_7125_xor__numerals_I2_J,axiom,
    ! [Y3: num] :
      ( ( bit_se6526347334894502574or_int @ one_one_int @ ( numeral_numeral_int @ ( bit1 @ Y3 ) ) )
      = ( numeral_numeral_int @ ( bit0 @ Y3 ) ) ) ).

% xor_numerals(2)
thf(fact_7126_xor__numerals_I5_J,axiom,
    ! [X4: num] :
      ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit0 @ X4 ) ) @ one_one_nat )
      = ( numeral_numeral_nat @ ( bit1 @ X4 ) ) ) ).

% xor_numerals(5)
thf(fact_7127_xor__numerals_I5_J,axiom,
    ! [X4: num] :
      ( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit0 @ X4 ) ) @ one_one_int )
      = ( numeral_numeral_int @ ( bit1 @ X4 ) ) ) ).

% xor_numerals(5)
thf(fact_7128_xor__numerals_I8_J,axiom,
    ! [X4: num] :
      ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit1 @ X4 ) ) @ one_one_nat )
      = ( numeral_numeral_nat @ ( bit0 @ X4 ) ) ) ).

% xor_numerals(8)
thf(fact_7129_xor__numerals_I8_J,axiom,
    ! [X4: num] :
      ( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit1 @ X4 ) ) @ one_one_int )
      = ( numeral_numeral_int @ ( bit0 @ X4 ) ) ) ).

% xor_numerals(8)
thf(fact_7130_xor__numerals_I7_J,axiom,
    ! [X4: num,Y3: num] :
      ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit1 @ X4 ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) )
      = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ X4 ) @ ( numeral_numeral_nat @ Y3 ) ) ) ) ).

% xor_numerals(7)
thf(fact_7131_xor__numerals_I7_J,axiom,
    ! [X4: num,Y3: num] :
      ( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit1 @ X4 ) ) @ ( numeral_numeral_int @ ( bit1 @ Y3 ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ X4 ) @ ( numeral_numeral_int @ Y3 ) ) ) ) ).

% xor_numerals(7)
thf(fact_7132_xor__nat__numerals_I1_J,axiom,
    ! [Y3: num] :
      ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) )
      = ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) ) ).

% xor_nat_numerals(1)
thf(fact_7133_xor__nat__numerals_I2_J,axiom,
    ! [Y3: num] :
      ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) )
      = ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) ) ).

% xor_nat_numerals(2)
thf(fact_7134_xor__nat__numerals_I3_J,axiom,
    ! [X4: num] :
      ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit0 @ X4 ) ) @ ( suc @ zero_zero_nat ) )
      = ( numeral_numeral_nat @ ( bit1 @ X4 ) ) ) ).

% xor_nat_numerals(3)
thf(fact_7135_xor__nat__numerals_I4_J,axiom,
    ! [X4: num] :
      ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit1 @ X4 ) ) @ ( suc @ zero_zero_nat ) )
      = ( numeral_numeral_nat @ ( bit0 @ X4 ) ) ) ).

% xor_nat_numerals(4)
thf(fact_7136_xor__numerals_I4_J,axiom,
    ! [X4: num,Y3: num] :
      ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit0 @ X4 ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) )
      = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ X4 ) @ ( numeral_numeral_nat @ Y3 ) ) ) ) ) ).

% xor_numerals(4)
thf(fact_7137_xor__numerals_I4_J,axiom,
    ! [X4: num,Y3: num] :
      ( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit0 @ X4 ) ) @ ( numeral_numeral_int @ ( bit1 @ Y3 ) ) )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ X4 ) @ ( numeral_numeral_int @ Y3 ) ) ) ) ) ).

% xor_numerals(4)
thf(fact_7138_xor__numerals_I6_J,axiom,
    ! [X4: num,Y3: num] :
      ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit1 @ X4 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) )
      = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ X4 ) @ ( numeral_numeral_nat @ Y3 ) ) ) ) ) ).

% xor_numerals(6)
thf(fact_7139_xor__numerals_I6_J,axiom,
    ! [X4: num,Y3: num] :
      ( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit1 @ X4 ) ) @ ( numeral_numeral_int @ ( bit0 @ Y3 ) ) )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ X4 ) @ ( numeral_numeral_int @ Y3 ) ) ) ) ) ).

% xor_numerals(6)
thf(fact_7140_power__inverse,axiom,
    ! [A: real,N: nat] :
      ( ( power_power_real @ ( inverse_inverse_real @ A ) @ N )
      = ( inverse_inverse_real @ ( power_power_real @ A @ N ) ) ) ).

% power_inverse
thf(fact_7141_power__inverse,axiom,
    ! [A: complex,N: nat] :
      ( ( power_power_complex @ ( invers8013647133539491842omplex @ A ) @ N )
      = ( invers8013647133539491842omplex @ ( power_power_complex @ A @ N ) ) ) ).

% power_inverse
thf(fact_7142_power__inverse,axiom,
    ! [A: rat,N: nat] :
      ( ( power_power_rat @ ( inverse_inverse_rat @ A ) @ N )
      = ( inverse_inverse_rat @ ( power_power_rat @ A @ N ) ) ) ).

% power_inverse
thf(fact_7143_mult__commute__imp__mult__inverse__commute,axiom,
    ! [Y3: real,X4: real] :
      ( ( ( times_times_real @ Y3 @ X4 )
        = ( times_times_real @ X4 @ Y3 ) )
     => ( ( times_times_real @ ( inverse_inverse_real @ Y3 ) @ X4 )
        = ( times_times_real @ X4 @ ( inverse_inverse_real @ Y3 ) ) ) ) ).

% mult_commute_imp_mult_inverse_commute
thf(fact_7144_mult__commute__imp__mult__inverse__commute,axiom,
    ! [Y3: complex,X4: complex] :
      ( ( ( times_times_complex @ Y3 @ X4 )
        = ( times_times_complex @ X4 @ Y3 ) )
     => ( ( times_times_complex @ ( invers8013647133539491842omplex @ Y3 ) @ X4 )
        = ( times_times_complex @ X4 @ ( invers8013647133539491842omplex @ Y3 ) ) ) ) ).

% mult_commute_imp_mult_inverse_commute
thf(fact_7145_mult__commute__imp__mult__inverse__commute,axiom,
    ! [Y3: rat,X4: rat] :
      ( ( ( times_times_rat @ Y3 @ X4 )
        = ( times_times_rat @ X4 @ Y3 ) )
     => ( ( times_times_rat @ ( inverse_inverse_rat @ Y3 ) @ X4 )
        = ( times_times_rat @ X4 @ ( inverse_inverse_rat @ Y3 ) ) ) ) ).

% mult_commute_imp_mult_inverse_commute
thf(fact_7146_of__nat__xor__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri1316708129612266289at_nat @ ( bit_se6528837805403552850or_nat @ M @ N ) )
      = ( bit_se6528837805403552850or_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).

% of_nat_xor_eq
thf(fact_7147_of__nat__xor__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri1314217659103216013at_int @ ( bit_se6528837805403552850or_nat @ M @ N ) )
      = ( bit_se6526347334894502574or_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).

% of_nat_xor_eq
thf(fact_7148_of__int__xor__eq,axiom,
    ! [K: int,L2: int] :
      ( ( ring_1_of_int_int @ ( bit_se6526347334894502574or_int @ K @ L2 ) )
      = ( bit_se6526347334894502574or_int @ ( ring_1_of_int_int @ K ) @ ( ring_1_of_int_int @ L2 ) ) ) ).

% of_int_xor_eq
thf(fact_7149_xor_Oleft__commute,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( ( bit_se6528837805403552850or_nat @ B2 @ ( bit_se6528837805403552850or_nat @ A @ C ) )
      = ( bit_se6528837805403552850or_nat @ A @ ( bit_se6528837805403552850or_nat @ B2 @ C ) ) ) ).

% xor.left_commute
thf(fact_7150_xor_Oleft__commute,axiom,
    ! [B2: int,A: int,C: int] :
      ( ( bit_se6526347334894502574or_int @ B2 @ ( bit_se6526347334894502574or_int @ A @ C ) )
      = ( bit_se6526347334894502574or_int @ A @ ( bit_se6526347334894502574or_int @ B2 @ C ) ) ) ).

% xor.left_commute
thf(fact_7151_xor_Ocommute,axiom,
    ( bit_se6528837805403552850or_nat
    = ( ^ [A4: nat,B3: nat] : ( bit_se6528837805403552850or_nat @ B3 @ A4 ) ) ) ).

% xor.commute
thf(fact_7152_xor_Ocommute,axiom,
    ( bit_se6526347334894502574or_int
    = ( ^ [A4: int,B3: int] : ( bit_se6526347334894502574or_int @ B3 @ A4 ) ) ) ).

% xor.commute
thf(fact_7153_xor_Oassoc,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( bit_se6528837805403552850or_nat @ ( bit_se6528837805403552850or_nat @ A @ B2 ) @ C )
      = ( bit_se6528837805403552850or_nat @ A @ ( bit_se6528837805403552850or_nat @ B2 @ C ) ) ) ).

% xor.assoc
thf(fact_7154_xor_Oassoc,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( bit_se6526347334894502574or_int @ ( bit_se6526347334894502574or_int @ A @ B2 ) @ C )
      = ( bit_se6526347334894502574or_int @ A @ ( bit_se6526347334894502574or_int @ B2 @ C ) ) ) ).

% xor.assoc
thf(fact_7155_inverse__eq__imp__eq,axiom,
    ! [A: real,B2: real] :
      ( ( ( inverse_inverse_real @ A )
        = ( inverse_inverse_real @ B2 ) )
     => ( A = B2 ) ) ).

% inverse_eq_imp_eq
thf(fact_7156_inverse__eq__imp__eq,axiom,
    ! [A: complex,B2: complex] :
      ( ( ( invers8013647133539491842omplex @ A )
        = ( invers8013647133539491842omplex @ B2 ) )
     => ( A = B2 ) ) ).

% inverse_eq_imp_eq
thf(fact_7157_inverse__eq__imp__eq,axiom,
    ! [A: rat,B2: rat] :
      ( ( ( inverse_inverse_rat @ A )
        = ( inverse_inverse_rat @ B2 ) )
     => ( A = B2 ) ) ).

% inverse_eq_imp_eq
thf(fact_7158_field__class_Ofield__inverse__zero,axiom,
    ( ( inverse_inverse_real @ zero_zero_real )
    = zero_zero_real ) ).

% field_class.field_inverse_zero
thf(fact_7159_field__class_Ofield__inverse__zero,axiom,
    ( ( invers8013647133539491842omplex @ zero_zero_complex )
    = zero_zero_complex ) ).

% field_class.field_inverse_zero
thf(fact_7160_field__class_Ofield__inverse__zero,axiom,
    ( ( inverse_inverse_rat @ zero_zero_rat )
    = zero_zero_rat ) ).

% field_class.field_inverse_zero
thf(fact_7161_inverse__zero__imp__zero,axiom,
    ! [A: real] :
      ( ( ( inverse_inverse_real @ A )
        = zero_zero_real )
     => ( A = zero_zero_real ) ) ).

% inverse_zero_imp_zero
thf(fact_7162_inverse__zero__imp__zero,axiom,
    ! [A: complex] :
      ( ( ( invers8013647133539491842omplex @ A )
        = zero_zero_complex )
     => ( A = zero_zero_complex ) ) ).

% inverse_zero_imp_zero
thf(fact_7163_inverse__zero__imp__zero,axiom,
    ! [A: rat] :
      ( ( ( inverse_inverse_rat @ A )
        = zero_zero_rat )
     => ( A = zero_zero_rat ) ) ).

% inverse_zero_imp_zero
thf(fact_7164_nonzero__inverse__eq__imp__eq,axiom,
    ! [A: real,B2: real] :
      ( ( ( inverse_inverse_real @ A )
        = ( inverse_inverse_real @ B2 ) )
     => ( ( A != zero_zero_real )
       => ( ( B2 != zero_zero_real )
         => ( A = B2 ) ) ) ) ).

% nonzero_inverse_eq_imp_eq
thf(fact_7165_nonzero__inverse__eq__imp__eq,axiom,
    ! [A: complex,B2: complex] :
      ( ( ( invers8013647133539491842omplex @ A )
        = ( invers8013647133539491842omplex @ B2 ) )
     => ( ( A != zero_zero_complex )
       => ( ( B2 != zero_zero_complex )
         => ( A = B2 ) ) ) ) ).

% nonzero_inverse_eq_imp_eq
thf(fact_7166_nonzero__inverse__eq__imp__eq,axiom,
    ! [A: rat,B2: rat] :
      ( ( ( inverse_inverse_rat @ A )
        = ( inverse_inverse_rat @ B2 ) )
     => ( ( A != zero_zero_rat )
       => ( ( B2 != zero_zero_rat )
         => ( A = B2 ) ) ) ) ).

% nonzero_inverse_eq_imp_eq
thf(fact_7167_nonzero__inverse__inverse__eq,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( ( inverse_inverse_real @ ( inverse_inverse_real @ A ) )
        = A ) ) ).

% nonzero_inverse_inverse_eq
thf(fact_7168_nonzero__inverse__inverse__eq,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( ( invers8013647133539491842omplex @ ( invers8013647133539491842omplex @ A ) )
        = A ) ) ).

% nonzero_inverse_inverse_eq
thf(fact_7169_nonzero__inverse__inverse__eq,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( ( inverse_inverse_rat @ ( inverse_inverse_rat @ A ) )
        = A ) ) ).

% nonzero_inverse_inverse_eq
thf(fact_7170_nonzero__imp__inverse__nonzero,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( ( inverse_inverse_real @ A )
       != zero_zero_real ) ) ).

% nonzero_imp_inverse_nonzero
thf(fact_7171_nonzero__imp__inverse__nonzero,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( ( invers8013647133539491842omplex @ A )
       != zero_zero_complex ) ) ).

% nonzero_imp_inverse_nonzero
thf(fact_7172_nonzero__imp__inverse__nonzero,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( ( inverse_inverse_rat @ A )
       != zero_zero_rat ) ) ).

% nonzero_imp_inverse_nonzero
thf(fact_7173_Ints__numeral,axiom,
    ! [N: num] : ( member_complex @ ( numera6690914467698888265omplex @ N ) @ ring_1_Ints_complex ) ).

% Ints_numeral
thf(fact_7174_Ints__numeral,axiom,
    ! [N: num] : ( member_real @ ( numeral_numeral_real @ N ) @ ring_1_Ints_real ) ).

% Ints_numeral
thf(fact_7175_Ints__numeral,axiom,
    ! [N: num] : ( member_rat @ ( numeral_numeral_rat @ N ) @ ring_1_Ints_rat ) ).

% Ints_numeral
thf(fact_7176_Ints__numeral,axiom,
    ! [N: num] : ( member_int @ ( numeral_numeral_int @ N ) @ ring_1_Ints_int ) ).

% Ints_numeral
thf(fact_7177_Ints__mult,axiom,
    ! [A: complex,B2: complex] :
      ( ( member_complex @ A @ ring_1_Ints_complex )
     => ( ( member_complex @ B2 @ ring_1_Ints_complex )
       => ( member_complex @ ( times_times_complex @ A @ B2 ) @ ring_1_Ints_complex ) ) ) ).

% Ints_mult
thf(fact_7178_Ints__mult,axiom,
    ! [A: real,B2: real] :
      ( ( member_real @ A @ ring_1_Ints_real )
     => ( ( member_real @ B2 @ ring_1_Ints_real )
       => ( member_real @ ( times_times_real @ A @ B2 ) @ ring_1_Ints_real ) ) ) ).

% Ints_mult
thf(fact_7179_Ints__mult,axiom,
    ! [A: rat,B2: rat] :
      ( ( member_rat @ A @ ring_1_Ints_rat )
     => ( ( member_rat @ B2 @ ring_1_Ints_rat )
       => ( member_rat @ ( times_times_rat @ A @ B2 ) @ ring_1_Ints_rat ) ) ) ).

% Ints_mult
thf(fact_7180_Ints__mult,axiom,
    ! [A: int,B2: int] :
      ( ( member_int @ A @ ring_1_Ints_int )
     => ( ( member_int @ B2 @ ring_1_Ints_int )
       => ( member_int @ ( times_times_int @ A @ B2 ) @ ring_1_Ints_int ) ) ) ).

% Ints_mult
thf(fact_7181_Ints__1,axiom,
    member_complex @ one_one_complex @ ring_1_Ints_complex ).

% Ints_1
thf(fact_7182_Ints__1,axiom,
    member_rat @ one_one_rat @ ring_1_Ints_rat ).

% Ints_1
thf(fact_7183_Ints__1,axiom,
    member_int @ one_one_int @ ring_1_Ints_int ).

% Ints_1
thf(fact_7184_Ints__1,axiom,
    member_real @ one_one_real @ ring_1_Ints_real ).

% Ints_1
thf(fact_7185_Ints__add,axiom,
    ! [A: complex,B2: complex] :
      ( ( member_complex @ A @ ring_1_Ints_complex )
     => ( ( member_complex @ B2 @ ring_1_Ints_complex )
       => ( member_complex @ ( plus_plus_complex @ A @ B2 ) @ ring_1_Ints_complex ) ) ) ).

% Ints_add
thf(fact_7186_Ints__add,axiom,
    ! [A: real,B2: real] :
      ( ( member_real @ A @ ring_1_Ints_real )
     => ( ( member_real @ B2 @ ring_1_Ints_real )
       => ( member_real @ ( plus_plus_real @ A @ B2 ) @ ring_1_Ints_real ) ) ) ).

% Ints_add
thf(fact_7187_Ints__add,axiom,
    ! [A: rat,B2: rat] :
      ( ( member_rat @ A @ ring_1_Ints_rat )
     => ( ( member_rat @ B2 @ ring_1_Ints_rat )
       => ( member_rat @ ( plus_plus_rat @ A @ B2 ) @ ring_1_Ints_rat ) ) ) ).

% Ints_add
thf(fact_7188_Ints__add,axiom,
    ! [A: int,B2: int] :
      ( ( member_int @ A @ ring_1_Ints_int )
     => ( ( member_int @ B2 @ ring_1_Ints_int )
       => ( member_int @ ( plus_plus_int @ A @ B2 ) @ ring_1_Ints_int ) ) ) ).

% Ints_add
thf(fact_7189_real__sqrt__inverse,axiom,
    ! [X4: real] :
      ( ( sqrt @ ( inverse_inverse_real @ X4 ) )
      = ( inverse_inverse_real @ ( sqrt @ X4 ) ) ) ).

% real_sqrt_inverse
thf(fact_7190_Ints__diff,axiom,
    ! [A: complex,B2: complex] :
      ( ( member_complex @ A @ ring_1_Ints_complex )
     => ( ( member_complex @ B2 @ ring_1_Ints_complex )
       => ( member_complex @ ( minus_minus_complex @ A @ B2 ) @ ring_1_Ints_complex ) ) ) ).

% Ints_diff
thf(fact_7191_Ints__diff,axiom,
    ! [A: real,B2: real] :
      ( ( member_real @ A @ ring_1_Ints_real )
     => ( ( member_real @ B2 @ ring_1_Ints_real )
       => ( member_real @ ( minus_minus_real @ A @ B2 ) @ ring_1_Ints_real ) ) ) ).

% Ints_diff
thf(fact_7192_Ints__diff,axiom,
    ! [A: rat,B2: rat] :
      ( ( member_rat @ A @ ring_1_Ints_rat )
     => ( ( member_rat @ B2 @ ring_1_Ints_rat )
       => ( member_rat @ ( minus_minus_rat @ A @ B2 ) @ ring_1_Ints_rat ) ) ) ).

% Ints_diff
thf(fact_7193_Ints__diff,axiom,
    ! [A: int,B2: int] :
      ( ( member_int @ A @ ring_1_Ints_int )
     => ( ( member_int @ B2 @ ring_1_Ints_int )
       => ( member_int @ ( minus_minus_int @ A @ B2 ) @ ring_1_Ints_int ) ) ) ).

% Ints_diff
thf(fact_7194_Ints__power,axiom,
    ! [A: real,N: nat] :
      ( ( member_real @ A @ ring_1_Ints_real )
     => ( member_real @ ( power_power_real @ A @ N ) @ ring_1_Ints_real ) ) ).

% Ints_power
thf(fact_7195_Ints__power,axiom,
    ! [A: int,N: nat] :
      ( ( member_int @ A @ ring_1_Ints_int )
     => ( member_int @ ( power_power_int @ A @ N ) @ ring_1_Ints_int ) ) ).

% Ints_power
thf(fact_7196_Ints__power,axiom,
    ! [A: complex,N: nat] :
      ( ( member_complex @ A @ ring_1_Ints_complex )
     => ( member_complex @ ( power_power_complex @ A @ N ) @ ring_1_Ints_complex ) ) ).

% Ints_power
thf(fact_7197_bit_Oconj__xor__distrib2,axiom,
    ! [Y3: int,Z: int,X4: int] :
      ( ( bit_se725231765392027082nd_int @ ( bit_se6526347334894502574or_int @ Y3 @ Z ) @ X4 )
      = ( bit_se6526347334894502574or_int @ ( bit_se725231765392027082nd_int @ Y3 @ X4 ) @ ( bit_se725231765392027082nd_int @ Z @ X4 ) ) ) ).

% bit.conj_xor_distrib2
thf(fact_7198_bit_Oconj__xor__distrib,axiom,
    ! [X4: int,Y3: int,Z: int] :
      ( ( bit_se725231765392027082nd_int @ X4 @ ( bit_se6526347334894502574or_int @ Y3 @ Z ) )
      = ( bit_se6526347334894502574or_int @ ( bit_se725231765392027082nd_int @ X4 @ Y3 ) @ ( bit_se725231765392027082nd_int @ X4 @ Z ) ) ) ).

% bit.conj_xor_distrib
thf(fact_7199_bit__xor__iff,axiom,
    ! [A: nat,B2: nat,N: nat] :
      ( ( bit_se1148574629649215175it_nat @ ( bit_se6528837805403552850or_nat @ A @ B2 ) @ N )
      = ( ( bit_se1148574629649215175it_nat @ A @ N )
       != ( bit_se1148574629649215175it_nat @ B2 @ N ) ) ) ).

% bit_xor_iff
thf(fact_7200_bit__xor__iff,axiom,
    ! [A: int,B2: int,N: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_se6526347334894502574or_int @ A @ B2 ) @ N )
      = ( ( bit_se1146084159140164899it_int @ A @ N )
       != ( bit_se1146084159140164899it_int @ B2 @ N ) ) ) ).

% bit_xor_iff
thf(fact_7201_norm__inverse__le__norm,axiom,
    ! [R2: real,X4: real] :
      ( ( ord_less_eq_real @ R2 @ ( real_V7735802525324610683m_real @ X4 ) )
     => ( ( ord_less_real @ zero_zero_real @ R2 )
       => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( inverse_inverse_real @ X4 ) ) @ ( inverse_inverse_real @ R2 ) ) ) ) ).

% norm_inverse_le_norm
thf(fact_7202_norm__inverse__le__norm,axiom,
    ! [R2: real,X4: complex] :
      ( ( ord_less_eq_real @ R2 @ ( real_V1022390504157884413omplex @ X4 ) )
     => ( ( ord_less_real @ zero_zero_real @ R2 )
       => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( invers8013647133539491842omplex @ X4 ) ) @ ( inverse_inverse_real @ R2 ) ) ) ) ).

% norm_inverse_le_norm
thf(fact_7203_positive__imp__inverse__positive,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ A ) ) ) ).

% positive_imp_inverse_positive
thf(fact_7204_positive__imp__inverse__positive,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ord_less_rat @ zero_zero_rat @ ( inverse_inverse_rat @ A ) ) ) ).

% positive_imp_inverse_positive
thf(fact_7205_negative__imp__inverse__negative,axiom,
    ! [A: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ord_less_real @ ( inverse_inverse_real @ A ) @ zero_zero_real ) ) ).

% negative_imp_inverse_negative
thf(fact_7206_negative__imp__inverse__negative,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ zero_zero_rat ) ) ).

% negative_imp_inverse_negative
thf(fact_7207_inverse__positive__imp__positive,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ A ) )
     => ( ( A != zero_zero_real )
       => ( ord_less_real @ zero_zero_real @ A ) ) ) ).

% inverse_positive_imp_positive
thf(fact_7208_inverse__positive__imp__positive,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( inverse_inverse_rat @ A ) )
     => ( ( A != zero_zero_rat )
       => ( ord_less_rat @ zero_zero_rat @ A ) ) ) ).

% inverse_positive_imp_positive
thf(fact_7209_inverse__negative__imp__negative,axiom,
    ! [A: real] :
      ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ zero_zero_real )
     => ( ( A != zero_zero_real )
       => ( ord_less_real @ A @ zero_zero_real ) ) ) ).

% inverse_negative_imp_negative
thf(fact_7210_inverse__negative__imp__negative,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ zero_zero_rat )
     => ( ( A != zero_zero_rat )
       => ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).

% inverse_negative_imp_negative
thf(fact_7211_less__imp__inverse__less__neg,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ( ord_less_real @ B2 @ zero_zero_real )
       => ( ord_less_real @ ( inverse_inverse_real @ B2 ) @ ( inverse_inverse_real @ A ) ) ) ) ).

% less_imp_inverse_less_neg
thf(fact_7212_less__imp__inverse__less__neg,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_rat @ A @ B2 )
     => ( ( ord_less_rat @ B2 @ zero_zero_rat )
       => ( ord_less_rat @ ( inverse_inverse_rat @ B2 ) @ ( inverse_inverse_rat @ A ) ) ) ) ).

% less_imp_inverse_less_neg
thf(fact_7213_inverse__less__imp__less__neg,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B2 ) )
     => ( ( ord_less_real @ B2 @ zero_zero_real )
       => ( ord_less_real @ B2 @ A ) ) ) ).

% inverse_less_imp_less_neg
thf(fact_7214_inverse__less__imp__less__neg,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B2 ) )
     => ( ( ord_less_rat @ B2 @ zero_zero_rat )
       => ( ord_less_rat @ B2 @ A ) ) ) ).

% inverse_less_imp_less_neg
thf(fact_7215_less__imp__inverse__less,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ( ord_less_real @ ( inverse_inverse_real @ B2 ) @ ( inverse_inverse_real @ A ) ) ) ) ).

% less_imp_inverse_less
thf(fact_7216_less__imp__inverse__less,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_rat @ A @ B2 )
     => ( ( ord_less_rat @ zero_zero_rat @ A )
       => ( ord_less_rat @ ( inverse_inverse_rat @ B2 ) @ ( inverse_inverse_rat @ A ) ) ) ) ).

% less_imp_inverse_less
thf(fact_7217_inverse__less__imp__less,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B2 ) )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ( ord_less_real @ B2 @ A ) ) ) ).

% inverse_less_imp_less
thf(fact_7218_inverse__less__imp__less,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B2 ) )
     => ( ( ord_less_rat @ zero_zero_rat @ A )
       => ( ord_less_rat @ B2 @ A ) ) ) ).

% inverse_less_imp_less
thf(fact_7219_nonzero__inverse__mult__distrib,axiom,
    ! [A: real,B2: real] :
      ( ( A != zero_zero_real )
     => ( ( B2 != zero_zero_real )
       => ( ( inverse_inverse_real @ ( times_times_real @ A @ B2 ) )
          = ( times_times_real @ ( inverse_inverse_real @ B2 ) @ ( inverse_inverse_real @ A ) ) ) ) ) ).

% nonzero_inverse_mult_distrib
thf(fact_7220_nonzero__inverse__mult__distrib,axiom,
    ! [A: complex,B2: complex] :
      ( ( A != zero_zero_complex )
     => ( ( B2 != zero_zero_complex )
       => ( ( invers8013647133539491842omplex @ ( times_times_complex @ A @ B2 ) )
          = ( times_times_complex @ ( invers8013647133539491842omplex @ B2 ) @ ( invers8013647133539491842omplex @ A ) ) ) ) ) ).

% nonzero_inverse_mult_distrib
thf(fact_7221_nonzero__inverse__mult__distrib,axiom,
    ! [A: rat,B2: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B2 != zero_zero_rat )
       => ( ( inverse_inverse_rat @ ( times_times_rat @ A @ B2 ) )
          = ( times_times_rat @ ( inverse_inverse_rat @ B2 ) @ ( inverse_inverse_rat @ A ) ) ) ) ) ).

% nonzero_inverse_mult_distrib
thf(fact_7222_nonzero__inverse__minus__eq,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( ( inverse_inverse_real @ ( uminus_uminus_real @ A ) )
        = ( uminus_uminus_real @ ( inverse_inverse_real @ A ) ) ) ) ).

% nonzero_inverse_minus_eq
thf(fact_7223_nonzero__inverse__minus__eq,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( ( invers8013647133539491842omplex @ ( uminus1482373934393186551omplex @ A ) )
        = ( uminus1482373934393186551omplex @ ( invers8013647133539491842omplex @ A ) ) ) ) ).

% nonzero_inverse_minus_eq
thf(fact_7224_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
thf(fact_7225_inverse__numeral__1,axiom,
    ( ( inverse_inverse_real @ ( numeral_numeral_real @ one ) )
    = ( numeral_numeral_real @ one ) ) ).

% inverse_numeral_1
thf(fact_7226_inverse__numeral__1,axiom,
    ( ( invers8013647133539491842omplex @ ( numera6690914467698888265omplex @ one ) )
    = ( numera6690914467698888265omplex @ one ) ) ).

% inverse_numeral_1
thf(fact_7227_inverse__numeral__1,axiom,
    ( ( inverse_inverse_rat @ ( numeral_numeral_rat @ one ) )
    = ( numeral_numeral_rat @ one ) ) ).

% inverse_numeral_1
thf(fact_7228_inverse__unique,axiom,
    ! [A: real,B2: real] :
      ( ( ( times_times_real @ A @ B2 )
        = one_one_real )
     => ( ( inverse_inverse_real @ A )
        = B2 ) ) ).

% inverse_unique
thf(fact_7229_inverse__unique,axiom,
    ! [A: complex,B2: complex] :
      ( ( ( times_times_complex @ A @ B2 )
        = one_one_complex )
     => ( ( invers8013647133539491842omplex @ A )
        = B2 ) ) ).

% inverse_unique
thf(fact_7230_inverse__unique,axiom,
    ! [A: rat,B2: rat] :
      ( ( ( times_times_rat @ A @ B2 )
        = one_one_rat )
     => ( ( inverse_inverse_rat @ A )
        = B2 ) ) ).

% inverse_unique
thf(fact_7231_divide__inverse__commute,axiom,
    ( divide_divide_real
    = ( ^ [A4: real,B3: real] : ( times_times_real @ ( inverse_inverse_real @ B3 ) @ A4 ) ) ) ).

% divide_inverse_commute
thf(fact_7232_divide__inverse__commute,axiom,
    ( divide1717551699836669952omplex
    = ( ^ [A4: complex,B3: complex] : ( times_times_complex @ ( invers8013647133539491842omplex @ B3 ) @ A4 ) ) ) ).

% divide_inverse_commute
thf(fact_7233_divide__inverse__commute,axiom,
    ( divide_divide_rat
    = ( ^ [A4: rat,B3: rat] : ( times_times_rat @ ( inverse_inverse_rat @ B3 ) @ A4 ) ) ) ).

% divide_inverse_commute
thf(fact_7234_divide__inverse,axiom,
    ( divide_divide_real
    = ( ^ [A4: real,B3: real] : ( times_times_real @ A4 @ ( inverse_inverse_real @ B3 ) ) ) ) ).

% divide_inverse
thf(fact_7235_divide__inverse,axiom,
    ( divide1717551699836669952omplex
    = ( ^ [A4: complex,B3: complex] : ( times_times_complex @ A4 @ ( invers8013647133539491842omplex @ B3 ) ) ) ) ).

% divide_inverse
thf(fact_7236_divide__inverse,axiom,
    ( divide_divide_rat
    = ( ^ [A4: rat,B3: rat] : ( times_times_rat @ A4 @ ( inverse_inverse_rat @ B3 ) ) ) ) ).

% divide_inverse
thf(fact_7237_field__class_Ofield__divide__inverse,axiom,
    ( divide_divide_real
    = ( ^ [A4: real,B3: real] : ( times_times_real @ A4 @ ( inverse_inverse_real @ B3 ) ) ) ) ).

% field_class.field_divide_inverse
thf(fact_7238_field__class_Ofield__divide__inverse,axiom,
    ( divide1717551699836669952omplex
    = ( ^ [A4: complex,B3: complex] : ( times_times_complex @ A4 @ ( invers8013647133539491842omplex @ B3 ) ) ) ) ).

% field_class.field_divide_inverse
thf(fact_7239_field__class_Ofield__divide__inverse,axiom,
    ( divide_divide_rat
    = ( ^ [A4: rat,B3: rat] : ( times_times_rat @ A4 @ ( inverse_inverse_rat @ B3 ) ) ) ) ).

% field_class.field_divide_inverse
thf(fact_7240_inverse__eq__divide,axiom,
    ( inverse_inverse_real
    = ( divide_divide_real @ one_one_real ) ) ).

% inverse_eq_divide
thf(fact_7241_inverse__eq__divide,axiom,
    ( invers8013647133539491842omplex
    = ( divide1717551699836669952omplex @ one_one_complex ) ) ).

% inverse_eq_divide
thf(fact_7242_inverse__eq__divide,axiom,
    ( inverse_inverse_rat
    = ( divide_divide_rat @ one_one_rat ) ) ).

% inverse_eq_divide
thf(fact_7243_power__mult__power__inverse__commute,axiom,
    ! [X4: real,M: nat,N: nat] :
      ( ( times_times_real @ ( power_power_real @ X4 @ M ) @ ( power_power_real @ ( inverse_inverse_real @ X4 ) @ N ) )
      = ( times_times_real @ ( power_power_real @ ( inverse_inverse_real @ X4 ) @ N ) @ ( power_power_real @ X4 @ M ) ) ) ).

% power_mult_power_inverse_commute
thf(fact_7244_power__mult__power__inverse__commute,axiom,
    ! [X4: complex,M: nat,N: nat] :
      ( ( times_times_complex @ ( power_power_complex @ X4 @ M ) @ ( power_power_complex @ ( invers8013647133539491842omplex @ X4 ) @ N ) )
      = ( times_times_complex @ ( power_power_complex @ ( invers8013647133539491842omplex @ X4 ) @ N ) @ ( power_power_complex @ X4 @ M ) ) ) ).

% power_mult_power_inverse_commute
thf(fact_7245_power__mult__power__inverse__commute,axiom,
    ! [X4: rat,M: nat,N: nat] :
      ( ( times_times_rat @ ( power_power_rat @ X4 @ M ) @ ( power_power_rat @ ( inverse_inverse_rat @ X4 ) @ N ) )
      = ( times_times_rat @ ( power_power_rat @ ( inverse_inverse_rat @ X4 ) @ N ) @ ( power_power_rat @ X4 @ M ) ) ) ).

% power_mult_power_inverse_commute
thf(fact_7246_power__mult__inverse__distrib,axiom,
    ! [X4: real,M: nat] :
      ( ( times_times_real @ ( power_power_real @ X4 @ M ) @ ( inverse_inverse_real @ X4 ) )
      = ( times_times_real @ ( inverse_inverse_real @ X4 ) @ ( power_power_real @ X4 @ M ) ) ) ).

% power_mult_inverse_distrib
thf(fact_7247_power__mult__inverse__distrib,axiom,
    ! [X4: complex,M: nat] :
      ( ( times_times_complex @ ( power_power_complex @ X4 @ M ) @ ( invers8013647133539491842omplex @ X4 ) )
      = ( times_times_complex @ ( invers8013647133539491842omplex @ X4 ) @ ( power_power_complex @ X4 @ M ) ) ) ).

% power_mult_inverse_distrib
thf(fact_7248_power__mult__inverse__distrib,axiom,
    ! [X4: rat,M: nat] :
      ( ( times_times_rat @ ( power_power_rat @ X4 @ M ) @ ( inverse_inverse_rat @ X4 ) )
      = ( times_times_rat @ ( inverse_inverse_rat @ X4 ) @ ( power_power_rat @ X4 @ M ) ) ) ).

% power_mult_inverse_distrib
thf(fact_7249_mult__inverse__of__nat__commute,axiom,
    ! [Xa: nat,X4: real] :
      ( ( times_times_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ Xa ) ) @ X4 )
      = ( times_times_real @ X4 @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ Xa ) ) ) ) ).

% mult_inverse_of_nat_commute
thf(fact_7250_mult__inverse__of__nat__commute,axiom,
    ! [Xa: nat,X4: complex] :
      ( ( times_times_complex @ ( invers8013647133539491842omplex @ ( semiri8010041392384452111omplex @ Xa ) ) @ X4 )
      = ( times_times_complex @ X4 @ ( invers8013647133539491842omplex @ ( semiri8010041392384452111omplex @ Xa ) ) ) ) ).

% mult_inverse_of_nat_commute
thf(fact_7251_mult__inverse__of__nat__commute,axiom,
    ! [Xa: nat,X4: rat] :
      ( ( times_times_rat @ ( inverse_inverse_rat @ ( semiri681578069525770553at_rat @ Xa ) ) @ X4 )
      = ( times_times_rat @ X4 @ ( inverse_inverse_rat @ ( semiri681578069525770553at_rat @ Xa ) ) ) ) ).

% mult_inverse_of_nat_commute
thf(fact_7252_nonzero__abs__inverse,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( ( abs_abs_real @ ( inverse_inverse_real @ A ) )
        = ( inverse_inverse_real @ ( abs_abs_real @ A ) ) ) ) ).

% nonzero_abs_inverse
thf(fact_7253_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
thf(fact_7254_mult__inverse__of__int__commute,axiom,
    ! [Xa: int,X4: real] :
      ( ( times_times_real @ ( inverse_inverse_real @ ( ring_1_of_int_real @ Xa ) ) @ X4 )
      = ( times_times_real @ X4 @ ( inverse_inverse_real @ ( ring_1_of_int_real @ Xa ) ) ) ) ).

% mult_inverse_of_int_commute
thf(fact_7255_mult__inverse__of__int__commute,axiom,
    ! [Xa: int,X4: complex] :
      ( ( times_times_complex @ ( invers8013647133539491842omplex @ ( ring_17405671764205052669omplex @ Xa ) ) @ X4 )
      = ( times_times_complex @ X4 @ ( invers8013647133539491842omplex @ ( ring_17405671764205052669omplex @ Xa ) ) ) ) ).

% mult_inverse_of_int_commute
thf(fact_7256_mult__inverse__of__int__commute,axiom,
    ! [Xa: int,X4: rat] :
      ( ( times_times_rat @ ( inverse_inverse_rat @ ( ring_1_of_int_rat @ Xa ) ) @ X4 )
      = ( times_times_rat @ X4 @ ( inverse_inverse_rat @ ( ring_1_of_int_rat @ Xa ) ) ) ) ).

% mult_inverse_of_int_commute
thf(fact_7257_divide__real__def,axiom,
    ( divide_divide_real
    = ( ^ [X: real,Y: real] : ( times_times_real @ X @ ( inverse_inverse_real @ Y ) ) ) ) ).

% divide_real_def
thf(fact_7258_Ints__double__eq__0__iff,axiom,
    ! [A: complex] :
      ( ( member_complex @ A @ ring_1_Ints_complex )
     => ( ( ( plus_plus_complex @ A @ A )
          = zero_zero_complex )
        = ( A = zero_zero_complex ) ) ) ).

% Ints_double_eq_0_iff
thf(fact_7259_Ints__double__eq__0__iff,axiom,
    ! [A: real] :
      ( ( member_real @ A @ ring_1_Ints_real )
     => ( ( ( plus_plus_real @ A @ A )
          = zero_zero_real )
        = ( A = zero_zero_real ) ) ) ).

% Ints_double_eq_0_iff
thf(fact_7260_Ints__double__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( member_rat @ A @ ring_1_Ints_rat )
     => ( ( ( plus_plus_rat @ A @ A )
          = zero_zero_rat )
        = ( A = zero_zero_rat ) ) ) ).

% Ints_double_eq_0_iff
thf(fact_7261_Ints__double__eq__0__iff,axiom,
    ! [A: int] :
      ( ( member_int @ A @ ring_1_Ints_int )
     => ( ( ( plus_plus_int @ A @ A )
          = zero_zero_int )
        = ( A = zero_zero_int ) ) ) ).

% Ints_double_eq_0_iff
thf(fact_7262_inverse__le__imp__le,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B2 ) )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ( ord_less_eq_real @ B2 @ A ) ) ) ).

% inverse_le_imp_le
thf(fact_7263_inverse__le__imp__le,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B2 ) )
     => ( ( ord_less_rat @ zero_zero_rat @ A )
       => ( ord_less_eq_rat @ B2 @ A ) ) ) ).

% inverse_le_imp_le
thf(fact_7264_le__imp__inverse__le,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ A @ B2 )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ( ord_less_eq_real @ ( inverse_inverse_real @ B2 ) @ ( inverse_inverse_real @ A ) ) ) ) ).

% le_imp_inverse_le
thf(fact_7265_le__imp__inverse__le,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_eq_rat @ A @ B2 )
     => ( ( ord_less_rat @ zero_zero_rat @ A )
       => ( ord_less_eq_rat @ ( inverse_inverse_rat @ B2 ) @ ( inverse_inverse_rat @ A ) ) ) ) ).

% le_imp_inverse_le
thf(fact_7266_inverse__le__imp__le__neg,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B2 ) )
     => ( ( ord_less_real @ B2 @ zero_zero_real )
       => ( ord_less_eq_real @ B2 @ A ) ) ) ).

% inverse_le_imp_le_neg
thf(fact_7267_inverse__le__imp__le__neg,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B2 ) )
     => ( ( ord_less_rat @ B2 @ zero_zero_rat )
       => ( ord_less_eq_rat @ B2 @ A ) ) ) ).

% inverse_le_imp_le_neg
thf(fact_7268_le__imp__inverse__le__neg,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ A @ B2 )
     => ( ( ord_less_real @ B2 @ zero_zero_real )
       => ( ord_less_eq_real @ ( inverse_inverse_real @ B2 ) @ ( inverse_inverse_real @ A ) ) ) ) ).

% le_imp_inverse_le_neg
thf(fact_7269_le__imp__inverse__le__neg,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_eq_rat @ A @ B2 )
     => ( ( ord_less_rat @ B2 @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( inverse_inverse_rat @ B2 ) @ ( inverse_inverse_rat @ A ) ) ) ) ).

% le_imp_inverse_le_neg
thf(fact_7270_inverse__le__1__iff,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ ( inverse_inverse_real @ X4 ) @ one_one_real )
      = ( ( ord_less_eq_real @ X4 @ zero_zero_real )
        | ( ord_less_eq_real @ one_one_real @ X4 ) ) ) ).

% inverse_le_1_iff
thf(fact_7271_inverse__le__1__iff,axiom,
    ! [X4: rat] :
      ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ X4 ) @ one_one_rat )
      = ( ( ord_less_eq_rat @ X4 @ zero_zero_rat )
        | ( ord_less_eq_rat @ one_one_rat @ X4 ) ) ) ).

% inverse_le_1_iff
thf(fact_7272_one__less__inverse__iff,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ one_one_real @ ( inverse_inverse_real @ X4 ) )
      = ( ( ord_less_real @ zero_zero_real @ X4 )
        & ( ord_less_real @ X4 @ one_one_real ) ) ) ).

% one_less_inverse_iff
thf(fact_7273_one__less__inverse__iff,axiom,
    ! [X4: rat] :
      ( ( ord_less_rat @ one_one_rat @ ( inverse_inverse_rat @ X4 ) )
      = ( ( ord_less_rat @ zero_zero_rat @ X4 )
        & ( ord_less_rat @ X4 @ one_one_rat ) ) ) ).

% one_less_inverse_iff
thf(fact_7274_one__less__inverse,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ A @ one_one_real )
       => ( ord_less_real @ one_one_real @ ( inverse_inverse_real @ A ) ) ) ) ).

% one_less_inverse
thf(fact_7275_one__less__inverse,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ A @ one_one_rat )
       => ( ord_less_rat @ one_one_rat @ ( inverse_inverse_rat @ A ) ) ) ) ).

% one_less_inverse
thf(fact_7276_field__class_Ofield__inverse,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( ( times_times_real @ ( inverse_inverse_real @ A ) @ A )
        = one_one_real ) ) ).

% field_class.field_inverse
thf(fact_7277_field__class_Ofield__inverse,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( ( times_times_complex @ ( invers8013647133539491842omplex @ A ) @ A )
        = one_one_complex ) ) ).

% field_class.field_inverse
thf(fact_7278_field__class_Ofield__inverse,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( ( times_times_rat @ ( inverse_inverse_rat @ A ) @ A )
        = one_one_rat ) ) ).

% field_class.field_inverse
thf(fact_7279_division__ring__inverse__add,axiom,
    ! [A: real,B2: real] :
      ( ( A != zero_zero_real )
     => ( ( B2 != zero_zero_real )
       => ( ( plus_plus_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B2 ) )
          = ( times_times_real @ ( times_times_real @ ( inverse_inverse_real @ A ) @ ( plus_plus_real @ A @ B2 ) ) @ ( inverse_inverse_real @ B2 ) ) ) ) ) ).

% division_ring_inverse_add
thf(fact_7280_division__ring__inverse__add,axiom,
    ! [A: complex,B2: complex] :
      ( ( A != zero_zero_complex )
     => ( ( B2 != zero_zero_complex )
       => ( ( plus_plus_complex @ ( invers8013647133539491842omplex @ A ) @ ( invers8013647133539491842omplex @ B2 ) )
          = ( times_times_complex @ ( times_times_complex @ ( invers8013647133539491842omplex @ A ) @ ( plus_plus_complex @ A @ B2 ) ) @ ( invers8013647133539491842omplex @ B2 ) ) ) ) ) ).

% division_ring_inverse_add
thf(fact_7281_division__ring__inverse__add,axiom,
    ! [A: rat,B2: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B2 != zero_zero_rat )
       => ( ( plus_plus_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B2 ) )
          = ( times_times_rat @ ( times_times_rat @ ( inverse_inverse_rat @ A ) @ ( plus_plus_rat @ A @ B2 ) ) @ ( inverse_inverse_rat @ B2 ) ) ) ) ) ).

% division_ring_inverse_add
thf(fact_7282_inverse__add,axiom,
    ! [A: real,B2: real] :
      ( ( A != zero_zero_real )
     => ( ( B2 != zero_zero_real )
       => ( ( plus_plus_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B2 ) )
          = ( times_times_real @ ( times_times_real @ ( plus_plus_real @ A @ B2 ) @ ( inverse_inverse_real @ A ) ) @ ( inverse_inverse_real @ B2 ) ) ) ) ) ).

% inverse_add
thf(fact_7283_inverse__add,axiom,
    ! [A: complex,B2: complex] :
      ( ( A != zero_zero_complex )
     => ( ( B2 != zero_zero_complex )
       => ( ( plus_plus_complex @ ( invers8013647133539491842omplex @ A ) @ ( invers8013647133539491842omplex @ B2 ) )
          = ( times_times_complex @ ( times_times_complex @ ( plus_plus_complex @ A @ B2 ) @ ( invers8013647133539491842omplex @ A ) ) @ ( invers8013647133539491842omplex @ B2 ) ) ) ) ) ).

% inverse_add
thf(fact_7284_inverse__add,axiom,
    ! [A: rat,B2: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B2 != zero_zero_rat )
       => ( ( plus_plus_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B2 ) )
          = ( times_times_rat @ ( times_times_rat @ ( plus_plus_rat @ A @ B2 ) @ ( inverse_inverse_rat @ A ) ) @ ( inverse_inverse_rat @ B2 ) ) ) ) ) ).

% inverse_add
thf(fact_7285_division__ring__inverse__diff,axiom,
    ! [A: real,B2: real] :
      ( ( A != zero_zero_real )
     => ( ( B2 != zero_zero_real )
       => ( ( minus_minus_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B2 ) )
          = ( times_times_real @ ( times_times_real @ ( inverse_inverse_real @ A ) @ ( minus_minus_real @ B2 @ A ) ) @ ( inverse_inverse_real @ B2 ) ) ) ) ) ).

% division_ring_inverse_diff
thf(fact_7286_division__ring__inverse__diff,axiom,
    ! [A: complex,B2: complex] :
      ( ( A != zero_zero_complex )
     => ( ( B2 != zero_zero_complex )
       => ( ( minus_minus_complex @ ( invers8013647133539491842omplex @ A ) @ ( invers8013647133539491842omplex @ B2 ) )
          = ( times_times_complex @ ( times_times_complex @ ( invers8013647133539491842omplex @ A ) @ ( minus_minus_complex @ B2 @ A ) ) @ ( invers8013647133539491842omplex @ B2 ) ) ) ) ) ).

% division_ring_inverse_diff
thf(fact_7287_division__ring__inverse__diff,axiom,
    ! [A: rat,B2: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B2 != zero_zero_rat )
       => ( ( minus_minus_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B2 ) )
          = ( times_times_rat @ ( times_times_rat @ ( inverse_inverse_rat @ A ) @ ( minus_minus_rat @ B2 @ A ) ) @ ( inverse_inverse_rat @ B2 ) ) ) ) ) ).

% division_ring_inverse_diff
thf(fact_7288_nonzero__inverse__eq__divide,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( ( inverse_inverse_real @ A )
        = ( divide_divide_real @ one_one_real @ A ) ) ) ).

% nonzero_inverse_eq_divide
thf(fact_7289_nonzero__inverse__eq__divide,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( ( invers8013647133539491842omplex @ A )
        = ( divide1717551699836669952omplex @ one_one_complex @ A ) ) ) ).

% nonzero_inverse_eq_divide
thf(fact_7290_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
thf(fact_7291_gbinomial__Suc__Suc,axiom,
    ! [A: complex,K: nat] :
      ( ( gbinomial_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( suc @ K ) )
      = ( plus_plus_complex @ ( gbinomial_complex @ A @ K ) @ ( gbinomial_complex @ A @ ( suc @ K ) ) ) ) ).

% gbinomial_Suc_Suc
thf(fact_7292_gbinomial__Suc__Suc,axiom,
    ! [A: real,K: nat] :
      ( ( gbinomial_real @ ( plus_plus_real @ A @ one_one_real ) @ ( suc @ K ) )
      = ( plus_plus_real @ ( gbinomial_real @ A @ K ) @ ( gbinomial_real @ A @ ( suc @ K ) ) ) ) ).

% gbinomial_Suc_Suc
thf(fact_7293_gbinomial__Suc__Suc,axiom,
    ! [A: rat,K: nat] :
      ( ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K ) )
      = ( plus_plus_rat @ ( gbinomial_rat @ A @ K ) @ ( gbinomial_rat @ A @ ( suc @ K ) ) ) ) ).

% gbinomial_Suc_Suc
thf(fact_7294_gbinomial__of__nat__symmetric,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( gbinomial_complex @ ( semiri8010041392384452111omplex @ N ) @ K )
        = ( gbinomial_complex @ ( semiri8010041392384452111omplex @ N ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).

% gbinomial_of_nat_symmetric
thf(fact_7295_gbinomial__of__nat__symmetric,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( gbinomial_real @ ( semiri5074537144036343181t_real @ N ) @ K )
        = ( gbinomial_real @ ( semiri5074537144036343181t_real @ N ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).

% gbinomial_of_nat_symmetric
thf(fact_7296_gbinomial__of__nat__symmetric,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( gbinomial_rat @ ( semiri681578069525770553at_rat @ N ) @ K )
        = ( gbinomial_rat @ ( semiri681578069525770553at_rat @ N ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).

% gbinomial_of_nat_symmetric
thf(fact_7297_inverse__powr,axiom,
    ! [Y3: real,A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
     => ( ( powr_real @ ( inverse_inverse_real @ Y3 ) @ A )
        = ( inverse_inverse_real @ ( powr_real @ Y3 @ A ) ) ) ) ).

% inverse_powr
thf(fact_7298_Ints__odd__nonzero,axiom,
    ! [A: complex] :
      ( ( member_complex @ A @ ring_1_Ints_complex )
     => ( ( plus_plus_complex @ ( plus_plus_complex @ one_one_complex @ A ) @ A )
       != zero_zero_complex ) ) ).

% Ints_odd_nonzero
thf(fact_7299_Ints__odd__nonzero,axiom,
    ! [A: real] :
      ( ( member_real @ A @ ring_1_Ints_real )
     => ( ( plus_plus_real @ ( plus_plus_real @ one_one_real @ A ) @ A )
       != zero_zero_real ) ) ).

% Ints_odd_nonzero
thf(fact_7300_Ints__odd__nonzero,axiom,
    ! [A: rat] :
      ( ( member_rat @ A @ ring_1_Ints_rat )
     => ( ( plus_plus_rat @ ( plus_plus_rat @ one_one_rat @ A ) @ A )
       != zero_zero_rat ) ) ).

% Ints_odd_nonzero
thf(fact_7301_Ints__odd__nonzero,axiom,
    ! [A: int] :
      ( ( member_int @ A @ ring_1_Ints_int )
     => ( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ A ) @ A )
       != zero_zero_int ) ) ).

% Ints_odd_nonzero
thf(fact_7302_of__int__divide__in__Ints,axiom,
    ! [B2: int,A: int] :
      ( ( dvd_dvd_int @ B2 @ A )
     => ( member_complex @ ( divide1717551699836669952omplex @ ( ring_17405671764205052669omplex @ A ) @ ( ring_17405671764205052669omplex @ B2 ) ) @ ring_1_Ints_complex ) ) ).

% of_int_divide_in_Ints
thf(fact_7303_of__int__divide__in__Ints,axiom,
    ! [B2: int,A: int] :
      ( ( dvd_dvd_int @ B2 @ A )
     => ( member_real @ ( divide_divide_real @ ( ring_1_of_int_real @ A ) @ ( ring_1_of_int_real @ B2 ) ) @ ring_1_Ints_real ) ) ).

% of_int_divide_in_Ints
thf(fact_7304_of__int__divide__in__Ints,axiom,
    ! [B2: int,A: int] :
      ( ( dvd_dvd_int @ B2 @ A )
     => ( member_rat @ ( divide_divide_rat @ ( ring_1_of_int_rat @ A ) @ ( ring_1_of_int_rat @ B2 ) ) @ ring_1_Ints_rat ) ) ).

% of_int_divide_in_Ints
thf(fact_7305_of__int__divide__in__Ints,axiom,
    ! [B2: int,A: int] :
      ( ( dvd_dvd_int @ B2 @ A )
     => ( member_int @ ( divide_divide_int @ ( ring_1_of_int_int @ A ) @ ( ring_1_of_int_int @ B2 ) ) @ ring_1_Ints_int ) ) ).

% of_int_divide_in_Ints
thf(fact_7306_inverse__le__iff,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B2 ) )
      = ( ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) )
         => ( ord_less_eq_real @ B2 @ A ) )
        & ( ( ord_less_eq_real @ ( times_times_real @ A @ B2 ) @ zero_zero_real )
         => ( ord_less_eq_real @ A @ B2 ) ) ) ) ).

% inverse_le_iff
thf(fact_7307_inverse__le__iff,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B2 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B2 ) )
         => ( ord_less_eq_rat @ B2 @ A ) )
        & ( ( ord_less_eq_rat @ ( times_times_rat @ A @ B2 ) @ zero_zero_rat )
         => ( ord_less_eq_rat @ A @ B2 ) ) ) ) ).

% inverse_le_iff
thf(fact_7308_inverse__less__iff,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B2 ) )
      = ( ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) )
         => ( ord_less_real @ B2 @ A ) )
        & ( ( ord_less_eq_real @ ( times_times_real @ A @ B2 ) @ zero_zero_real )
         => ( ord_less_real @ A @ B2 ) ) ) ) ).

% inverse_less_iff
thf(fact_7309_inverse__less__iff,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B2 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B2 ) )
         => ( ord_less_rat @ B2 @ A ) )
        & ( ( ord_less_eq_rat @ ( times_times_rat @ A @ B2 ) @ zero_zero_rat )
         => ( ord_less_rat @ A @ B2 ) ) ) ) ).

% inverse_less_iff
thf(fact_7310_one__le__inverse,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ A @ one_one_real )
       => ( ord_less_eq_real @ one_one_real @ ( inverse_inverse_real @ A ) ) ) ) ).

% one_le_inverse
thf(fact_7311_one__le__inverse,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ A @ one_one_rat )
       => ( ord_less_eq_rat @ one_one_rat @ ( inverse_inverse_rat @ A ) ) ) ) ).

% one_le_inverse
thf(fact_7312_inverse__less__1__iff,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ ( inverse_inverse_real @ X4 ) @ one_one_real )
      = ( ( ord_less_eq_real @ X4 @ zero_zero_real )
        | ( ord_less_real @ one_one_real @ X4 ) ) ) ).

% inverse_less_1_iff
thf(fact_7313_inverse__less__1__iff,axiom,
    ! [X4: rat] :
      ( ( ord_less_rat @ ( inverse_inverse_rat @ X4 ) @ one_one_rat )
      = ( ( ord_less_eq_rat @ X4 @ zero_zero_rat )
        | ( ord_less_rat @ one_one_rat @ X4 ) ) ) ).

% inverse_less_1_iff
thf(fact_7314_one__le__inverse__iff,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ one_one_real @ ( inverse_inverse_real @ X4 ) )
      = ( ( ord_less_real @ zero_zero_real @ X4 )
        & ( ord_less_eq_real @ X4 @ one_one_real ) ) ) ).

% one_le_inverse_iff
thf(fact_7315_one__le__inverse__iff,axiom,
    ! [X4: rat] :
      ( ( ord_less_eq_rat @ one_one_rat @ ( inverse_inverse_rat @ X4 ) )
      = ( ( ord_less_rat @ zero_zero_rat @ X4 )
        & ( ord_less_eq_rat @ X4 @ one_one_rat ) ) ) ).

% one_le_inverse_iff
thf(fact_7316_inverse__diff__inverse,axiom,
    ! [A: real,B2: real] :
      ( ( A != zero_zero_real )
     => ( ( B2 != zero_zero_real )
       => ( ( minus_minus_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B2 ) )
          = ( uminus_uminus_real @ ( times_times_real @ ( times_times_real @ ( inverse_inverse_real @ A ) @ ( minus_minus_real @ A @ B2 ) ) @ ( inverse_inverse_real @ B2 ) ) ) ) ) ) ).

% inverse_diff_inverse
thf(fact_7317_inverse__diff__inverse,axiom,
    ! [A: complex,B2: complex] :
      ( ( A != zero_zero_complex )
     => ( ( B2 != zero_zero_complex )
       => ( ( minus_minus_complex @ ( invers8013647133539491842omplex @ A ) @ ( invers8013647133539491842omplex @ B2 ) )
          = ( uminus1482373934393186551omplex @ ( times_times_complex @ ( times_times_complex @ ( invers8013647133539491842omplex @ A ) @ ( minus_minus_complex @ A @ B2 ) ) @ ( invers8013647133539491842omplex @ B2 ) ) ) ) ) ) ).

% inverse_diff_inverse
thf(fact_7318_inverse__diff__inverse,axiom,
    ! [A: rat,B2: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B2 != zero_zero_rat )
       => ( ( minus_minus_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B2 ) )
          = ( uminus_uminus_rat @ ( times_times_rat @ ( times_times_rat @ ( inverse_inverse_rat @ A ) @ ( minus_minus_rat @ A @ B2 ) ) @ ( inverse_inverse_rat @ B2 ) ) ) ) ) ) ).

% inverse_diff_inverse
thf(fact_7319_reals__Archimedean,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ? [N4: nat] : ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N4 ) ) ) @ X4 ) ) ).

% reals_Archimedean
thf(fact_7320_reals__Archimedean,axiom,
    ! [X4: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ X4 )
     => ? [N4: nat] : ( ord_less_rat @ ( inverse_inverse_rat @ ( semiri681578069525770553at_rat @ ( suc @ N4 ) ) ) @ X4 ) ) ).

% reals_Archimedean
thf(fact_7321_gbinomial__addition__formula,axiom,
    ! [A: complex,K: nat] :
      ( ( gbinomial_complex @ A @ ( suc @ K ) )
      = ( plus_plus_complex @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ ( suc @ K ) ) @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ K ) ) ) ).

% gbinomial_addition_formula
thf(fact_7322_gbinomial__addition__formula,axiom,
    ! [A: real,K: nat] :
      ( ( gbinomial_real @ A @ ( suc @ K ) )
      = ( plus_plus_real @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ ( suc @ K ) ) @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ K ) ) ) ).

% gbinomial_addition_formula
thf(fact_7323_gbinomial__addition__formula,axiom,
    ! [A: rat,K: nat] :
      ( ( gbinomial_rat @ A @ ( suc @ K ) )
      = ( plus_plus_rat @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ ( suc @ K ) ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ K ) ) ) ).

% gbinomial_addition_formula
thf(fact_7324_gbinomial__absorb__comp,axiom,
    ! [A: complex,K: nat] :
      ( ( times_times_complex @ ( minus_minus_complex @ A @ ( semiri8010041392384452111omplex @ K ) ) @ ( gbinomial_complex @ A @ K ) )
      = ( times_times_complex @ A @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ K ) ) ) ).

% gbinomial_absorb_comp
thf(fact_7325_gbinomial__absorb__comp,axiom,
    ! [A: real,K: nat] :
      ( ( times_times_real @ ( minus_minus_real @ A @ ( semiri5074537144036343181t_real @ K ) ) @ ( gbinomial_real @ A @ K ) )
      = ( times_times_real @ A @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ K ) ) ) ).

% gbinomial_absorb_comp
thf(fact_7326_gbinomial__absorb__comp,axiom,
    ! [A: rat,K: nat] :
      ( ( times_times_rat @ ( minus_minus_rat @ A @ ( semiri681578069525770553at_rat @ K ) ) @ ( gbinomial_rat @ A @ K ) )
      = ( times_times_rat @ A @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ K ) ) ) ).

% gbinomial_absorb_comp
thf(fact_7327_gbinomial__ge__n__over__k__pow__k,axiom,
    ! [K: nat,A: real] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ K ) @ A )
     => ( ord_less_eq_real @ ( power_power_real @ ( divide_divide_real @ A @ ( semiri5074537144036343181t_real @ K ) ) @ K ) @ ( gbinomial_real @ A @ K ) ) ) ).

% gbinomial_ge_n_over_k_pow_k
thf(fact_7328_gbinomial__ge__n__over__k__pow__k,axiom,
    ! [K: nat,A: rat] :
      ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ K ) @ A )
     => ( ord_less_eq_rat @ ( power_power_rat @ ( divide_divide_rat @ A @ ( semiri681578069525770553at_rat @ K ) ) @ K ) @ ( gbinomial_rat @ A @ K ) ) ) ).

% gbinomial_ge_n_over_k_pow_k
thf(fact_7329_gbinomial__mult__1_H,axiom,
    ! [A: complex,K: nat] :
      ( ( times_times_complex @ ( gbinomial_complex @ A @ K ) @ A )
      = ( plus_plus_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ K ) @ ( gbinomial_complex @ A @ K ) ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) @ ( gbinomial_complex @ A @ ( suc @ K ) ) ) ) ) ).

% gbinomial_mult_1'
thf(fact_7330_gbinomial__mult__1_H,axiom,
    ! [A: real,K: nat] :
      ( ( times_times_real @ ( gbinomial_real @ A @ K ) @ A )
      = ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ K ) @ ( gbinomial_real @ A @ K ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) @ ( gbinomial_real @ A @ ( suc @ K ) ) ) ) ) ).

% gbinomial_mult_1'
thf(fact_7331_gbinomial__mult__1_H,axiom,
    ! [A: rat,K: nat] :
      ( ( times_times_rat @ ( gbinomial_rat @ A @ K ) @ A )
      = ( plus_plus_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ K ) @ ( gbinomial_rat @ A @ K ) ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) @ ( gbinomial_rat @ A @ ( suc @ K ) ) ) ) ) ).

% gbinomial_mult_1'
thf(fact_7332_gbinomial__mult__1,axiom,
    ! [A: complex,K: nat] :
      ( ( times_times_complex @ A @ ( gbinomial_complex @ A @ K ) )
      = ( plus_plus_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ K ) @ ( gbinomial_complex @ A @ K ) ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) @ ( gbinomial_complex @ A @ ( suc @ K ) ) ) ) ) ).

% gbinomial_mult_1
thf(fact_7333_gbinomial__mult__1,axiom,
    ! [A: real,K: nat] :
      ( ( times_times_real @ A @ ( gbinomial_real @ A @ K ) )
      = ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ K ) @ ( gbinomial_real @ A @ K ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) @ ( gbinomial_real @ A @ ( suc @ K ) ) ) ) ) ).

% gbinomial_mult_1
thf(fact_7334_gbinomial__mult__1,axiom,
    ! [A: rat,K: nat] :
      ( ( times_times_rat @ A @ ( gbinomial_rat @ A @ K ) )
      = ( plus_plus_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ K ) @ ( gbinomial_rat @ A @ K ) ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) @ ( gbinomial_rat @ A @ ( suc @ K ) ) ) ) ) ).

% gbinomial_mult_1
thf(fact_7335_forall__pos__mono__1,axiom,
    ! [P: real > $o,E: real] :
      ( ! [D3: real,E2: real] :
          ( ( ord_less_real @ D3 @ E2 )
         => ( ( P @ D3 )
           => ( P @ E2 ) ) )
     => ( ! [N4: nat] : ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N4 ) ) ) )
       => ( ( ord_less_real @ zero_zero_real @ E )
         => ( P @ E ) ) ) ) ).

% forall_pos_mono_1
thf(fact_7336_forall__pos__mono,axiom,
    ! [P: real > $o,E: real] :
      ( ! [D3: real,E2: real] :
          ( ( ord_less_real @ D3 @ E2 )
         => ( ( P @ D3 )
           => ( P @ E2 ) ) )
     => ( ! [N4: nat] :
            ( ( N4 != zero_zero_nat )
           => ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N4 ) ) ) )
       => ( ( ord_less_real @ zero_zero_real @ E )
         => ( P @ E ) ) ) ) ).

% forall_pos_mono
thf(fact_7337_real__arch__inverse,axiom,
    ! [E: real] :
      ( ( ord_less_real @ zero_zero_real @ E )
      = ( ? [N2: nat] :
            ( ( N2 != zero_zero_nat )
            & ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N2 ) ) )
            & ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N2 ) ) @ E ) ) ) ) ).

% real_arch_inverse
thf(fact_7338_even__xor__iff,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se3222712562003087583nteger @ A @ B2 ) )
      = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
        = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) ) ) ).

% even_xor_iff
thf(fact_7339_even__xor__iff,axiom,
    ! [A: nat,B2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ A @ B2 ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
        = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) ) ).

% even_xor_iff
thf(fact_7340_even__xor__iff,axiom,
    ! [A: int,B2: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ A @ B2 ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
        = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) ) ).

% even_xor_iff
thf(fact_7341_sqrt__divide__self__eq,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( divide_divide_real @ ( sqrt @ X4 ) @ X4 )
        = ( inverse_inverse_real @ ( sqrt @ X4 ) ) ) ) ).

% sqrt_divide_self_eq
thf(fact_7342_ln__inverse,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( ln_ln_real @ ( inverse_inverse_real @ X4 ) )
        = ( uminus_uminus_real @ ( ln_ln_real @ X4 ) ) ) ) ).

% ln_inverse
thf(fact_7343_Ints__odd__less__0,axiom,
    ! [A: real] :
      ( ( member_real @ A @ ring_1_Ints_real )
     => ( ( ord_less_real @ ( plus_plus_real @ ( plus_plus_real @ one_one_real @ A ) @ A ) @ zero_zero_real )
        = ( ord_less_real @ A @ zero_zero_real ) ) ) ).

% Ints_odd_less_0
thf(fact_7344_Ints__odd__less__0,axiom,
    ! [A: rat] :
      ( ( member_rat @ A @ ring_1_Ints_rat )
     => ( ( ord_less_rat @ ( plus_plus_rat @ ( plus_plus_rat @ one_one_rat @ A ) @ A ) @ zero_zero_rat )
        = ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).

% Ints_odd_less_0
thf(fact_7345_Ints__odd__less__0,axiom,
    ! [A: int] :
      ( ( member_int @ A @ ring_1_Ints_int )
     => ( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ A ) @ A ) @ zero_zero_int )
        = ( ord_less_int @ A @ zero_zero_int ) ) ) ).

% Ints_odd_less_0
thf(fact_7346_Ints__nonzero__abs__ge1,axiom,
    ! [X4: code_integer] :
      ( ( member_Code_integer @ X4 @ ring_11222124179247155820nteger )
     => ( ( X4 != zero_z3403309356797280102nteger )
       => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( abs_abs_Code_integer @ X4 ) ) ) ) ).

% Ints_nonzero_abs_ge1
thf(fact_7347_Ints__nonzero__abs__ge1,axiom,
    ! [X4: real] :
      ( ( member_real @ X4 @ ring_1_Ints_real )
     => ( ( X4 != zero_zero_real )
       => ( ord_less_eq_real @ one_one_real @ ( abs_abs_real @ X4 ) ) ) ) ).

% Ints_nonzero_abs_ge1
thf(fact_7348_Ints__nonzero__abs__ge1,axiom,
    ! [X4: rat] :
      ( ( member_rat @ X4 @ ring_1_Ints_rat )
     => ( ( X4 != zero_zero_rat )
       => ( ord_less_eq_rat @ one_one_rat @ ( abs_abs_rat @ X4 ) ) ) ) ).

% Ints_nonzero_abs_ge1
thf(fact_7349_Ints__nonzero__abs__ge1,axiom,
    ! [X4: int] :
      ( ( member_int @ X4 @ ring_1_Ints_int )
     => ( ( X4 != zero_zero_int )
       => ( ord_less_eq_int @ one_one_int @ ( abs_abs_int @ X4 ) ) ) ) ).

% Ints_nonzero_abs_ge1
thf(fact_7350_Ints__nonzero__abs__less1,axiom,
    ! [X4: code_integer] :
      ( ( member_Code_integer @ X4 @ ring_11222124179247155820nteger )
     => ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ X4 ) @ one_one_Code_integer )
       => ( X4 = zero_z3403309356797280102nteger ) ) ) ).

% Ints_nonzero_abs_less1
thf(fact_7351_Ints__nonzero__abs__less1,axiom,
    ! [X4: real] :
      ( ( member_real @ X4 @ ring_1_Ints_real )
     => ( ( ord_less_real @ ( abs_abs_real @ X4 ) @ one_one_real )
       => ( X4 = zero_zero_real ) ) ) ).

% Ints_nonzero_abs_less1
thf(fact_7352_Ints__nonzero__abs__less1,axiom,
    ! [X4: rat] :
      ( ( member_rat @ X4 @ ring_1_Ints_rat )
     => ( ( ord_less_rat @ ( abs_abs_rat @ X4 ) @ one_one_rat )
       => ( X4 = zero_zero_rat ) ) ) ).

% Ints_nonzero_abs_less1
thf(fact_7353_Ints__nonzero__abs__less1,axiom,
    ! [X4: int] :
      ( ( member_int @ X4 @ ring_1_Ints_int )
     => ( ( ord_less_int @ ( abs_abs_int @ X4 ) @ one_one_int )
       => ( X4 = zero_zero_int ) ) ) ).

% Ints_nonzero_abs_less1
thf(fact_7354_Ints__eq__abs__less1,axiom,
    ! [X4: code_integer,Y3: code_integer] :
      ( ( member_Code_integer @ X4 @ ring_11222124179247155820nteger )
     => ( ( member_Code_integer @ Y3 @ ring_11222124179247155820nteger )
       => ( ( X4 = Y3 )
          = ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X4 @ Y3 ) ) @ one_one_Code_integer ) ) ) ) ).

% Ints_eq_abs_less1
thf(fact_7355_Ints__eq__abs__less1,axiom,
    ! [X4: real,Y3: real] :
      ( ( member_real @ X4 @ ring_1_Ints_real )
     => ( ( member_real @ Y3 @ ring_1_Ints_real )
       => ( ( X4 = Y3 )
          = ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X4 @ Y3 ) ) @ one_one_real ) ) ) ) ).

% Ints_eq_abs_less1
thf(fact_7356_Ints__eq__abs__less1,axiom,
    ! [X4: rat,Y3: rat] :
      ( ( member_rat @ X4 @ ring_1_Ints_rat )
     => ( ( member_rat @ Y3 @ ring_1_Ints_rat )
       => ( ( X4 = Y3 )
          = ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X4 @ Y3 ) ) @ one_one_rat ) ) ) ) ).

% Ints_eq_abs_less1
thf(fact_7357_Ints__eq__abs__less1,axiom,
    ! [X4: int,Y3: int] :
      ( ( member_int @ X4 @ ring_1_Ints_int )
     => ( ( member_int @ Y3 @ ring_1_Ints_int )
       => ( ( X4 = Y3 )
          = ( ord_less_int @ ( abs_abs_int @ ( minus_minus_int @ X4 @ Y3 ) ) @ one_one_int ) ) ) ) ).

% Ints_eq_abs_less1
thf(fact_7358_sin__times__pi__eq__0,axiom,
    ! [X4: real] :
      ( ( ( sin_real @ ( times_times_real @ X4 @ pi ) )
        = zero_zero_real )
      = ( member_real @ X4 @ ring_1_Ints_real ) ) ).

% sin_times_pi_eq_0
thf(fact_7359_ex__inverse__of__nat__less,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ? [N4: nat] :
          ( ( ord_less_nat @ zero_zero_nat @ N4 )
          & ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N4 ) ) @ X4 ) ) ) ).

% ex_inverse_of_nat_less
thf(fact_7360_ex__inverse__of__nat__less,axiom,
    ! [X4: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ X4 )
     => ? [N4: nat] :
          ( ( ord_less_nat @ zero_zero_nat @ N4 )
          & ( ord_less_rat @ ( inverse_inverse_rat @ ( semiri681578069525770553at_rat @ N4 ) ) @ X4 ) ) ) ).

% ex_inverse_of_nat_less
thf(fact_7361_power__diff__conv__inverse,axiom,
    ! [X4: real,M: nat,N: nat] :
      ( ( X4 != zero_zero_real )
     => ( ( ord_less_eq_nat @ M @ N )
       => ( ( power_power_real @ X4 @ ( minus_minus_nat @ N @ M ) )
          = ( times_times_real @ ( power_power_real @ X4 @ N ) @ ( power_power_real @ ( inverse_inverse_real @ X4 ) @ M ) ) ) ) ) ).

% power_diff_conv_inverse
thf(fact_7362_power__diff__conv__inverse,axiom,
    ! [X4: complex,M: nat,N: nat] :
      ( ( X4 != zero_zero_complex )
     => ( ( ord_less_eq_nat @ M @ N )
       => ( ( power_power_complex @ X4 @ ( minus_minus_nat @ N @ M ) )
          = ( times_times_complex @ ( power_power_complex @ X4 @ N ) @ ( power_power_complex @ ( invers8013647133539491842omplex @ X4 ) @ M ) ) ) ) ) ).

% power_diff_conv_inverse
thf(fact_7363_power__diff__conv__inverse,axiom,
    ! [X4: rat,M: nat,N: nat] :
      ( ( X4 != zero_zero_rat )
     => ( ( ord_less_eq_nat @ M @ N )
       => ( ( power_power_rat @ X4 @ ( minus_minus_nat @ N @ M ) )
          = ( times_times_rat @ ( power_power_rat @ X4 @ N ) @ ( power_power_rat @ ( inverse_inverse_rat @ X4 ) @ M ) ) ) ) ) ).

% power_diff_conv_inverse
thf(fact_7364_Suc__times__gbinomial,axiom,
    ! [K: nat,A: complex] :
      ( ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) @ ( gbinomial_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( suc @ K ) ) )
      = ( times_times_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( gbinomial_complex @ A @ K ) ) ) ).

% Suc_times_gbinomial
thf(fact_7365_Suc__times__gbinomial,axiom,
    ! [K: nat,A: real] :
      ( ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) @ ( gbinomial_real @ ( plus_plus_real @ A @ one_one_real ) @ ( suc @ K ) ) )
      = ( times_times_real @ ( plus_plus_real @ A @ one_one_real ) @ ( gbinomial_real @ A @ K ) ) ) ).

% Suc_times_gbinomial
thf(fact_7366_Suc__times__gbinomial,axiom,
    ! [K: nat,A: rat] :
      ( ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) @ ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K ) ) )
      = ( times_times_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( gbinomial_rat @ A @ K ) ) ) ).

% Suc_times_gbinomial
thf(fact_7367_gbinomial__absorption,axiom,
    ! [K: nat,A: complex] :
      ( ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) @ ( gbinomial_complex @ A @ ( suc @ K ) ) )
      = ( times_times_complex @ A @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ K ) ) ) ).

% gbinomial_absorption
thf(fact_7368_gbinomial__absorption,axiom,
    ! [K: nat,A: real] :
      ( ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) @ ( gbinomial_real @ A @ ( suc @ K ) ) )
      = ( times_times_real @ A @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ K ) ) ) ).

% gbinomial_absorption
thf(fact_7369_gbinomial__absorption,axiom,
    ! [K: nat,A: rat] :
      ( ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) @ ( gbinomial_rat @ A @ ( suc @ K ) ) )
      = ( times_times_rat @ A @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ K ) ) ) ).

% gbinomial_absorption
thf(fact_7370_gbinomial__trinomial__revision,axiom,
    ! [K: nat,M: nat,A: complex] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( times_times_complex @ ( gbinomial_complex @ A @ M ) @ ( gbinomial_complex @ ( semiri8010041392384452111omplex @ M ) @ K ) )
        = ( times_times_complex @ ( gbinomial_complex @ A @ K ) @ ( gbinomial_complex @ ( minus_minus_complex @ A @ ( semiri8010041392384452111omplex @ K ) ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ).

% gbinomial_trinomial_revision
thf(fact_7371_gbinomial__trinomial__revision,axiom,
    ! [K: nat,M: nat,A: real] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( times_times_real @ ( gbinomial_real @ A @ M ) @ ( gbinomial_real @ ( semiri5074537144036343181t_real @ M ) @ K ) )
        = ( times_times_real @ ( gbinomial_real @ A @ K ) @ ( gbinomial_real @ ( minus_minus_real @ A @ ( semiri5074537144036343181t_real @ K ) ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ).

% gbinomial_trinomial_revision
thf(fact_7372_gbinomial__trinomial__revision,axiom,
    ! [K: nat,M: nat,A: rat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( times_times_rat @ ( gbinomial_rat @ A @ M ) @ ( gbinomial_rat @ ( semiri681578069525770553at_rat @ M ) @ K ) )
        = ( times_times_rat @ ( gbinomial_rat @ A @ K ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ ( semiri681578069525770553at_rat @ K ) ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ).

% gbinomial_trinomial_revision
thf(fact_7373_log__inverse,axiom,
    ! [A: real,X4: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X4 )
         => ( ( log @ A @ ( inverse_inverse_real @ X4 ) )
            = ( uminus_uminus_real @ ( log @ A @ X4 ) ) ) ) ) ) ).

% log_inverse
thf(fact_7374_frac__neg,axiom,
    ! [X4: real] :
      ( ( ( member_real @ X4 @ ring_1_Ints_real )
       => ( ( archim2898591450579166408c_real @ ( uminus_uminus_real @ X4 ) )
          = zero_zero_real ) )
      & ( ~ ( member_real @ X4 @ ring_1_Ints_real )
       => ( ( archim2898591450579166408c_real @ ( uminus_uminus_real @ X4 ) )
          = ( minus_minus_real @ one_one_real @ ( archim2898591450579166408c_real @ X4 ) ) ) ) ) ).

% frac_neg
thf(fact_7375_frac__neg,axiom,
    ! [X4: rat] :
      ( ( ( member_rat @ X4 @ ring_1_Ints_rat )
       => ( ( archimedean_frac_rat @ ( uminus_uminus_rat @ X4 ) )
          = zero_zero_rat ) )
      & ( ~ ( member_rat @ X4 @ ring_1_Ints_rat )
       => ( ( archimedean_frac_rat @ ( uminus_uminus_rat @ X4 ) )
          = ( minus_minus_rat @ one_one_rat @ ( archimedean_frac_rat @ X4 ) ) ) ) ) ).

% frac_neg
thf(fact_7376_gbinomial__factors,axiom,
    ! [A: complex,K: nat] :
      ( ( gbinomial_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( suc @ K ) )
      = ( times_times_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) ) @ ( gbinomial_complex @ A @ K ) ) ) ).

% gbinomial_factors
thf(fact_7377_gbinomial__factors,axiom,
    ! [A: real,K: nat] :
      ( ( gbinomial_real @ ( plus_plus_real @ A @ one_one_real ) @ ( suc @ K ) )
      = ( times_times_real @ ( divide_divide_real @ ( plus_plus_real @ A @ one_one_real ) @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) ) @ ( gbinomial_real @ A @ K ) ) ) ).

% gbinomial_factors
thf(fact_7378_gbinomial__factors,axiom,
    ! [A: rat,K: nat] :
      ( ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K ) )
      = ( times_times_rat @ ( divide_divide_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) ) @ ( gbinomial_rat @ A @ K ) ) ) ).

% gbinomial_factors
thf(fact_7379_gbinomial__rec,axiom,
    ! [A: complex,K: nat] :
      ( ( gbinomial_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( suc @ K ) )
      = ( times_times_complex @ ( gbinomial_complex @ A @ K ) @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) ) ) ) ).

% gbinomial_rec
thf(fact_7380_gbinomial__rec,axiom,
    ! [A: real,K: nat] :
      ( ( gbinomial_real @ ( plus_plus_real @ A @ one_one_real ) @ ( suc @ K ) )
      = ( times_times_real @ ( gbinomial_real @ A @ K ) @ ( divide_divide_real @ ( plus_plus_real @ A @ one_one_real ) @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) ) ) ) ).

% gbinomial_rec
thf(fact_7381_gbinomial__rec,axiom,
    ! [A: rat,K: nat] :
      ( ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K ) )
      = ( times_times_rat @ ( gbinomial_rat @ A @ K ) @ ( divide_divide_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) ) ) ) ).

% gbinomial_rec
thf(fact_7382_gbinomial__negated__upper,axiom,
    ( gbinomial_complex
    = ( ^ [A4: complex,K3: nat] : ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K3 ) @ ( gbinomial_complex @ ( minus_minus_complex @ ( minus_minus_complex @ ( semiri8010041392384452111omplex @ K3 ) @ A4 ) @ one_one_complex ) @ K3 ) ) ) ) ).

% gbinomial_negated_upper
thf(fact_7383_gbinomial__negated__upper,axiom,
    ( gbinomial_real
    = ( ^ [A4: real,K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( gbinomial_real @ ( minus_minus_real @ ( minus_minus_real @ ( semiri5074537144036343181t_real @ K3 ) @ A4 ) @ one_one_real ) @ K3 ) ) ) ) ).

% gbinomial_negated_upper
thf(fact_7384_gbinomial__negated__upper,axiom,
    ( gbinomial_rat
    = ( ^ [A4: rat,K3: nat] : ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K3 ) @ ( gbinomial_rat @ ( minus_minus_rat @ ( minus_minus_rat @ ( semiri681578069525770553at_rat @ K3 ) @ A4 ) @ one_one_rat ) @ K3 ) ) ) ) ).

% gbinomial_negated_upper
thf(fact_7385_gbinomial__index__swap,axiom,
    ! [K: nat,N: nat] :
      ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K ) @ ( gbinomial_complex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ one_one_complex ) @ K ) )
      = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( gbinomial_complex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ K ) ) @ one_one_complex ) @ N ) ) ) ).

% gbinomial_index_swap
thf(fact_7386_gbinomial__index__swap,axiom,
    ! [K: nat,N: nat] :
      ( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K ) @ ( gbinomial_real @ ( minus_minus_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ one_one_real ) @ K ) )
      = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( gbinomial_real @ ( minus_minus_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ K ) ) @ one_one_real ) @ N ) ) ) ).

% gbinomial_index_swap
thf(fact_7387_gbinomial__index__swap,axiom,
    ! [K: nat,N: nat] :
      ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K ) @ ( gbinomial_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ one_one_rat ) @ K ) )
      = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( gbinomial_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ K ) ) @ one_one_rat ) @ N ) ) ) ).

% gbinomial_index_swap
thf(fact_7388_exp__plus__inverse__exp,axiom,
    ! [X4: real] : ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( plus_plus_real @ ( exp_real @ X4 ) @ ( inverse_inverse_real @ ( exp_real @ X4 ) ) ) ) ).

% exp_plus_inverse_exp
thf(fact_7389_le__mult__floor__Ints,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( member_real @ A @ ring_1_Ints_real )
       => ( ord_less_eq_real @ ( ring_1_of_int_real @ ( times_times_int @ ( archim6058952711729229775r_real @ A ) @ ( archim6058952711729229775r_real @ B2 ) ) ) @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ ( times_times_real @ A @ B2 ) ) ) ) ) ) ).

% le_mult_floor_Ints
thf(fact_7390_le__mult__floor__Ints,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( member_real @ A @ ring_1_Ints_real )
       => ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( times_times_int @ ( archim6058952711729229775r_real @ A ) @ ( archim6058952711729229775r_real @ B2 ) ) ) @ ( ring_1_of_int_rat @ ( archim6058952711729229775r_real @ ( times_times_real @ A @ B2 ) ) ) ) ) ) ).

% le_mult_floor_Ints
thf(fact_7391_le__mult__floor__Ints,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( member_real @ A @ ring_1_Ints_real )
       => ( ord_less_eq_int @ ( ring_1_of_int_int @ ( times_times_int @ ( archim6058952711729229775r_real @ A ) @ ( archim6058952711729229775r_real @ B2 ) ) ) @ ( ring_1_of_int_int @ ( archim6058952711729229775r_real @ ( times_times_real @ A @ B2 ) ) ) ) ) ) ).

% le_mult_floor_Ints
thf(fact_7392_le__mult__floor__Ints,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( member_rat @ A @ ring_1_Ints_rat )
       => ( ord_less_eq_real @ ( ring_1_of_int_real @ ( times_times_int @ ( archim3151403230148437115or_rat @ A ) @ ( archim3151403230148437115or_rat @ B2 ) ) ) @ ( ring_1_of_int_real @ ( archim3151403230148437115or_rat @ ( times_times_rat @ A @ B2 ) ) ) ) ) ) ).

% le_mult_floor_Ints
thf(fact_7393_le__mult__floor__Ints,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( member_rat @ A @ ring_1_Ints_rat )
       => ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( times_times_int @ ( archim3151403230148437115or_rat @ A ) @ ( archim3151403230148437115or_rat @ B2 ) ) ) @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ ( times_times_rat @ A @ B2 ) ) ) ) ) ) ).

% le_mult_floor_Ints
thf(fact_7394_le__mult__floor__Ints,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( member_rat @ A @ ring_1_Ints_rat )
       => ( ord_less_eq_int @ ( ring_1_of_int_int @ ( times_times_int @ ( archim3151403230148437115or_rat @ A ) @ ( archim3151403230148437115or_rat @ B2 ) ) ) @ ( ring_1_of_int_int @ ( archim3151403230148437115or_rat @ ( times_times_rat @ A @ B2 ) ) ) ) ) ) ).

% le_mult_floor_Ints
thf(fact_7395_frac__unique__iff,axiom,
    ! [X4: real,A: real] :
      ( ( ( archim2898591450579166408c_real @ X4 )
        = A )
      = ( ( member_real @ ( minus_minus_real @ X4 @ A ) @ ring_1_Ints_real )
        & ( ord_less_eq_real @ zero_zero_real @ A )
        & ( ord_less_real @ A @ one_one_real ) ) ) ).

% frac_unique_iff
thf(fact_7396_frac__unique__iff,axiom,
    ! [X4: rat,A: rat] :
      ( ( ( archimedean_frac_rat @ X4 )
        = A )
      = ( ( member_rat @ ( minus_minus_rat @ X4 @ A ) @ ring_1_Ints_rat )
        & ( ord_less_eq_rat @ zero_zero_rat @ A )
        & ( ord_less_rat @ A @ one_one_rat ) ) ) ).

% frac_unique_iff
thf(fact_7397_mult__ceiling__le__Ints,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( member_real @ A @ ring_1_Ints_real )
       => ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ ( times_times_real @ A @ B2 ) ) ) @ ( ring_1_of_int_real @ ( times_times_int @ ( archim7802044766580827645g_real @ A ) @ ( archim7802044766580827645g_real @ B2 ) ) ) ) ) ) ).

% mult_ceiling_le_Ints
thf(fact_7398_mult__ceiling__le__Ints,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( member_real @ A @ ring_1_Ints_real )
       => ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim7802044766580827645g_real @ ( times_times_real @ A @ B2 ) ) ) @ ( ring_1_of_int_rat @ ( times_times_int @ ( archim7802044766580827645g_real @ A ) @ ( archim7802044766580827645g_real @ B2 ) ) ) ) ) ) ).

% mult_ceiling_le_Ints
thf(fact_7399_mult__ceiling__le__Ints,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( member_real @ A @ ring_1_Ints_real )
       => ( ord_less_eq_int @ ( ring_1_of_int_int @ ( archim7802044766580827645g_real @ ( times_times_real @ A @ B2 ) ) ) @ ( ring_1_of_int_int @ ( times_times_int @ ( archim7802044766580827645g_real @ A ) @ ( archim7802044766580827645g_real @ B2 ) ) ) ) ) ) ).

% mult_ceiling_le_Ints
thf(fact_7400_mult__ceiling__le__Ints,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( member_rat @ A @ ring_1_Ints_rat )
       => ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim2889992004027027881ng_rat @ ( times_times_rat @ A @ B2 ) ) ) @ ( ring_1_of_int_real @ ( times_times_int @ ( archim2889992004027027881ng_rat @ A ) @ ( archim2889992004027027881ng_rat @ B2 ) ) ) ) ) ) ).

% mult_ceiling_le_Ints
thf(fact_7401_mult__ceiling__le__Ints,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( member_rat @ A @ ring_1_Ints_rat )
       => ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ ( times_times_rat @ A @ B2 ) ) ) @ ( ring_1_of_int_rat @ ( times_times_int @ ( archim2889992004027027881ng_rat @ A ) @ ( archim2889992004027027881ng_rat @ B2 ) ) ) ) ) ) ).

% mult_ceiling_le_Ints
thf(fact_7402_mult__ceiling__le__Ints,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( member_rat @ A @ ring_1_Ints_rat )
       => ( ord_less_eq_int @ ( ring_1_of_int_int @ ( archim2889992004027027881ng_rat @ ( times_times_rat @ A @ B2 ) ) ) @ ( ring_1_of_int_int @ ( times_times_int @ ( archim2889992004027027881ng_rat @ A ) @ ( archim2889992004027027881ng_rat @ B2 ) ) ) ) ) ) ).

% mult_ceiling_le_Ints
thf(fact_7403_gbinomial__minus,axiom,
    ! [A: complex,K: nat] :
      ( ( gbinomial_complex @ ( uminus1482373934393186551omplex @ A ) @ K )
      = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K ) @ ( gbinomial_complex @ ( minus_minus_complex @ ( plus_plus_complex @ A @ ( semiri8010041392384452111omplex @ K ) ) @ one_one_complex ) @ K ) ) ) ).

% gbinomial_minus
thf(fact_7404_gbinomial__minus,axiom,
    ! [A: real,K: nat] :
      ( ( gbinomial_real @ ( uminus_uminus_real @ A ) @ K )
      = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K ) @ ( gbinomial_real @ ( minus_minus_real @ ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ K ) ) @ one_one_real ) @ K ) ) ) ).

% gbinomial_minus
thf(fact_7405_gbinomial__minus,axiom,
    ! [A: rat,K: nat] :
      ( ( gbinomial_rat @ ( uminus_uminus_rat @ A ) @ K )
      = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K ) @ ( gbinomial_rat @ ( minus_minus_rat @ ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ K ) ) @ one_one_rat ) @ K ) ) ) ).

% gbinomial_minus
thf(fact_7406_plus__inverse__ge__2,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( plus_plus_real @ X4 @ ( inverse_inverse_real @ X4 ) ) ) ) ).

% plus_inverse_ge_2
thf(fact_7407_real__inv__sqrt__pow2,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( power_power_real @ ( inverse_inverse_real @ ( sqrt @ X4 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( inverse_inverse_real @ X4 ) ) ) ).

% real_inv_sqrt_pow2
thf(fact_7408_gbinomial__reduce__nat,axiom,
    ! [K: nat,A: complex] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( gbinomial_complex @ A @ K )
        = ( plus_plus_complex @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ ( minus_minus_nat @ K @ one_one_nat ) ) @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ K ) ) ) ) ).

% gbinomial_reduce_nat
thf(fact_7409_gbinomial__reduce__nat,axiom,
    ! [K: nat,A: real] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( gbinomial_real @ A @ K )
        = ( plus_plus_real @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ ( minus_minus_nat @ K @ one_one_nat ) ) @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ K ) ) ) ) ).

% gbinomial_reduce_nat
thf(fact_7410_gbinomial__reduce__nat,axiom,
    ! [K: nat,A: rat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( gbinomial_rat @ A @ K )
        = ( plus_plus_rat @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ K ) ) ) ) ).

% gbinomial_reduce_nat
thf(fact_7411_gbinomial__pochhammer,axiom,
    ( gbinomial_complex
    = ( ^ [A4: complex,K3: nat] : ( divide1717551699836669952omplex @ ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K3 ) @ ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ A4 ) @ K3 ) ) @ ( semiri5044797733671781792omplex @ K3 ) ) ) ) ).

% gbinomial_pochhammer
thf(fact_7412_gbinomial__pochhammer,axiom,
    ( gbinomial_rat
    = ( ^ [A4: rat,K3: nat] : ( divide_divide_rat @ ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K3 ) @ ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ A4 ) @ K3 ) ) @ ( semiri773545260158071498ct_rat @ K3 ) ) ) ) ).

% gbinomial_pochhammer
thf(fact_7413_gbinomial__pochhammer,axiom,
    ( gbinomial_real
    = ( ^ [A4: real,K3: nat] : ( divide_divide_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ A4 ) @ K3 ) ) @ ( semiri2265585572941072030t_real @ K3 ) ) ) ) ).

% gbinomial_pochhammer
thf(fact_7414_gbinomial__pochhammer_H,axiom,
    ( gbinomial_complex
    = ( ^ [A4: complex,K3: nat] : ( divide1717551699836669952omplex @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( minus_minus_complex @ A4 @ ( semiri8010041392384452111omplex @ K3 ) ) @ one_one_complex ) @ K3 ) @ ( semiri5044797733671781792omplex @ K3 ) ) ) ) ).

% gbinomial_pochhammer'
thf(fact_7415_gbinomial__pochhammer_H,axiom,
    ( gbinomial_rat
    = ( ^ [A4: rat,K3: nat] : ( divide_divide_rat @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ A4 @ ( semiri681578069525770553at_rat @ K3 ) ) @ one_one_rat ) @ K3 ) @ ( semiri773545260158071498ct_rat @ K3 ) ) ) ) ).

% gbinomial_pochhammer'
thf(fact_7416_gbinomial__pochhammer_H,axiom,
    ( gbinomial_real
    = ( ^ [A4: real,K3: nat] : ( divide_divide_real @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( minus_minus_real @ A4 @ ( semiri5074537144036343181t_real @ K3 ) ) @ one_one_real ) @ K3 ) @ ( semiri2265585572941072030t_real @ K3 ) ) ) ) ).

% gbinomial_pochhammer'
thf(fact_7417_tan__cot,axiom,
    ! [X4: real] :
      ( ( tan_real @ ( minus_minus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X4 ) )
      = ( inverse_inverse_real @ ( tan_real @ X4 ) ) ) ).

% tan_cot
thf(fact_7418_sin__integer__2pi,axiom,
    ! [N: real] :
      ( ( member_real @ N @ ring_1_Ints_real )
     => ( ( sin_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ N ) )
        = zero_zero_real ) ) ).

% sin_integer_2pi
thf(fact_7419_cos__integer__2pi,axiom,
    ! [N: real] :
      ( ( member_real @ N @ ring_1_Ints_real )
     => ( ( cos_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ N ) )
        = one_one_real ) ) ).

% cos_integer_2pi
thf(fact_7420_xor__nat__unfold,axiom,
    ( bit_se6528837805403552850or_nat
    = ( ^ [M6: nat,N2: nat] : ( if_nat @ ( M6 = zero_zero_nat ) @ N2 @ ( if_nat @ ( N2 = zero_zero_nat ) @ M6 @ ( plus_plus_nat @ ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( divide_divide_nat @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% xor_nat_unfold
thf(fact_7421_xor__nat__rec,axiom,
    ( bit_se6528837805403552850or_nat
    = ( ^ [M6: nat,N2: nat] :
          ( plus_plus_nat
          @ ( zero_n2687167440665602831ol_nat
            @ ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M6 ) )
             != ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
          @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( divide_divide_nat @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% xor_nat_rec
thf(fact_7422_real__le__x__sinh,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ord_less_eq_real @ X4 @ ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ X4 ) @ ( inverse_inverse_real @ ( exp_real @ X4 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% real_le_x_sinh
thf(fact_7423_xor__one__eq,axiom,
    ! [A: code_integer] :
      ( ( bit_se3222712562003087583nteger @ A @ one_one_Code_integer )
      = ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A @ ( zero_n356916108424825756nteger @ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) )
        @ ( zero_n356916108424825756nteger
          @ ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ) ).

% xor_one_eq
thf(fact_7424_xor__one__eq,axiom,
    ! [A: nat] :
      ( ( bit_se6528837805403552850or_nat @ A @ one_one_nat )
      = ( minus_minus_nat @ ( plus_plus_nat @ A @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) )
        @ ( zero_n2687167440665602831ol_nat
          @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ) ).

% xor_one_eq
thf(fact_7425_xor__one__eq,axiom,
    ! [A: int] :
      ( ( bit_se6526347334894502574or_int @ A @ one_one_int )
      = ( minus_minus_int @ ( plus_plus_int @ A @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) )
        @ ( zero_n2684676970156552555ol_int
          @ ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ) ).

% xor_one_eq
thf(fact_7426_one__xor__eq,axiom,
    ! [A: code_integer] :
      ( ( bit_se3222712562003087583nteger @ one_one_Code_integer @ A )
      = ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A @ ( zero_n356916108424825756nteger @ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) )
        @ ( zero_n356916108424825756nteger
          @ ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ) ).

% one_xor_eq
thf(fact_7427_one__xor__eq,axiom,
    ! [A: nat] :
      ( ( bit_se6528837805403552850or_nat @ one_one_nat @ A )
      = ( minus_minus_nat @ ( plus_plus_nat @ A @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) )
        @ ( zero_n2687167440665602831ol_nat
          @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ) ).

% one_xor_eq
thf(fact_7428_one__xor__eq,axiom,
    ! [A: int] :
      ( ( bit_se6526347334894502574or_int @ one_one_int @ A )
      = ( minus_minus_int @ ( plus_plus_int @ A @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) )
        @ ( zero_n2684676970156552555ol_int
          @ ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ) ).

% one_xor_eq
thf(fact_7429_real__le__abs__sinh,axiom,
    ! [X4: real] : ( ord_less_eq_real @ ( abs_abs_real @ X4 ) @ ( abs_abs_real @ ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ X4 ) @ ( inverse_inverse_real @ ( exp_real @ X4 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% real_le_abs_sinh
thf(fact_7430_tan__sec,axiom,
    ! [X4: real] :
      ( ( ( cos_real @ X4 )
       != zero_zero_real )
     => ( ( plus_plus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = ( power_power_real @ ( inverse_inverse_real @ ( cos_real @ X4 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% tan_sec
thf(fact_7431_tan__sec,axiom,
    ! [X4: complex] :
      ( ( ( cos_complex @ X4 )
       != zero_zero_complex )
     => ( ( plus_plus_complex @ one_one_complex @ ( power_power_complex @ ( tan_complex @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = ( power_power_complex @ ( invers8013647133539491842omplex @ ( cos_complex @ X4 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% tan_sec
thf(fact_7432_sinh__ln__real,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( sinh_real @ ( ln_ln_real @ X4 ) )
        = ( divide_divide_real @ ( minus_minus_real @ X4 @ ( inverse_inverse_real @ X4 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% sinh_ln_real
thf(fact_7433_cosh__ln__real,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( cosh_real @ ( ln_ln_real @ X4 ) )
        = ( divide_divide_real @ ( plus_plus_real @ X4 @ ( inverse_inverse_real @ X4 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% cosh_ln_real
thf(fact_7434_cosh__zero__iff,axiom,
    ! [X4: real] :
      ( ( ( cosh_real @ X4 )
        = zero_zero_real )
      = ( ( power_power_real @ ( exp_real @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( uminus_uminus_real @ one_one_real ) ) ) ).

% cosh_zero_iff
thf(fact_7435_cosh__zero__iff,axiom,
    ! [X4: complex] :
      ( ( ( cosh_complex @ X4 )
        = zero_zero_complex )
      = ( ( power_power_complex @ ( exp_complex @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ).

% cosh_zero_iff
thf(fact_7436_psubsetI,axiom,
    ! [A3: set_nat,B4: set_nat] :
      ( ( ord_less_eq_set_nat @ A3 @ B4 )
     => ( ( A3 != B4 )
       => ( ord_less_set_nat @ A3 @ B4 ) ) ) ).

% psubsetI
thf(fact_7437_push__bit__numeral__minus__1,axiom,
    ! [N: num] :
      ( ( bit_se7788150548672797655nteger @ ( numeral_numeral_nat @ N ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ N ) ) ) ) ).

% push_bit_numeral_minus_1
thf(fact_7438_push__bit__numeral__minus__1,axiom,
    ! [N: num] :
      ( ( bit_se545348938243370406it_int @ ( numeral_numeral_nat @ N ) @ ( uminus_uminus_int @ one_one_int ) )
      = ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ N ) ) ) ) ).

% push_bit_numeral_minus_1
thf(fact_7439_push__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se545348938243370406it_int @ N @ K ) )
      = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).

% push_bit_nonnegative_int_iff
thf(fact_7440_push__bit__negative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_int @ ( bit_se545348938243370406it_int @ N @ K ) @ zero_zero_int )
      = ( ord_less_int @ K @ zero_zero_int ) ) ).

% push_bit_negative_int_iff
thf(fact_7441_push__bit__of__0,axiom,
    ! [N: nat] :
      ( ( bit_se545348938243370406it_int @ N @ zero_zero_int )
      = zero_zero_int ) ).

% push_bit_of_0
thf(fact_7442_push__bit__of__0,axiom,
    ! [N: nat] :
      ( ( bit_se547839408752420682it_nat @ N @ zero_zero_nat )
      = zero_zero_nat ) ).

% push_bit_of_0
thf(fact_7443_push__bit__eq__0__iff,axiom,
    ! [N: nat,A: int] :
      ( ( ( bit_se545348938243370406it_int @ N @ A )
        = zero_zero_int )
      = ( A = zero_zero_int ) ) ).

% push_bit_eq_0_iff
thf(fact_7444_push__bit__eq__0__iff,axiom,
    ! [N: nat,A: nat] :
      ( ( ( bit_se547839408752420682it_nat @ N @ A )
        = zero_zero_nat )
      = ( A = zero_zero_nat ) ) ).

% push_bit_eq_0_iff
thf(fact_7445_push__bit__push__bit,axiom,
    ! [M: nat,N: nat,A: int] :
      ( ( bit_se545348938243370406it_int @ M @ ( bit_se545348938243370406it_int @ N @ A ) )
      = ( bit_se545348938243370406it_int @ ( plus_plus_nat @ M @ N ) @ A ) ) ).

% push_bit_push_bit
thf(fact_7446_push__bit__push__bit,axiom,
    ! [M: nat,N: nat,A: nat] :
      ( ( bit_se547839408752420682it_nat @ M @ ( bit_se547839408752420682it_nat @ N @ A ) )
      = ( bit_se547839408752420682it_nat @ ( plus_plus_nat @ M @ N ) @ A ) ) ).

% push_bit_push_bit
thf(fact_7447_push__bit__and,axiom,
    ! [N: nat,A: int,B2: int] :
      ( ( bit_se545348938243370406it_int @ N @ ( bit_se725231765392027082nd_int @ A @ B2 ) )
      = ( bit_se725231765392027082nd_int @ ( bit_se545348938243370406it_int @ N @ A ) @ ( bit_se545348938243370406it_int @ N @ B2 ) ) ) ).

% push_bit_and
thf(fact_7448_push__bit__and,axiom,
    ! [N: nat,A: nat,B2: nat] :
      ( ( bit_se547839408752420682it_nat @ N @ ( bit_se727722235901077358nd_nat @ A @ B2 ) )
      = ( bit_se727722235901077358nd_nat @ ( bit_se547839408752420682it_nat @ N @ A ) @ ( bit_se547839408752420682it_nat @ N @ B2 ) ) ) ).

% push_bit_and
thf(fact_7449_push__bit__xor,axiom,
    ! [N: nat,A: int,B2: int] :
      ( ( bit_se545348938243370406it_int @ N @ ( bit_se6526347334894502574or_int @ A @ B2 ) )
      = ( bit_se6526347334894502574or_int @ ( bit_se545348938243370406it_int @ N @ A ) @ ( bit_se545348938243370406it_int @ N @ B2 ) ) ) ).

% push_bit_xor
thf(fact_7450_push__bit__xor,axiom,
    ! [N: nat,A: nat,B2: nat] :
      ( ( bit_se547839408752420682it_nat @ N @ ( bit_se6528837805403552850or_nat @ A @ B2 ) )
      = ( bit_se6528837805403552850or_nat @ ( bit_se547839408752420682it_nat @ N @ A ) @ ( bit_se547839408752420682it_nat @ N @ B2 ) ) ) ).

% push_bit_xor
thf(fact_7451_concat__bit__of__zero__1,axiom,
    ! [N: nat,L2: int] :
      ( ( bit_concat_bit @ N @ zero_zero_int @ L2 )
      = ( bit_se545348938243370406it_int @ N @ L2 ) ) ).

% concat_bit_of_zero_1
thf(fact_7452_sinh__real__less__iff,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_real @ ( sinh_real @ X4 ) @ ( sinh_real @ Y3 ) )
      = ( ord_less_real @ X4 @ Y3 ) ) ).

% sinh_real_less_iff
thf(fact_7453_sinh__real__le__iff,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_eq_real @ ( sinh_real @ X4 ) @ ( sinh_real @ Y3 ) )
      = ( ord_less_eq_real @ X4 @ Y3 ) ) ).

% sinh_real_le_iff
thf(fact_7454_cosh__0,axiom,
    ( ( cosh_complex @ zero_zero_complex )
    = one_one_complex ) ).

% cosh_0
thf(fact_7455_cosh__0,axiom,
    ( ( cosh_real @ zero_zero_real )
    = one_one_real ) ).

% cosh_0
thf(fact_7456_sinh__real__pos__iff,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ ( sinh_real @ X4 ) )
      = ( ord_less_real @ zero_zero_real @ X4 ) ) ).

% sinh_real_pos_iff
thf(fact_7457_sinh__real__neg__iff,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ ( sinh_real @ X4 ) @ zero_zero_real )
      = ( ord_less_real @ X4 @ zero_zero_real ) ) ).

% sinh_real_neg_iff
thf(fact_7458_sinh__real__nonneg__iff,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( sinh_real @ X4 ) )
      = ( ord_less_eq_real @ zero_zero_real @ X4 ) ) ).

% sinh_real_nonneg_iff
thf(fact_7459_sinh__real__nonpos__iff,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ ( sinh_real @ X4 ) @ zero_zero_real )
      = ( ord_less_eq_real @ X4 @ zero_zero_real ) ) ).

% sinh_real_nonpos_iff
thf(fact_7460_xor__nonnegative__int__iff,axiom,
    ! [K: int,L2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se6526347334894502574or_int @ K @ L2 ) )
      = ( ( ord_less_eq_int @ zero_zero_int @ K )
        = ( ord_less_eq_int @ zero_zero_int @ L2 ) ) ) ).

% xor_nonnegative_int_iff
thf(fact_7461_xor__negative__int__iff,axiom,
    ! [K: int,L2: int] :
      ( ( ord_less_int @ ( bit_se6526347334894502574or_int @ K @ L2 ) @ zero_zero_int )
      = ( ( ord_less_int @ K @ zero_zero_int )
       != ( ord_less_int @ L2 @ zero_zero_int ) ) ) ).

% xor_negative_int_iff
thf(fact_7462_push__bit__Suc__numeral,axiom,
    ! [N: nat,K: num] :
      ( ( bit_se545348938243370406it_int @ ( suc @ N ) @ ( numeral_numeral_int @ K ) )
      = ( bit_se545348938243370406it_int @ N @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) ) ).

% push_bit_Suc_numeral
thf(fact_7463_push__bit__Suc__numeral,axiom,
    ! [N: nat,K: num] :
      ( ( bit_se547839408752420682it_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
      = ( bit_se547839408752420682it_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).

% push_bit_Suc_numeral
thf(fact_7464_push__bit__Suc__minus__numeral,axiom,
    ! [N: nat,K: num] :
      ( ( bit_se7788150548672797655nteger @ ( suc @ N ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
      = ( bit_se7788150548672797655nteger @ N @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ K ) ) ) ) ) ).

% push_bit_Suc_minus_numeral
thf(fact_7465_push__bit__Suc__minus__numeral,axiom,
    ! [N: nat,K: num] :
      ( ( bit_se545348938243370406it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
      = ( bit_se545348938243370406it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) ) ) ).

% push_bit_Suc_minus_numeral
thf(fact_7466_push__bit__numeral,axiom,
    ! [L2: num,K: num] :
      ( ( bit_se545348938243370406it_int @ ( numeral_numeral_nat @ L2 ) @ ( numeral_numeral_int @ K ) )
      = ( bit_se545348938243370406it_int @ ( pred_numeral @ L2 ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) ) ).

% push_bit_numeral
thf(fact_7467_push__bit__numeral,axiom,
    ! [L2: num,K: num] :
      ( ( bit_se547839408752420682it_nat @ ( numeral_numeral_nat @ L2 ) @ ( numeral_numeral_nat @ K ) )
      = ( bit_se547839408752420682it_nat @ ( pred_numeral @ L2 ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).

% push_bit_numeral
thf(fact_7468_push__bit__of__Suc__0,axiom,
    ! [N: nat] :
      ( ( bit_se547839408752420682it_nat @ N @ ( suc @ zero_zero_nat ) )
      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).

% push_bit_of_Suc_0
thf(fact_7469_push__bit__Suc,axiom,
    ! [N: nat,A: int] :
      ( ( bit_se545348938243370406it_int @ ( suc @ N ) @ A )
      = ( bit_se545348938243370406it_int @ N @ ( times_times_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).

% push_bit_Suc
thf(fact_7470_push__bit__Suc,axiom,
    ! [N: nat,A: nat] :
      ( ( bit_se547839408752420682it_nat @ ( suc @ N ) @ A )
      = ( bit_se547839408752420682it_nat @ N @ ( times_times_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% push_bit_Suc
thf(fact_7471_push__bit__of__1,axiom,
    ! [N: nat] :
      ( ( bit_se545348938243370406it_int @ N @ one_one_int )
      = ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).

% push_bit_of_1
thf(fact_7472_push__bit__of__1,axiom,
    ! [N: nat] :
      ( ( bit_se547839408752420682it_nat @ N @ one_one_nat )
      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).

% push_bit_of_1
thf(fact_7473_even__push__bit__iff,axiom,
    ! [N: nat,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se7788150548672797655nteger @ N @ A ) )
      = ( ( N != zero_zero_nat )
        | ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).

% even_push_bit_iff
thf(fact_7474_even__push__bit__iff,axiom,
    ! [N: nat,A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se545348938243370406it_int @ N @ A ) )
      = ( ( N != zero_zero_nat )
        | ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).

% even_push_bit_iff
thf(fact_7475_even__push__bit__iff,axiom,
    ! [N: nat,A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se547839408752420682it_nat @ N @ A ) )
      = ( ( N != zero_zero_nat )
        | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).

% even_push_bit_iff
thf(fact_7476_push__bit__minus__numeral,axiom,
    ! [L2: num,K: num] :
      ( ( bit_se7788150548672797655nteger @ ( numeral_numeral_nat @ L2 ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
      = ( bit_se7788150548672797655nteger @ ( pred_numeral @ L2 ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ K ) ) ) ) ) ).

% push_bit_minus_numeral
thf(fact_7477_push__bit__minus__numeral,axiom,
    ! [L2: num,K: num] :
      ( ( bit_se545348938243370406it_int @ ( numeral_numeral_nat @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
      = ( bit_se545348938243370406it_int @ ( pred_numeral @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) ) ) ).

% push_bit_minus_numeral
thf(fact_7478_flip__bit__int__def,axiom,
    ( bit_se2159334234014336723it_int
    = ( ^ [N2: nat,K3: int] : ( bit_se6526347334894502574or_int @ K3 @ ( bit_se545348938243370406it_int @ N2 @ one_one_int ) ) ) ) ).

% flip_bit_int_def
thf(fact_7479_push__bit__add,axiom,
    ! [N: nat,A: int,B2: int] :
      ( ( bit_se545348938243370406it_int @ N @ ( plus_plus_int @ A @ B2 ) )
      = ( plus_plus_int @ ( bit_se545348938243370406it_int @ N @ A ) @ ( bit_se545348938243370406it_int @ N @ B2 ) ) ) ).

% push_bit_add
thf(fact_7480_push__bit__add,axiom,
    ! [N: nat,A: nat,B2: nat] :
      ( ( bit_se547839408752420682it_nat @ N @ ( plus_plus_nat @ A @ B2 ) )
      = ( plus_plus_nat @ ( bit_se547839408752420682it_nat @ N @ A ) @ ( bit_se547839408752420682it_nat @ N @ B2 ) ) ) ).

% push_bit_add
thf(fact_7481_cosh__plus__sinh,axiom,
    ! [X4: complex] :
      ( ( plus_plus_complex @ ( cosh_complex @ X4 ) @ ( sinh_complex @ X4 ) )
      = ( exp_complex @ X4 ) ) ).

% cosh_plus_sinh
thf(fact_7482_cosh__plus__sinh,axiom,
    ! [X4: real] :
      ( ( plus_plus_real @ ( cosh_real @ X4 ) @ ( sinh_real @ X4 ) )
      = ( exp_real @ X4 ) ) ).

% cosh_plus_sinh
thf(fact_7483_sinh__plus__cosh,axiom,
    ! [X4: complex] :
      ( ( plus_plus_complex @ ( sinh_complex @ X4 ) @ ( cosh_complex @ X4 ) )
      = ( exp_complex @ X4 ) ) ).

% sinh_plus_cosh
thf(fact_7484_sinh__plus__cosh,axiom,
    ! [X4: real] :
      ( ( plus_plus_real @ ( sinh_real @ X4 ) @ ( cosh_real @ X4 ) )
      = ( exp_real @ X4 ) ) ).

% sinh_plus_cosh
thf(fact_7485_psubsetD,axiom,
    ! [A3: set_Extended_enat,B4: set_Extended_enat,C: extended_enat] :
      ( ( ord_le2529575680413868914d_enat @ A3 @ B4 )
     => ( ( member_Extended_enat @ C @ A3 )
       => ( member_Extended_enat @ C @ B4 ) ) ) ).

% psubsetD
thf(fact_7486_psubsetD,axiom,
    ! [A3: set_complex,B4: set_complex,C: complex] :
      ( ( ord_less_set_complex @ A3 @ B4 )
     => ( ( member_complex @ C @ A3 )
       => ( member_complex @ C @ B4 ) ) ) ).

% psubsetD
thf(fact_7487_psubsetD,axiom,
    ! [A3: set_real,B4: set_real,C: real] :
      ( ( ord_less_set_real @ A3 @ B4 )
     => ( ( member_real @ C @ A3 )
       => ( member_real @ C @ B4 ) ) ) ).

% psubsetD
thf(fact_7488_psubsetD,axiom,
    ! [A3: set_nat,B4: set_nat,C: nat] :
      ( ( ord_less_set_nat @ A3 @ B4 )
     => ( ( member_nat @ C @ A3 )
       => ( member_nat @ C @ B4 ) ) ) ).

% psubsetD
thf(fact_7489_psubsetD,axiom,
    ! [A3: set_int,B4: set_int,C: int] :
      ( ( ord_less_set_int @ A3 @ B4 )
     => ( ( member_int @ C @ A3 )
       => ( member_int @ C @ B4 ) ) ) ).

% psubsetD
thf(fact_7490_sinh__le__cosh__real,axiom,
    ! [X4: real] : ( ord_less_eq_real @ ( sinh_real @ X4 ) @ ( cosh_real @ X4 ) ) ).

% sinh_le_cosh_real
thf(fact_7491_sinh__less__cosh__real,axiom,
    ! [X4: real] : ( ord_less_real @ ( sinh_real @ X4 ) @ ( cosh_real @ X4 ) ) ).

% sinh_less_cosh_real
thf(fact_7492_push__bit__of__int,axiom,
    ! [N: nat,K: int] :
      ( ( bit_se545348938243370406it_int @ N @ ( ring_1_of_int_int @ K ) )
      = ( ring_1_of_int_int @ ( bit_se545348938243370406it_int @ N @ K ) ) ) ).

% push_bit_of_int
thf(fact_7493_of__nat__push__bit,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri1314217659103216013at_int @ ( bit_se547839408752420682it_nat @ M @ N ) )
      = ( bit_se545348938243370406it_int @ M @ ( semiri1314217659103216013at_int @ N ) ) ) ).

% of_nat_push_bit
thf(fact_7494_of__nat__push__bit,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri1316708129612266289at_nat @ ( bit_se547839408752420682it_nat @ M @ N ) )
      = ( bit_se547839408752420682it_nat @ M @ ( semiri1316708129612266289at_nat @ N ) ) ) ).

% of_nat_push_bit
thf(fact_7495_push__bit__of__nat,axiom,
    ! [N: nat,M: nat] :
      ( ( bit_se545348938243370406it_int @ N @ ( semiri1314217659103216013at_int @ M ) )
      = ( semiri1314217659103216013at_int @ ( bit_se547839408752420682it_nat @ N @ M ) ) ) ).

% push_bit_of_nat
thf(fact_7496_push__bit__of__nat,axiom,
    ! [N: nat,M: nat] :
      ( ( bit_se547839408752420682it_nat @ N @ ( semiri1316708129612266289at_nat @ M ) )
      = ( semiri1316708129612266289at_nat @ ( bit_se547839408752420682it_nat @ N @ M ) ) ) ).

% push_bit_of_nat
thf(fact_7497_psubset__imp__ex__mem,axiom,
    ! [A3: set_Extended_enat,B4: set_Extended_enat] :
      ( ( ord_le2529575680413868914d_enat @ A3 @ B4 )
     => ? [B5: extended_enat] : ( member_Extended_enat @ B5 @ ( minus_925952699566721837d_enat @ B4 @ A3 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_7498_psubset__imp__ex__mem,axiom,
    ! [A3: set_complex,B4: set_complex] :
      ( ( ord_less_set_complex @ A3 @ B4 )
     => ? [B5: complex] : ( member_complex @ B5 @ ( minus_811609699411566653omplex @ B4 @ A3 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_7499_psubset__imp__ex__mem,axiom,
    ! [A3: set_real,B4: set_real] :
      ( ( ord_less_set_real @ A3 @ B4 )
     => ? [B5: real] : ( member_real @ B5 @ ( minus_minus_set_real @ B4 @ A3 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_7500_psubset__imp__ex__mem,axiom,
    ! [A3: set_int,B4: set_int] :
      ( ( ord_less_set_int @ A3 @ B4 )
     => ? [B5: int] : ( member_int @ B5 @ ( minus_minus_set_int @ B4 @ A3 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_7501_psubset__imp__ex__mem,axiom,
    ! [A3: set_nat,B4: set_nat] :
      ( ( ord_less_set_nat @ A3 @ B4 )
     => ? [B5: nat] : ( member_nat @ B5 @ ( minus_minus_set_nat @ B4 @ A3 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_7502_sinh__diff,axiom,
    ! [X4: real,Y3: real] :
      ( ( sinh_real @ ( minus_minus_real @ X4 @ Y3 ) )
      = ( minus_minus_real @ ( times_times_real @ ( sinh_real @ X4 ) @ ( cosh_real @ Y3 ) ) @ ( times_times_real @ ( cosh_real @ X4 ) @ ( sinh_real @ Y3 ) ) ) ) ).

% sinh_diff
thf(fact_7503_cosh__diff,axiom,
    ! [X4: real,Y3: real] :
      ( ( cosh_real @ ( minus_minus_real @ X4 @ Y3 ) )
      = ( minus_minus_real @ ( times_times_real @ ( cosh_real @ X4 ) @ ( cosh_real @ Y3 ) ) @ ( times_times_real @ ( sinh_real @ X4 ) @ ( sinh_real @ Y3 ) ) ) ) ).

% cosh_diff
thf(fact_7504_sinh__add,axiom,
    ! [X4: real,Y3: real] :
      ( ( sinh_real @ ( plus_plus_real @ X4 @ Y3 ) )
      = ( plus_plus_real @ ( times_times_real @ ( sinh_real @ X4 ) @ ( cosh_real @ Y3 ) ) @ ( times_times_real @ ( cosh_real @ X4 ) @ ( sinh_real @ Y3 ) ) ) ) ).

% sinh_add
thf(fact_7505_cosh__add,axiom,
    ! [X4: real,Y3: real] :
      ( ( cosh_real @ ( plus_plus_real @ X4 @ Y3 ) )
      = ( plus_plus_real @ ( times_times_real @ ( cosh_real @ X4 ) @ ( cosh_real @ Y3 ) ) @ ( times_times_real @ ( sinh_real @ X4 ) @ ( sinh_real @ Y3 ) ) ) ) ).

% cosh_add
thf(fact_7506_push__bit__nat__eq,axiom,
    ! [N: nat,K: int] :
      ( ( bit_se547839408752420682it_nat @ N @ ( nat2 @ K ) )
      = ( nat2 @ ( bit_se545348938243370406it_int @ N @ K ) ) ) ).

% push_bit_nat_eq
thf(fact_7507_push__bit__minus,axiom,
    ! [N: nat,A: code_integer] :
      ( ( bit_se7788150548672797655nteger @ N @ ( uminus1351360451143612070nteger @ A ) )
      = ( uminus1351360451143612070nteger @ ( bit_se7788150548672797655nteger @ N @ A ) ) ) ).

% push_bit_minus
thf(fact_7508_push__bit__minus,axiom,
    ! [N: nat,A: int] :
      ( ( bit_se545348938243370406it_int @ N @ ( uminus_uminus_int @ A ) )
      = ( uminus_uminus_int @ ( bit_se545348938243370406it_int @ N @ A ) ) ) ).

% push_bit_minus
thf(fact_7509_tanh__def,axiom,
    ( tanh_complex
    = ( ^ [X: complex] : ( divide1717551699836669952omplex @ ( sinh_complex @ X ) @ ( cosh_complex @ X ) ) ) ) ).

% tanh_def
thf(fact_7510_tanh__def,axiom,
    ( tanh_real
    = ( ^ [X: real] : ( divide_divide_real @ ( sinh_real @ X ) @ ( cosh_real @ X ) ) ) ) ).

% tanh_def
thf(fact_7511_bit__xor__int__iff,axiom,
    ! [K: int,L2: int,N: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_se6526347334894502574or_int @ K @ L2 ) @ N )
      = ( ( bit_se1146084159140164899it_int @ K @ N )
       != ( bit_se1146084159140164899it_int @ L2 @ N ) ) ) ).

% bit_xor_int_iff
thf(fact_7512_sinh__minus__cosh,axiom,
    ! [X4: real] :
      ( ( minus_minus_real @ ( sinh_real @ X4 ) @ ( cosh_real @ X4 ) )
      = ( uminus_uminus_real @ ( exp_real @ ( uminus_uminus_real @ X4 ) ) ) ) ).

% sinh_minus_cosh
thf(fact_7513_sinh__minus__cosh,axiom,
    ! [X4: complex] :
      ( ( minus_minus_complex @ ( sinh_complex @ X4 ) @ ( cosh_complex @ X4 ) )
      = ( uminus1482373934393186551omplex @ ( exp_complex @ ( uminus1482373934393186551omplex @ X4 ) ) ) ) ).

% sinh_minus_cosh
thf(fact_7514_cosh__minus__sinh,axiom,
    ! [X4: real] :
      ( ( minus_minus_real @ ( cosh_real @ X4 ) @ ( sinh_real @ X4 ) )
      = ( exp_real @ ( uminus_uminus_real @ X4 ) ) ) ).

% cosh_minus_sinh
thf(fact_7515_cosh__minus__sinh,axiom,
    ! [X4: complex] :
      ( ( minus_minus_complex @ ( cosh_complex @ X4 ) @ ( sinh_complex @ X4 ) )
      = ( exp_complex @ ( uminus1482373934393186551omplex @ X4 ) ) ) ).

% cosh_minus_sinh
thf(fact_7516_XOR__lower,axiom,
    ! [X4: int,Y3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X4 )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
       => ( ord_less_eq_int @ zero_zero_int @ ( bit_se6526347334894502574or_int @ X4 @ Y3 ) ) ) ) ).

% XOR_lower
thf(fact_7517_cosh__real__pos,axiom,
    ! [X4: real] : ( ord_less_real @ zero_zero_real @ ( cosh_real @ X4 ) ) ).

% cosh_real_pos
thf(fact_7518_cosh__real__nonneg,axiom,
    ! [X4: real] : ( ord_less_eq_real @ zero_zero_real @ ( cosh_real @ X4 ) ) ).

% cosh_real_nonneg
thf(fact_7519_cosh__real__nonneg__le__iff,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
       => ( ( ord_less_eq_real @ ( cosh_real @ X4 ) @ ( cosh_real @ Y3 ) )
          = ( ord_less_eq_real @ X4 @ Y3 ) ) ) ) ).

% cosh_real_nonneg_le_iff
thf(fact_7520_cosh__real__nonpos__le__iff,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_eq_real @ X4 @ zero_zero_real )
     => ( ( ord_less_eq_real @ Y3 @ zero_zero_real )
       => ( ( ord_less_eq_real @ ( cosh_real @ X4 ) @ ( cosh_real @ Y3 ) )
          = ( ord_less_eq_real @ Y3 @ X4 ) ) ) ) ).

% cosh_real_nonpos_le_iff
thf(fact_7521_cosh__real__ge__1,axiom,
    ! [X4: real] : ( ord_less_eq_real @ one_one_real @ ( cosh_real @ X4 ) ) ).

% cosh_real_ge_1
thf(fact_7522_sinh__double,axiom,
    ! [X4: complex] :
      ( ( sinh_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X4 ) )
      = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sinh_complex @ X4 ) ) @ ( cosh_complex @ X4 ) ) ) ).

% sinh_double
thf(fact_7523_sinh__double,axiom,
    ! [X4: real] :
      ( ( sinh_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X4 ) )
      = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sinh_real @ X4 ) ) @ ( cosh_real @ X4 ) ) ) ).

% sinh_double
thf(fact_7524_push__bit__take__bit,axiom,
    ! [M: nat,N: nat,A: int] :
      ( ( bit_se545348938243370406it_int @ M @ ( bit_se2923211474154528505it_int @ N @ A ) )
      = ( bit_se2923211474154528505it_int @ ( plus_plus_nat @ M @ N ) @ ( bit_se545348938243370406it_int @ M @ A ) ) ) ).

% push_bit_take_bit
thf(fact_7525_push__bit__take__bit,axiom,
    ! [M: nat,N: nat,A: nat] :
      ( ( bit_se547839408752420682it_nat @ M @ ( bit_se2925701944663578781it_nat @ N @ A ) )
      = ( bit_se2925701944663578781it_nat @ ( plus_plus_nat @ M @ N ) @ ( bit_se547839408752420682it_nat @ M @ A ) ) ) ).

% push_bit_take_bit
thf(fact_7526_take__bit__push__bit,axiom,
    ! [M: nat,N: nat,A: int] :
      ( ( bit_se2923211474154528505it_int @ M @ ( bit_se545348938243370406it_int @ N @ A ) )
      = ( bit_se545348938243370406it_int @ N @ ( bit_se2923211474154528505it_int @ ( minus_minus_nat @ M @ N ) @ A ) ) ) ).

% take_bit_push_bit
thf(fact_7527_take__bit__push__bit,axiom,
    ! [M: nat,N: nat,A: nat] :
      ( ( bit_se2925701944663578781it_nat @ M @ ( bit_se547839408752420682it_nat @ N @ A ) )
      = ( bit_se547839408752420682it_nat @ N @ ( bit_se2925701944663578781it_nat @ ( minus_minus_nat @ M @ N ) @ A ) ) ) ).

% take_bit_push_bit
thf(fact_7528_divide__complex__def,axiom,
    ( divide1717551699836669952omplex
    = ( ^ [X: complex,Y: complex] : ( times_times_complex @ X @ ( invers8013647133539491842omplex @ Y ) ) ) ) ).

% divide_complex_def
thf(fact_7529_flip__bit__nat__def,axiom,
    ( bit_se2161824704523386999it_nat
    = ( ^ [M6: nat,N2: nat] : ( bit_se6528837805403552850or_nat @ N2 @ ( bit_se547839408752420682it_nat @ M6 @ one_one_nat ) ) ) ) ).

% flip_bit_nat_def
thf(fact_7530_cosh__real__strict__mono,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_real @ X4 @ Y3 )
       => ( ord_less_real @ ( cosh_real @ X4 ) @ ( cosh_real @ Y3 ) ) ) ) ).

% cosh_real_strict_mono
thf(fact_7531_cosh__real__nonneg__less__iff,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
       => ( ( ord_less_real @ ( cosh_real @ X4 ) @ ( cosh_real @ Y3 ) )
          = ( ord_less_real @ X4 @ Y3 ) ) ) ) ).

% cosh_real_nonneg_less_iff
thf(fact_7532_cosh__real__nonpos__less__iff,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_eq_real @ X4 @ zero_zero_real )
     => ( ( ord_less_eq_real @ Y3 @ zero_zero_real )
       => ( ( ord_less_real @ ( cosh_real @ X4 ) @ ( cosh_real @ Y3 ) )
          = ( ord_less_real @ Y3 @ X4 ) ) ) ) ).

% cosh_real_nonpos_less_iff
thf(fact_7533_bit__push__bit__iff__int,axiom,
    ! [M: nat,K: int,N: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_se545348938243370406it_int @ M @ K ) @ N )
      = ( ( ord_less_eq_nat @ M @ N )
        & ( bit_se1146084159140164899it_int @ K @ ( minus_minus_nat @ N @ M ) ) ) ) ).

% bit_push_bit_iff_int
thf(fact_7534_cosh__square__eq,axiom,
    ! [X4: real] :
      ( ( power_power_real @ ( cosh_real @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_real @ ( power_power_real @ ( sinh_real @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ).

% cosh_square_eq
thf(fact_7535_cosh__square__eq,axiom,
    ! [X4: complex] :
      ( ( power_power_complex @ ( cosh_complex @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_complex @ ( power_power_complex @ ( sinh_complex @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_complex ) ) ).

% cosh_square_eq
thf(fact_7536_sinh__square__eq,axiom,
    ! [X4: complex] :
      ( ( power_power_complex @ ( sinh_complex @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_complex @ ( power_power_complex @ ( cosh_complex @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_complex ) ) ).

% sinh_square_eq
thf(fact_7537_sinh__square__eq,axiom,
    ! [X4: real] :
      ( ( power_power_real @ ( sinh_real @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_real @ ( power_power_real @ ( cosh_real @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ).

% sinh_square_eq
thf(fact_7538_hyperbolic__pythagoras,axiom,
    ! [X4: complex] :
      ( ( minus_minus_complex @ ( power_power_complex @ ( cosh_complex @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( sinh_complex @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = one_one_complex ) ).

% hyperbolic_pythagoras
thf(fact_7539_hyperbolic__pythagoras,axiom,
    ! [X4: real] :
      ( ( minus_minus_real @ ( power_power_real @ ( cosh_real @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( sinh_real @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = one_one_real ) ).

% hyperbolic_pythagoras
thf(fact_7540_xor__nat__def,axiom,
    ( bit_se6528837805403552850or_nat
    = ( ^ [M6: nat,N2: nat] : ( nat2 @ ( bit_se6526347334894502574or_int @ ( semiri1314217659103216013at_int @ M6 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).

% xor_nat_def
thf(fact_7541_psubsetE,axiom,
    ! [A3: set_nat,B4: set_nat] :
      ( ( ord_less_set_nat @ A3 @ B4 )
     => ~ ( ( ord_less_eq_set_nat @ A3 @ B4 )
         => ( ord_less_eq_set_nat @ B4 @ A3 ) ) ) ).

% psubsetE
thf(fact_7542_psubset__eq,axiom,
    ( ord_less_set_nat
    = ( ^ [A6: set_nat,B7: set_nat] :
          ( ( ord_less_eq_set_nat @ A6 @ B7 )
          & ( A6 != B7 ) ) ) ) ).

% psubset_eq
thf(fact_7543_psubset__imp__subset,axiom,
    ! [A3: set_nat,B4: set_nat] :
      ( ( ord_less_set_nat @ A3 @ B4 )
     => ( ord_less_eq_set_nat @ A3 @ B4 ) ) ).

% psubset_imp_subset
thf(fact_7544_psubset__subset__trans,axiom,
    ! [A3: set_nat,B4: set_nat,C5: set_nat] :
      ( ( ord_less_set_nat @ A3 @ B4 )
     => ( ( ord_less_eq_set_nat @ B4 @ C5 )
       => ( ord_less_set_nat @ A3 @ C5 ) ) ) ).

% psubset_subset_trans
thf(fact_7545_subset__not__subset__eq,axiom,
    ( ord_less_set_nat
    = ( ^ [A6: set_nat,B7: set_nat] :
          ( ( ord_less_eq_set_nat @ A6 @ B7 )
          & ~ ( ord_less_eq_set_nat @ B7 @ A6 ) ) ) ) ).

% subset_not_subset_eq
thf(fact_7546_subset__psubset__trans,axiom,
    ! [A3: set_nat,B4: set_nat,C5: set_nat] :
      ( ( ord_less_eq_set_nat @ A3 @ B4 )
     => ( ( ord_less_set_nat @ B4 @ C5 )
       => ( ord_less_set_nat @ A3 @ C5 ) ) ) ).

% subset_psubset_trans
thf(fact_7547_subset__iff__psubset__eq,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A6: set_nat,B7: set_nat] :
          ( ( ord_less_set_nat @ A6 @ B7 )
          | ( A6 = B7 ) ) ) ) ).

% subset_iff_psubset_eq
thf(fact_7548_double__diff,axiom,
    ! [A3: set_nat,B4: set_nat,C5: set_nat] :
      ( ( ord_less_eq_set_nat @ A3 @ B4 )
     => ( ( ord_less_eq_set_nat @ B4 @ C5 )
       => ( ( minus_minus_set_nat @ B4 @ ( minus_minus_set_nat @ C5 @ A3 ) )
          = A3 ) ) ) ).

% double_diff
thf(fact_7549_Diff__subset,axiom,
    ! [A3: set_nat,B4: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A3 @ B4 ) @ A3 ) ).

% Diff_subset
thf(fact_7550_Diff__mono,axiom,
    ! [A3: set_nat,C5: set_nat,D4: set_nat,B4: set_nat] :
      ( ( ord_less_eq_set_nat @ A3 @ C5 )
     => ( ( ord_less_eq_set_nat @ D4 @ B4 )
       => ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A3 @ B4 ) @ ( minus_minus_set_nat @ C5 @ D4 ) ) ) ) ).

% Diff_mono
thf(fact_7551_bit__push__bit__iff__nat,axiom,
    ! [M: nat,Q2: nat,N: nat] :
      ( ( bit_se1148574629649215175it_nat @ ( bit_se547839408752420682it_nat @ M @ Q2 ) @ N )
      = ( ( ord_less_eq_nat @ M @ N )
        & ( bit_se1148574629649215175it_nat @ Q2 @ ( minus_minus_nat @ N @ M ) ) ) ) ).

% bit_push_bit_iff_nat
thf(fact_7552_concat__bit__eq,axiom,
    ( bit_concat_bit
    = ( ^ [N2: nat,K3: int,L: int] : ( plus_plus_int @ ( bit_se2923211474154528505it_int @ N2 @ K3 ) @ ( bit_se545348938243370406it_int @ N2 @ L ) ) ) ) ).

% concat_bit_eq
thf(fact_7553_arcosh__cosh__real,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( arcosh_real @ ( cosh_real @ X4 ) )
        = X4 ) ) ).

% arcosh_cosh_real
thf(fact_7554_flip__bit__eq__xor,axiom,
    ( bit_se2159334234014336723it_int
    = ( ^ [N2: nat,A4: int] : ( bit_se6526347334894502574or_int @ A4 @ ( bit_se545348938243370406it_int @ N2 @ one_one_int ) ) ) ) ).

% flip_bit_eq_xor
thf(fact_7555_flip__bit__eq__xor,axiom,
    ( bit_se2161824704523386999it_nat
    = ( ^ [N2: nat,A4: nat] : ( bit_se6528837805403552850or_nat @ A4 @ ( bit_se547839408752420682it_nat @ N2 @ one_one_nat ) ) ) ) ).

% flip_bit_eq_xor
thf(fact_7556_cosh__double,axiom,
    ! [X4: complex] :
      ( ( cosh_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X4 ) )
      = ( plus_plus_complex @ ( power_power_complex @ ( cosh_complex @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( sinh_complex @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% cosh_double
thf(fact_7557_cosh__double,axiom,
    ! [X4: real] :
      ( ( cosh_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X4 ) )
      = ( plus_plus_real @ ( power_power_real @ ( cosh_real @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( sinh_real @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% cosh_double
thf(fact_7558_push__bit__double,axiom,
    ! [N: nat,A: int] :
      ( ( bit_se545348938243370406it_int @ N @ ( times_times_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
      = ( times_times_int @ ( bit_se545348938243370406it_int @ N @ A ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% push_bit_double
thf(fact_7559_push__bit__double,axiom,
    ! [N: nat,A: nat] :
      ( ( bit_se547839408752420682it_nat @ N @ ( times_times_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( times_times_nat @ ( bit_se547839408752420682it_nat @ N @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% push_bit_double
thf(fact_7560_bit__iff__and__push__bit__not__eq__0,axiom,
    ( bit_se1146084159140164899it_int
    = ( ^ [A4: int,N2: nat] :
          ( ( bit_se725231765392027082nd_int @ A4 @ ( bit_se545348938243370406it_int @ N2 @ one_one_int ) )
         != zero_zero_int ) ) ) ).

% bit_iff_and_push_bit_not_eq_0
thf(fact_7561_bit__iff__and__push__bit__not__eq__0,axiom,
    ( bit_se1148574629649215175it_nat
    = ( ^ [A4: nat,N2: nat] :
          ( ( bit_se727722235901077358nd_nat @ A4 @ ( bit_se547839408752420682it_nat @ N2 @ one_one_nat ) )
         != zero_zero_nat ) ) ) ).

% bit_iff_and_push_bit_not_eq_0
thf(fact_7562_push__bit__int__def,axiom,
    ( bit_se545348938243370406it_int
    = ( ^ [N2: nat,K3: int] : ( times_times_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% push_bit_int_def
thf(fact_7563_push__bit__nat__def,axiom,
    ( bit_se547839408752420682it_nat
    = ( ^ [N2: nat,M6: nat] : ( times_times_nat @ M6 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% push_bit_nat_def
thf(fact_7564_push__bit__eq__mult,axiom,
    ( bit_se545348938243370406it_int
    = ( ^ [N2: nat,A4: int] : ( times_times_int @ A4 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% push_bit_eq_mult
thf(fact_7565_push__bit__eq__mult,axiom,
    ( bit_se547839408752420682it_nat
    = ( ^ [N2: nat,A4: nat] : ( times_times_nat @ A4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% push_bit_eq_mult
thf(fact_7566_exp__dvdE,axiom,
    ! [N: nat,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) @ A )
     => ~ ! [B5: code_integer] :
            ( A
           != ( bit_se7788150548672797655nteger @ N @ B5 ) ) ) ).

% exp_dvdE
thf(fact_7567_exp__dvdE,axiom,
    ! [N: nat,A: int] :
      ( ( dvd_dvd_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ A )
     => ~ ! [B5: int] :
            ( A
           != ( bit_se545348938243370406it_int @ N @ B5 ) ) ) ).

% exp_dvdE
thf(fact_7568_exp__dvdE,axiom,
    ! [N: nat,A: nat] :
      ( ( dvd_dvd_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ A )
     => ~ ! [B5: nat] :
            ( A
           != ( bit_se547839408752420682it_nat @ N @ B5 ) ) ) ).

% exp_dvdE
thf(fact_7569_push__bit__minus__one,axiom,
    ! [N: nat] :
      ( ( bit_se545348938243370406it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
      = ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).

% push_bit_minus_one
thf(fact_7570_tanh__add,axiom,
    ! [X4: complex,Y3: complex] :
      ( ( ( cosh_complex @ X4 )
       != zero_zero_complex )
     => ( ( ( cosh_complex @ Y3 )
         != zero_zero_complex )
       => ( ( tanh_complex @ ( plus_plus_complex @ X4 @ Y3 ) )
          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( tanh_complex @ X4 ) @ ( tanh_complex @ Y3 ) ) @ ( plus_plus_complex @ one_one_complex @ ( times_times_complex @ ( tanh_complex @ X4 ) @ ( tanh_complex @ Y3 ) ) ) ) ) ) ) ).

% tanh_add
thf(fact_7571_tanh__add,axiom,
    ! [X4: real,Y3: real] :
      ( ( ( cosh_real @ X4 )
       != zero_zero_real )
     => ( ( ( cosh_real @ Y3 )
         != zero_zero_real )
       => ( ( tanh_real @ ( plus_plus_real @ X4 @ Y3 ) )
          = ( divide_divide_real @ ( plus_plus_real @ ( tanh_real @ X4 ) @ ( tanh_real @ Y3 ) ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( tanh_real @ X4 ) @ ( tanh_real @ Y3 ) ) ) ) ) ) ) ).

% tanh_add
thf(fact_7572_XOR__upper,axiom,
    ! [X4: int,N: nat,Y3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X4 )
     => ( ( ord_less_int @ X4 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
       => ( ( ord_less_int @ Y3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
         => ( ord_less_int @ ( bit_se6526347334894502574or_int @ X4 @ Y3 ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).

% XOR_upper
thf(fact_7573_cosh__field__def,axiom,
    ( cosh_real
    = ( ^ [Z5: real] : ( divide_divide_real @ ( plus_plus_real @ ( exp_real @ Z5 ) @ ( exp_real @ ( uminus_uminus_real @ Z5 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% cosh_field_def
thf(fact_7574_cosh__field__def,axiom,
    ( cosh_complex
    = ( ^ [Z5: complex] : ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( exp_complex @ Z5 ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ Z5 ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ).

% cosh_field_def
thf(fact_7575_complex__inverse,axiom,
    ! [A: real,B2: real] :
      ( ( invers8013647133539491842omplex @ ( complex2 @ A @ B2 ) )
      = ( complex2 @ ( divide_divide_real @ A @ ( plus_plus_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ B2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( divide_divide_real @ ( uminus_uminus_real @ B2 ) @ ( plus_plus_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ B2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% complex_inverse
thf(fact_7576_xor__int__rec,axiom,
    ( bit_se6526347334894502574or_int
    = ( ^ [K3: int,L: int] :
          ( plus_plus_int
          @ ( zero_n2684676970156552555ol_int
            @ ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 ) )
             != ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) ) )
          @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% xor_int_rec
thf(fact_7577_sinh__field__def,axiom,
    ( sinh_real
    = ( ^ [Z5: real] : ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ Z5 ) @ ( exp_real @ ( uminus_uminus_real @ Z5 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% sinh_field_def
thf(fact_7578_sinh__field__def,axiom,
    ( sinh_complex
    = ( ^ [Z5: complex] : ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( exp_complex @ Z5 ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ Z5 ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ).

% sinh_field_def
thf(fact_7579_xor__int__unfold,axiom,
    ( bit_se6526347334894502574or_int
    = ( ^ [K3: int,L: int] :
          ( if_int
          @ ( K3
            = ( uminus_uminus_int @ one_one_int ) )
          @ ( bit_ri7919022796975470100ot_int @ L )
          @ ( if_int
            @ ( L
              = ( uminus_uminus_int @ one_one_int ) )
            @ ( bit_ri7919022796975470100ot_int @ K3 )
            @ ( if_int @ ( K3 = zero_zero_int ) @ L @ ( if_int @ ( L = zero_zero_int ) @ K3 @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ).

% xor_int_unfold
thf(fact_7580_True,axiom,
    ord_less_nat @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ treeList ) ).

% True
thf(fact_7581_Cauchy__iff2,axiom,
    ( topolo4055970368930404560y_real
    = ( ^ [X6: nat > real] :
        ! [J3: nat] :
        ? [M8: nat] :
        ! [M6: nat] :
          ( ( ord_less_eq_nat @ M8 @ M6 )
         => ! [N2: nat] :
              ( ( ord_less_eq_nat @ M8 @ N2 )
             => ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ ( X6 @ M6 ) @ ( X6 @ N2 ) ) ) @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ J3 ) ) ) ) ) ) ) ) ).

% Cauchy_iff2
thf(fact_7582_dual__order_Orefl,axiom,
    ! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).

% dual_order.refl
thf(fact_7583_dual__order_Orefl,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ A @ A ) ).

% dual_order.refl
thf(fact_7584_dual__order_Orefl,axiom,
    ! [A: num] : ( ord_less_eq_num @ A @ A ) ).

% dual_order.refl
thf(fact_7585_dual__order_Orefl,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).

% dual_order.refl
thf(fact_7586_dual__order_Orefl,axiom,
    ! [A: int] : ( ord_less_eq_int @ A @ A ) ).

% dual_order.refl
thf(fact_7587_order__refl,axiom,
    ! [X4: set_nat] : ( ord_less_eq_set_nat @ X4 @ X4 ) ).

% order_refl
thf(fact_7588_order__refl,axiom,
    ! [X4: rat] : ( ord_less_eq_rat @ X4 @ X4 ) ).

% order_refl
thf(fact_7589_order__refl,axiom,
    ! [X4: num] : ( ord_less_eq_num @ X4 @ X4 ) ).

% order_refl
thf(fact_7590_order__refl,axiom,
    ! [X4: nat] : ( ord_less_eq_nat @ X4 @ X4 ) ).

% order_refl
thf(fact_7591_order__refl,axiom,
    ! [X4: int] : ( ord_less_eq_int @ X4 @ X4 ) ).

% order_refl
thf(fact_7592__C5_Ohyps_C_I4_J,axiom,
    ( ( size_s6755466524823107622T_VEBT @ treeList )
    = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) ) ).

% "5.hyps"(4)
thf(fact_7593_DiffI,axiom,
    ! [C: extended_enat,A3: set_Extended_enat,B4: set_Extended_enat] :
      ( ( member_Extended_enat @ C @ A3 )
     => ( ~ ( member_Extended_enat @ C @ B4 )
       => ( member_Extended_enat @ C @ ( minus_925952699566721837d_enat @ A3 @ B4 ) ) ) ) ).

% DiffI
thf(fact_7594_DiffI,axiom,
    ! [C: complex,A3: set_complex,B4: set_complex] :
      ( ( member_complex @ C @ A3 )
     => ( ~ ( member_complex @ C @ B4 )
       => ( member_complex @ C @ ( minus_811609699411566653omplex @ A3 @ B4 ) ) ) ) ).

% DiffI
thf(fact_7595_DiffI,axiom,
    ! [C: real,A3: set_real,B4: set_real] :
      ( ( member_real @ C @ A3 )
     => ( ~ ( member_real @ C @ B4 )
       => ( member_real @ C @ ( minus_minus_set_real @ A3 @ B4 ) ) ) ) ).

% DiffI
thf(fact_7596_DiffI,axiom,
    ! [C: int,A3: set_int,B4: set_int] :
      ( ( member_int @ C @ A3 )
     => ( ~ ( member_int @ C @ B4 )
       => ( member_int @ C @ ( minus_minus_set_int @ A3 @ B4 ) ) ) ) ).

% DiffI
thf(fact_7597_DiffI,axiom,
    ! [C: nat,A3: set_nat,B4: set_nat] :
      ( ( member_nat @ C @ A3 )
     => ( ~ ( member_nat @ C @ B4 )
       => ( member_nat @ C @ ( minus_minus_set_nat @ A3 @ B4 ) ) ) ) ).

% DiffI
thf(fact_7598_Diff__iff,axiom,
    ! [C: extended_enat,A3: set_Extended_enat,B4: set_Extended_enat] :
      ( ( member_Extended_enat @ C @ ( minus_925952699566721837d_enat @ A3 @ B4 ) )
      = ( ( member_Extended_enat @ C @ A3 )
        & ~ ( member_Extended_enat @ C @ B4 ) ) ) ).

% Diff_iff
thf(fact_7599_Diff__iff,axiom,
    ! [C: complex,A3: set_complex,B4: set_complex] :
      ( ( member_complex @ C @ ( minus_811609699411566653omplex @ A3 @ B4 ) )
      = ( ( member_complex @ C @ A3 )
        & ~ ( member_complex @ C @ B4 ) ) ) ).

% Diff_iff
thf(fact_7600_Diff__iff,axiom,
    ! [C: real,A3: set_real,B4: set_real] :
      ( ( member_real @ C @ ( minus_minus_set_real @ A3 @ B4 ) )
      = ( ( member_real @ C @ A3 )
        & ~ ( member_real @ C @ B4 ) ) ) ).

% Diff_iff
thf(fact_7601_Diff__iff,axiom,
    ! [C: int,A3: set_int,B4: set_int] :
      ( ( member_int @ C @ ( minus_minus_set_int @ A3 @ B4 ) )
      = ( ( member_int @ C @ A3 )
        & ~ ( member_int @ C @ B4 ) ) ) ).

% Diff_iff
thf(fact_7602_Diff__iff,axiom,
    ! [C: nat,A3: set_nat,B4: set_nat] :
      ( ( member_nat @ C @ ( minus_minus_set_nat @ A3 @ B4 ) )
      = ( ( member_nat @ C @ A3 )
        & ~ ( member_nat @ C @ B4 ) ) ) ).

% Diff_iff
thf(fact_7603_Diff__idemp,axiom,
    ! [A3: set_nat,B4: set_nat] :
      ( ( minus_minus_set_nat @ ( minus_minus_set_nat @ A3 @ B4 ) @ B4 )
      = ( minus_minus_set_nat @ A3 @ B4 ) ) ).

% Diff_idemp
thf(fact_7604_bit_Ocompl__eq__compl__iff,axiom,
    ! [X4: int,Y3: int] :
      ( ( ( bit_ri7919022796975470100ot_int @ X4 )
        = ( bit_ri7919022796975470100ot_int @ Y3 ) )
      = ( X4 = Y3 ) ) ).

% bit.compl_eq_compl_iff
thf(fact_7605_bit_Odouble__compl,axiom,
    ! [X4: int] :
      ( ( bit_ri7919022796975470100ot_int @ ( bit_ri7919022796975470100ot_int @ X4 ) )
      = X4 ) ).

% bit.double_compl
thf(fact_7606_bit_Oxor__compl__right,axiom,
    ! [X4: int,Y3: int] :
      ( ( bit_se6526347334894502574or_int @ X4 @ ( bit_ri7919022796975470100ot_int @ Y3 ) )
      = ( bit_ri7919022796975470100ot_int @ ( bit_se6526347334894502574or_int @ X4 @ Y3 ) ) ) ).

% bit.xor_compl_right
thf(fact_7607_bit_Oxor__compl__left,axiom,
    ! [X4: int,Y3: int] :
      ( ( bit_se6526347334894502574or_int @ ( bit_ri7919022796975470100ot_int @ X4 ) @ Y3 )
      = ( bit_ri7919022796975470100ot_int @ ( bit_se6526347334894502574or_int @ X4 @ Y3 ) ) ) ).

% bit.xor_compl_left
thf(fact_7608_bit_Oconj__cancel__right,axiom,
    ! [X4: int] :
      ( ( bit_se725231765392027082nd_int @ X4 @ ( bit_ri7919022796975470100ot_int @ X4 ) )
      = zero_zero_int ) ).

% bit.conj_cancel_right
thf(fact_7609_bit_Oconj__cancel__left,axiom,
    ! [X4: int] :
      ( ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ X4 ) @ X4 )
      = zero_zero_int ) ).

% bit.conj_cancel_left
thf(fact_7610_bit_Ocompl__zero,axiom,
    ( ( bit_ri7632146776885996613nteger @ zero_z3403309356797280102nteger )
    = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% bit.compl_zero
thf(fact_7611_bit_Ocompl__zero,axiom,
    ( ( bit_ri7919022796975470100ot_int @ zero_zero_int )
    = ( uminus_uminus_int @ one_one_int ) ) ).

% bit.compl_zero
thf(fact_7612_bit_Ocompl__one,axiom,
    ( ( bit_ri7632146776885996613nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = zero_z3403309356797280102nteger ) ).

% bit.compl_one
thf(fact_7613_bit_Ocompl__one,axiom,
    ( ( bit_ri7919022796975470100ot_int @ ( uminus_uminus_int @ one_one_int ) )
    = zero_zero_int ) ).

% bit.compl_one
thf(fact_7614_bit_Oxor__one__left,axiom,
    ! [X4: code_integer] :
      ( ( bit_se3222712562003087583nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ X4 )
      = ( bit_ri7632146776885996613nteger @ X4 ) ) ).

% bit.xor_one_left
thf(fact_7615_bit_Oxor__one__left,axiom,
    ! [X4: int] :
      ( ( bit_se6526347334894502574or_int @ ( uminus_uminus_int @ one_one_int ) @ X4 )
      = ( bit_ri7919022796975470100ot_int @ X4 ) ) ).

% bit.xor_one_left
thf(fact_7616_bit_Oxor__one__right,axiom,
    ! [X4: code_integer] :
      ( ( bit_se3222712562003087583nteger @ X4 @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( bit_ri7632146776885996613nteger @ X4 ) ) ).

% bit.xor_one_right
thf(fact_7617_bit_Oxor__one__right,axiom,
    ! [X4: int] :
      ( ( bit_se6526347334894502574or_int @ X4 @ ( uminus_uminus_int @ one_one_int ) )
      = ( bit_ri7919022796975470100ot_int @ X4 ) ) ).

% bit.xor_one_right
thf(fact_7618_bit_Oxor__cancel__left,axiom,
    ! [X4: code_integer] :
      ( ( bit_se3222712562003087583nteger @ ( bit_ri7632146776885996613nteger @ X4 ) @ X4 )
      = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% bit.xor_cancel_left
thf(fact_7619_bit_Oxor__cancel__left,axiom,
    ! [X4: int] :
      ( ( bit_se6526347334894502574or_int @ ( bit_ri7919022796975470100ot_int @ X4 ) @ X4 )
      = ( uminus_uminus_int @ one_one_int ) ) ).

% bit.xor_cancel_left
thf(fact_7620_bit_Oxor__cancel__right,axiom,
    ! [X4: code_integer] :
      ( ( bit_se3222712562003087583nteger @ X4 @ ( bit_ri7632146776885996613nteger @ X4 ) )
      = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% bit.xor_cancel_right
thf(fact_7621_bit_Oxor__cancel__right,axiom,
    ! [X4: int] :
      ( ( bit_se6526347334894502574or_int @ X4 @ ( bit_ri7919022796975470100ot_int @ X4 ) )
      = ( uminus_uminus_int @ one_one_int ) ) ).

% bit.xor_cancel_right
thf(fact_7622_not__negative__int__iff,axiom,
    ! [K: int] :
      ( ( ord_less_int @ ( bit_ri7919022796975470100ot_int @ K ) @ zero_zero_int )
      = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).

% not_negative_int_iff
thf(fact_7623_not__nonnegative__int__iff,axiom,
    ! [K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_ri7919022796975470100ot_int @ K ) )
      = ( ord_less_int @ K @ zero_zero_int ) ) ).

% not_nonnegative_int_iff
thf(fact_7624_minus__not__numeral__eq,axiom,
    ! [N: num] :
      ( ( uminus1351360451143612070nteger @ ( bit_ri7632146776885996613nteger @ ( numera6620942414471956472nteger @ N ) ) )
      = ( numera6620942414471956472nteger @ ( inc @ N ) ) ) ).

% minus_not_numeral_eq
thf(fact_7625_minus__not__numeral__eq,axiom,
    ! [N: num] :
      ( ( uminus_uminus_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
      = ( numeral_numeral_int @ ( inc @ N ) ) ) ).

% minus_not_numeral_eq
thf(fact_7626_even__not__iff,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri7632146776885996613nteger @ A ) )
      = ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).

% even_not_iff
thf(fact_7627_even__not__iff,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri7919022796975470100ot_int @ A ) )
      = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).

% even_not_iff
thf(fact_7628_push__bit__minus__one__eq__not__mask,axiom,
    ! [N: nat] :
      ( ( bit_se7788150548672797655nteger @ N @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( bit_ri7632146776885996613nteger @ ( bit_se2119862282449309892nteger @ N ) ) ) ).

% push_bit_minus_one_eq_not_mask
thf(fact_7629_push__bit__minus__one__eq__not__mask,axiom,
    ! [N: nat] :
      ( ( bit_se545348938243370406it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
      = ( bit_ri7919022796975470100ot_int @ ( bit_se2000444600071755411sk_int @ N ) ) ) ).

% push_bit_minus_one_eq_not_mask
thf(fact_7630_not__one__eq,axiom,
    ( ( bit_ri7632146776885996613nteger @ one_one_Code_integer )
    = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% not_one_eq
thf(fact_7631_not__one__eq,axiom,
    ( ( bit_ri7919022796975470100ot_int @ one_one_int )
    = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% not_one_eq
thf(fact_7632_length__induct,axiom,
    ! [P: list_VEBT_VEBT > $o,Xs2: list_VEBT_VEBT] :
      ( ! [Xs3: list_VEBT_VEBT] :
          ( ! [Ys: list_VEBT_VEBT] :
              ( ( ord_less_nat @ ( size_s6755466524823107622T_VEBT @ Ys ) @ ( size_s6755466524823107622T_VEBT @ Xs3 ) )
             => ( P @ Ys ) )
         => ( P @ Xs3 ) )
     => ( P @ Xs2 ) ) ).

% length_induct
thf(fact_7633_length__induct,axiom,
    ! [P: list_o > $o,Xs2: list_o] :
      ( ! [Xs3: list_o] :
          ( ! [Ys: list_o] :
              ( ( ord_less_nat @ ( size_size_list_o @ Ys ) @ ( size_size_list_o @ Xs3 ) )
             => ( P @ Ys ) )
         => ( P @ Xs3 ) )
     => ( P @ Xs2 ) ) ).

% length_induct
thf(fact_7634_length__induct,axiom,
    ! [P: list_nat > $o,Xs2: list_nat] :
      ( ! [Xs3: list_nat] :
          ( ! [Ys: list_nat] :
              ( ( ord_less_nat @ ( size_size_list_nat @ Ys ) @ ( size_size_list_nat @ Xs3 ) )
             => ( P @ Ys ) )
         => ( P @ Xs3 ) )
     => ( P @ Xs2 ) ) ).

% length_induct
thf(fact_7635_length__induct,axiom,
    ! [P: list_int > $o,Xs2: list_int] :
      ( ! [Xs3: list_int] :
          ( ! [Ys: list_int] :
              ( ( ord_less_nat @ ( size_size_list_int @ Ys ) @ ( size_size_list_int @ Xs3 ) )
             => ( P @ Ys ) )
         => ( P @ Xs3 ) )
     => ( P @ Xs2 ) ) ).

% length_induct
thf(fact_7636_DiffE,axiom,
    ! [C: extended_enat,A3: set_Extended_enat,B4: set_Extended_enat] :
      ( ( member_Extended_enat @ C @ ( minus_925952699566721837d_enat @ A3 @ B4 ) )
     => ~ ( ( member_Extended_enat @ C @ A3 )
         => ( member_Extended_enat @ C @ B4 ) ) ) ).

% DiffE
thf(fact_7637_DiffE,axiom,
    ! [C: complex,A3: set_complex,B4: set_complex] :
      ( ( member_complex @ C @ ( minus_811609699411566653omplex @ A3 @ B4 ) )
     => ~ ( ( member_complex @ C @ A3 )
         => ( member_complex @ C @ B4 ) ) ) ).

% DiffE
thf(fact_7638_DiffE,axiom,
    ! [C: real,A3: set_real,B4: set_real] :
      ( ( member_real @ C @ ( minus_minus_set_real @ A3 @ B4 ) )
     => ~ ( ( member_real @ C @ A3 )
         => ( member_real @ C @ B4 ) ) ) ).

% DiffE
thf(fact_7639_DiffE,axiom,
    ! [C: int,A3: set_int,B4: set_int] :
      ( ( member_int @ C @ ( minus_minus_set_int @ A3 @ B4 ) )
     => ~ ( ( member_int @ C @ A3 )
         => ( member_int @ C @ B4 ) ) ) ).

% DiffE
thf(fact_7640_DiffE,axiom,
    ! [C: nat,A3: set_nat,B4: set_nat] :
      ( ( member_nat @ C @ ( minus_minus_set_nat @ A3 @ B4 ) )
     => ~ ( ( member_nat @ C @ A3 )
         => ( member_nat @ C @ B4 ) ) ) ).

% DiffE
thf(fact_7641_DiffD1,axiom,
    ! [C: extended_enat,A3: set_Extended_enat,B4: set_Extended_enat] :
      ( ( member_Extended_enat @ C @ ( minus_925952699566721837d_enat @ A3 @ B4 ) )
     => ( member_Extended_enat @ C @ A3 ) ) ).

% DiffD1
thf(fact_7642_DiffD1,axiom,
    ! [C: complex,A3: set_complex,B4: set_complex] :
      ( ( member_complex @ C @ ( minus_811609699411566653omplex @ A3 @ B4 ) )
     => ( member_complex @ C @ A3 ) ) ).

% DiffD1
thf(fact_7643_DiffD1,axiom,
    ! [C: real,A3: set_real,B4: set_real] :
      ( ( member_real @ C @ ( minus_minus_set_real @ A3 @ B4 ) )
     => ( member_real @ C @ A3 ) ) ).

% DiffD1
thf(fact_7644_DiffD1,axiom,
    ! [C: int,A3: set_int,B4: set_int] :
      ( ( member_int @ C @ ( minus_minus_set_int @ A3 @ B4 ) )
     => ( member_int @ C @ A3 ) ) ).

% DiffD1
thf(fact_7645_DiffD1,axiom,
    ! [C: nat,A3: set_nat,B4: set_nat] :
      ( ( member_nat @ C @ ( minus_minus_set_nat @ A3 @ B4 ) )
     => ( member_nat @ C @ A3 ) ) ).

% DiffD1
thf(fact_7646_DiffD2,axiom,
    ! [C: extended_enat,A3: set_Extended_enat,B4: set_Extended_enat] :
      ( ( member_Extended_enat @ C @ ( minus_925952699566721837d_enat @ A3 @ B4 ) )
     => ~ ( member_Extended_enat @ C @ B4 ) ) ).

% DiffD2
thf(fact_7647_DiffD2,axiom,
    ! [C: complex,A3: set_complex,B4: set_complex] :
      ( ( member_complex @ C @ ( minus_811609699411566653omplex @ A3 @ B4 ) )
     => ~ ( member_complex @ C @ B4 ) ) ).

% DiffD2
thf(fact_7648_DiffD2,axiom,
    ! [C: real,A3: set_real,B4: set_real] :
      ( ( member_real @ C @ ( minus_minus_set_real @ A3 @ B4 ) )
     => ~ ( member_real @ C @ B4 ) ) ).

% DiffD2
thf(fact_7649_DiffD2,axiom,
    ! [C: int,A3: set_int,B4: set_int] :
      ( ( member_int @ C @ ( minus_minus_set_int @ A3 @ B4 ) )
     => ~ ( member_int @ C @ B4 ) ) ).

% DiffD2
thf(fact_7650_DiffD2,axiom,
    ! [C: nat,A3: set_nat,B4: set_nat] :
      ( ( member_nat @ C @ ( minus_minus_set_nat @ A3 @ B4 ) )
     => ~ ( member_nat @ C @ B4 ) ) ).

% DiffD2
thf(fact_7651_take__bit__not__iff,axiom,
    ! [N: nat,A: int,B2: int] :
      ( ( ( bit_se2923211474154528505it_int @ N @ ( bit_ri7919022796975470100ot_int @ A ) )
        = ( bit_se2923211474154528505it_int @ N @ ( bit_ri7919022796975470100ot_int @ B2 ) ) )
      = ( ( bit_se2923211474154528505it_int @ N @ A )
        = ( bit_se2923211474154528505it_int @ N @ B2 ) ) ) ).

% take_bit_not_iff
thf(fact_7652_take__bit__not__take__bit,axiom,
    ! [N: nat,A: int] :
      ( ( bit_se2923211474154528505it_int @ N @ ( bit_ri7919022796975470100ot_int @ ( bit_se2923211474154528505it_int @ N @ A ) ) )
      = ( bit_se2923211474154528505it_int @ N @ ( bit_ri7919022796975470100ot_int @ A ) ) ) ).

% take_bit_not_take_bit
thf(fact_7653_bit__not__int__iff,axiom,
    ! [K: int,N: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_ri7919022796975470100ot_int @ K ) @ N )
      = ( ~ ( bit_se1146084159140164899it_int @ K @ N ) ) ) ).

% bit_not_int_iff
thf(fact_7654_of__int__not__eq,axiom,
    ! [K: int] :
      ( ( ring_1_of_int_int @ ( bit_ri7919022796975470100ot_int @ K ) )
      = ( bit_ri7919022796975470100ot_int @ ( ring_1_of_int_int @ K ) ) ) ).

% of_int_not_eq
thf(fact_7655_size__neq__size__imp__neq,axiom,
    ! [X4: list_VEBT_VEBT,Y3: list_VEBT_VEBT] :
      ( ( ( size_s6755466524823107622T_VEBT @ X4 )
       != ( size_s6755466524823107622T_VEBT @ Y3 ) )
     => ( X4 != Y3 ) ) ).

% size_neq_size_imp_neq
thf(fact_7656_size__neq__size__imp__neq,axiom,
    ! [X4: list_o,Y3: list_o] :
      ( ( ( size_size_list_o @ X4 )
       != ( size_size_list_o @ Y3 ) )
     => ( X4 != Y3 ) ) ).

% size_neq_size_imp_neq
thf(fact_7657_size__neq__size__imp__neq,axiom,
    ! [X4: list_nat,Y3: list_nat] :
      ( ( ( size_size_list_nat @ X4 )
       != ( size_size_list_nat @ Y3 ) )
     => ( X4 != Y3 ) ) ).

% size_neq_size_imp_neq
thf(fact_7658_size__neq__size__imp__neq,axiom,
    ! [X4: list_int,Y3: list_int] :
      ( ( ( size_size_list_int @ X4 )
       != ( size_size_list_int @ Y3 ) )
     => ( X4 != Y3 ) ) ).

% size_neq_size_imp_neq
thf(fact_7659_size__neq__size__imp__neq,axiom,
    ! [X4: num,Y3: num] :
      ( ( ( size_size_num @ X4 )
       != ( size_size_num @ Y3 ) )
     => ( X4 != Y3 ) ) ).

% size_neq_size_imp_neq
thf(fact_7660_of__int__not__numeral,axiom,
    ! [K: num] :
      ( ( ring_1_of_int_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ K ) ) )
      = ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ K ) ) ) ).

% of_int_not_numeral
thf(fact_7661_not__add__distrib,axiom,
    ! [A: int,B2: int] :
      ( ( bit_ri7919022796975470100ot_int @ ( plus_plus_int @ A @ B2 ) )
      = ( minus_minus_int @ ( bit_ri7919022796975470100ot_int @ A ) @ B2 ) ) ).

% not_add_distrib
thf(fact_7662_not__diff__distrib,axiom,
    ! [A: int,B2: int] :
      ( ( bit_ri7919022796975470100ot_int @ ( minus_minus_int @ A @ B2 ) )
      = ( plus_plus_int @ ( bit_ri7919022796975470100ot_int @ A ) @ B2 ) ) ).

% not_diff_distrib
thf(fact_7663_minus__eq__not__plus__1,axiom,
    ( uminus1351360451143612070nteger
    = ( ^ [A4: code_integer] : ( plus_p5714425477246183910nteger @ ( bit_ri7632146776885996613nteger @ A4 ) @ one_one_Code_integer ) ) ) ).

% minus_eq_not_plus_1
thf(fact_7664_minus__eq__not__plus__1,axiom,
    ( uminus_uminus_int
    = ( ^ [A4: int] : ( plus_plus_int @ ( bit_ri7919022796975470100ot_int @ A4 ) @ one_one_int ) ) ) ).

% minus_eq_not_plus_1
thf(fact_7665_not__eq__complement,axiom,
    ( bit_ri7632146776885996613nteger
    = ( ^ [A4: code_integer] : ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ A4 ) @ one_one_Code_integer ) ) ) ).

% not_eq_complement
thf(fact_7666_not__eq__complement,axiom,
    ( bit_ri7919022796975470100ot_int
    = ( ^ [A4: int] : ( minus_minus_int @ ( uminus_uminus_int @ A4 ) @ one_one_int ) ) ) ).

% not_eq_complement
thf(fact_7667_minus__eq__not__minus__1,axiom,
    ( uminus1351360451143612070nteger
    = ( ^ [A4: code_integer] : ( bit_ri7632146776885996613nteger @ ( minus_8373710615458151222nteger @ A4 @ one_one_Code_integer ) ) ) ) ).

% minus_eq_not_minus_1
thf(fact_7668_minus__eq__not__minus__1,axiom,
    ( uminus_uminus_int
    = ( ^ [A4: int] : ( bit_ri7919022796975470100ot_int @ ( minus_minus_int @ A4 @ one_one_int ) ) ) ) ).

% minus_eq_not_minus_1
thf(fact_7669_not__int__def,axiom,
    ( bit_ri7919022796975470100ot_int
    = ( ^ [K3: int] : ( minus_minus_int @ ( uminus_uminus_int @ K3 ) @ one_one_int ) ) ) ).

% not_int_def
thf(fact_7670_and__not__numerals_I1_J,axiom,
    ( ( bit_se725231765392027082nd_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
    = zero_zero_int ) ).

% and_not_numerals(1)
thf(fact_7671_disjunctive__diff,axiom,
    ! [B2: int,A: int] :
      ( ! [N4: nat] :
          ( ( bit_se1146084159140164899it_int @ B2 @ N4 )
         => ( bit_se1146084159140164899it_int @ A @ N4 ) )
     => ( ( minus_minus_int @ A @ B2 )
        = ( bit_se725231765392027082nd_int @ A @ ( bit_ri7919022796975470100ot_int @ B2 ) ) ) ) ).

% disjunctive_diff
thf(fact_7672_nle__le,axiom,
    ! [A: rat,B2: rat] :
      ( ( ~ ( ord_less_eq_rat @ A @ B2 ) )
      = ( ( ord_less_eq_rat @ B2 @ A )
        & ( B2 != A ) ) ) ).

% nle_le
thf(fact_7673_nle__le,axiom,
    ! [A: num,B2: num] :
      ( ( ~ ( ord_less_eq_num @ A @ B2 ) )
      = ( ( ord_less_eq_num @ B2 @ A )
        & ( B2 != A ) ) ) ).

% nle_le
thf(fact_7674_nle__le,axiom,
    ! [A: nat,B2: nat] :
      ( ( ~ ( ord_less_eq_nat @ A @ B2 ) )
      = ( ( ord_less_eq_nat @ B2 @ A )
        & ( B2 != A ) ) ) ).

% nle_le
thf(fact_7675_nle__le,axiom,
    ! [A: int,B2: int] :
      ( ( ~ ( ord_less_eq_int @ A @ B2 ) )
      = ( ( ord_less_eq_int @ B2 @ A )
        & ( B2 != A ) ) ) ).

% nle_le
thf(fact_7676_le__cases3,axiom,
    ! [X4: rat,Y3: rat,Z: rat] :
      ( ( ( ord_less_eq_rat @ X4 @ Y3 )
       => ~ ( ord_less_eq_rat @ Y3 @ Z ) )
     => ( ( ( ord_less_eq_rat @ Y3 @ X4 )
         => ~ ( ord_less_eq_rat @ X4 @ Z ) )
       => ( ( ( ord_less_eq_rat @ X4 @ Z )
           => ~ ( ord_less_eq_rat @ Z @ Y3 ) )
         => ( ( ( ord_less_eq_rat @ Z @ Y3 )
             => ~ ( ord_less_eq_rat @ Y3 @ X4 ) )
           => ( ( ( ord_less_eq_rat @ Y3 @ Z )
               => ~ ( ord_less_eq_rat @ Z @ X4 ) )
             => ~ ( ( ord_less_eq_rat @ Z @ X4 )
                 => ~ ( ord_less_eq_rat @ X4 @ Y3 ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_7677_le__cases3,axiom,
    ! [X4: num,Y3: num,Z: num] :
      ( ( ( ord_less_eq_num @ X4 @ Y3 )
       => ~ ( ord_less_eq_num @ Y3 @ Z ) )
     => ( ( ( ord_less_eq_num @ Y3 @ X4 )
         => ~ ( ord_less_eq_num @ X4 @ Z ) )
       => ( ( ( ord_less_eq_num @ X4 @ Z )
           => ~ ( ord_less_eq_num @ Z @ Y3 ) )
         => ( ( ( ord_less_eq_num @ Z @ Y3 )
             => ~ ( ord_less_eq_num @ Y3 @ X4 ) )
           => ( ( ( ord_less_eq_num @ Y3 @ Z )
               => ~ ( ord_less_eq_num @ Z @ X4 ) )
             => ~ ( ( ord_less_eq_num @ Z @ X4 )
                 => ~ ( ord_less_eq_num @ X4 @ Y3 ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_7678_le__cases3,axiom,
    ! [X4: nat,Y3: nat,Z: nat] :
      ( ( ( ord_less_eq_nat @ X4 @ Y3 )
       => ~ ( ord_less_eq_nat @ Y3 @ Z ) )
     => ( ( ( ord_less_eq_nat @ Y3 @ X4 )
         => ~ ( ord_less_eq_nat @ X4 @ Z ) )
       => ( ( ( ord_less_eq_nat @ X4 @ Z )
           => ~ ( ord_less_eq_nat @ Z @ Y3 ) )
         => ( ( ( ord_less_eq_nat @ Z @ Y3 )
             => ~ ( ord_less_eq_nat @ Y3 @ X4 ) )
           => ( ( ( ord_less_eq_nat @ Y3 @ Z )
               => ~ ( ord_less_eq_nat @ Z @ X4 ) )
             => ~ ( ( ord_less_eq_nat @ Z @ X4 )
                 => ~ ( ord_less_eq_nat @ X4 @ Y3 ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_7679_le__cases3,axiom,
    ! [X4: int,Y3: int,Z: int] :
      ( ( ( ord_less_eq_int @ X4 @ Y3 )
       => ~ ( ord_less_eq_int @ Y3 @ Z ) )
     => ( ( ( ord_less_eq_int @ Y3 @ X4 )
         => ~ ( ord_less_eq_int @ X4 @ Z ) )
       => ( ( ( ord_less_eq_int @ X4 @ Z )
           => ~ ( ord_less_eq_int @ Z @ Y3 ) )
         => ( ( ( ord_less_eq_int @ Z @ Y3 )
             => ~ ( ord_less_eq_int @ Y3 @ X4 ) )
           => ( ( ( ord_less_eq_int @ Y3 @ Z )
               => ~ ( ord_less_eq_int @ Z @ X4 ) )
             => ~ ( ( ord_less_eq_int @ Z @ X4 )
                 => ~ ( ord_less_eq_int @ X4 @ Y3 ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_7680_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y6: set_nat,Z4: set_nat] : Y6 = Z4 )
    = ( ^ [X: set_nat,Y: set_nat] :
          ( ( ord_less_eq_set_nat @ X @ Y )
          & ( ord_less_eq_set_nat @ Y @ X ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_7681_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y6: rat,Z4: rat] : Y6 = Z4 )
    = ( ^ [X: rat,Y: rat] :
          ( ( ord_less_eq_rat @ X @ Y )
          & ( ord_less_eq_rat @ Y @ X ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_7682_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y6: num,Z4: num] : Y6 = Z4 )
    = ( ^ [X: num,Y: num] :
          ( ( ord_less_eq_num @ X @ Y )
          & ( ord_less_eq_num @ Y @ X ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_7683_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y6: nat,Z4: nat] : Y6 = Z4 )
    = ( ^ [X: nat,Y: nat] :
          ( ( ord_less_eq_nat @ X @ Y )
          & ( ord_less_eq_nat @ Y @ X ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_7684_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y6: int,Z4: int] : Y6 = Z4 )
    = ( ^ [X: int,Y: int] :
          ( ( ord_less_eq_int @ X @ Y )
          & ( ord_less_eq_int @ Y @ X ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_7685_ord__eq__le__trans,axiom,
    ! [A: set_nat,B2: set_nat,C: set_nat] :
      ( ( A = B2 )
     => ( ( ord_less_eq_set_nat @ B2 @ C )
       => ( ord_less_eq_set_nat @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_7686_ord__eq__le__trans,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( A = B2 )
     => ( ( ord_less_eq_rat @ B2 @ C )
       => ( ord_less_eq_rat @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_7687_ord__eq__le__trans,axiom,
    ! [A: num,B2: num,C: num] :
      ( ( A = B2 )
     => ( ( ord_less_eq_num @ B2 @ C )
       => ( ord_less_eq_num @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_7688_ord__eq__le__trans,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( A = B2 )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ord_less_eq_nat @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_7689_ord__eq__le__trans,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( A = B2 )
     => ( ( ord_less_eq_int @ B2 @ C )
       => ( ord_less_eq_int @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_7690_ord__le__eq__trans,axiom,
    ! [A: set_nat,B2: set_nat,C: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B2 )
     => ( ( B2 = C )
       => ( ord_less_eq_set_nat @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_7691_ord__le__eq__trans,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B2 )
     => ( ( B2 = C )
       => ( ord_less_eq_rat @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_7692_ord__le__eq__trans,axiom,
    ! [A: num,B2: num,C: num] :
      ( ( ord_less_eq_num @ A @ B2 )
     => ( ( B2 = C )
       => ( ord_less_eq_num @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_7693_ord__le__eq__trans,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( B2 = C )
       => ( ord_less_eq_nat @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_7694_ord__le__eq__trans,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( ord_less_eq_int @ A @ B2 )
     => ( ( B2 = C )
       => ( ord_less_eq_int @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_7695_order__antisym,axiom,
    ! [X4: set_nat,Y3: set_nat] :
      ( ( ord_less_eq_set_nat @ X4 @ Y3 )
     => ( ( ord_less_eq_set_nat @ Y3 @ X4 )
       => ( X4 = Y3 ) ) ) ).

% order_antisym
thf(fact_7696_order__antisym,axiom,
    ! [X4: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ X4 @ Y3 )
     => ( ( ord_less_eq_rat @ Y3 @ X4 )
       => ( X4 = Y3 ) ) ) ).

% order_antisym
thf(fact_7697_order__antisym,axiom,
    ! [X4: num,Y3: num] :
      ( ( ord_less_eq_num @ X4 @ Y3 )
     => ( ( ord_less_eq_num @ Y3 @ X4 )
       => ( X4 = Y3 ) ) ) ).

% order_antisym
thf(fact_7698_order__antisym,axiom,
    ! [X4: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ X4 @ Y3 )
     => ( ( ord_less_eq_nat @ Y3 @ X4 )
       => ( X4 = Y3 ) ) ) ).

% order_antisym
thf(fact_7699_order__antisym,axiom,
    ! [X4: int,Y3: int] :
      ( ( ord_less_eq_int @ X4 @ Y3 )
     => ( ( ord_less_eq_int @ Y3 @ X4 )
       => ( X4 = Y3 ) ) ) ).

% order_antisym
thf(fact_7700_order_Otrans,axiom,
    ! [A: set_nat,B2: set_nat,C: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B2 )
     => ( ( ord_less_eq_set_nat @ B2 @ C )
       => ( ord_less_eq_set_nat @ A @ C ) ) ) ).

% order.trans
thf(fact_7701_order_Otrans,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B2 )
     => ( ( ord_less_eq_rat @ B2 @ C )
       => ( ord_less_eq_rat @ A @ C ) ) ) ).

% order.trans
thf(fact_7702_order_Otrans,axiom,
    ! [A: num,B2: num,C: num] :
      ( ( ord_less_eq_num @ A @ B2 )
     => ( ( ord_less_eq_num @ B2 @ C )
       => ( ord_less_eq_num @ A @ C ) ) ) ).

% order.trans
thf(fact_7703_order_Otrans,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ord_less_eq_nat @ A @ C ) ) ) ).

% order.trans
thf(fact_7704_order_Otrans,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( ord_less_eq_int @ A @ B2 )
     => ( ( ord_less_eq_int @ B2 @ C )
       => ( ord_less_eq_int @ A @ C ) ) ) ).

% order.trans
thf(fact_7705_order__trans,axiom,
    ! [X4: set_nat,Y3: set_nat,Z: set_nat] :
      ( ( ord_less_eq_set_nat @ X4 @ Y3 )
     => ( ( ord_less_eq_set_nat @ Y3 @ Z )
       => ( ord_less_eq_set_nat @ X4 @ Z ) ) ) ).

% order_trans
thf(fact_7706_order__trans,axiom,
    ! [X4: rat,Y3: rat,Z: rat] :
      ( ( ord_less_eq_rat @ X4 @ Y3 )
     => ( ( ord_less_eq_rat @ Y3 @ Z )
       => ( ord_less_eq_rat @ X4 @ Z ) ) ) ).

% order_trans
thf(fact_7707_order__trans,axiom,
    ! [X4: num,Y3: num,Z: num] :
      ( ( ord_less_eq_num @ X4 @ Y3 )
     => ( ( ord_less_eq_num @ Y3 @ Z )
       => ( ord_less_eq_num @ X4 @ Z ) ) ) ).

% order_trans
thf(fact_7708_order__trans,axiom,
    ! [X4: nat,Y3: nat,Z: nat] :
      ( ( ord_less_eq_nat @ X4 @ Y3 )
     => ( ( ord_less_eq_nat @ Y3 @ Z )
       => ( ord_less_eq_nat @ X4 @ Z ) ) ) ).

% order_trans
thf(fact_7709_order__trans,axiom,
    ! [X4: int,Y3: int,Z: int] :
      ( ( ord_less_eq_int @ X4 @ Y3 )
     => ( ( ord_less_eq_int @ Y3 @ Z )
       => ( ord_less_eq_int @ X4 @ Z ) ) ) ).

% order_trans
thf(fact_7710_linorder__wlog,axiom,
    ! [P: rat > rat > $o,A: rat,B2: rat] :
      ( ! [A5: rat,B5: rat] :
          ( ( ord_less_eq_rat @ A5 @ B5 )
         => ( P @ A5 @ B5 ) )
     => ( ! [A5: rat,B5: rat] :
            ( ( P @ B5 @ A5 )
           => ( P @ A5 @ B5 ) )
       => ( P @ A @ B2 ) ) ) ).

% linorder_wlog
thf(fact_7711_linorder__wlog,axiom,
    ! [P: num > num > $o,A: num,B2: num] :
      ( ! [A5: num,B5: num] :
          ( ( ord_less_eq_num @ A5 @ B5 )
         => ( P @ A5 @ B5 ) )
     => ( ! [A5: num,B5: num] :
            ( ( P @ B5 @ A5 )
           => ( P @ A5 @ B5 ) )
       => ( P @ A @ B2 ) ) ) ).

% linorder_wlog
thf(fact_7712_linorder__wlog,axiom,
    ! [P: nat > nat > $o,A: nat,B2: nat] :
      ( ! [A5: nat,B5: nat] :
          ( ( ord_less_eq_nat @ A5 @ B5 )
         => ( P @ A5 @ B5 ) )
     => ( ! [A5: nat,B5: nat] :
            ( ( P @ B5 @ A5 )
           => ( P @ A5 @ B5 ) )
       => ( P @ A @ B2 ) ) ) ).

% linorder_wlog
thf(fact_7713_linorder__wlog,axiom,
    ! [P: int > int > $o,A: int,B2: int] :
      ( ! [A5: int,B5: int] :
          ( ( ord_less_eq_int @ A5 @ B5 )
         => ( P @ A5 @ B5 ) )
     => ( ! [A5: int,B5: int] :
            ( ( P @ B5 @ A5 )
           => ( P @ A5 @ B5 ) )
       => ( P @ A @ B2 ) ) ) ).

% linorder_wlog
thf(fact_7714_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y6: set_nat,Z4: set_nat] : Y6 = Z4 )
    = ( ^ [A4: set_nat,B3: set_nat] :
          ( ( ord_less_eq_set_nat @ B3 @ A4 )
          & ( ord_less_eq_set_nat @ A4 @ B3 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_7715_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y6: rat,Z4: rat] : Y6 = Z4 )
    = ( ^ [A4: rat,B3: rat] :
          ( ( ord_less_eq_rat @ B3 @ A4 )
          & ( ord_less_eq_rat @ A4 @ B3 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_7716_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y6: num,Z4: num] : Y6 = Z4 )
    = ( ^ [A4: num,B3: num] :
          ( ( ord_less_eq_num @ B3 @ A4 )
          & ( ord_less_eq_num @ A4 @ B3 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_7717_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y6: nat,Z4: nat] : Y6 = Z4 )
    = ( ^ [A4: nat,B3: nat] :
          ( ( ord_less_eq_nat @ B3 @ A4 )
          & ( ord_less_eq_nat @ A4 @ B3 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_7718_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y6: int,Z4: int] : Y6 = Z4 )
    = ( ^ [A4: int,B3: int] :
          ( ( ord_less_eq_int @ B3 @ A4 )
          & ( ord_less_eq_int @ A4 @ B3 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_7719_dual__order_Oantisym,axiom,
    ! [B2: set_nat,A: set_nat] :
      ( ( ord_less_eq_set_nat @ B2 @ A )
     => ( ( ord_less_eq_set_nat @ A @ B2 )
       => ( A = B2 ) ) ) ).

% dual_order.antisym
thf(fact_7720_dual__order_Oantisym,axiom,
    ! [B2: rat,A: rat] :
      ( ( ord_less_eq_rat @ B2 @ A )
     => ( ( ord_less_eq_rat @ A @ B2 )
       => ( A = B2 ) ) ) ).

% dual_order.antisym
thf(fact_7721_dual__order_Oantisym,axiom,
    ! [B2: num,A: num] :
      ( ( ord_less_eq_num @ B2 @ A )
     => ( ( ord_less_eq_num @ A @ B2 )
       => ( A = B2 ) ) ) ).

% dual_order.antisym
thf(fact_7722_dual__order_Oantisym,axiom,
    ! [B2: nat,A: nat] :
      ( ( ord_less_eq_nat @ B2 @ A )
     => ( ( ord_less_eq_nat @ A @ B2 )
       => ( A = B2 ) ) ) ).

% dual_order.antisym
thf(fact_7723_dual__order_Oantisym,axiom,
    ! [B2: int,A: int] :
      ( ( ord_less_eq_int @ B2 @ A )
     => ( ( ord_less_eq_int @ A @ B2 )
       => ( A = B2 ) ) ) ).

% dual_order.antisym
thf(fact_7724_dual__order_Otrans,axiom,
    ! [B2: set_nat,A: set_nat,C: set_nat] :
      ( ( ord_less_eq_set_nat @ B2 @ A )
     => ( ( ord_less_eq_set_nat @ C @ B2 )
       => ( ord_less_eq_set_nat @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_7725_dual__order_Otrans,axiom,
    ! [B2: rat,A: rat,C: rat] :
      ( ( ord_less_eq_rat @ B2 @ A )
     => ( ( ord_less_eq_rat @ C @ B2 )
       => ( ord_less_eq_rat @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_7726_dual__order_Otrans,axiom,
    ! [B2: num,A: num,C: num] :
      ( ( ord_less_eq_num @ B2 @ A )
     => ( ( ord_less_eq_num @ C @ B2 )
       => ( ord_less_eq_num @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_7727_dual__order_Otrans,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( ( ord_less_eq_nat @ B2 @ A )
     => ( ( ord_less_eq_nat @ C @ B2 )
       => ( ord_less_eq_nat @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_7728_dual__order_Otrans,axiom,
    ! [B2: int,A: int,C: int] :
      ( ( ord_less_eq_int @ B2 @ A )
     => ( ( ord_less_eq_int @ C @ B2 )
       => ( ord_less_eq_int @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_7729_antisym,axiom,
    ! [A: set_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B2 )
     => ( ( ord_less_eq_set_nat @ B2 @ A )
       => ( A = B2 ) ) ) ).

% antisym
thf(fact_7730_antisym,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_eq_rat @ A @ B2 )
     => ( ( ord_less_eq_rat @ B2 @ A )
       => ( A = B2 ) ) ) ).

% antisym
thf(fact_7731_antisym,axiom,
    ! [A: num,B2: num] :
      ( ( ord_less_eq_num @ A @ B2 )
     => ( ( ord_less_eq_num @ B2 @ A )
       => ( A = B2 ) ) ) ).

% antisym
thf(fact_7732_antisym,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( ord_less_eq_nat @ B2 @ A )
       => ( A = B2 ) ) ) ).

% antisym
thf(fact_7733_antisym,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ A @ B2 )
     => ( ( ord_less_eq_int @ B2 @ A )
       => ( A = B2 ) ) ) ).

% antisym
thf(fact_7734_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y6: set_nat,Z4: set_nat] : Y6 = Z4 )
    = ( ^ [A4: set_nat,B3: set_nat] :
          ( ( ord_less_eq_set_nat @ A4 @ B3 )
          & ( ord_less_eq_set_nat @ B3 @ A4 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_7735_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y6: rat,Z4: rat] : Y6 = Z4 )
    = ( ^ [A4: rat,B3: rat] :
          ( ( ord_less_eq_rat @ A4 @ B3 )
          & ( ord_less_eq_rat @ B3 @ A4 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_7736_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y6: num,Z4: num] : Y6 = Z4 )
    = ( ^ [A4: num,B3: num] :
          ( ( ord_less_eq_num @ A4 @ B3 )
          & ( ord_less_eq_num @ B3 @ A4 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_7737_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y6: nat,Z4: nat] : Y6 = Z4 )
    = ( ^ [A4: nat,B3: nat] :
          ( ( ord_less_eq_nat @ A4 @ B3 )
          & ( ord_less_eq_nat @ B3 @ A4 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_7738_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y6: int,Z4: int] : Y6 = Z4 )
    = ( ^ [A4: int,B3: int] :
          ( ( ord_less_eq_int @ A4 @ B3 )
          & ( ord_less_eq_int @ B3 @ A4 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_7739_order__subst1,axiom,
    ! [A: rat,F: rat > rat,B2: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ ( F @ B2 ) )
     => ( ( ord_less_eq_rat @ B2 @ C )
       => ( ! [X3: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y4 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_7740_order__subst1,axiom,
    ! [A: rat,F: num > rat,B2: num,C: num] :
      ( ( ord_less_eq_rat @ A @ ( F @ B2 ) )
     => ( ( ord_less_eq_num @ B2 @ C )
       => ( ! [X3: num,Y4: num] :
              ( ( ord_less_eq_num @ X3 @ Y4 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_7741_order__subst1,axiom,
    ! [A: rat,F: nat > rat,B2: nat,C: nat] :
      ( ( ord_less_eq_rat @ A @ ( F @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y4 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_7742_order__subst1,axiom,
    ! [A: rat,F: int > rat,B2: int,C: int] :
      ( ( ord_less_eq_rat @ A @ ( F @ B2 ) )
     => ( ( ord_less_eq_int @ B2 @ C )
       => ( ! [X3: int,Y4: int] :
              ( ( ord_less_eq_int @ X3 @ Y4 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_7743_order__subst1,axiom,
    ! [A: num,F: rat > num,B2: rat,C: rat] :
      ( ( ord_less_eq_num @ A @ ( F @ B2 ) )
     => ( ( ord_less_eq_rat @ B2 @ C )
       => ( ! [X3: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y4 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_7744_order__subst1,axiom,
    ! [A: num,F: num > num,B2: num,C: num] :
      ( ( ord_less_eq_num @ A @ ( F @ B2 ) )
     => ( ( ord_less_eq_num @ B2 @ C )
       => ( ! [X3: num,Y4: num] :
              ( ( ord_less_eq_num @ X3 @ Y4 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_7745_order__subst1,axiom,
    ! [A: num,F: nat > num,B2: nat,C: nat] :
      ( ( ord_less_eq_num @ A @ ( F @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y4 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_7746_order__subst1,axiom,
    ! [A: num,F: int > num,B2: int,C: int] :
      ( ( ord_less_eq_num @ A @ ( F @ B2 ) )
     => ( ( ord_less_eq_int @ B2 @ C )
       => ( ! [X3: int,Y4: int] :
              ( ( ord_less_eq_int @ X3 @ Y4 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_7747_order__subst1,axiom,
    ! [A: nat,F: rat > nat,B2: rat,C: rat] :
      ( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
     => ( ( ord_less_eq_rat @ B2 @ C )
       => ( ! [X3: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_7748_order__subst1,axiom,
    ! [A: nat,F: num > nat,B2: num,C: num] :
      ( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
     => ( ( ord_less_eq_num @ B2 @ C )
       => ( ! [X3: num,Y4: num] :
              ( ( ord_less_eq_num @ X3 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_7749_order__subst2,axiom,
    ! [A: rat,B2: rat,F: rat > rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B2 )
     => ( ( ord_less_eq_rat @ ( F @ B2 ) @ C )
       => ( ! [X3: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y4 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_7750_order__subst2,axiom,
    ! [A: rat,B2: rat,F: rat > num,C: num] :
      ( ( ord_less_eq_rat @ A @ B2 )
     => ( ( ord_less_eq_num @ ( F @ B2 ) @ C )
       => ( ! [X3: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y4 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_7751_order__subst2,axiom,
    ! [A: rat,B2: rat,F: rat > nat,C: nat] :
      ( ( ord_less_eq_rat @ A @ B2 )
     => ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
       => ( ! [X3: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_7752_order__subst2,axiom,
    ! [A: rat,B2: rat,F: rat > int,C: int] :
      ( ( ord_less_eq_rat @ A @ B2 )
     => ( ( ord_less_eq_int @ ( F @ B2 ) @ C )
       => ( ! [X3: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y4 )
             => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_7753_order__subst2,axiom,
    ! [A: num,B2: num,F: num > rat,C: rat] :
      ( ( ord_less_eq_num @ A @ B2 )
     => ( ( ord_less_eq_rat @ ( F @ B2 ) @ C )
       => ( ! [X3: num,Y4: num] :
              ( ( ord_less_eq_num @ X3 @ Y4 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_7754_order__subst2,axiom,
    ! [A: num,B2: num,F: num > num,C: num] :
      ( ( ord_less_eq_num @ A @ B2 )
     => ( ( ord_less_eq_num @ ( F @ B2 ) @ C )
       => ( ! [X3: num,Y4: num] :
              ( ( ord_less_eq_num @ X3 @ Y4 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_7755_order__subst2,axiom,
    ! [A: num,B2: num,F: num > nat,C: nat] :
      ( ( ord_less_eq_num @ A @ B2 )
     => ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
       => ( ! [X3: num,Y4: num] :
              ( ( ord_less_eq_num @ X3 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_7756_order__subst2,axiom,
    ! [A: num,B2: num,F: num > int,C: int] :
      ( ( ord_less_eq_num @ A @ B2 )
     => ( ( ord_less_eq_int @ ( F @ B2 ) @ C )
       => ( ! [X3: num,Y4: num] :
              ( ( ord_less_eq_num @ X3 @ Y4 )
             => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_7757_order__subst2,axiom,
    ! [A: nat,B2: nat,F: nat > rat,C: rat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( ord_less_eq_rat @ ( F @ B2 ) @ C )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y4 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_7758_order__subst2,axiom,
    ! [A: nat,B2: nat,F: nat > num,C: num] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( ord_less_eq_num @ ( F @ B2 ) @ C )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y4 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_7759_order__eq__refl,axiom,
    ! [X4: set_nat,Y3: set_nat] :
      ( ( X4 = Y3 )
     => ( ord_less_eq_set_nat @ X4 @ Y3 ) ) ).

% order_eq_refl
thf(fact_7760_order__eq__refl,axiom,
    ! [X4: rat,Y3: rat] :
      ( ( X4 = Y3 )
     => ( ord_less_eq_rat @ X4 @ Y3 ) ) ).

% order_eq_refl
thf(fact_7761_order__eq__refl,axiom,
    ! [X4: num,Y3: num] :
      ( ( X4 = Y3 )
     => ( ord_less_eq_num @ X4 @ Y3 ) ) ).

% order_eq_refl
thf(fact_7762_order__eq__refl,axiom,
    ! [X4: nat,Y3: nat] :
      ( ( X4 = Y3 )
     => ( ord_less_eq_nat @ X4 @ Y3 ) ) ).

% order_eq_refl
thf(fact_7763_order__eq__refl,axiom,
    ! [X4: int,Y3: int] :
      ( ( X4 = Y3 )
     => ( ord_less_eq_int @ X4 @ Y3 ) ) ).

% order_eq_refl
thf(fact_7764_linorder__linear,axiom,
    ! [X4: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ X4 @ Y3 )
      | ( ord_less_eq_rat @ Y3 @ X4 ) ) ).

% linorder_linear
thf(fact_7765_linorder__linear,axiom,
    ! [X4: num,Y3: num] :
      ( ( ord_less_eq_num @ X4 @ Y3 )
      | ( ord_less_eq_num @ Y3 @ X4 ) ) ).

% linorder_linear
thf(fact_7766_linorder__linear,axiom,
    ! [X4: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ X4 @ Y3 )
      | ( ord_less_eq_nat @ Y3 @ X4 ) ) ).

% linorder_linear
thf(fact_7767_linorder__linear,axiom,
    ! [X4: int,Y3: int] :
      ( ( ord_less_eq_int @ X4 @ Y3 )
      | ( ord_less_eq_int @ Y3 @ X4 ) ) ).

% linorder_linear
thf(fact_7768_ord__eq__le__subst,axiom,
    ! [A: rat,F: rat > rat,B2: rat,C: rat] :
      ( ( A
        = ( F @ B2 ) )
     => ( ( ord_less_eq_rat @ B2 @ C )
       => ( ! [X3: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y4 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_7769_ord__eq__le__subst,axiom,
    ! [A: num,F: rat > num,B2: rat,C: rat] :
      ( ( A
        = ( F @ B2 ) )
     => ( ( ord_less_eq_rat @ B2 @ C )
       => ( ! [X3: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y4 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_7770_ord__eq__le__subst,axiom,
    ! [A: nat,F: rat > nat,B2: rat,C: rat] :
      ( ( A
        = ( F @ B2 ) )
     => ( ( ord_less_eq_rat @ B2 @ C )
       => ( ! [X3: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_7771_ord__eq__le__subst,axiom,
    ! [A: int,F: rat > int,B2: rat,C: rat] :
      ( ( A
        = ( F @ B2 ) )
     => ( ( ord_less_eq_rat @ B2 @ C )
       => ( ! [X3: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y4 )
             => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_7772_ord__eq__le__subst,axiom,
    ! [A: rat,F: num > rat,B2: num,C: num] :
      ( ( A
        = ( F @ B2 ) )
     => ( ( ord_less_eq_num @ B2 @ C )
       => ( ! [X3: num,Y4: num] :
              ( ( ord_less_eq_num @ X3 @ Y4 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_7773_ord__eq__le__subst,axiom,
    ! [A: num,F: num > num,B2: num,C: num] :
      ( ( A
        = ( F @ B2 ) )
     => ( ( ord_less_eq_num @ B2 @ C )
       => ( ! [X3: num,Y4: num] :
              ( ( ord_less_eq_num @ X3 @ Y4 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_7774_ord__eq__le__subst,axiom,
    ! [A: nat,F: num > nat,B2: num,C: num] :
      ( ( A
        = ( F @ B2 ) )
     => ( ( ord_less_eq_num @ B2 @ C )
       => ( ! [X3: num,Y4: num] :
              ( ( ord_less_eq_num @ X3 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_7775_ord__eq__le__subst,axiom,
    ! [A: int,F: num > int,B2: num,C: num] :
      ( ( A
        = ( F @ B2 ) )
     => ( ( ord_less_eq_num @ B2 @ C )
       => ( ! [X3: num,Y4: num] :
              ( ( ord_less_eq_num @ X3 @ Y4 )
             => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_7776_ord__eq__le__subst,axiom,
    ! [A: rat,F: nat > rat,B2: nat,C: nat] :
      ( ( A
        = ( F @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y4 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_7777_ord__eq__le__subst,axiom,
    ! [A: num,F: nat > num,B2: nat,C: nat] :
      ( ( A
        = ( F @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y4 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_7778_ord__le__eq__subst,axiom,
    ! [A: rat,B2: rat,F: rat > rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X3: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y4 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_7779_ord__le__eq__subst,axiom,
    ! [A: rat,B2: rat,F: rat > num,C: num] :
      ( ( ord_less_eq_rat @ A @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X3: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y4 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_7780_ord__le__eq__subst,axiom,
    ! [A: rat,B2: rat,F: rat > nat,C: nat] :
      ( ( ord_less_eq_rat @ A @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X3: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_7781_ord__le__eq__subst,axiom,
    ! [A: rat,B2: rat,F: rat > int,C: int] :
      ( ( ord_less_eq_rat @ A @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X3: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y4 )
             => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_7782_ord__le__eq__subst,axiom,
    ! [A: num,B2: num,F: num > rat,C: rat] :
      ( ( ord_less_eq_num @ A @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X3: num,Y4: num] :
              ( ( ord_less_eq_num @ X3 @ Y4 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_7783_ord__le__eq__subst,axiom,
    ! [A: num,B2: num,F: num > num,C: num] :
      ( ( ord_less_eq_num @ A @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X3: num,Y4: num] :
              ( ( ord_less_eq_num @ X3 @ Y4 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_7784_ord__le__eq__subst,axiom,
    ! [A: num,B2: num,F: num > nat,C: nat] :
      ( ( ord_less_eq_num @ A @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X3: num,Y4: num] :
              ( ( ord_less_eq_num @ X3 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_7785_ord__le__eq__subst,axiom,
    ! [A: num,B2: num,F: num > int,C: int] :
      ( ( ord_less_eq_num @ A @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X3: num,Y4: num] :
              ( ( ord_less_eq_num @ X3 @ Y4 )
             => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_7786_ord__le__eq__subst,axiom,
    ! [A: nat,B2: nat,F: nat > rat,C: rat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y4 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_7787_ord__le__eq__subst,axiom,
    ! [A: nat,B2: nat,F: nat > num,C: num] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y4 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_7788_linorder__le__cases,axiom,
    ! [X4: rat,Y3: rat] :
      ( ~ ( ord_less_eq_rat @ X4 @ Y3 )
     => ( ord_less_eq_rat @ Y3 @ X4 ) ) ).

% linorder_le_cases
thf(fact_7789_linorder__le__cases,axiom,
    ! [X4: num,Y3: num] :
      ( ~ ( ord_less_eq_num @ X4 @ Y3 )
     => ( ord_less_eq_num @ Y3 @ X4 ) ) ).

% linorder_le_cases
thf(fact_7790_linorder__le__cases,axiom,
    ! [X4: nat,Y3: nat] :
      ( ~ ( ord_less_eq_nat @ X4 @ Y3 )
     => ( ord_less_eq_nat @ Y3 @ X4 ) ) ).

% linorder_le_cases
thf(fact_7791_linorder__le__cases,axiom,
    ! [X4: int,Y3: int] :
      ( ~ ( ord_less_eq_int @ X4 @ Y3 )
     => ( ord_less_eq_int @ Y3 @ X4 ) ) ).

% linorder_le_cases
thf(fact_7792_order__antisym__conv,axiom,
    ! [Y3: set_nat,X4: set_nat] :
      ( ( ord_less_eq_set_nat @ Y3 @ X4 )
     => ( ( ord_less_eq_set_nat @ X4 @ Y3 )
        = ( X4 = Y3 ) ) ) ).

% order_antisym_conv
thf(fact_7793_order__antisym__conv,axiom,
    ! [Y3: rat,X4: rat] :
      ( ( ord_less_eq_rat @ Y3 @ X4 )
     => ( ( ord_less_eq_rat @ X4 @ Y3 )
        = ( X4 = Y3 ) ) ) ).

% order_antisym_conv
thf(fact_7794_order__antisym__conv,axiom,
    ! [Y3: num,X4: num] :
      ( ( ord_less_eq_num @ Y3 @ X4 )
     => ( ( ord_less_eq_num @ X4 @ Y3 )
        = ( X4 = Y3 ) ) ) ).

% order_antisym_conv
thf(fact_7795_order__antisym__conv,axiom,
    ! [Y3: nat,X4: nat] :
      ( ( ord_less_eq_nat @ Y3 @ X4 )
     => ( ( ord_less_eq_nat @ X4 @ Y3 )
        = ( X4 = Y3 ) ) ) ).

% order_antisym_conv
thf(fact_7796_order__antisym__conv,axiom,
    ! [Y3: int,X4: int] :
      ( ( ord_less_eq_int @ Y3 @ X4 )
     => ( ( ord_less_eq_int @ X4 @ Y3 )
        = ( X4 = Y3 ) ) ) ).

% order_antisym_conv
thf(fact_7797_lt__ex,axiom,
    ! [X4: real] :
    ? [Y4: real] : ( ord_less_real @ Y4 @ X4 ) ).

% lt_ex
thf(fact_7798_lt__ex,axiom,
    ! [X4: rat] :
    ? [Y4: rat] : ( ord_less_rat @ Y4 @ X4 ) ).

% lt_ex
thf(fact_7799_lt__ex,axiom,
    ! [X4: int] :
    ? [Y4: int] : ( ord_less_int @ Y4 @ X4 ) ).

% lt_ex
thf(fact_7800_gt__ex,axiom,
    ! [X4: real] :
    ? [X_1: real] : ( ord_less_real @ X4 @ X_1 ) ).

% gt_ex
thf(fact_7801_gt__ex,axiom,
    ! [X4: rat] :
    ? [X_1: rat] : ( ord_less_rat @ X4 @ X_1 ) ).

% gt_ex
thf(fact_7802_gt__ex,axiom,
    ! [X4: nat] :
    ? [X_1: nat] : ( ord_less_nat @ X4 @ X_1 ) ).

% gt_ex
thf(fact_7803_gt__ex,axiom,
    ! [X4: int] :
    ? [X_1: int] : ( ord_less_int @ X4 @ X_1 ) ).

% gt_ex
thf(fact_7804_dense,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_real @ X4 @ Y3 )
     => ? [Z2: real] :
          ( ( ord_less_real @ X4 @ Z2 )
          & ( ord_less_real @ Z2 @ Y3 ) ) ) ).

% dense
thf(fact_7805_dense,axiom,
    ! [X4: rat,Y3: rat] :
      ( ( ord_less_rat @ X4 @ Y3 )
     => ? [Z2: rat] :
          ( ( ord_less_rat @ X4 @ Z2 )
          & ( ord_less_rat @ Z2 @ Y3 ) ) ) ).

% dense
thf(fact_7806_less__imp__neq,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_real @ X4 @ Y3 )
     => ( X4 != Y3 ) ) ).

% less_imp_neq
thf(fact_7807_less__imp__neq,axiom,
    ! [X4: rat,Y3: rat] :
      ( ( ord_less_rat @ X4 @ Y3 )
     => ( X4 != Y3 ) ) ).

% less_imp_neq
thf(fact_7808_less__imp__neq,axiom,
    ! [X4: num,Y3: num] :
      ( ( ord_less_num @ X4 @ Y3 )
     => ( X4 != Y3 ) ) ).

% less_imp_neq
thf(fact_7809_less__imp__neq,axiom,
    ! [X4: nat,Y3: nat] :
      ( ( ord_less_nat @ X4 @ Y3 )
     => ( X4 != Y3 ) ) ).

% less_imp_neq
thf(fact_7810_less__imp__neq,axiom,
    ! [X4: int,Y3: int] :
      ( ( ord_less_int @ X4 @ Y3 )
     => ( X4 != Y3 ) ) ).

% less_imp_neq
thf(fact_7811_order_Oasym,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ A @ B2 )
     => ~ ( ord_less_real @ B2 @ A ) ) ).

% order.asym
thf(fact_7812_order_Oasym,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_rat @ A @ B2 )
     => ~ ( ord_less_rat @ B2 @ A ) ) ).

% order.asym
thf(fact_7813_order_Oasym,axiom,
    ! [A: num,B2: num] :
      ( ( ord_less_num @ A @ B2 )
     => ~ ( ord_less_num @ B2 @ A ) ) ).

% order.asym
thf(fact_7814_order_Oasym,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ~ ( ord_less_nat @ B2 @ A ) ) ).

% order.asym
thf(fact_7815_order_Oasym,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_int @ A @ B2 )
     => ~ ( ord_less_int @ B2 @ A ) ) ).

% order.asym
thf(fact_7816_ord__eq__less__trans,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( A = B2 )
     => ( ( ord_less_real @ B2 @ C )
       => ( ord_less_real @ A @ C ) ) ) ).

% ord_eq_less_trans
thf(fact_7817_ord__eq__less__trans,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( A = B2 )
     => ( ( ord_less_rat @ B2 @ C )
       => ( ord_less_rat @ A @ C ) ) ) ).

% ord_eq_less_trans
thf(fact_7818_ord__eq__less__trans,axiom,
    ! [A: num,B2: num,C: num] :
      ( ( A = B2 )
     => ( ( ord_less_num @ B2 @ C )
       => ( ord_less_num @ A @ C ) ) ) ).

% ord_eq_less_trans
thf(fact_7819_ord__eq__less__trans,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( A = B2 )
     => ( ( ord_less_nat @ B2 @ C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% ord_eq_less_trans
thf(fact_7820_ord__eq__less__trans,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( A = B2 )
     => ( ( ord_less_int @ B2 @ C )
       => ( ord_less_int @ A @ C ) ) ) ).

% ord_eq_less_trans
thf(fact_7821_ord__less__eq__trans,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ( B2 = C )
       => ( ord_less_real @ A @ C ) ) ) ).

% ord_less_eq_trans
thf(fact_7822_ord__less__eq__trans,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( ord_less_rat @ A @ B2 )
     => ( ( B2 = C )
       => ( ord_less_rat @ A @ C ) ) ) ).

% ord_less_eq_trans
thf(fact_7823_ord__less__eq__trans,axiom,
    ! [A: num,B2: num,C: num] :
      ( ( ord_less_num @ A @ B2 )
     => ( ( B2 = C )
       => ( ord_less_num @ A @ C ) ) ) ).

% ord_less_eq_trans
thf(fact_7824_ord__less__eq__trans,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( ( B2 = C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% ord_less_eq_trans
thf(fact_7825_ord__less__eq__trans,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( ord_less_int @ A @ B2 )
     => ( ( B2 = C )
       => ( ord_less_int @ A @ C ) ) ) ).

% ord_less_eq_trans
thf(fact_7826_less__induct,axiom,
    ! [P: nat > $o,A: nat] :
      ( ! [X3: nat] :
          ( ! [Y5: nat] :
              ( ( ord_less_nat @ Y5 @ X3 )
             => ( P @ Y5 ) )
         => ( P @ X3 ) )
     => ( P @ A ) ) ).

% less_induct
thf(fact_7827_antisym__conv3,axiom,
    ! [Y3: real,X4: real] :
      ( ~ ( ord_less_real @ Y3 @ X4 )
     => ( ( ~ ( ord_less_real @ X4 @ Y3 ) )
        = ( X4 = Y3 ) ) ) ).

% antisym_conv3
thf(fact_7828_antisym__conv3,axiom,
    ! [Y3: rat,X4: rat] :
      ( ~ ( ord_less_rat @ Y3 @ X4 )
     => ( ( ~ ( ord_less_rat @ X4 @ Y3 ) )
        = ( X4 = Y3 ) ) ) ).

% antisym_conv3
thf(fact_7829_antisym__conv3,axiom,
    ! [Y3: num,X4: num] :
      ( ~ ( ord_less_num @ Y3 @ X4 )
     => ( ( ~ ( ord_less_num @ X4 @ Y3 ) )
        = ( X4 = Y3 ) ) ) ).

% antisym_conv3
thf(fact_7830_antisym__conv3,axiom,
    ! [Y3: nat,X4: nat] :
      ( ~ ( ord_less_nat @ Y3 @ X4 )
     => ( ( ~ ( ord_less_nat @ X4 @ Y3 ) )
        = ( X4 = Y3 ) ) ) ).

% antisym_conv3
thf(fact_7831_antisym__conv3,axiom,
    ! [Y3: int,X4: int] :
      ( ~ ( ord_less_int @ Y3 @ X4 )
     => ( ( ~ ( ord_less_int @ X4 @ Y3 ) )
        = ( X4 = Y3 ) ) ) ).

% antisym_conv3
thf(fact_7832_linorder__cases,axiom,
    ! [X4: real,Y3: real] :
      ( ~ ( ord_less_real @ X4 @ Y3 )
     => ( ( X4 != Y3 )
       => ( ord_less_real @ Y3 @ X4 ) ) ) ).

% linorder_cases
thf(fact_7833_linorder__cases,axiom,
    ! [X4: rat,Y3: rat] :
      ( ~ ( ord_less_rat @ X4 @ Y3 )
     => ( ( X4 != Y3 )
       => ( ord_less_rat @ Y3 @ X4 ) ) ) ).

% linorder_cases
thf(fact_7834_linorder__cases,axiom,
    ! [X4: num,Y3: num] :
      ( ~ ( ord_less_num @ X4 @ Y3 )
     => ( ( X4 != Y3 )
       => ( ord_less_num @ Y3 @ X4 ) ) ) ).

% linorder_cases
thf(fact_7835_linorder__cases,axiom,
    ! [X4: nat,Y3: nat] :
      ( ~ ( ord_less_nat @ X4 @ Y3 )
     => ( ( X4 != Y3 )
       => ( ord_less_nat @ Y3 @ X4 ) ) ) ).

% linorder_cases
thf(fact_7836_linorder__cases,axiom,
    ! [X4: int,Y3: int] :
      ( ~ ( ord_less_int @ X4 @ Y3 )
     => ( ( X4 != Y3 )
       => ( ord_less_int @ Y3 @ X4 ) ) ) ).

% linorder_cases
thf(fact_7837_dual__order_Oasym,axiom,
    ! [B2: real,A: real] :
      ( ( ord_less_real @ B2 @ A )
     => ~ ( ord_less_real @ A @ B2 ) ) ).

% dual_order.asym
thf(fact_7838_dual__order_Oasym,axiom,
    ! [B2: rat,A: rat] :
      ( ( ord_less_rat @ B2 @ A )
     => ~ ( ord_less_rat @ A @ B2 ) ) ).

% dual_order.asym
thf(fact_7839_dual__order_Oasym,axiom,
    ! [B2: num,A: num] :
      ( ( ord_less_num @ B2 @ A )
     => ~ ( ord_less_num @ A @ B2 ) ) ).

% dual_order.asym
thf(fact_7840_dual__order_Oasym,axiom,
    ! [B2: nat,A: nat] :
      ( ( ord_less_nat @ B2 @ A )
     => ~ ( ord_less_nat @ A @ B2 ) ) ).

% dual_order.asym
thf(fact_7841_dual__order_Oasym,axiom,
    ! [B2: int,A: int] :
      ( ( ord_less_int @ B2 @ A )
     => ~ ( ord_less_int @ A @ B2 ) ) ).

% dual_order.asym
thf(fact_7842_dual__order_Oirrefl,axiom,
    ! [A: real] :
      ~ ( ord_less_real @ A @ A ) ).

% dual_order.irrefl
thf(fact_7843_dual__order_Oirrefl,axiom,
    ! [A: rat] :
      ~ ( ord_less_rat @ A @ A ) ).

% dual_order.irrefl
thf(fact_7844_dual__order_Oirrefl,axiom,
    ! [A: num] :
      ~ ( ord_less_num @ A @ A ) ).

% dual_order.irrefl
thf(fact_7845_dual__order_Oirrefl,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ A ) ).

% dual_order.irrefl
thf(fact_7846_dual__order_Oirrefl,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ A @ A ) ).

% dual_order.irrefl
thf(fact_7847_exists__least__iff,axiom,
    ( ( ^ [P5: nat > $o] :
        ? [X7: nat] : ( P5 @ X7 ) )
    = ( ^ [P6: nat > $o] :
        ? [N2: nat] :
          ( ( P6 @ N2 )
          & ! [M6: nat] :
              ( ( ord_less_nat @ M6 @ N2 )
             => ~ ( P6 @ M6 ) ) ) ) ) ).

% exists_least_iff
thf(fact_7848_linorder__less__wlog,axiom,
    ! [P: real > real > $o,A: real,B2: real] :
      ( ! [A5: real,B5: real] :
          ( ( ord_less_real @ A5 @ B5 )
         => ( P @ A5 @ B5 ) )
     => ( ! [A5: real] : ( P @ A5 @ A5 )
       => ( ! [A5: real,B5: real] :
              ( ( P @ B5 @ A5 )
             => ( P @ A5 @ B5 ) )
         => ( P @ A @ B2 ) ) ) ) ).

% linorder_less_wlog
thf(fact_7849_linorder__less__wlog,axiom,
    ! [P: rat > rat > $o,A: rat,B2: rat] :
      ( ! [A5: rat,B5: rat] :
          ( ( ord_less_rat @ A5 @ B5 )
         => ( P @ A5 @ B5 ) )
     => ( ! [A5: rat] : ( P @ A5 @ A5 )
       => ( ! [A5: rat,B5: rat] :
              ( ( P @ B5 @ A5 )
             => ( P @ A5 @ B5 ) )
         => ( P @ A @ B2 ) ) ) ) ).

% linorder_less_wlog
thf(fact_7850_linorder__less__wlog,axiom,
    ! [P: num > num > $o,A: num,B2: num] :
      ( ! [A5: num,B5: num] :
          ( ( ord_less_num @ A5 @ B5 )
         => ( P @ A5 @ B5 ) )
     => ( ! [A5: num] : ( P @ A5 @ A5 )
       => ( ! [A5: num,B5: num] :
              ( ( P @ B5 @ A5 )
             => ( P @ A5 @ B5 ) )
         => ( P @ A @ B2 ) ) ) ) ).

% linorder_less_wlog
thf(fact_7851_linorder__less__wlog,axiom,
    ! [P: nat > nat > $o,A: nat,B2: nat] :
      ( ! [A5: nat,B5: nat] :
          ( ( ord_less_nat @ A5 @ B5 )
         => ( P @ A5 @ B5 ) )
     => ( ! [A5: nat] : ( P @ A5 @ A5 )
       => ( ! [A5: nat,B5: nat] :
              ( ( P @ B5 @ A5 )
             => ( P @ A5 @ B5 ) )
         => ( P @ A @ B2 ) ) ) ) ).

% linorder_less_wlog
thf(fact_7852_linorder__less__wlog,axiom,
    ! [P: int > int > $o,A: int,B2: int] :
      ( ! [A5: int,B5: int] :
          ( ( ord_less_int @ A5 @ B5 )
         => ( P @ A5 @ B5 ) )
     => ( ! [A5: int] : ( P @ A5 @ A5 )
       => ( ! [A5: int,B5: int] :
              ( ( P @ B5 @ A5 )
             => ( P @ A5 @ B5 ) )
         => ( P @ A @ B2 ) ) ) ) ).

% linorder_less_wlog
thf(fact_7853_order_Ostrict__trans,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ( ord_less_real @ B2 @ C )
       => ( ord_less_real @ A @ C ) ) ) ).

% order.strict_trans
thf(fact_7854_order_Ostrict__trans,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( ord_less_rat @ A @ B2 )
     => ( ( ord_less_rat @ B2 @ C )
       => ( ord_less_rat @ A @ C ) ) ) ).

% order.strict_trans
thf(fact_7855_order_Ostrict__trans,axiom,
    ! [A: num,B2: num,C: num] :
      ( ( ord_less_num @ A @ B2 )
     => ( ( ord_less_num @ B2 @ C )
       => ( ord_less_num @ A @ C ) ) ) ).

% order.strict_trans
thf(fact_7856_order_Ostrict__trans,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( ( ord_less_nat @ B2 @ C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% order.strict_trans
thf(fact_7857_order_Ostrict__trans,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( ord_less_int @ A @ B2 )
     => ( ( ord_less_int @ B2 @ C )
       => ( ord_less_int @ A @ C ) ) ) ).

% order.strict_trans
thf(fact_7858_not__less__iff__gr__or__eq,axiom,
    ! [X4: real,Y3: real] :
      ( ( ~ ( ord_less_real @ X4 @ Y3 ) )
      = ( ( ord_less_real @ Y3 @ X4 )
        | ( X4 = Y3 ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_7859_not__less__iff__gr__or__eq,axiom,
    ! [X4: rat,Y3: rat] :
      ( ( ~ ( ord_less_rat @ X4 @ Y3 ) )
      = ( ( ord_less_rat @ Y3 @ X4 )
        | ( X4 = Y3 ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_7860_not__less__iff__gr__or__eq,axiom,
    ! [X4: num,Y3: num] :
      ( ( ~ ( ord_less_num @ X4 @ Y3 ) )
      = ( ( ord_less_num @ Y3 @ X4 )
        | ( X4 = Y3 ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_7861_not__less__iff__gr__or__eq,axiom,
    ! [X4: nat,Y3: nat] :
      ( ( ~ ( ord_less_nat @ X4 @ Y3 ) )
      = ( ( ord_less_nat @ Y3 @ X4 )
        | ( X4 = Y3 ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_7862_not__less__iff__gr__or__eq,axiom,
    ! [X4: int,Y3: int] :
      ( ( ~ ( ord_less_int @ X4 @ Y3 ) )
      = ( ( ord_less_int @ Y3 @ X4 )
        | ( X4 = Y3 ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_7863_dual__order_Ostrict__trans,axiom,
    ! [B2: real,A: real,C: real] :
      ( ( ord_less_real @ B2 @ A )
     => ( ( ord_less_real @ C @ B2 )
       => ( ord_less_real @ C @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_7864_dual__order_Ostrict__trans,axiom,
    ! [B2: rat,A: rat,C: rat] :
      ( ( ord_less_rat @ B2 @ A )
     => ( ( ord_less_rat @ C @ B2 )
       => ( ord_less_rat @ C @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_7865_dual__order_Ostrict__trans,axiom,
    ! [B2: num,A: num,C: num] :
      ( ( ord_less_num @ B2 @ A )
     => ( ( ord_less_num @ C @ B2 )
       => ( ord_less_num @ C @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_7866_dual__order_Ostrict__trans,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( ( ord_less_nat @ B2 @ A )
     => ( ( ord_less_nat @ C @ B2 )
       => ( ord_less_nat @ C @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_7867_dual__order_Ostrict__trans,axiom,
    ! [B2: int,A: int,C: int] :
      ( ( ord_less_int @ B2 @ A )
     => ( ( ord_less_int @ C @ B2 )
       => ( ord_less_int @ C @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_7868_order_Ostrict__implies__not__eq,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ A @ B2 )
     => ( A != B2 ) ) ).

% order.strict_implies_not_eq
thf(fact_7869_order_Ostrict__implies__not__eq,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_rat @ A @ B2 )
     => ( A != B2 ) ) ).

% order.strict_implies_not_eq
thf(fact_7870_order_Ostrict__implies__not__eq,axiom,
    ! [A: num,B2: num] :
      ( ( ord_less_num @ A @ B2 )
     => ( A != B2 ) ) ).

% order.strict_implies_not_eq
thf(fact_7871_order_Ostrict__implies__not__eq,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( A != B2 ) ) ).

% order.strict_implies_not_eq
thf(fact_7872_order_Ostrict__implies__not__eq,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_int @ A @ B2 )
     => ( A != B2 ) ) ).

% order.strict_implies_not_eq
thf(fact_7873_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B2: real,A: real] :
      ( ( ord_less_real @ B2 @ A )
     => ( A != B2 ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_7874_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B2: rat,A: rat] :
      ( ( ord_less_rat @ B2 @ A )
     => ( A != B2 ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_7875_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B2: num,A: num] :
      ( ( ord_less_num @ B2 @ A )
     => ( A != B2 ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_7876_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B2: nat,A: nat] :
      ( ( ord_less_nat @ B2 @ A )
     => ( A != B2 ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_7877_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B2: int,A: int] :
      ( ( ord_less_int @ B2 @ A )
     => ( A != B2 ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_7878_linorder__neqE,axiom,
    ! [X4: real,Y3: real] :
      ( ( X4 != Y3 )
     => ( ~ ( ord_less_real @ X4 @ Y3 )
       => ( ord_less_real @ Y3 @ X4 ) ) ) ).

% linorder_neqE
thf(fact_7879_linorder__neqE,axiom,
    ! [X4: rat,Y3: rat] :
      ( ( X4 != Y3 )
     => ( ~ ( ord_less_rat @ X4 @ Y3 )
       => ( ord_less_rat @ Y3 @ X4 ) ) ) ).

% linorder_neqE
thf(fact_7880_linorder__neqE,axiom,
    ! [X4: num,Y3: num] :
      ( ( X4 != Y3 )
     => ( ~ ( ord_less_num @ X4 @ Y3 )
       => ( ord_less_num @ Y3 @ X4 ) ) ) ).

% linorder_neqE
thf(fact_7881_linorder__neqE,axiom,
    ! [X4: nat,Y3: nat] :
      ( ( X4 != Y3 )
     => ( ~ ( ord_less_nat @ X4 @ Y3 )
       => ( ord_less_nat @ Y3 @ X4 ) ) ) ).

% linorder_neqE
thf(fact_7882_linorder__neqE,axiom,
    ! [X4: int,Y3: int] :
      ( ( X4 != Y3 )
     => ( ~ ( ord_less_int @ X4 @ Y3 )
       => ( ord_less_int @ Y3 @ X4 ) ) ) ).

% linorder_neqE
thf(fact_7883_order__less__asym,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_real @ X4 @ Y3 )
     => ~ ( ord_less_real @ Y3 @ X4 ) ) ).

% order_less_asym
thf(fact_7884_order__less__asym,axiom,
    ! [X4: rat,Y3: rat] :
      ( ( ord_less_rat @ X4 @ Y3 )
     => ~ ( ord_less_rat @ Y3 @ X4 ) ) ).

% order_less_asym
thf(fact_7885_order__less__asym,axiom,
    ! [X4: num,Y3: num] :
      ( ( ord_less_num @ X4 @ Y3 )
     => ~ ( ord_less_num @ Y3 @ X4 ) ) ).

% order_less_asym
thf(fact_7886_order__less__asym,axiom,
    ! [X4: nat,Y3: nat] :
      ( ( ord_less_nat @ X4 @ Y3 )
     => ~ ( ord_less_nat @ Y3 @ X4 ) ) ).

% order_less_asym
thf(fact_7887_order__less__asym,axiom,
    ! [X4: int,Y3: int] :
      ( ( ord_less_int @ X4 @ Y3 )
     => ~ ( ord_less_int @ Y3 @ X4 ) ) ).

% order_less_asym
thf(fact_7888_linorder__neq__iff,axiom,
    ! [X4: real,Y3: real] :
      ( ( X4 != Y3 )
      = ( ( ord_less_real @ X4 @ Y3 )
        | ( ord_less_real @ Y3 @ X4 ) ) ) ).

% linorder_neq_iff
thf(fact_7889_linorder__neq__iff,axiom,
    ! [X4: rat,Y3: rat] :
      ( ( X4 != Y3 )
      = ( ( ord_less_rat @ X4 @ Y3 )
        | ( ord_less_rat @ Y3 @ X4 ) ) ) ).

% linorder_neq_iff
thf(fact_7890_linorder__neq__iff,axiom,
    ! [X4: num,Y3: num] :
      ( ( X4 != Y3 )
      = ( ( ord_less_num @ X4 @ Y3 )
        | ( ord_less_num @ Y3 @ X4 ) ) ) ).

% linorder_neq_iff
thf(fact_7891_linorder__neq__iff,axiom,
    ! [X4: nat,Y3: nat] :
      ( ( X4 != Y3 )
      = ( ( ord_less_nat @ X4 @ Y3 )
        | ( ord_less_nat @ Y3 @ X4 ) ) ) ).

% linorder_neq_iff
thf(fact_7892_linorder__neq__iff,axiom,
    ! [X4: int,Y3: int] :
      ( ( X4 != Y3 )
      = ( ( ord_less_int @ X4 @ Y3 )
        | ( ord_less_int @ Y3 @ X4 ) ) ) ).

% linorder_neq_iff
thf(fact_7893_order__less__asym_H,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ A @ B2 )
     => ~ ( ord_less_real @ B2 @ A ) ) ).

% order_less_asym'
thf(fact_7894_order__less__asym_H,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_rat @ A @ B2 )
     => ~ ( ord_less_rat @ B2 @ A ) ) ).

% order_less_asym'
thf(fact_7895_order__less__asym_H,axiom,
    ! [A: num,B2: num] :
      ( ( ord_less_num @ A @ B2 )
     => ~ ( ord_less_num @ B2 @ A ) ) ).

% order_less_asym'
thf(fact_7896_order__less__asym_H,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ~ ( ord_less_nat @ B2 @ A ) ) ).

% order_less_asym'
thf(fact_7897_order__less__asym_H,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_int @ A @ B2 )
     => ~ ( ord_less_int @ B2 @ A ) ) ).

% order_less_asym'
thf(fact_7898_order__less__trans,axiom,
    ! [X4: real,Y3: real,Z: real] :
      ( ( ord_less_real @ X4 @ Y3 )
     => ( ( ord_less_real @ Y3 @ Z )
       => ( ord_less_real @ X4 @ Z ) ) ) ).

% order_less_trans
thf(fact_7899_order__less__trans,axiom,
    ! [X4: rat,Y3: rat,Z: rat] :
      ( ( ord_less_rat @ X4 @ Y3 )
     => ( ( ord_less_rat @ Y3 @ Z )
       => ( ord_less_rat @ X4 @ Z ) ) ) ).

% order_less_trans
thf(fact_7900_order__less__trans,axiom,
    ! [X4: num,Y3: num,Z: num] :
      ( ( ord_less_num @ X4 @ Y3 )
     => ( ( ord_less_num @ Y3 @ Z )
       => ( ord_less_num @ X4 @ Z ) ) ) ).

% order_less_trans
thf(fact_7901_order__less__trans,axiom,
    ! [X4: nat,Y3: nat,Z: nat] :
      ( ( ord_less_nat @ X4 @ Y3 )
     => ( ( ord_less_nat @ Y3 @ Z )
       => ( ord_less_nat @ X4 @ Z ) ) ) ).

% order_less_trans
thf(fact_7902_order__less__trans,axiom,
    ! [X4: int,Y3: int,Z: int] :
      ( ( ord_less_int @ X4 @ Y3 )
     => ( ( ord_less_int @ Y3 @ Z )
       => ( ord_less_int @ X4 @ Z ) ) ) ).

% order_less_trans
thf(fact_7903_ord__eq__less__subst,axiom,
    ! [A: real,F: real > real,B2: real,C: real] :
      ( ( A
        = ( F @ B2 ) )
     => ( ( ord_less_real @ B2 @ C )
       => ( ! [X3: real,Y4: real] :
              ( ( ord_less_real @ X3 @ Y4 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_7904_ord__eq__less__subst,axiom,
    ! [A: rat,F: real > rat,B2: real,C: real] :
      ( ( A
        = ( F @ B2 ) )
     => ( ( ord_less_real @ B2 @ C )
       => ( ! [X3: real,Y4: real] :
              ( ( ord_less_real @ X3 @ Y4 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_7905_ord__eq__less__subst,axiom,
    ! [A: num,F: real > num,B2: real,C: real] :
      ( ( A
        = ( F @ B2 ) )
     => ( ( ord_less_real @ B2 @ C )
       => ( ! [X3: real,Y4: real] :
              ( ( ord_less_real @ X3 @ Y4 )
             => ( ord_less_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_7906_ord__eq__less__subst,axiom,
    ! [A: nat,F: real > nat,B2: real,C: real] :
      ( ( A
        = ( F @ B2 ) )
     => ( ( ord_less_real @ B2 @ C )
       => ( ! [X3: real,Y4: real] :
              ( ( ord_less_real @ X3 @ Y4 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_7907_ord__eq__less__subst,axiom,
    ! [A: int,F: real > int,B2: real,C: real] :
      ( ( A
        = ( F @ B2 ) )
     => ( ( ord_less_real @ B2 @ C )
       => ( ! [X3: real,Y4: real] :
              ( ( ord_less_real @ X3 @ Y4 )
             => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_7908_ord__eq__less__subst,axiom,
    ! [A: real,F: rat > real,B2: rat,C: rat] :
      ( ( A
        = ( F @ B2 ) )
     => ( ( ord_less_rat @ B2 @ C )
       => ( ! [X3: rat,Y4: rat] :
              ( ( ord_less_rat @ X3 @ Y4 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_7909_ord__eq__less__subst,axiom,
    ! [A: rat,F: rat > rat,B2: rat,C: rat] :
      ( ( A
        = ( F @ B2 ) )
     => ( ( ord_less_rat @ B2 @ C )
       => ( ! [X3: rat,Y4: rat] :
              ( ( ord_less_rat @ X3 @ Y4 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_7910_ord__eq__less__subst,axiom,
    ! [A: num,F: rat > num,B2: rat,C: rat] :
      ( ( A
        = ( F @ B2 ) )
     => ( ( ord_less_rat @ B2 @ C )
       => ( ! [X3: rat,Y4: rat] :
              ( ( ord_less_rat @ X3 @ Y4 )
             => ( ord_less_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_7911_ord__eq__less__subst,axiom,
    ! [A: nat,F: rat > nat,B2: rat,C: rat] :
      ( ( A
        = ( F @ B2 ) )
     => ( ( ord_less_rat @ B2 @ C )
       => ( ! [X3: rat,Y4: rat] :
              ( ( ord_less_rat @ X3 @ Y4 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_7912_ord__eq__less__subst,axiom,
    ! [A: int,F: rat > int,B2: rat,C: rat] :
      ( ( A
        = ( F @ B2 ) )
     => ( ( ord_less_rat @ B2 @ C )
       => ( ! [X3: rat,Y4: rat] :
              ( ( ord_less_rat @ X3 @ Y4 )
             => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_7913_ord__less__eq__subst,axiom,
    ! [A: real,B2: real,F: real > real,C: real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X3: real,Y4: real] :
              ( ( ord_less_real @ X3 @ Y4 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_7914_ord__less__eq__subst,axiom,
    ! [A: real,B2: real,F: real > rat,C: rat] :
      ( ( ord_less_real @ A @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X3: real,Y4: real] :
              ( ( ord_less_real @ X3 @ Y4 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_7915_ord__less__eq__subst,axiom,
    ! [A: real,B2: real,F: real > num,C: num] :
      ( ( ord_less_real @ A @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X3: real,Y4: real] :
              ( ( ord_less_real @ X3 @ Y4 )
             => ( ord_less_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_7916_ord__less__eq__subst,axiom,
    ! [A: real,B2: real,F: real > nat,C: nat] :
      ( ( ord_less_real @ A @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X3: real,Y4: real] :
              ( ( ord_less_real @ X3 @ Y4 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_7917_ord__less__eq__subst,axiom,
    ! [A: real,B2: real,F: real > int,C: int] :
      ( ( ord_less_real @ A @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X3: real,Y4: real] :
              ( ( ord_less_real @ X3 @ Y4 )
             => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_7918_ord__less__eq__subst,axiom,
    ! [A: rat,B2: rat,F: rat > real,C: real] :
      ( ( ord_less_rat @ A @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X3: rat,Y4: rat] :
              ( ( ord_less_rat @ X3 @ Y4 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_7919_ord__less__eq__subst,axiom,
    ! [A: rat,B2: rat,F: rat > rat,C: rat] :
      ( ( ord_less_rat @ A @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X3: rat,Y4: rat] :
              ( ( ord_less_rat @ X3 @ Y4 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_7920_ord__less__eq__subst,axiom,
    ! [A: rat,B2: rat,F: rat > num,C: num] :
      ( ( ord_less_rat @ A @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X3: rat,Y4: rat] :
              ( ( ord_less_rat @ X3 @ Y4 )
             => ( ord_less_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_7921_ord__less__eq__subst,axiom,
    ! [A: rat,B2: rat,F: rat > nat,C: nat] :
      ( ( ord_less_rat @ A @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X3: rat,Y4: rat] :
              ( ( ord_less_rat @ X3 @ Y4 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_7922_ord__less__eq__subst,axiom,
    ! [A: rat,B2: rat,F: rat > int,C: int] :
      ( ( ord_less_rat @ A @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X3: rat,Y4: rat] :
              ( ( ord_less_rat @ X3 @ Y4 )
             => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_7923_order__less__irrefl,axiom,
    ! [X4: real] :
      ~ ( ord_less_real @ X4 @ X4 ) ).

% order_less_irrefl
thf(fact_7924_order__less__irrefl,axiom,
    ! [X4: rat] :
      ~ ( ord_less_rat @ X4 @ X4 ) ).

% order_less_irrefl
thf(fact_7925_order__less__irrefl,axiom,
    ! [X4: num] :
      ~ ( ord_less_num @ X4 @ X4 ) ).

% order_less_irrefl
thf(fact_7926_order__less__irrefl,axiom,
    ! [X4: nat] :
      ~ ( ord_less_nat @ X4 @ X4 ) ).

% order_less_irrefl
thf(fact_7927_order__less__irrefl,axiom,
    ! [X4: int] :
      ~ ( ord_less_int @ X4 @ X4 ) ).

% order_less_irrefl
thf(fact_7928_order__less__subst1,axiom,
    ! [A: real,F: real > real,B2: real,C: real] :
      ( ( ord_less_real @ A @ ( F @ B2 ) )
     => ( ( ord_less_real @ B2 @ C )
       => ( ! [X3: real,Y4: real] :
              ( ( ord_less_real @ X3 @ Y4 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_7929_order__less__subst1,axiom,
    ! [A: real,F: rat > real,B2: rat,C: rat] :
      ( ( ord_less_real @ A @ ( F @ B2 ) )
     => ( ( ord_less_rat @ B2 @ C )
       => ( ! [X3: rat,Y4: rat] :
              ( ( ord_less_rat @ X3 @ Y4 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_7930_order__less__subst1,axiom,
    ! [A: real,F: num > real,B2: num,C: num] :
      ( ( ord_less_real @ A @ ( F @ B2 ) )
     => ( ( ord_less_num @ B2 @ C )
       => ( ! [X3: num,Y4: num] :
              ( ( ord_less_num @ X3 @ Y4 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_7931_order__less__subst1,axiom,
    ! [A: real,F: nat > real,B2: nat,C: nat] :
      ( ( ord_less_real @ A @ ( F @ B2 ) )
     => ( ( ord_less_nat @ B2 @ C )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_nat @ X3 @ Y4 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_7932_order__less__subst1,axiom,
    ! [A: real,F: int > real,B2: int,C: int] :
      ( ( ord_less_real @ A @ ( F @ B2 ) )
     => ( ( ord_less_int @ B2 @ C )
       => ( ! [X3: int,Y4: int] :
              ( ( ord_less_int @ X3 @ Y4 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_7933_order__less__subst1,axiom,
    ! [A: rat,F: real > rat,B2: real,C: real] :
      ( ( ord_less_rat @ A @ ( F @ B2 ) )
     => ( ( ord_less_real @ B2 @ C )
       => ( ! [X3: real,Y4: real] :
              ( ( ord_less_real @ X3 @ Y4 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_7934_order__less__subst1,axiom,
    ! [A: rat,F: rat > rat,B2: rat,C: rat] :
      ( ( ord_less_rat @ A @ ( F @ B2 ) )
     => ( ( ord_less_rat @ B2 @ C )
       => ( ! [X3: rat,Y4: rat] :
              ( ( ord_less_rat @ X3 @ Y4 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_7935_order__less__subst1,axiom,
    ! [A: rat,F: num > rat,B2: num,C: num] :
      ( ( ord_less_rat @ A @ ( F @ B2 ) )
     => ( ( ord_less_num @ B2 @ C )
       => ( ! [X3: num,Y4: num] :
              ( ( ord_less_num @ X3 @ Y4 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_7936_order__less__subst1,axiom,
    ! [A: rat,F: nat > rat,B2: nat,C: nat] :
      ( ( ord_less_rat @ A @ ( F @ B2 ) )
     => ( ( ord_less_nat @ B2 @ C )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_nat @ X3 @ Y4 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_7937_order__less__subst1,axiom,
    ! [A: rat,F: int > rat,B2: int,C: int] :
      ( ( ord_less_rat @ A @ ( F @ B2 ) )
     => ( ( ord_less_int @ B2 @ C )
       => ( ! [X3: int,Y4: int] :
              ( ( ord_less_int @ X3 @ Y4 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_7938_order__less__subst2,axiom,
    ! [A: real,B2: real,F: real > real,C: real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ( ord_less_real @ ( F @ B2 ) @ C )
       => ( ! [X3: real,Y4: real] :
              ( ( ord_less_real @ X3 @ Y4 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_7939_order__less__subst2,axiom,
    ! [A: real,B2: real,F: real > rat,C: rat] :
      ( ( ord_less_real @ A @ B2 )
     => ( ( ord_less_rat @ ( F @ B2 ) @ C )
       => ( ! [X3: real,Y4: real] :
              ( ( ord_less_real @ X3 @ Y4 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_7940_order__less__subst2,axiom,
    ! [A: real,B2: real,F: real > num,C: num] :
      ( ( ord_less_real @ A @ B2 )
     => ( ( ord_less_num @ ( F @ B2 ) @ C )
       => ( ! [X3: real,Y4: real] :
              ( ( ord_less_real @ X3 @ Y4 )
             => ( ord_less_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_7941_order__less__subst2,axiom,
    ! [A: real,B2: real,F: real > nat,C: nat] :
      ( ( ord_less_real @ A @ B2 )
     => ( ( ord_less_nat @ ( F @ B2 ) @ C )
       => ( ! [X3: real,Y4: real] :
              ( ( ord_less_real @ X3 @ Y4 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_7942_order__less__subst2,axiom,
    ! [A: real,B2: real,F: real > int,C: int] :
      ( ( ord_less_real @ A @ B2 )
     => ( ( ord_less_int @ ( F @ B2 ) @ C )
       => ( ! [X3: real,Y4: real] :
              ( ( ord_less_real @ X3 @ Y4 )
             => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_7943_order__less__subst2,axiom,
    ! [A: rat,B2: rat,F: rat > real,C: real] :
      ( ( ord_less_rat @ A @ B2 )
     => ( ( ord_less_real @ ( F @ B2 ) @ C )
       => ( ! [X3: rat,Y4: rat] :
              ( ( ord_less_rat @ X3 @ Y4 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_7944_order__less__subst2,axiom,
    ! [A: rat,B2: rat,F: rat > rat,C: rat] :
      ( ( ord_less_rat @ A @ B2 )
     => ( ( ord_less_rat @ ( F @ B2 ) @ C )
       => ( ! [X3: rat,Y4: rat] :
              ( ( ord_less_rat @ X3 @ Y4 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_7945_order__less__subst2,axiom,
    ! [A: rat,B2: rat,F: rat > num,C: num] :
      ( ( ord_less_rat @ A @ B2 )
     => ( ( ord_less_num @ ( F @ B2 ) @ C )
       => ( ! [X3: rat,Y4: rat] :
              ( ( ord_less_rat @ X3 @ Y4 )
             => ( ord_less_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_7946_order__less__subst2,axiom,
    ! [A: rat,B2: rat,F: rat > nat,C: nat] :
      ( ( ord_less_rat @ A @ B2 )
     => ( ( ord_less_nat @ ( F @ B2 ) @ C )
       => ( ! [X3: rat,Y4: rat] :
              ( ( ord_less_rat @ X3 @ Y4 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_7947_order__less__subst2,axiom,
    ! [A: rat,B2: rat,F: rat > int,C: int] :
      ( ( ord_less_rat @ A @ B2 )
     => ( ( ord_less_int @ ( F @ B2 ) @ C )
       => ( ! [X3: rat,Y4: rat] :
              ( ( ord_less_rat @ X3 @ Y4 )
             => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_7948_order__less__not__sym,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_real @ X4 @ Y3 )
     => ~ ( ord_less_real @ Y3 @ X4 ) ) ).

% order_less_not_sym
thf(fact_7949_order__less__not__sym,axiom,
    ! [X4: rat,Y3: rat] :
      ( ( ord_less_rat @ X4 @ Y3 )
     => ~ ( ord_less_rat @ Y3 @ X4 ) ) ).

% order_less_not_sym
thf(fact_7950_order__less__not__sym,axiom,
    ! [X4: num,Y3: num] :
      ( ( ord_less_num @ X4 @ Y3 )
     => ~ ( ord_less_num @ Y3 @ X4 ) ) ).

% order_less_not_sym
thf(fact_7951_order__less__not__sym,axiom,
    ! [X4: nat,Y3: nat] :
      ( ( ord_less_nat @ X4 @ Y3 )
     => ~ ( ord_less_nat @ Y3 @ X4 ) ) ).

% order_less_not_sym
thf(fact_7952_order__less__not__sym,axiom,
    ! [X4: int,Y3: int] :
      ( ( ord_less_int @ X4 @ Y3 )
     => ~ ( ord_less_int @ Y3 @ X4 ) ) ).

% order_less_not_sym
thf(fact_7953_order__less__imp__triv,axiom,
    ! [X4: real,Y3: real,P: $o] :
      ( ( ord_less_real @ X4 @ Y3 )
     => ( ( ord_less_real @ Y3 @ X4 )
       => P ) ) ).

% order_less_imp_triv
thf(fact_7954_order__less__imp__triv,axiom,
    ! [X4: rat,Y3: rat,P: $o] :
      ( ( ord_less_rat @ X4 @ Y3 )
     => ( ( ord_less_rat @ Y3 @ X4 )
       => P ) ) ).

% order_less_imp_triv
thf(fact_7955_order__less__imp__triv,axiom,
    ! [X4: num,Y3: num,P: $o] :
      ( ( ord_less_num @ X4 @ Y3 )
     => ( ( ord_less_num @ Y3 @ X4 )
       => P ) ) ).

% order_less_imp_triv
thf(fact_7956_order__less__imp__triv,axiom,
    ! [X4: nat,Y3: nat,P: $o] :
      ( ( ord_less_nat @ X4 @ Y3 )
     => ( ( ord_less_nat @ Y3 @ X4 )
       => P ) ) ).

% order_less_imp_triv
thf(fact_7957_order__less__imp__triv,axiom,
    ! [X4: int,Y3: int,P: $o] :
      ( ( ord_less_int @ X4 @ Y3 )
     => ( ( ord_less_int @ Y3 @ X4 )
       => P ) ) ).

% order_less_imp_triv
thf(fact_7958_linorder__less__linear,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_real @ X4 @ Y3 )
      | ( X4 = Y3 )
      | ( ord_less_real @ Y3 @ X4 ) ) ).

% linorder_less_linear
thf(fact_7959_linorder__less__linear,axiom,
    ! [X4: rat,Y3: rat] :
      ( ( ord_less_rat @ X4 @ Y3 )
      | ( X4 = Y3 )
      | ( ord_less_rat @ Y3 @ X4 ) ) ).

% linorder_less_linear
thf(fact_7960_linorder__less__linear,axiom,
    ! [X4: num,Y3: num] :
      ( ( ord_less_num @ X4 @ Y3 )
      | ( X4 = Y3 )
      | ( ord_less_num @ Y3 @ X4 ) ) ).

% linorder_less_linear
thf(fact_7961_linorder__less__linear,axiom,
    ! [X4: nat,Y3: nat] :
      ( ( ord_less_nat @ X4 @ Y3 )
      | ( X4 = Y3 )
      | ( ord_less_nat @ Y3 @ X4 ) ) ).

% linorder_less_linear
thf(fact_7962_linorder__less__linear,axiom,
    ! [X4: int,Y3: int] :
      ( ( ord_less_int @ X4 @ Y3 )
      | ( X4 = Y3 )
      | ( ord_less_int @ Y3 @ X4 ) ) ).

% linorder_less_linear
thf(fact_7963_order__less__imp__not__eq,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_real @ X4 @ Y3 )
     => ( X4 != Y3 ) ) ).

% order_less_imp_not_eq
thf(fact_7964_order__less__imp__not__eq,axiom,
    ! [X4: rat,Y3: rat] :
      ( ( ord_less_rat @ X4 @ Y3 )
     => ( X4 != Y3 ) ) ).

% order_less_imp_not_eq
thf(fact_7965_order__less__imp__not__eq,axiom,
    ! [X4: num,Y3: num] :
      ( ( ord_less_num @ X4 @ Y3 )
     => ( X4 != Y3 ) ) ).

% order_less_imp_not_eq
thf(fact_7966_order__less__imp__not__eq,axiom,
    ! [X4: nat,Y3: nat] :
      ( ( ord_less_nat @ X4 @ Y3 )
     => ( X4 != Y3 ) ) ).

% order_less_imp_not_eq
thf(fact_7967_order__less__imp__not__eq,axiom,
    ! [X4: int,Y3: int] :
      ( ( ord_less_int @ X4 @ Y3 )
     => ( X4 != Y3 ) ) ).

% order_less_imp_not_eq
thf(fact_7968_order__less__imp__not__eq2,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_real @ X4 @ Y3 )
     => ( Y3 != X4 ) ) ).

% order_less_imp_not_eq2
thf(fact_7969_order__less__imp__not__eq2,axiom,
    ! [X4: rat,Y3: rat] :
      ( ( ord_less_rat @ X4 @ Y3 )
     => ( Y3 != X4 ) ) ).

% order_less_imp_not_eq2
thf(fact_7970_order__less__imp__not__eq2,axiom,
    ! [X4: num,Y3: num] :
      ( ( ord_less_num @ X4 @ Y3 )
     => ( Y3 != X4 ) ) ).

% order_less_imp_not_eq2
thf(fact_7971_order__less__imp__not__eq2,axiom,
    ! [X4: nat,Y3: nat] :
      ( ( ord_less_nat @ X4 @ Y3 )
     => ( Y3 != X4 ) ) ).

% order_less_imp_not_eq2
thf(fact_7972_order__less__imp__not__eq2,axiom,
    ! [X4: int,Y3: int] :
      ( ( ord_less_int @ X4 @ Y3 )
     => ( Y3 != X4 ) ) ).

% order_less_imp_not_eq2
thf(fact_7973_order__less__imp__not__less,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_real @ X4 @ Y3 )
     => ~ ( ord_less_real @ Y3 @ X4 ) ) ).

% order_less_imp_not_less
thf(fact_7974_order__less__imp__not__less,axiom,
    ! [X4: rat,Y3: rat] :
      ( ( ord_less_rat @ X4 @ Y3 )
     => ~ ( ord_less_rat @ Y3 @ X4 ) ) ).

% order_less_imp_not_less
thf(fact_7975_order__less__imp__not__less,axiom,
    ! [X4: num,Y3: num] :
      ( ( ord_less_num @ X4 @ Y3 )
     => ~ ( ord_less_num @ Y3 @ X4 ) ) ).

% order_less_imp_not_less
thf(fact_7976_order__less__imp__not__less,axiom,
    ! [X4: nat,Y3: nat] :
      ( ( ord_less_nat @ X4 @ Y3 )
     => ~ ( ord_less_nat @ Y3 @ X4 ) ) ).

% order_less_imp_not_less
thf(fact_7977_order__less__imp__not__less,axiom,
    ! [X4: int,Y3: int] :
      ( ( ord_less_int @ X4 @ Y3 )
     => ~ ( ord_less_int @ Y3 @ X4 ) ) ).

% order_less_imp_not_less
thf(fact_7978_take__bit__not__eq__mask__diff,axiom,
    ! [N: nat,A: int] :
      ( ( bit_se2923211474154528505it_int @ N @ ( bit_ri7919022796975470100ot_int @ A ) )
      = ( minus_minus_int @ ( bit_se2000444600071755411sk_int @ N ) @ ( bit_se2923211474154528505it_int @ N @ A ) ) ) ).

% take_bit_not_eq_mask_diff
thf(fact_7979_minus__numeral__inc__eq,axiom,
    ! [N: num] :
      ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( inc @ N ) ) )
      = ( bit_ri7632146776885996613nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).

% minus_numeral_inc_eq
thf(fact_7980_minus__numeral__inc__eq,axiom,
    ! [N: num] :
      ( ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ N ) ) )
      = ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ).

% minus_numeral_inc_eq
thf(fact_7981_unset__bit__int__def,axiom,
    ( bit_se4203085406695923979it_int
    = ( ^ [N2: nat,K3: int] : ( bit_se725231765392027082nd_int @ K3 @ ( bit_ri7919022796975470100ot_int @ ( bit_se545348938243370406it_int @ N2 @ one_one_int ) ) ) ) ) ).

% unset_bit_int_def
thf(fact_7982_not__int__div__2,axiom,
    ! [K: int] :
      ( ( divide_divide_int @ ( bit_ri7919022796975470100ot_int @ K ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
      = ( bit_ri7919022796975470100ot_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).

% not_int_div_2
thf(fact_7983_even__not__iff__int,axiom,
    ! [K: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri7919022796975470100ot_int @ K ) )
      = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ) ).

% even_not_iff_int
thf(fact_7984_not__numeral__Bit0__eq,axiom,
    ! [N: num] :
      ( ( bit_ri7632146776885996613nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) )
      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) ) ) ).

% not_numeral_Bit0_eq
thf(fact_7985_not__numeral__Bit0__eq,axiom,
    ! [N: num] :
      ( ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) ) ).

% not_numeral_Bit0_eq
thf(fact_7986_and__not__numerals_I2_J,axiom,
    ! [N: num] :
      ( ( bit_se725231765392027082nd_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
      = one_one_int ) ).

% and_not_numerals(2)
thf(fact_7987_and__not__numerals_I4_J,axiom,
    ! [M: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
      = ( numeral_numeral_int @ ( bit0 @ M ) ) ) ).

% and_not_numerals(4)
thf(fact_7988_bit__minus__int__iff,axiom,
    ! [K: int,N: nat] :
      ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ K ) @ N )
      = ( bit_se1146084159140164899it_int @ ( bit_ri7919022796975470100ot_int @ ( minus_minus_int @ K @ one_one_int ) ) @ N ) ) ).

% bit_minus_int_iff
thf(fact_7989_take__bit__not__mask__eq__0,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( bit_se2923211474154528505it_int @ M @ ( bit_ri7919022796975470100ot_int @ ( bit_se2000444600071755411sk_int @ N ) ) )
        = zero_zero_int ) ) ).

% take_bit_not_mask_eq_0
thf(fact_7990_push__bit__mask__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( bit_se545348938243370406it_int @ M @ ( bit_se2000444600071755411sk_int @ N ) )
      = ( bit_se725231765392027082nd_int @ ( bit_se2000444600071755411sk_int @ ( plus_plus_nat @ N @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( bit_se2000444600071755411sk_int @ M ) ) ) ) ).

% push_bit_mask_eq
thf(fact_7991_unset__bit__eq__and__not,axiom,
    ( bit_se4203085406695923979it_int
    = ( ^ [N2: nat,A4: int] : ( bit_se725231765392027082nd_int @ A4 @ ( bit_ri7919022796975470100ot_int @ ( bit_se545348938243370406it_int @ N2 @ one_one_int ) ) ) ) ) ).

% unset_bit_eq_and_not
thf(fact_7992_and__not__numerals_I5_J,axiom,
    ! [M: num,N: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).

% and_not_numerals(5)
thf(fact_7993_and__not__numerals_I7_J,axiom,
    ! [M: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
      = ( numeral_numeral_int @ ( bit0 @ M ) ) ) ).

% and_not_numerals(7)
thf(fact_7994_and__not__numerals_I3_J,axiom,
    ! [N: num] :
      ( ( bit_se725231765392027082nd_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
      = zero_zero_int ) ).

% and_not_numerals(3)
thf(fact_7995_and__not__numerals_I9_J,axiom,
    ! [M: num,N: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).

% and_not_numerals(9)
thf(fact_7996_and__not__numerals_I6_J,axiom,
    ! [M: num,N: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).

% and_not_numerals(6)
thf(fact_7997_bit__not__iff__eq,axiom,
    ! [A: int,N: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_ri7919022796975470100ot_int @ A ) @ N )
      = ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
         != zero_zero_int )
        & ~ ( bit_se1146084159140164899it_int @ A @ N ) ) ) ).

% bit_not_iff_eq
thf(fact_7998_minus__exp__eq__not__mask,axiom,
    ! [N: nat] :
      ( ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
      = ( bit_ri7632146776885996613nteger @ ( bit_se2119862282449309892nteger @ N ) ) ) ).

% minus_exp_eq_not_mask
thf(fact_7999_minus__exp__eq__not__mask,axiom,
    ! [N: nat] :
      ( ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
      = ( bit_ri7919022796975470100ot_int @ ( bit_se2000444600071755411sk_int @ N ) ) ) ).

% minus_exp_eq_not_mask
thf(fact_8000_CauchyD,axiom,
    ! [X8: nat > complex,E: real] :
      ( ( topolo6517432010174082258omplex @ X8 )
     => ( ( ord_less_real @ zero_zero_real @ E )
       => ? [M9: nat] :
          ! [M5: nat] :
            ( ( ord_less_eq_nat @ M9 @ M5 )
           => ! [N6: nat] :
                ( ( ord_less_eq_nat @ M9 @ N6 )
               => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( X8 @ M5 ) @ ( X8 @ N6 ) ) ) @ E ) ) ) ) ) ).

% CauchyD
thf(fact_8001_CauchyD,axiom,
    ! [X8: nat > real,E: real] :
      ( ( topolo4055970368930404560y_real @ X8 )
     => ( ( ord_less_real @ zero_zero_real @ E )
       => ? [M9: nat] :
          ! [M5: nat] :
            ( ( ord_less_eq_nat @ M9 @ M5 )
           => ! [N6: nat] :
                ( ( ord_less_eq_nat @ M9 @ N6 )
               => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( X8 @ M5 ) @ ( X8 @ N6 ) ) ) @ E ) ) ) ) ) ).

% CauchyD
thf(fact_8002_CauchyI,axiom,
    ! [X8: nat > complex] :
      ( ! [E2: real] :
          ( ( ord_less_real @ zero_zero_real @ E2 )
         => ? [M10: nat] :
            ! [M4: nat] :
              ( ( ord_less_eq_nat @ M10 @ M4 )
             => ! [N4: nat] :
                  ( ( ord_less_eq_nat @ M10 @ N4 )
                 => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( X8 @ M4 ) @ ( X8 @ N4 ) ) ) @ E2 ) ) ) )
     => ( topolo6517432010174082258omplex @ X8 ) ) ).

% CauchyI
thf(fact_8003_CauchyI,axiom,
    ! [X8: nat > real] :
      ( ! [E2: real] :
          ( ( ord_less_real @ zero_zero_real @ E2 )
         => ? [M10: nat] :
            ! [M4: nat] :
              ( ( ord_less_eq_nat @ M10 @ M4 )
             => ! [N4: nat] :
                  ( ( ord_less_eq_nat @ M10 @ N4 )
                 => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( X8 @ M4 ) @ ( X8 @ N4 ) ) ) @ E2 ) ) ) )
     => ( topolo4055970368930404560y_real @ X8 ) ) ).

% CauchyI
thf(fact_8004_Cauchy__iff,axiom,
    ( topolo6517432010174082258omplex
    = ( ^ [X6: nat > complex] :
        ! [E3: real] :
          ( ( ord_less_real @ zero_zero_real @ E3 )
         => ? [M8: nat] :
            ! [M6: nat] :
              ( ( ord_less_eq_nat @ M8 @ M6 )
             => ! [N2: nat] :
                  ( ( ord_less_eq_nat @ M8 @ N2 )
                 => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( X6 @ M6 ) @ ( X6 @ N2 ) ) ) @ E3 ) ) ) ) ) ) ).

% Cauchy_iff
thf(fact_8005_Cauchy__iff,axiom,
    ( topolo4055970368930404560y_real
    = ( ^ [X6: nat > real] :
        ! [E3: real] :
          ( ( ord_less_real @ zero_zero_real @ E3 )
         => ? [M8: nat] :
            ! [M6: nat] :
              ( ( ord_less_eq_nat @ M8 @ M6 )
             => ! [N2: nat] :
                  ( ( ord_less_eq_nat @ M8 @ N2 )
                 => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( X6 @ M6 ) @ ( X6 @ N2 ) ) ) @ E3 ) ) ) ) ) ) ).

% Cauchy_iff
thf(fact_8006_leD,axiom,
    ! [Y3: real,X4: real] :
      ( ( ord_less_eq_real @ Y3 @ X4 )
     => ~ ( ord_less_real @ X4 @ Y3 ) ) ).

% leD
thf(fact_8007_leD,axiom,
    ! [Y3: set_nat,X4: set_nat] :
      ( ( ord_less_eq_set_nat @ Y3 @ X4 )
     => ~ ( ord_less_set_nat @ X4 @ Y3 ) ) ).

% leD
thf(fact_8008_leD,axiom,
    ! [Y3: rat,X4: rat] :
      ( ( ord_less_eq_rat @ Y3 @ X4 )
     => ~ ( ord_less_rat @ X4 @ Y3 ) ) ).

% leD
thf(fact_8009_leD,axiom,
    ! [Y3: num,X4: num] :
      ( ( ord_less_eq_num @ Y3 @ X4 )
     => ~ ( ord_less_num @ X4 @ Y3 ) ) ).

% leD
thf(fact_8010_leD,axiom,
    ! [Y3: nat,X4: nat] :
      ( ( ord_less_eq_nat @ Y3 @ X4 )
     => ~ ( ord_less_nat @ X4 @ Y3 ) ) ).

% leD
thf(fact_8011_leD,axiom,
    ! [Y3: int,X4: int] :
      ( ( ord_less_eq_int @ Y3 @ X4 )
     => ~ ( ord_less_int @ X4 @ Y3 ) ) ).

% leD
thf(fact_8012_leI,axiom,
    ! [X4: real,Y3: real] :
      ( ~ ( ord_less_real @ X4 @ Y3 )
     => ( ord_less_eq_real @ Y3 @ X4 ) ) ).

% leI
thf(fact_8013_leI,axiom,
    ! [X4: rat,Y3: rat] :
      ( ~ ( ord_less_rat @ X4 @ Y3 )
     => ( ord_less_eq_rat @ Y3 @ X4 ) ) ).

% leI
thf(fact_8014_leI,axiom,
    ! [X4: num,Y3: num] :
      ( ~ ( ord_less_num @ X4 @ Y3 )
     => ( ord_less_eq_num @ Y3 @ X4 ) ) ).

% leI
thf(fact_8015_leI,axiom,
    ! [X4: nat,Y3: nat] :
      ( ~ ( ord_less_nat @ X4 @ Y3 )
     => ( ord_less_eq_nat @ Y3 @ X4 ) ) ).

% leI
thf(fact_8016_leI,axiom,
    ! [X4: int,Y3: int] :
      ( ~ ( ord_less_int @ X4 @ Y3 )
     => ( ord_less_eq_int @ Y3 @ X4 ) ) ).

% leI
thf(fact_8017_nless__le,axiom,
    ! [A: real,B2: real] :
      ( ( ~ ( ord_less_real @ A @ B2 ) )
      = ( ~ ( ord_less_eq_real @ A @ B2 )
        | ( A = B2 ) ) ) ).

% nless_le
thf(fact_8018_nless__le,axiom,
    ! [A: set_nat,B2: set_nat] :
      ( ( ~ ( ord_less_set_nat @ A @ B2 ) )
      = ( ~ ( ord_less_eq_set_nat @ A @ B2 )
        | ( A = B2 ) ) ) ).

% nless_le
thf(fact_8019_nless__le,axiom,
    ! [A: rat,B2: rat] :
      ( ( ~ ( ord_less_rat @ A @ B2 ) )
      = ( ~ ( ord_less_eq_rat @ A @ B2 )
        | ( A = B2 ) ) ) ).

% nless_le
thf(fact_8020_nless__le,axiom,
    ! [A: num,B2: num] :
      ( ( ~ ( ord_less_num @ A @ B2 ) )
      = ( ~ ( ord_less_eq_num @ A @ B2 )
        | ( A = B2 ) ) ) ).

% nless_le
thf(fact_8021_nless__le,axiom,
    ! [A: nat,B2: nat] :
      ( ( ~ ( ord_less_nat @ A @ B2 ) )
      = ( ~ ( ord_less_eq_nat @ A @ B2 )
        | ( A = B2 ) ) ) ).

% nless_le
thf(fact_8022_nless__le,axiom,
    ! [A: int,B2: int] :
      ( ( ~ ( ord_less_int @ A @ B2 ) )
      = ( ~ ( ord_less_eq_int @ A @ B2 )
        | ( A = B2 ) ) ) ).

% nless_le
thf(fact_8023_antisym__conv1,axiom,
    ! [X4: real,Y3: real] :
      ( ~ ( ord_less_real @ X4 @ Y3 )
     => ( ( ord_less_eq_real @ X4 @ Y3 )
        = ( X4 = Y3 ) ) ) ).

% antisym_conv1
thf(fact_8024_antisym__conv1,axiom,
    ! [X4: set_nat,Y3: set_nat] :
      ( ~ ( ord_less_set_nat @ X4 @ Y3 )
     => ( ( ord_less_eq_set_nat @ X4 @ Y3 )
        = ( X4 = Y3 ) ) ) ).

% antisym_conv1
thf(fact_8025_antisym__conv1,axiom,
    ! [X4: rat,Y3: rat] :
      ( ~ ( ord_less_rat @ X4 @ Y3 )
     => ( ( ord_less_eq_rat @ X4 @ Y3 )
        = ( X4 = Y3 ) ) ) ).

% antisym_conv1
thf(fact_8026_antisym__conv1,axiom,
    ! [X4: num,Y3: num] :
      ( ~ ( ord_less_num @ X4 @ Y3 )
     => ( ( ord_less_eq_num @ X4 @ Y3 )
        = ( X4 = Y3 ) ) ) ).

% antisym_conv1
thf(fact_8027_antisym__conv1,axiom,
    ! [X4: nat,Y3: nat] :
      ( ~ ( ord_less_nat @ X4 @ Y3 )
     => ( ( ord_less_eq_nat @ X4 @ Y3 )
        = ( X4 = Y3 ) ) ) ).

% antisym_conv1
thf(fact_8028_antisym__conv1,axiom,
    ! [X4: int,Y3: int] :
      ( ~ ( ord_less_int @ X4 @ Y3 )
     => ( ( ord_less_eq_int @ X4 @ Y3 )
        = ( X4 = Y3 ) ) ) ).

% antisym_conv1
thf(fact_8029_antisym__conv2,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_eq_real @ X4 @ Y3 )
     => ( ( ~ ( ord_less_real @ X4 @ Y3 ) )
        = ( X4 = Y3 ) ) ) ).

% antisym_conv2
thf(fact_8030_antisym__conv2,axiom,
    ! [X4: set_nat,Y3: set_nat] :
      ( ( ord_less_eq_set_nat @ X4 @ Y3 )
     => ( ( ~ ( ord_less_set_nat @ X4 @ Y3 ) )
        = ( X4 = Y3 ) ) ) ).

% antisym_conv2
thf(fact_8031_antisym__conv2,axiom,
    ! [X4: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ X4 @ Y3 )
     => ( ( ~ ( ord_less_rat @ X4 @ Y3 ) )
        = ( X4 = Y3 ) ) ) ).

% antisym_conv2
thf(fact_8032_antisym__conv2,axiom,
    ! [X4: num,Y3: num] :
      ( ( ord_less_eq_num @ X4 @ Y3 )
     => ( ( ~ ( ord_less_num @ X4 @ Y3 ) )
        = ( X4 = Y3 ) ) ) ).

% antisym_conv2
thf(fact_8033_antisym__conv2,axiom,
    ! [X4: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ X4 @ Y3 )
     => ( ( ~ ( ord_less_nat @ X4 @ Y3 ) )
        = ( X4 = Y3 ) ) ) ).

% antisym_conv2
thf(fact_8034_antisym__conv2,axiom,
    ! [X4: int,Y3: int] :
      ( ( ord_less_eq_int @ X4 @ Y3 )
     => ( ( ~ ( ord_less_int @ X4 @ Y3 ) )
        = ( X4 = Y3 ) ) ) ).

% antisym_conv2
thf(fact_8035_dense__ge,axiom,
    ! [Z: real,Y3: real] :
      ( ! [X3: real] :
          ( ( ord_less_real @ Z @ X3 )
         => ( ord_less_eq_real @ Y3 @ X3 ) )
     => ( ord_less_eq_real @ Y3 @ Z ) ) ).

% dense_ge
thf(fact_8036_dense__ge,axiom,
    ! [Z: rat,Y3: rat] :
      ( ! [X3: rat] :
          ( ( ord_less_rat @ Z @ X3 )
         => ( ord_less_eq_rat @ Y3 @ X3 ) )
     => ( ord_less_eq_rat @ Y3 @ Z ) ) ).

% dense_ge
thf(fact_8037_dense__le,axiom,
    ! [Y3: real,Z: real] :
      ( ! [X3: real] :
          ( ( ord_less_real @ X3 @ Y3 )
         => ( ord_less_eq_real @ X3 @ Z ) )
     => ( ord_less_eq_real @ Y3 @ Z ) ) ).

% dense_le
thf(fact_8038_dense__le,axiom,
    ! [Y3: rat,Z: rat] :
      ( ! [X3: rat] :
          ( ( ord_less_rat @ X3 @ Y3 )
         => ( ord_less_eq_rat @ X3 @ Z ) )
     => ( ord_less_eq_rat @ Y3 @ Z ) ) ).

% dense_le
thf(fact_8039_less__le__not__le,axiom,
    ( ord_less_real
    = ( ^ [X: real,Y: real] :
          ( ( ord_less_eq_real @ X @ Y )
          & ~ ( ord_less_eq_real @ Y @ X ) ) ) ) ).

% less_le_not_le
thf(fact_8040_less__le__not__le,axiom,
    ( ord_less_set_nat
    = ( ^ [X: set_nat,Y: set_nat] :
          ( ( ord_less_eq_set_nat @ X @ Y )
          & ~ ( ord_less_eq_set_nat @ Y @ X ) ) ) ) ).

% less_le_not_le
thf(fact_8041_less__le__not__le,axiom,
    ( ord_less_rat
    = ( ^ [X: rat,Y: rat] :
          ( ( ord_less_eq_rat @ X @ Y )
          & ~ ( ord_less_eq_rat @ Y @ X ) ) ) ) ).

% less_le_not_le
thf(fact_8042_less__le__not__le,axiom,
    ( ord_less_num
    = ( ^ [X: num,Y: num] :
          ( ( ord_less_eq_num @ X @ Y )
          & ~ ( ord_less_eq_num @ Y @ X ) ) ) ) ).

% less_le_not_le
thf(fact_8043_less__le__not__le,axiom,
    ( ord_less_nat
    = ( ^ [X: nat,Y: nat] :
          ( ( ord_less_eq_nat @ X @ Y )
          & ~ ( ord_less_eq_nat @ Y @ X ) ) ) ) ).

% less_le_not_le
thf(fact_8044_less__le__not__le,axiom,
    ( ord_less_int
    = ( ^ [X: int,Y: int] :
          ( ( ord_less_eq_int @ X @ Y )
          & ~ ( ord_less_eq_int @ Y @ X ) ) ) ) ).

% less_le_not_le
thf(fact_8045_not__le__imp__less,axiom,
    ! [Y3: real,X4: real] :
      ( ~ ( ord_less_eq_real @ Y3 @ X4 )
     => ( ord_less_real @ X4 @ Y3 ) ) ).

% not_le_imp_less
thf(fact_8046_not__le__imp__less,axiom,
    ! [Y3: rat,X4: rat] :
      ( ~ ( ord_less_eq_rat @ Y3 @ X4 )
     => ( ord_less_rat @ X4 @ Y3 ) ) ).

% not_le_imp_less
thf(fact_8047_not__le__imp__less,axiom,
    ! [Y3: num,X4: num] :
      ( ~ ( ord_less_eq_num @ Y3 @ X4 )
     => ( ord_less_num @ X4 @ Y3 ) ) ).

% not_le_imp_less
thf(fact_8048_not__le__imp__less,axiom,
    ! [Y3: nat,X4: nat] :
      ( ~ ( ord_less_eq_nat @ Y3 @ X4 )
     => ( ord_less_nat @ X4 @ Y3 ) ) ).

% not_le_imp_less
thf(fact_8049_not__le__imp__less,axiom,
    ! [Y3: int,X4: int] :
      ( ~ ( ord_less_eq_int @ Y3 @ X4 )
     => ( ord_less_int @ X4 @ Y3 ) ) ).

% not_le_imp_less
thf(fact_8050_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_real
    = ( ^ [A4: real,B3: real] :
          ( ( ord_less_real @ A4 @ B3 )
          | ( A4 = B3 ) ) ) ) ).

% order.order_iff_strict
thf(fact_8051_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A4: set_nat,B3: set_nat] :
          ( ( ord_less_set_nat @ A4 @ B3 )
          | ( A4 = B3 ) ) ) ) ).

% order.order_iff_strict
thf(fact_8052_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_rat
    = ( ^ [A4: rat,B3: rat] :
          ( ( ord_less_rat @ A4 @ B3 )
          | ( A4 = B3 ) ) ) ) ).

% order.order_iff_strict
thf(fact_8053_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_num
    = ( ^ [A4: num,B3: num] :
          ( ( ord_less_num @ A4 @ B3 )
          | ( A4 = B3 ) ) ) ) ).

% order.order_iff_strict
thf(fact_8054_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [A4: nat,B3: nat] :
          ( ( ord_less_nat @ A4 @ B3 )
          | ( A4 = B3 ) ) ) ) ).

% order.order_iff_strict
thf(fact_8055_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_int
    = ( ^ [A4: int,B3: int] :
          ( ( ord_less_int @ A4 @ B3 )
          | ( A4 = B3 ) ) ) ) ).

% order.order_iff_strict
thf(fact_8056_order_Ostrict__iff__order,axiom,
    ( ord_less_real
    = ( ^ [A4: real,B3: real] :
          ( ( ord_less_eq_real @ A4 @ B3 )
          & ( A4 != B3 ) ) ) ) ).

% order.strict_iff_order
thf(fact_8057_order_Ostrict__iff__order,axiom,
    ( ord_less_set_nat
    = ( ^ [A4: set_nat,B3: set_nat] :
          ( ( ord_less_eq_set_nat @ A4 @ B3 )
          & ( A4 != B3 ) ) ) ) ).

% order.strict_iff_order
thf(fact_8058_order_Ostrict__iff__order,axiom,
    ( ord_less_rat
    = ( ^ [A4: rat,B3: rat] :
          ( ( ord_less_eq_rat @ A4 @ B3 )
          & ( A4 != B3 ) ) ) ) ).

% order.strict_iff_order
thf(fact_8059_order_Ostrict__iff__order,axiom,
    ( ord_less_num
    = ( ^ [A4: num,B3: num] :
          ( ( ord_less_eq_num @ A4 @ B3 )
          & ( A4 != B3 ) ) ) ) ).

% order.strict_iff_order
thf(fact_8060_order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [A4: nat,B3: nat] :
          ( ( ord_less_eq_nat @ A4 @ B3 )
          & ( A4 != B3 ) ) ) ) ).

% order.strict_iff_order
thf(fact_8061_order_Ostrict__iff__order,axiom,
    ( ord_less_int
    = ( ^ [A4: int,B3: int] :
          ( ( ord_less_eq_int @ A4 @ B3 )
          & ( A4 != B3 ) ) ) ) ).

% order.strict_iff_order
thf(fact_8062_order_Ostrict__trans1,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ord_less_eq_real @ A @ B2 )
     => ( ( ord_less_real @ B2 @ C )
       => ( ord_less_real @ A @ C ) ) ) ).

% order.strict_trans1
thf(fact_8063_order_Ostrict__trans1,axiom,
    ! [A: set_nat,B2: set_nat,C: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B2 )
     => ( ( ord_less_set_nat @ B2 @ C )
       => ( ord_less_set_nat @ A @ C ) ) ) ).

% order.strict_trans1
thf(fact_8064_order_Ostrict__trans1,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B2 )
     => ( ( ord_less_rat @ B2 @ C )
       => ( ord_less_rat @ A @ C ) ) ) ).

% order.strict_trans1
thf(fact_8065_order_Ostrict__trans1,axiom,
    ! [A: num,B2: num,C: num] :
      ( ( ord_less_eq_num @ A @ B2 )
     => ( ( ord_less_num @ B2 @ C )
       => ( ord_less_num @ A @ C ) ) ) ).

% order.strict_trans1
thf(fact_8066_order_Ostrict__trans1,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( ord_less_nat @ B2 @ C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% order.strict_trans1
thf(fact_8067_order_Ostrict__trans1,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( ord_less_eq_int @ A @ B2 )
     => ( ( ord_less_int @ B2 @ C )
       => ( ord_less_int @ A @ C ) ) ) ).

% order.strict_trans1
thf(fact_8068_order_Ostrict__trans2,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ( ord_less_eq_real @ B2 @ C )
       => ( ord_less_real @ A @ C ) ) ) ).

% order.strict_trans2
thf(fact_8069_order_Ostrict__trans2,axiom,
    ! [A: set_nat,B2: set_nat,C: set_nat] :
      ( ( ord_less_set_nat @ A @ B2 )
     => ( ( ord_less_eq_set_nat @ B2 @ C )
       => ( ord_less_set_nat @ A @ C ) ) ) ).

% order.strict_trans2
thf(fact_8070_order_Ostrict__trans2,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( ord_less_rat @ A @ B2 )
     => ( ( ord_less_eq_rat @ B2 @ C )
       => ( ord_less_rat @ A @ C ) ) ) ).

% order.strict_trans2
thf(fact_8071_order_Ostrict__trans2,axiom,
    ! [A: num,B2: num,C: num] :
      ( ( ord_less_num @ A @ B2 )
     => ( ( ord_less_eq_num @ B2 @ C )
       => ( ord_less_num @ A @ C ) ) ) ).

% order.strict_trans2
thf(fact_8072_order_Ostrict__trans2,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% order.strict_trans2
thf(fact_8073_order_Ostrict__trans2,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( ord_less_int @ A @ B2 )
     => ( ( ord_less_eq_int @ B2 @ C )
       => ( ord_less_int @ A @ C ) ) ) ).

% order.strict_trans2
thf(fact_8074_order_Ostrict__iff__not,axiom,
    ( ord_less_real
    = ( ^ [A4: real,B3: real] :
          ( ( ord_less_eq_real @ A4 @ B3 )
          & ~ ( ord_less_eq_real @ B3 @ A4 ) ) ) ) ).

% order.strict_iff_not
thf(fact_8075_order_Ostrict__iff__not,axiom,
    ( ord_less_set_nat
    = ( ^ [A4: set_nat,B3: set_nat] :
          ( ( ord_less_eq_set_nat @ A4 @ B3 )
          & ~ ( ord_less_eq_set_nat @ B3 @ A4 ) ) ) ) ).

% order.strict_iff_not
thf(fact_8076_order_Ostrict__iff__not,axiom,
    ( ord_less_rat
    = ( ^ [A4: rat,B3: rat] :
          ( ( ord_less_eq_rat @ A4 @ B3 )
          & ~ ( ord_less_eq_rat @ B3 @ A4 ) ) ) ) ).

% order.strict_iff_not
thf(fact_8077_order_Ostrict__iff__not,axiom,
    ( ord_less_num
    = ( ^ [A4: num,B3: num] :
          ( ( ord_less_eq_num @ A4 @ B3 )
          & ~ ( ord_less_eq_num @ B3 @ A4 ) ) ) ) ).

% order.strict_iff_not
thf(fact_8078_order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [A4: nat,B3: nat] :
          ( ( ord_less_eq_nat @ A4 @ B3 )
          & ~ ( ord_less_eq_nat @ B3 @ A4 ) ) ) ) ).

% order.strict_iff_not
thf(fact_8079_order_Ostrict__iff__not,axiom,
    ( ord_less_int
    = ( ^ [A4: int,B3: int] :
          ( ( ord_less_eq_int @ A4 @ B3 )
          & ~ ( ord_less_eq_int @ B3 @ A4 ) ) ) ) ).

% order.strict_iff_not
thf(fact_8080_dense__ge__bounded,axiom,
    ! [Z: real,X4: real,Y3: real] :
      ( ( ord_less_real @ Z @ X4 )
     => ( ! [W3: real] :
            ( ( ord_less_real @ Z @ W3 )
           => ( ( ord_less_real @ W3 @ X4 )
             => ( ord_less_eq_real @ Y3 @ W3 ) ) )
       => ( ord_less_eq_real @ Y3 @ Z ) ) ) ).

% dense_ge_bounded
thf(fact_8081_dense__ge__bounded,axiom,
    ! [Z: rat,X4: rat,Y3: rat] :
      ( ( ord_less_rat @ Z @ X4 )
     => ( ! [W3: rat] :
            ( ( ord_less_rat @ Z @ W3 )
           => ( ( ord_less_rat @ W3 @ X4 )
             => ( ord_less_eq_rat @ Y3 @ W3 ) ) )
       => ( ord_less_eq_rat @ Y3 @ Z ) ) ) ).

% dense_ge_bounded
thf(fact_8082_dense__le__bounded,axiom,
    ! [X4: real,Y3: real,Z: real] :
      ( ( ord_less_real @ X4 @ Y3 )
     => ( ! [W3: real] :
            ( ( ord_less_real @ X4 @ W3 )
           => ( ( ord_less_real @ W3 @ Y3 )
             => ( ord_less_eq_real @ W3 @ Z ) ) )
       => ( ord_less_eq_real @ Y3 @ Z ) ) ) ).

% dense_le_bounded
thf(fact_8083_dense__le__bounded,axiom,
    ! [X4: rat,Y3: rat,Z: rat] :
      ( ( ord_less_rat @ X4 @ Y3 )
     => ( ! [W3: rat] :
            ( ( ord_less_rat @ X4 @ W3 )
           => ( ( ord_less_rat @ W3 @ Y3 )
             => ( ord_less_eq_rat @ W3 @ Z ) ) )
       => ( ord_less_eq_rat @ Y3 @ Z ) ) ) ).

% dense_le_bounded
thf(fact_8084_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_real
    = ( ^ [B3: real,A4: real] :
          ( ( ord_less_real @ B3 @ A4 )
          | ( A4 = B3 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_8085_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [B3: set_nat,A4: set_nat] :
          ( ( ord_less_set_nat @ B3 @ A4 )
          | ( A4 = B3 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_8086_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_rat
    = ( ^ [B3: rat,A4: rat] :
          ( ( ord_less_rat @ B3 @ A4 )
          | ( A4 = B3 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_8087_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_num
    = ( ^ [B3: num,A4: num] :
          ( ( ord_less_num @ B3 @ A4 )
          | ( A4 = B3 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_8088_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [B3: nat,A4: nat] :
          ( ( ord_less_nat @ B3 @ A4 )
          | ( A4 = B3 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_8089_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_int
    = ( ^ [B3: int,A4: int] :
          ( ( ord_less_int @ B3 @ A4 )
          | ( A4 = B3 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_8090_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_real
    = ( ^ [B3: real,A4: real] :
          ( ( ord_less_eq_real @ B3 @ A4 )
          & ( A4 != B3 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_8091_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_set_nat
    = ( ^ [B3: set_nat,A4: set_nat] :
          ( ( ord_less_eq_set_nat @ B3 @ A4 )
          & ( A4 != B3 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_8092_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_rat
    = ( ^ [B3: rat,A4: rat] :
          ( ( ord_less_eq_rat @ B3 @ A4 )
          & ( A4 != B3 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_8093_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_num
    = ( ^ [B3: num,A4: num] :
          ( ( ord_less_eq_num @ B3 @ A4 )
          & ( A4 != B3 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_8094_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [B3: nat,A4: nat] :
          ( ( ord_less_eq_nat @ B3 @ A4 )
          & ( A4 != B3 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_8095_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_int
    = ( ^ [B3: int,A4: int] :
          ( ( ord_less_eq_int @ B3 @ A4 )
          & ( A4 != B3 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_8096_dual__order_Ostrict__trans1,axiom,
    ! [B2: real,A: real,C: real] :
      ( ( ord_less_eq_real @ B2 @ A )
     => ( ( ord_less_real @ C @ B2 )
       => ( ord_less_real @ C @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_8097_dual__order_Ostrict__trans1,axiom,
    ! [B2: set_nat,A: set_nat,C: set_nat] :
      ( ( ord_less_eq_set_nat @ B2 @ A )
     => ( ( ord_less_set_nat @ C @ B2 )
       => ( ord_less_set_nat @ C @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_8098_dual__order_Ostrict__trans1,axiom,
    ! [B2: rat,A: rat,C: rat] :
      ( ( ord_less_eq_rat @ B2 @ A )
     => ( ( ord_less_rat @ C @ B2 )
       => ( ord_less_rat @ C @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_8099_dual__order_Ostrict__trans1,axiom,
    ! [B2: num,A: num,C: num] :
      ( ( ord_less_eq_num @ B2 @ A )
     => ( ( ord_less_num @ C @ B2 )
       => ( ord_less_num @ C @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_8100_dual__order_Ostrict__trans1,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( ( ord_less_eq_nat @ B2 @ A )
     => ( ( ord_less_nat @ C @ B2 )
       => ( ord_less_nat @ C @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_8101_dual__order_Ostrict__trans1,axiom,
    ! [B2: int,A: int,C: int] :
      ( ( ord_less_eq_int @ B2 @ A )
     => ( ( ord_less_int @ C @ B2 )
       => ( ord_less_int @ C @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_8102_dual__order_Ostrict__trans2,axiom,
    ! [B2: real,A: real,C: real] :
      ( ( ord_less_real @ B2 @ A )
     => ( ( ord_less_eq_real @ C @ B2 )
       => ( ord_less_real @ C @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_8103_dual__order_Ostrict__trans2,axiom,
    ! [B2: set_nat,A: set_nat,C: set_nat] :
      ( ( ord_less_set_nat @ B2 @ A )
     => ( ( ord_less_eq_set_nat @ C @ B2 )
       => ( ord_less_set_nat @ C @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_8104_dual__order_Ostrict__trans2,axiom,
    ! [B2: rat,A: rat,C: rat] :
      ( ( ord_less_rat @ B2 @ A )
     => ( ( ord_less_eq_rat @ C @ B2 )
       => ( ord_less_rat @ C @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_8105_dual__order_Ostrict__trans2,axiom,
    ! [B2: num,A: num,C: num] :
      ( ( ord_less_num @ B2 @ A )
     => ( ( ord_less_eq_num @ C @ B2 )
       => ( ord_less_num @ C @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_8106_dual__order_Ostrict__trans2,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( ( ord_less_nat @ B2 @ A )
     => ( ( ord_less_eq_nat @ C @ B2 )
       => ( ord_less_nat @ C @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_8107_dual__order_Ostrict__trans2,axiom,
    ! [B2: int,A: int,C: int] :
      ( ( ord_less_int @ B2 @ A )
     => ( ( ord_less_eq_int @ C @ B2 )
       => ( ord_less_int @ C @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_8108_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_real
    = ( ^ [B3: real,A4: real] :
          ( ( ord_less_eq_real @ B3 @ A4 )
          & ~ ( ord_less_eq_real @ A4 @ B3 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_8109_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_set_nat
    = ( ^ [B3: set_nat,A4: set_nat] :
          ( ( ord_less_eq_set_nat @ B3 @ A4 )
          & ~ ( ord_less_eq_set_nat @ A4 @ B3 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_8110_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_rat
    = ( ^ [B3: rat,A4: rat] :
          ( ( ord_less_eq_rat @ B3 @ A4 )
          & ~ ( ord_less_eq_rat @ A4 @ B3 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_8111_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_num
    = ( ^ [B3: num,A4: num] :
          ( ( ord_less_eq_num @ B3 @ A4 )
          & ~ ( ord_less_eq_num @ A4 @ B3 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_8112_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [B3: nat,A4: nat] :
          ( ( ord_less_eq_nat @ B3 @ A4 )
          & ~ ( ord_less_eq_nat @ A4 @ B3 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_8113_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_int
    = ( ^ [B3: int,A4: int] :
          ( ( ord_less_eq_int @ B3 @ A4 )
          & ~ ( ord_less_eq_int @ A4 @ B3 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_8114_order_Ostrict__implies__order,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ord_less_eq_real @ A @ B2 ) ) ).

% order.strict_implies_order
thf(fact_8115_order_Ostrict__implies__order,axiom,
    ! [A: set_nat,B2: set_nat] :
      ( ( ord_less_set_nat @ A @ B2 )
     => ( ord_less_eq_set_nat @ A @ B2 ) ) ).

% order.strict_implies_order
thf(fact_8116_order_Ostrict__implies__order,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_rat @ A @ B2 )
     => ( ord_less_eq_rat @ A @ B2 ) ) ).

% order.strict_implies_order
thf(fact_8117_order_Ostrict__implies__order,axiom,
    ! [A: num,B2: num] :
      ( ( ord_less_num @ A @ B2 )
     => ( ord_less_eq_num @ A @ B2 ) ) ).

% order.strict_implies_order
thf(fact_8118_order_Ostrict__implies__order,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( ord_less_eq_nat @ A @ B2 ) ) ).

% order.strict_implies_order
thf(fact_8119_order_Ostrict__implies__order,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_int @ A @ B2 )
     => ( ord_less_eq_int @ A @ B2 ) ) ).

% order.strict_implies_order
thf(fact_8120_dual__order_Ostrict__implies__order,axiom,
    ! [B2: real,A: real] :
      ( ( ord_less_real @ B2 @ A )
     => ( ord_less_eq_real @ B2 @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_8121_dual__order_Ostrict__implies__order,axiom,
    ! [B2: set_nat,A: set_nat] :
      ( ( ord_less_set_nat @ B2 @ A )
     => ( ord_less_eq_set_nat @ B2 @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_8122_dual__order_Ostrict__implies__order,axiom,
    ! [B2: rat,A: rat] :
      ( ( ord_less_rat @ B2 @ A )
     => ( ord_less_eq_rat @ B2 @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_8123_dual__order_Ostrict__implies__order,axiom,
    ! [B2: num,A: num] :
      ( ( ord_less_num @ B2 @ A )
     => ( ord_less_eq_num @ B2 @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_8124_dual__order_Ostrict__implies__order,axiom,
    ! [B2: nat,A: nat] :
      ( ( ord_less_nat @ B2 @ A )
     => ( ord_less_eq_nat @ B2 @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_8125_dual__order_Ostrict__implies__order,axiom,
    ! [B2: int,A: int] :
      ( ( ord_less_int @ B2 @ A )
     => ( ord_less_eq_int @ B2 @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_8126_order__le__less,axiom,
    ( ord_less_eq_real
    = ( ^ [X: real,Y: real] :
          ( ( ord_less_real @ X @ Y )
          | ( X = Y ) ) ) ) ).

% order_le_less
thf(fact_8127_order__le__less,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [X: set_nat,Y: set_nat] :
          ( ( ord_less_set_nat @ X @ Y )
          | ( X = Y ) ) ) ) ).

% order_le_less
thf(fact_8128_order__le__less,axiom,
    ( ord_less_eq_rat
    = ( ^ [X: rat,Y: rat] :
          ( ( ord_less_rat @ X @ Y )
          | ( X = Y ) ) ) ) ).

% order_le_less
thf(fact_8129_order__le__less,axiom,
    ( ord_less_eq_num
    = ( ^ [X: num,Y: num] :
          ( ( ord_less_num @ X @ Y )
          | ( X = Y ) ) ) ) ).

% order_le_less
thf(fact_8130_order__le__less,axiom,
    ( ord_less_eq_nat
    = ( ^ [X: nat,Y: nat] :
          ( ( ord_less_nat @ X @ Y )
          | ( X = Y ) ) ) ) ).

% order_le_less
thf(fact_8131_order__le__less,axiom,
    ( ord_less_eq_int
    = ( ^ [X: int,Y: int] :
          ( ( ord_less_int @ X @ Y )
          | ( X = Y ) ) ) ) ).

% order_le_less
thf(fact_8132_order__less__le,axiom,
    ( ord_less_real
    = ( ^ [X: real,Y: real] :
          ( ( ord_less_eq_real @ X @ Y )
          & ( X != Y ) ) ) ) ).

% order_less_le
thf(fact_8133_order__less__le,axiom,
    ( ord_less_set_nat
    = ( ^ [X: set_nat,Y: set_nat] :
          ( ( ord_less_eq_set_nat @ X @ Y )
          & ( X != Y ) ) ) ) ).

% order_less_le
thf(fact_8134_order__less__le,axiom,
    ( ord_less_rat
    = ( ^ [X: rat,Y: rat] :
          ( ( ord_less_eq_rat @ X @ Y )
          & ( X != Y ) ) ) ) ).

% order_less_le
thf(fact_8135_order__less__le,axiom,
    ( ord_less_num
    = ( ^ [X: num,Y: num] :
          ( ( ord_less_eq_num @ X @ Y )
          & ( X != Y ) ) ) ) ).

% order_less_le
thf(fact_8136_order__less__le,axiom,
    ( ord_less_nat
    = ( ^ [X: nat,Y: nat] :
          ( ( ord_less_eq_nat @ X @ Y )
          & ( X != Y ) ) ) ) ).

% order_less_le
thf(fact_8137_order__less__le,axiom,
    ( ord_less_int
    = ( ^ [X: int,Y: int] :
          ( ( ord_less_eq_int @ X @ Y )
          & ( X != Y ) ) ) ) ).

% order_less_le
thf(fact_8138_linorder__not__le,axiom,
    ! [X4: real,Y3: real] :
      ( ( ~ ( ord_less_eq_real @ X4 @ Y3 ) )
      = ( ord_less_real @ Y3 @ X4 ) ) ).

% linorder_not_le
thf(fact_8139_linorder__not__le,axiom,
    ! [X4: rat,Y3: rat] :
      ( ( ~ ( ord_less_eq_rat @ X4 @ Y3 ) )
      = ( ord_less_rat @ Y3 @ X4 ) ) ).

% linorder_not_le
thf(fact_8140_linorder__not__le,axiom,
    ! [X4: num,Y3: num] :
      ( ( ~ ( ord_less_eq_num @ X4 @ Y3 ) )
      = ( ord_less_num @ Y3 @ X4 ) ) ).

% linorder_not_le
thf(fact_8141_linorder__not__le,axiom,
    ! [X4: nat,Y3: nat] :
      ( ( ~ ( ord_less_eq_nat @ X4 @ Y3 ) )
      = ( ord_less_nat @ Y3 @ X4 ) ) ).

% linorder_not_le
thf(fact_8142_linorder__not__le,axiom,
    ! [X4: int,Y3: int] :
      ( ( ~ ( ord_less_eq_int @ X4 @ Y3 ) )
      = ( ord_less_int @ Y3 @ X4 ) ) ).

% linorder_not_le
thf(fact_8143_linorder__not__less,axiom,
    ! [X4: real,Y3: real] :
      ( ( ~ ( ord_less_real @ X4 @ Y3 ) )
      = ( ord_less_eq_real @ Y3 @ X4 ) ) ).

% linorder_not_less
thf(fact_8144_linorder__not__less,axiom,
    ! [X4: rat,Y3: rat] :
      ( ( ~ ( ord_less_rat @ X4 @ Y3 ) )
      = ( ord_less_eq_rat @ Y3 @ X4 ) ) ).

% linorder_not_less
thf(fact_8145_linorder__not__less,axiom,
    ! [X4: num,Y3: num] :
      ( ( ~ ( ord_less_num @ X4 @ Y3 ) )
      = ( ord_less_eq_num @ Y3 @ X4 ) ) ).

% linorder_not_less
thf(fact_8146_linorder__not__less,axiom,
    ! [X4: nat,Y3: nat] :
      ( ( ~ ( ord_less_nat @ X4 @ Y3 ) )
      = ( ord_less_eq_nat @ Y3 @ X4 ) ) ).

% linorder_not_less
thf(fact_8147_linorder__not__less,axiom,
    ! [X4: int,Y3: int] :
      ( ( ~ ( ord_less_int @ X4 @ Y3 ) )
      = ( ord_less_eq_int @ Y3 @ X4 ) ) ).

% linorder_not_less
thf(fact_8148_order__less__imp__le,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_real @ X4 @ Y3 )
     => ( ord_less_eq_real @ X4 @ Y3 ) ) ).

% order_less_imp_le
thf(fact_8149_order__less__imp__le,axiom,
    ! [X4: set_nat,Y3: set_nat] :
      ( ( ord_less_set_nat @ X4 @ Y3 )
     => ( ord_less_eq_set_nat @ X4 @ Y3 ) ) ).

% order_less_imp_le
thf(fact_8150_order__less__imp__le,axiom,
    ! [X4: rat,Y3: rat] :
      ( ( ord_less_rat @ X4 @ Y3 )
     => ( ord_less_eq_rat @ X4 @ Y3 ) ) ).

% order_less_imp_le
thf(fact_8151_order__less__imp__le,axiom,
    ! [X4: num,Y3: num] :
      ( ( ord_less_num @ X4 @ Y3 )
     => ( ord_less_eq_num @ X4 @ Y3 ) ) ).

% order_less_imp_le
thf(fact_8152_order__less__imp__le,axiom,
    ! [X4: nat,Y3: nat] :
      ( ( ord_less_nat @ X4 @ Y3 )
     => ( ord_less_eq_nat @ X4 @ Y3 ) ) ).

% order_less_imp_le
thf(fact_8153_order__less__imp__le,axiom,
    ! [X4: int,Y3: int] :
      ( ( ord_less_int @ X4 @ Y3 )
     => ( ord_less_eq_int @ X4 @ Y3 ) ) ).

% order_less_imp_le
thf(fact_8154_order__le__neq__trans,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ A @ B2 )
     => ( ( A != B2 )
       => ( ord_less_real @ A @ B2 ) ) ) ).

% order_le_neq_trans
thf(fact_8155_order__le__neq__trans,axiom,
    ! [A: set_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B2 )
     => ( ( A != B2 )
       => ( ord_less_set_nat @ A @ B2 ) ) ) ).

% order_le_neq_trans
thf(fact_8156_order__le__neq__trans,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_eq_rat @ A @ B2 )
     => ( ( A != B2 )
       => ( ord_less_rat @ A @ B2 ) ) ) ).

% order_le_neq_trans
thf(fact_8157_order__le__neq__trans,axiom,
    ! [A: num,B2: num] :
      ( ( ord_less_eq_num @ A @ B2 )
     => ( ( A != B2 )
       => ( ord_less_num @ A @ B2 ) ) ) ).

% order_le_neq_trans
thf(fact_8158_order__le__neq__trans,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( A != B2 )
       => ( ord_less_nat @ A @ B2 ) ) ) ).

% order_le_neq_trans
thf(fact_8159_order__le__neq__trans,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ A @ B2 )
     => ( ( A != B2 )
       => ( ord_less_int @ A @ B2 ) ) ) ).

% order_le_neq_trans
thf(fact_8160_order__neq__le__trans,axiom,
    ! [A: real,B2: real] :
      ( ( A != B2 )
     => ( ( ord_less_eq_real @ A @ B2 )
       => ( ord_less_real @ A @ B2 ) ) ) ).

% order_neq_le_trans
thf(fact_8161_order__neq__le__trans,axiom,
    ! [A: set_nat,B2: set_nat] :
      ( ( A != B2 )
     => ( ( ord_less_eq_set_nat @ A @ B2 )
       => ( ord_less_set_nat @ A @ B2 ) ) ) ).

% order_neq_le_trans
thf(fact_8162_order__neq__le__trans,axiom,
    ! [A: rat,B2: rat] :
      ( ( A != B2 )
     => ( ( ord_less_eq_rat @ A @ B2 )
       => ( ord_less_rat @ A @ B2 ) ) ) ).

% order_neq_le_trans
thf(fact_8163_order__neq__le__trans,axiom,
    ! [A: num,B2: num] :
      ( ( A != B2 )
     => ( ( ord_less_eq_num @ A @ B2 )
       => ( ord_less_num @ A @ B2 ) ) ) ).

% order_neq_le_trans
thf(fact_8164_order__neq__le__trans,axiom,
    ! [A: nat,B2: nat] :
      ( ( A != B2 )
     => ( ( ord_less_eq_nat @ A @ B2 )
       => ( ord_less_nat @ A @ B2 ) ) ) ).

% order_neq_le_trans
thf(fact_8165_order__neq__le__trans,axiom,
    ! [A: int,B2: int] :
      ( ( A != B2 )
     => ( ( ord_less_eq_int @ A @ B2 )
       => ( ord_less_int @ A @ B2 ) ) ) ).

% order_neq_le_trans
thf(fact_8166_order__le__less__trans,axiom,
    ! [X4: real,Y3: real,Z: real] :
      ( ( ord_less_eq_real @ X4 @ Y3 )
     => ( ( ord_less_real @ Y3 @ Z )
       => ( ord_less_real @ X4 @ Z ) ) ) ).

% order_le_less_trans
thf(fact_8167_order__le__less__trans,axiom,
    ! [X4: set_nat,Y3: set_nat,Z: set_nat] :
      ( ( ord_less_eq_set_nat @ X4 @ Y3 )
     => ( ( ord_less_set_nat @ Y3 @ Z )
       => ( ord_less_set_nat @ X4 @ Z ) ) ) ).

% order_le_less_trans
thf(fact_8168_order__le__less__trans,axiom,
    ! [X4: rat,Y3: rat,Z: rat] :
      ( ( ord_less_eq_rat @ X4 @ Y3 )
     => ( ( ord_less_rat @ Y3 @ Z )
       => ( ord_less_rat @ X4 @ Z ) ) ) ).

% order_le_less_trans
thf(fact_8169_order__le__less__trans,axiom,
    ! [X4: num,Y3: num,Z: num] :
      ( ( ord_less_eq_num @ X4 @ Y3 )
     => ( ( ord_less_num @ Y3 @ Z )
       => ( ord_less_num @ X4 @ Z ) ) ) ).

% order_le_less_trans
thf(fact_8170_order__le__less__trans,axiom,
    ! [X4: nat,Y3: nat,Z: nat] :
      ( ( ord_less_eq_nat @ X4 @ Y3 )
     => ( ( ord_less_nat @ Y3 @ Z )
       => ( ord_less_nat @ X4 @ Z ) ) ) ).

% order_le_less_trans
thf(fact_8171_order__le__less__trans,axiom,
    ! [X4: int,Y3: int,Z: int] :
      ( ( ord_less_eq_int @ X4 @ Y3 )
     => ( ( ord_less_int @ Y3 @ Z )
       => ( ord_less_int @ X4 @ Z ) ) ) ).

% order_le_less_trans
thf(fact_8172_order__less__le__trans,axiom,
    ! [X4: real,Y3: real,Z: real] :
      ( ( ord_less_real @ X4 @ Y3 )
     => ( ( ord_less_eq_real @ Y3 @ Z )
       => ( ord_less_real @ X4 @ Z ) ) ) ).

% order_less_le_trans
thf(fact_8173_order__less__le__trans,axiom,
    ! [X4: set_nat,Y3: set_nat,Z: set_nat] :
      ( ( ord_less_set_nat @ X4 @ Y3 )
     => ( ( ord_less_eq_set_nat @ Y3 @ Z )
       => ( ord_less_set_nat @ X4 @ Z ) ) ) ).

% order_less_le_trans
thf(fact_8174_order__less__le__trans,axiom,
    ! [X4: rat,Y3: rat,Z: rat] :
      ( ( ord_less_rat @ X4 @ Y3 )
     => ( ( ord_less_eq_rat @ Y3 @ Z )
       => ( ord_less_rat @ X4 @ Z ) ) ) ).

% order_less_le_trans
thf(fact_8175_order__less__le__trans,axiom,
    ! [X4: num,Y3: num,Z: num] :
      ( ( ord_less_num @ X4 @ Y3 )
     => ( ( ord_less_eq_num @ Y3 @ Z )
       => ( ord_less_num @ X4 @ Z ) ) ) ).

% order_less_le_trans
thf(fact_8176_order__less__le__trans,axiom,
    ! [X4: nat,Y3: nat,Z: nat] :
      ( ( ord_less_nat @ X4 @ Y3 )
     => ( ( ord_less_eq_nat @ Y3 @ Z )
       => ( ord_less_nat @ X4 @ Z ) ) ) ).

% order_less_le_trans
thf(fact_8177_order__less__le__trans,axiom,
    ! [X4: int,Y3: int,Z: int] :
      ( ( ord_less_int @ X4 @ Y3 )
     => ( ( ord_less_eq_int @ Y3 @ Z )
       => ( ord_less_int @ X4 @ Z ) ) ) ).

% order_less_le_trans
thf(fact_8178_order__le__less__subst1,axiom,
    ! [A: real,F: real > real,B2: real,C: real] :
      ( ( ord_less_eq_real @ A @ ( F @ B2 ) )
     => ( ( ord_less_real @ B2 @ C )
       => ( ! [X3: real,Y4: real] :
              ( ( ord_less_real @ X3 @ Y4 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_8179_order__le__less__subst1,axiom,
    ! [A: real,F: rat > real,B2: rat,C: rat] :
      ( ( ord_less_eq_real @ A @ ( F @ B2 ) )
     => ( ( ord_less_rat @ B2 @ C )
       => ( ! [X3: rat,Y4: rat] :
              ( ( ord_less_rat @ X3 @ Y4 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_8180_order__le__less__subst1,axiom,
    ! [A: real,F: num > real,B2: num,C: num] :
      ( ( ord_less_eq_real @ A @ ( F @ B2 ) )
     => ( ( ord_less_num @ B2 @ C )
       => ( ! [X3: num,Y4: num] :
              ( ( ord_less_num @ X3 @ Y4 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_8181_order__le__less__subst1,axiom,
    ! [A: real,F: nat > real,B2: nat,C: nat] :
      ( ( ord_less_eq_real @ A @ ( F @ B2 ) )
     => ( ( ord_less_nat @ B2 @ C )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_nat @ X3 @ Y4 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_8182_order__le__less__subst1,axiom,
    ! [A: real,F: int > real,B2: int,C: int] :
      ( ( ord_less_eq_real @ A @ ( F @ B2 ) )
     => ( ( ord_less_int @ B2 @ C )
       => ( ! [X3: int,Y4: int] :
              ( ( ord_less_int @ X3 @ Y4 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_8183_order__le__less__subst1,axiom,
    ! [A: rat,F: real > rat,B2: real,C: real] :
      ( ( ord_less_eq_rat @ A @ ( F @ B2 ) )
     => ( ( ord_less_real @ B2 @ C )
       => ( ! [X3: real,Y4: real] :
              ( ( ord_less_real @ X3 @ Y4 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_8184_order__le__less__subst1,axiom,
    ! [A: rat,F: rat > rat,B2: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ ( F @ B2 ) )
     => ( ( ord_less_rat @ B2 @ C )
       => ( ! [X3: rat,Y4: rat] :
              ( ( ord_less_rat @ X3 @ Y4 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_8185_order__le__less__subst1,axiom,
    ! [A: rat,F: num > rat,B2: num,C: num] :
      ( ( ord_less_eq_rat @ A @ ( F @ B2 ) )
     => ( ( ord_less_num @ B2 @ C )
       => ( ! [X3: num,Y4: num] :
              ( ( ord_less_num @ X3 @ Y4 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_8186_order__le__less__subst1,axiom,
    ! [A: rat,F: nat > rat,B2: nat,C: nat] :
      ( ( ord_less_eq_rat @ A @ ( F @ B2 ) )
     => ( ( ord_less_nat @ B2 @ C )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_nat @ X3 @ Y4 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_8187_order__le__less__subst1,axiom,
    ! [A: rat,F: int > rat,B2: int,C: int] :
      ( ( ord_less_eq_rat @ A @ ( F @ B2 ) )
     => ( ( ord_less_int @ B2 @ C )
       => ( ! [X3: int,Y4: int] :
              ( ( ord_less_int @ X3 @ Y4 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_8188_order__le__less__subst2,axiom,
    ! [A: rat,B2: rat,F: rat > real,C: real] :
      ( ( ord_less_eq_rat @ A @ B2 )
     => ( ( ord_less_real @ ( F @ B2 ) @ C )
       => ( ! [X3: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y4 )
             => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_8189_order__le__less__subst2,axiom,
    ! [A: rat,B2: rat,F: rat > rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B2 )
     => ( ( ord_less_rat @ ( F @ B2 ) @ C )
       => ( ! [X3: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y4 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_8190_order__le__less__subst2,axiom,
    ! [A: rat,B2: rat,F: rat > num,C: num] :
      ( ( ord_less_eq_rat @ A @ B2 )
     => ( ( ord_less_num @ ( F @ B2 ) @ C )
       => ( ! [X3: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y4 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_8191_order__le__less__subst2,axiom,
    ! [A: rat,B2: rat,F: rat > nat,C: nat] :
      ( ( ord_less_eq_rat @ A @ B2 )
     => ( ( ord_less_nat @ ( F @ B2 ) @ C )
       => ( ! [X3: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_8192_order__le__less__subst2,axiom,
    ! [A: rat,B2: rat,F: rat > int,C: int] :
      ( ( ord_less_eq_rat @ A @ B2 )
     => ( ( ord_less_int @ ( F @ B2 ) @ C )
       => ( ! [X3: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y4 )
             => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_8193_order__le__less__subst2,axiom,
    ! [A: num,B2: num,F: num > real,C: real] :
      ( ( ord_less_eq_num @ A @ B2 )
     => ( ( ord_less_real @ ( F @ B2 ) @ C )
       => ( ! [X3: num,Y4: num] :
              ( ( ord_less_eq_num @ X3 @ Y4 )
             => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_8194_order__le__less__subst2,axiom,
    ! [A: num,B2: num,F: num > rat,C: rat] :
      ( ( ord_less_eq_num @ A @ B2 )
     => ( ( ord_less_rat @ ( F @ B2 ) @ C )
       => ( ! [X3: num,Y4: num] :
              ( ( ord_less_eq_num @ X3 @ Y4 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_8195_order__le__less__subst2,axiom,
    ! [A: num,B2: num,F: num > num,C: num] :
      ( ( ord_less_eq_num @ A @ B2 )
     => ( ( ord_less_num @ ( F @ B2 ) @ C )
       => ( ! [X3: num,Y4: num] :
              ( ( ord_less_eq_num @ X3 @ Y4 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_8196_order__le__less__subst2,axiom,
    ! [A: num,B2: num,F: num > nat,C: nat] :
      ( ( ord_less_eq_num @ A @ B2 )
     => ( ( ord_less_nat @ ( F @ B2 ) @ C )
       => ( ! [X3: num,Y4: num] :
              ( ( ord_less_eq_num @ X3 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_8197_order__le__less__subst2,axiom,
    ! [A: num,B2: num,F: num > int,C: int] :
      ( ( ord_less_eq_num @ A @ B2 )
     => ( ( ord_less_int @ ( F @ B2 ) @ C )
       => ( ! [X3: num,Y4: num] :
              ( ( ord_less_eq_num @ X3 @ Y4 )
             => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_8198_order__less__le__subst1,axiom,
    ! [A: real,F: rat > real,B2: rat,C: rat] :
      ( ( ord_less_real @ A @ ( F @ B2 ) )
     => ( ( ord_less_eq_rat @ B2 @ C )
       => ( ! [X3: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y4 )
             => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_8199_order__less__le__subst1,axiom,
    ! [A: rat,F: rat > rat,B2: rat,C: rat] :
      ( ( ord_less_rat @ A @ ( F @ B2 ) )
     => ( ( ord_less_eq_rat @ B2 @ C )
       => ( ! [X3: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y4 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_8200_order__less__le__subst1,axiom,
    ! [A: num,F: rat > num,B2: rat,C: rat] :
      ( ( ord_less_num @ A @ ( F @ B2 ) )
     => ( ( ord_less_eq_rat @ B2 @ C )
       => ( ! [X3: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y4 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_8201_order__less__le__subst1,axiom,
    ! [A: nat,F: rat > nat,B2: rat,C: rat] :
      ( ( ord_less_nat @ A @ ( F @ B2 ) )
     => ( ( ord_less_eq_rat @ B2 @ C )
       => ( ! [X3: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_8202_order__less__le__subst1,axiom,
    ! [A: int,F: rat > int,B2: rat,C: rat] :
      ( ( ord_less_int @ A @ ( F @ B2 ) )
     => ( ( ord_less_eq_rat @ B2 @ C )
       => ( ! [X3: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y4 )
             => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_8203_order__less__le__subst1,axiom,
    ! [A: real,F: num > real,B2: num,C: num] :
      ( ( ord_less_real @ A @ ( F @ B2 ) )
     => ( ( ord_less_eq_num @ B2 @ C )
       => ( ! [X3: num,Y4: num] :
              ( ( ord_less_eq_num @ X3 @ Y4 )
             => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_8204_order__less__le__subst1,axiom,
    ! [A: rat,F: num > rat,B2: num,C: num] :
      ( ( ord_less_rat @ A @ ( F @ B2 ) )
     => ( ( ord_less_eq_num @ B2 @ C )
       => ( ! [X3: num,Y4: num] :
              ( ( ord_less_eq_num @ X3 @ Y4 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_8205_order__less__le__subst1,axiom,
    ! [A: num,F: num > num,B2: num,C: num] :
      ( ( ord_less_num @ A @ ( F @ B2 ) )
     => ( ( ord_less_eq_num @ B2 @ C )
       => ( ! [X3: num,Y4: num] :
              ( ( ord_less_eq_num @ X3 @ Y4 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_8206_order__less__le__subst1,axiom,
    ! [A: nat,F: num > nat,B2: num,C: num] :
      ( ( ord_less_nat @ A @ ( F @ B2 ) )
     => ( ( ord_less_eq_num @ B2 @ C )
       => ( ! [X3: num,Y4: num] :
              ( ( ord_less_eq_num @ X3 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_8207_order__less__le__subst1,axiom,
    ! [A: int,F: num > int,B2: num,C: num] :
      ( ( ord_less_int @ A @ ( F @ B2 ) )
     => ( ( ord_less_eq_num @ B2 @ C )
       => ( ! [X3: num,Y4: num] :
              ( ( ord_less_eq_num @ X3 @ Y4 )
             => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_8208_order__less__le__subst2,axiom,
    ! [A: real,B2: real,F: real > real,C: real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
       => ( ! [X3: real,Y4: real] :
              ( ( ord_less_real @ X3 @ Y4 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_8209_order__less__le__subst2,axiom,
    ! [A: rat,B2: rat,F: rat > real,C: real] :
      ( ( ord_less_rat @ A @ B2 )
     => ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
       => ( ! [X3: rat,Y4: rat] :
              ( ( ord_less_rat @ X3 @ Y4 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_8210_order__less__le__subst2,axiom,
    ! [A: num,B2: num,F: num > real,C: real] :
      ( ( ord_less_num @ A @ B2 )
     => ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
       => ( ! [X3: num,Y4: num] :
              ( ( ord_less_num @ X3 @ Y4 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_8211_order__less__le__subst2,axiom,
    ! [A: nat,B2: nat,F: nat > real,C: real] :
      ( ( ord_less_nat @ A @ B2 )
     => ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_nat @ X3 @ Y4 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_8212_order__less__le__subst2,axiom,
    ! [A: int,B2: int,F: int > real,C: real] :
      ( ( ord_less_int @ A @ B2 )
     => ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
       => ( ! [X3: int,Y4: int] :
              ( ( ord_less_int @ X3 @ Y4 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_8213_order__less__le__subst2,axiom,
    ! [A: real,B2: real,F: real > rat,C: rat] :
      ( ( ord_less_real @ A @ B2 )
     => ( ( ord_less_eq_rat @ ( F @ B2 ) @ C )
       => ( ! [X3: real,Y4: real] :
              ( ( ord_less_real @ X3 @ Y4 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_8214_order__less__le__subst2,axiom,
    ! [A: rat,B2: rat,F: rat > rat,C: rat] :
      ( ( ord_less_rat @ A @ B2 )
     => ( ( ord_less_eq_rat @ ( F @ B2 ) @ C )
       => ( ! [X3: rat,Y4: rat] :
              ( ( ord_less_rat @ X3 @ Y4 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_8215_order__less__le__subst2,axiom,
    ! [A: num,B2: num,F: num > rat,C: rat] :
      ( ( ord_less_num @ A @ B2 )
     => ( ( ord_less_eq_rat @ ( F @ B2 ) @ C )
       => ( ! [X3: num,Y4: num] :
              ( ( ord_less_num @ X3 @ Y4 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_8216_order__less__le__subst2,axiom,
    ! [A: nat,B2: nat,F: nat > rat,C: rat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( ( ord_less_eq_rat @ ( F @ B2 ) @ C )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_nat @ X3 @ Y4 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_8217_order__less__le__subst2,axiom,
    ! [A: int,B2: int,F: int > rat,C: rat] :
      ( ( ord_less_int @ A @ B2 )
     => ( ( ord_less_eq_rat @ ( F @ B2 ) @ C )
       => ( ! [X3: int,Y4: int] :
              ( ( ord_less_int @ X3 @ Y4 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_8218_linorder__le__less__linear,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_eq_real @ X4 @ Y3 )
      | ( ord_less_real @ Y3 @ X4 ) ) ).

% linorder_le_less_linear
thf(fact_8219_linorder__le__less__linear,axiom,
    ! [X4: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ X4 @ Y3 )
      | ( ord_less_rat @ Y3 @ X4 ) ) ).

% linorder_le_less_linear
thf(fact_8220_linorder__le__less__linear,axiom,
    ! [X4: num,Y3: num] :
      ( ( ord_less_eq_num @ X4 @ Y3 )
      | ( ord_less_num @ Y3 @ X4 ) ) ).

% linorder_le_less_linear
thf(fact_8221_linorder__le__less__linear,axiom,
    ! [X4: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ X4 @ Y3 )
      | ( ord_less_nat @ Y3 @ X4 ) ) ).

% linorder_le_less_linear
thf(fact_8222_linorder__le__less__linear,axiom,
    ! [X4: int,Y3: int] :
      ( ( ord_less_eq_int @ X4 @ Y3 )
      | ( ord_less_int @ Y3 @ X4 ) ) ).

% linorder_le_less_linear
thf(fact_8223_order__le__imp__less__or__eq,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_eq_real @ X4 @ Y3 )
     => ( ( ord_less_real @ X4 @ Y3 )
        | ( X4 = Y3 ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_8224_order__le__imp__less__or__eq,axiom,
    ! [X4: set_nat,Y3: set_nat] :
      ( ( ord_less_eq_set_nat @ X4 @ Y3 )
     => ( ( ord_less_set_nat @ X4 @ Y3 )
        | ( X4 = Y3 ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_8225_order__le__imp__less__or__eq,axiom,
    ! [X4: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ X4 @ Y3 )
     => ( ( ord_less_rat @ X4 @ Y3 )
        | ( X4 = Y3 ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_8226_order__le__imp__less__or__eq,axiom,
    ! [X4: num,Y3: num] :
      ( ( ord_less_eq_num @ X4 @ Y3 )
     => ( ( ord_less_num @ X4 @ Y3 )
        | ( X4 = Y3 ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_8227_order__le__imp__less__or__eq,axiom,
    ! [X4: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ X4 @ Y3 )
     => ( ( ord_less_nat @ X4 @ Y3 )
        | ( X4 = Y3 ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_8228_order__le__imp__less__or__eq,axiom,
    ! [X4: int,Y3: int] :
      ( ( ord_less_eq_int @ X4 @ Y3 )
     => ( ( ord_less_int @ X4 @ Y3 )
        | ( X4 = Y3 ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_8229_and__not__numerals_I8_J,axiom,
    ! [M: num,N: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ) ).

% and_not_numerals(8)
thf(fact_8230_not__int__rec,axiom,
    ( bit_ri7919022796975470100ot_int
    = ( ^ [K3: int] : ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri7919022796975470100ot_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% not_int_rec
thf(fact_8231__C5_Ohyps_C_I11_J,axiom,
    ( ( mi != ma )
   => ! [I4: nat] :
        ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
       => ( ( ( ( vEBT_VEBT_high @ ma @ na )
              = I4 )
           => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ I4 ) @ ( vEBT_VEBT_low @ ma @ na ) ) )
          & ! [X5: nat] :
              ( ( ( ( vEBT_VEBT_high @ X5 @ na )
                  = I4 )
                & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ I4 ) @ ( vEBT_VEBT_low @ X5 @ na ) ) )
             => ( ( ord_less_nat @ mi @ X5 )
                & ( ord_less_eq_nat @ X5 @ ma ) ) ) ) ) ) ).

% "5.hyps"(11)
thf(fact_8232__C20_C,axiom,
    ( ( za = mi )
    | ( za = ma )
    | ( ( ord_less_nat @ ( vEBT_VEBT_high @ za @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ treeList ) )
      & ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ za @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ za @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% "20"
thf(fact_8233_abd,axiom,
    ( ( za = ma )
   => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ za @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ za @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% abd
thf(fact_8234_aaa,axiom,
    ( ( ( vEBT_VEBT_high @ za @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
   => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ za @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% aaa
thf(fact_8235__C33_C,axiom,
    ~ ? [U2: nat] :
        ( ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ U2 )
        & ( ord_less_nat @ ( vEBT_VEBT_low @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ U2 ) ) ).

% "33"
thf(fact_8236__092_060open_062_092_060exists_062miny_O_Aboth__member__options_A_ItreeList_A_B_Asc_J_Aminy_092_060close_062,axiom,
    ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ sc ) @ X_1 ) ).

% \<open>\<exists>miny. both_member_options (treeList ! sc) miny\<close>
thf(fact_8237_abf,axiom,
    vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ za @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ za @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% abf
thf(fact_8238_abe,axiom,
    vEBT_vebt_member @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ za @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ za @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% abe
thf(fact_8239_aa,axiom,
    ( ( za = ma )
   => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ za @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ za @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% aa
thf(fact_8240_nth__equalityI,axiom,
    ! [Xs2: list_VEBT_VEBT,Ys2: list_VEBT_VEBT] :
      ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
        = ( size_s6755466524823107622T_VEBT @ Ys2 ) )
     => ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
           => ( ( nth_VEBT_VEBT @ Xs2 @ I3 )
              = ( nth_VEBT_VEBT @ Ys2 @ I3 ) ) )
       => ( Xs2 = Ys2 ) ) ) ).

% nth_equalityI
thf(fact_8241_nth__equalityI,axiom,
    ! [Xs2: list_o,Ys2: list_o] :
      ( ( ( size_size_list_o @ Xs2 )
        = ( size_size_list_o @ Ys2 ) )
     => ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_o @ Xs2 ) )
           => ( ( nth_o @ Xs2 @ I3 )
              = ( nth_o @ Ys2 @ I3 ) ) )
       => ( Xs2 = Ys2 ) ) ) ).

% nth_equalityI
thf(fact_8242_nth__equalityI,axiom,
    ! [Xs2: list_nat,Ys2: list_nat] :
      ( ( ( size_size_list_nat @ Xs2 )
        = ( size_size_list_nat @ Ys2 ) )
     => ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs2 ) )
           => ( ( nth_nat @ Xs2 @ I3 )
              = ( nth_nat @ Ys2 @ I3 ) ) )
       => ( Xs2 = Ys2 ) ) ) ).

% nth_equalityI
thf(fact_8243_nth__equalityI,axiom,
    ! [Xs2: list_int,Ys2: list_int] :
      ( ( ( size_size_list_int @ Xs2 )
        = ( size_size_list_int @ Ys2 ) )
     => ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs2 ) )
           => ( ( nth_int @ Xs2 @ I3 )
              = ( nth_int @ Ys2 @ I3 ) ) )
       => ( Xs2 = Ys2 ) ) ) ).

% nth_equalityI
thf(fact_8244_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > vEBT_VEBT > $o] :
      ( ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ K )
           => ? [X6: vEBT_VEBT] : ( P @ I2 @ X6 ) ) )
      = ( ? [Xs: list_VEBT_VEBT] :
            ( ( ( size_s6755466524823107622T_VEBT @ Xs )
              = K )
            & ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ K )
               => ( P @ I2 @ ( nth_VEBT_VEBT @ Xs @ I2 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_8245_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > $o > $o] :
      ( ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ K )
           => ? [X6: $o] : ( P @ I2 @ X6 ) ) )
      = ( ? [Xs: list_o] :
            ( ( ( size_size_list_o @ Xs )
              = K )
            & ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ K )
               => ( P @ I2 @ ( nth_o @ Xs @ I2 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_8246_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > nat > $o] :
      ( ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ K )
           => ? [X6: nat] : ( P @ I2 @ X6 ) ) )
      = ( ? [Xs: list_nat] :
            ( ( ( size_size_list_nat @ Xs )
              = K )
            & ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ K )
               => ( P @ I2 @ ( nth_nat @ Xs @ I2 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_8247_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > int > $o] :
      ( ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ K )
           => ? [X6: int] : ( P @ I2 @ X6 ) ) )
      = ( ? [Xs: list_int] :
            ( ( ( size_size_list_int @ Xs )
              = K )
            & ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ K )
               => ( P @ I2 @ ( nth_int @ Xs @ I2 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_8248_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y6: list_VEBT_VEBT,Z4: list_VEBT_VEBT] : Y6 = Z4 )
    = ( ^ [Xs: list_VEBT_VEBT,Ys3: list_VEBT_VEBT] :
          ( ( ( size_s6755466524823107622T_VEBT @ Xs )
            = ( size_s6755466524823107622T_VEBT @ Ys3 ) )
          & ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
             => ( ( nth_VEBT_VEBT @ Xs @ I2 )
                = ( nth_VEBT_VEBT @ Ys3 @ I2 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_8249_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y6: list_o,Z4: list_o] : Y6 = Z4 )
    = ( ^ [Xs: list_o,Ys3: list_o] :
          ( ( ( size_size_list_o @ Xs )
            = ( size_size_list_o @ Ys3 ) )
          & ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs ) )
             => ( ( nth_o @ Xs @ I2 )
                = ( nth_o @ Ys3 @ I2 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_8250_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y6: list_nat,Z4: list_nat] : Y6 = Z4 )
    = ( ^ [Xs: list_nat,Ys3: list_nat] :
          ( ( ( size_size_list_nat @ Xs )
            = ( size_size_list_nat @ Ys3 ) )
          & ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
             => ( ( nth_nat @ Xs @ I2 )
                = ( nth_nat @ Ys3 @ I2 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_8251_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y6: list_int,Z4: list_int] : Y6 = Z4 )
    = ( ^ [Xs: list_int,Ys3: list_int] :
          ( ( ( size_size_list_int @ Xs )
            = ( size_size_list_int @ Ys3 ) )
          & ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs ) )
             => ( ( nth_int @ Xs @ I2 )
                = ( nth_int @ Ys3 @ I2 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_8252_num_Osize_I4_J,axiom,
    ( ( size_size_num @ one )
    = zero_zero_nat ) ).

% num.size(4)
thf(fact_8253_num_Osize_I5_J,axiom,
    ! [X2: num] :
      ( ( size_size_num @ ( bit0 @ X2 ) )
      = ( plus_plus_nat @ ( size_size_num @ X2 ) @ ( suc @ zero_zero_nat ) ) ) ).

% num.size(5)
thf(fact_8254_num_Osize_I6_J,axiom,
    ! [X32: num] :
      ( ( size_size_num @ ( bit1 @ X32 ) )
      = ( plus_plus_nat @ ( size_size_num @ X32 ) @ ( suc @ zero_zero_nat ) ) ) ).

% num.size(6)
thf(fact_8255_in__children__def,axiom,
    ( vEBT_V5917875025757280293ildren
    = ( ^ [N2: nat,TreeList: list_VEBT_VEBT,X: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ N2 ) ) @ ( vEBT_VEBT_low @ X @ N2 ) ) ) ) ).

% in_children_def
thf(fact_8256_abc,axiom,
    vEBT_invar_vebt @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ za @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ na ).

% abc
thf(fact_8257__092_060open_062invar__vebt_A_ItreeList_A_B_Ahigh_Ax_A_Ideg_Adiv_A2_J_J_An_092_060close_062,axiom,
    vEBT_invar_vebt @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ na ).

% \<open>invar_vebt (treeList ! high x (deg div 2)) n\<close>
thf(fact_8258_abh,axiom,
    vEBT_vebt_member @ summary @ ( vEBT_VEBT_high @ za @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% abh
thf(fact_8259_valid__0__not,axiom,
    ! [T: vEBT_VEBT] :
      ~ ( vEBT_invar_vebt @ T @ zero_zero_nat ) ).

% valid_0_not
thf(fact_8260_valid__tree__deg__neq__0,axiom,
    ! [T: vEBT_VEBT] :
      ~ ( vEBT_invar_vebt @ T @ zero_zero_nat ) ).

% valid_tree_deg_neq_0
thf(fact_8261__C5_Ohyps_C_I2_J,axiom,
    vEBT_invar_vebt @ summary @ m ).

% "5.hyps"(2)
thf(fact_8262_deg__not__0,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% deg_not_0
thf(fact_8263_both__member__options__equiv__member,axiom,
    ! [T: vEBT_VEBT,N: nat,X4: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_V8194947554948674370ptions @ T @ X4 )
        = ( vEBT_vebt_member @ T @ X4 ) ) ) ).

% both_member_options_equiv_member
thf(fact_8264_valid__member__both__member__options,axiom,
    ! [T: vEBT_VEBT,N: nat,X4: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_V8194947554948674370ptions @ T @ X4 )
       => ( vEBT_vebt_member @ T @ X4 ) ) ) ).

% valid_member_both_member_options
thf(fact_8265__092_060open_062both__member__options_Asummary_Asc_092_060close_062,axiom,
    vEBT_V8194947554948674370ptions @ summary @ sc ).

% \<open>both_member_options summary sc\<close>
thf(fact_8266__092_060open_062vebt__member_Asummary_Asc_092_060close_062,axiom,
    vEBT_vebt_member @ summary @ sc ).

% \<open>vebt_member summary sc\<close>
thf(fact_8267_member__correct,axiom,
    ! [T: vEBT_VEBT,N: nat,X4: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_vebt_member @ T @ X4 )
        = ( member_nat @ X4 @ ( vEBT_set_vebt @ T ) ) ) ) ).

% member_correct
thf(fact_8268_member__bound,axiom,
    ! [Tree: vEBT_VEBT,X4: nat,N: nat] :
      ( ( vEBT_vebt_member @ Tree @ X4 )
     => ( ( vEBT_invar_vebt @ Tree @ N )
       => ( ord_less_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% member_bound
thf(fact_8269_abg,axiom,
    vEBT_V8194947554948674370ptions @ summary @ ( vEBT_VEBT_high @ za @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% abg
thf(fact_8270__092_060open_062length_AtreeList_A_061_A2_A_094_Am_A_092_060and_062_Ainvar__vebt_Asummary_Am_092_060close_062,axiom,
    ( ( ( size_s6755466524823107622T_VEBT @ treeList )
      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
    & ( vEBT_invar_vebt @ summary @ m ) ) ).

% \<open>length treeList = 2 ^ m \<and> invar_vebt summary m\<close>
thf(fact_8271__C5_Ohyps_C_I7_J,axiom,
    ! [I4: nat] :
      ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
     => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ I4 ) @ X6 ) )
        = ( vEBT_V8194947554948674370ptions @ summary @ I4 ) ) ) ).

% "5.hyps"(7)
thf(fact_8272_post__member__pre__member,axiom,
    ! [T: vEBT_VEBT,N: nat,X4: nat,Y3: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ord_less_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
       => ( ( ord_less_nat @ Y3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
         => ( ( vEBT_vebt_member @ ( vEBT_vebt_insert @ T @ X4 ) @ Y3 )
           => ( ( vEBT_vebt_member @ T @ Y3 )
              | ( X4 = Y3 ) ) ) ) ) ) ).

% post_member_pre_member
thf(fact_8273_valid__insert__both__member__options__pres,axiom,
    ! [T: vEBT_VEBT,N: nat,X4: nat,Y3: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ord_less_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
       => ( ( ord_less_nat @ Y3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
         => ( ( vEBT_V8194947554948674370ptions @ T @ X4 )
           => ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_insert @ T @ Y3 ) @ X4 ) ) ) ) ) ).

% valid_insert_both_member_options_pres
thf(fact_8274_valid__insert__both__member__options__add,axiom,
    ! [T: vEBT_VEBT,N: nat,X4: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ord_less_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
       => ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_insert @ T @ X4 ) @ X4 ) ) ) ).

% valid_insert_both_member_options_add
thf(fact_8275_both__member__options__ding,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat,X4: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ N )
     => ( ( ord_less_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
       => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
         => ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ X4 ) ) ) ) ).

% both_member_options_ding
thf(fact_8276_deg__deg__n,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ N )
     => ( Deg = N ) ) ).

% deg_deg_n
thf(fact_8277_deg__SUcn__Node,axiom,
    ! [Tree: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ Tree @ ( suc @ ( suc @ N ) ) )
     => ? [Info2: option4927543243414619207at_nat,TreeList3: list_VEBT_VEBT,S3: vEBT_VEBT] :
          ( Tree
          = ( vEBT_Node @ Info2 @ ( suc @ ( suc @ N ) ) @ TreeList3 @ S3 ) ) ) ).

% deg_SUcn_Node
thf(fact_8278__C5_Ohyps_C_I8_J,axiom,
    ( ( mi = ma )
   => ! [X5: vEBT_VEBT] :
        ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ treeList ) )
       => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) ) ) ).

% "5.hyps"(8)
thf(fact_8279_set__vebt__set__vebt_H__valid,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_set_vebt @ T )
        = ( vEBT_VEBT_set_vebt @ T ) ) ) ).

% set_vebt_set_vebt'_valid
thf(fact_8280_buildup__gives__valid,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( vEBT_invar_vebt @ ( vEBT_vebt_buildup @ N ) @ N ) ) ).

% buildup_gives_valid
thf(fact_8281__092_060open_062is__succ__in__set_A_Iset__vebt_H_Asummary_J_A_Ihigh_Ax_A_Ideg_Adiv_A2_J_J_Asc_092_060close_062,axiom,
    vEBT_is_succ_in_set @ ( vEBT_VEBT_set_vebt @ summary ) @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ sc ).

% \<open>is_succ_in_set (set_vebt' summary) (high x (deg div 2)) sc\<close>
thf(fact_8282_set__n__deg__not__0,axiom,
    ! [TreeList2: list_VEBT_VEBT,N: nat,M: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
         => ( vEBT_invar_vebt @ X3 @ N ) )
     => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
       => ( ord_less_eq_nat @ one_one_nat @ N ) ) ) ).

% set_n_deg_not_0
thf(fact_8283_inthall,axiom,
    ! [Xs2: list_Extended_enat,P: extended_enat > $o,N: nat] :
      ( ! [X3: extended_enat] :
          ( ( member_Extended_enat @ X3 @ ( set_Extended_enat2 @ Xs2 ) )
         => ( P @ X3 ) )
     => ( ( ord_less_nat @ N @ ( size_s3941691890525107288d_enat @ Xs2 ) )
       => ( P @ ( nth_Extended_enat @ Xs2 @ N ) ) ) ) ).

% inthall
thf(fact_8284_inthall,axiom,
    ! [Xs2: list_complex,P: complex > $o,N: nat] :
      ( ! [X3: complex] :
          ( ( member_complex @ X3 @ ( set_complex2 @ Xs2 ) )
         => ( P @ X3 ) )
     => ( ( ord_less_nat @ N @ ( size_s3451745648224563538omplex @ Xs2 ) )
       => ( P @ ( nth_complex @ Xs2 @ N ) ) ) ) ).

% inthall
thf(fact_8285_inthall,axiom,
    ! [Xs2: list_real,P: real > $o,N: nat] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ ( set_real2 @ Xs2 ) )
         => ( P @ X3 ) )
     => ( ( ord_less_nat @ N @ ( size_size_list_real @ Xs2 ) )
       => ( P @ ( nth_real @ Xs2 @ N ) ) ) ) ).

% inthall
thf(fact_8286_inthall,axiom,
    ! [Xs2: list_VEBT_VEBT,P: vEBT_VEBT > $o,N: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs2 ) )
         => ( P @ X3 ) )
     => ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
       => ( P @ ( nth_VEBT_VEBT @ Xs2 @ N ) ) ) ) ).

% inthall
thf(fact_8287_inthall,axiom,
    ! [Xs2: list_o,P: $o > $o,N: nat] :
      ( ! [X3: $o] :
          ( ( member_o @ X3 @ ( set_o2 @ Xs2 ) )
         => ( P @ X3 ) )
     => ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs2 ) )
       => ( P @ ( nth_o @ Xs2 @ N ) ) ) ) ).

% inthall
thf(fact_8288_inthall,axiom,
    ! [Xs2: list_nat,P: nat > $o,N: nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
         => ( P @ X3 ) )
     => ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
       => ( P @ ( nth_nat @ Xs2 @ N ) ) ) ) ).

% inthall
thf(fact_8289_inthall,axiom,
    ! [Xs2: list_int,P: int > $o,N: nat] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ ( set_int2 @ Xs2 ) )
         => ( P @ X3 ) )
     => ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs2 ) )
       => ( P @ ( nth_int @ Xs2 @ N ) ) ) ) ).

% inthall
thf(fact_8290_succ__member,axiom,
    ! [T: vEBT_VEBT,X4: nat,Y3: nat] :
      ( ( vEBT_is_succ_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X4 @ Y3 )
      = ( ( vEBT_vebt_member @ T @ Y3 )
        & ( ord_less_nat @ X4 @ Y3 )
        & ! [Z5: nat] :
            ( ( ( vEBT_vebt_member @ T @ Z5 )
              & ( ord_less_nat @ X4 @ Z5 ) )
           => ( ord_less_eq_nat @ Y3 @ Z5 ) ) ) ) ).

% succ_member
thf(fact_8291_length__pos__if__in__set,axiom,
    ! [X4: extended_enat,Xs2: list_Extended_enat] :
      ( ( member_Extended_enat @ X4 @ ( set_Extended_enat2 @ Xs2 ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_s3941691890525107288d_enat @ Xs2 ) ) ) ).

% length_pos_if_in_set
thf(fact_8292_length__pos__if__in__set,axiom,
    ! [X4: complex,Xs2: list_complex] :
      ( ( member_complex @ X4 @ ( set_complex2 @ Xs2 ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_s3451745648224563538omplex @ Xs2 ) ) ) ).

% length_pos_if_in_set
thf(fact_8293_length__pos__if__in__set,axiom,
    ! [X4: real,Xs2: list_real] :
      ( ( member_real @ X4 @ ( set_real2 @ Xs2 ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_real @ Xs2 ) ) ) ).

% length_pos_if_in_set
thf(fact_8294_length__pos__if__in__set,axiom,
    ! [X4: vEBT_VEBT,Xs2: list_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ Xs2 ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) ) ) ).

% length_pos_if_in_set
thf(fact_8295_length__pos__if__in__set,axiom,
    ! [X4: $o,Xs2: list_o] :
      ( ( member_o @ X4 @ ( set_o2 @ Xs2 ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_o @ Xs2 ) ) ) ).

% length_pos_if_in_set
thf(fact_8296_length__pos__if__in__set,axiom,
    ! [X4: nat,Xs2: list_nat] :
      ( ( member_nat @ X4 @ ( set_nat2 @ Xs2 ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs2 ) ) ) ).

% length_pos_if_in_set
thf(fact_8297_length__pos__if__in__set,axiom,
    ! [X4: int,Xs2: list_int] :
      ( ( member_int @ X4 @ ( set_int2 @ Xs2 ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_int @ Xs2 ) ) ) ).

% length_pos_if_in_set
thf(fact_8298_nth__mem,axiom,
    ! [N: nat,Xs2: list_Extended_enat] :
      ( ( ord_less_nat @ N @ ( size_s3941691890525107288d_enat @ Xs2 ) )
     => ( member_Extended_enat @ ( nth_Extended_enat @ Xs2 @ N ) @ ( set_Extended_enat2 @ Xs2 ) ) ) ).

% nth_mem
thf(fact_8299_nth__mem,axiom,
    ! [N: nat,Xs2: list_complex] :
      ( ( ord_less_nat @ N @ ( size_s3451745648224563538omplex @ Xs2 ) )
     => ( member_complex @ ( nth_complex @ Xs2 @ N ) @ ( set_complex2 @ Xs2 ) ) ) ).

% nth_mem
thf(fact_8300_nth__mem,axiom,
    ! [N: nat,Xs2: list_real] :
      ( ( ord_less_nat @ N @ ( size_size_list_real @ Xs2 ) )
     => ( member_real @ ( nth_real @ Xs2 @ N ) @ ( set_real2 @ Xs2 ) ) ) ).

% nth_mem
thf(fact_8301_nth__mem,axiom,
    ! [N: nat,Xs2: list_VEBT_VEBT] :
      ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( member_VEBT_VEBT @ ( nth_VEBT_VEBT @ Xs2 @ N ) @ ( set_VEBT_VEBT2 @ Xs2 ) ) ) ).

% nth_mem
thf(fact_8302_nth__mem,axiom,
    ! [N: nat,Xs2: list_o] :
      ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs2 ) )
     => ( member_o @ ( nth_o @ Xs2 @ N ) @ ( set_o2 @ Xs2 ) ) ) ).

% nth_mem
thf(fact_8303_nth__mem,axiom,
    ! [N: nat,Xs2: list_nat] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
     => ( member_nat @ ( nth_nat @ Xs2 @ N ) @ ( set_nat2 @ Xs2 ) ) ) ).

% nth_mem
thf(fact_8304_nth__mem,axiom,
    ! [N: nat,Xs2: list_int] :
      ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs2 ) )
     => ( member_int @ ( nth_int @ Xs2 @ N ) @ ( set_int2 @ Xs2 ) ) ) ).

% nth_mem
thf(fact_8305_list__ball__nth,axiom,
    ! [N: nat,Xs2: list_VEBT_VEBT,P: vEBT_VEBT > $o] :
      ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( ! [X3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs2 ) )
           => ( P @ X3 ) )
       => ( P @ ( nth_VEBT_VEBT @ Xs2 @ N ) ) ) ) ).

% list_ball_nth
thf(fact_8306_list__ball__nth,axiom,
    ! [N: nat,Xs2: list_o,P: $o > $o] :
      ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs2 ) )
     => ( ! [X3: $o] :
            ( ( member_o @ X3 @ ( set_o2 @ Xs2 ) )
           => ( P @ X3 ) )
       => ( P @ ( nth_o @ Xs2 @ N ) ) ) ) ).

% list_ball_nth
thf(fact_8307_list__ball__nth,axiom,
    ! [N: nat,Xs2: list_nat,P: nat > $o] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
           => ( P @ X3 ) )
       => ( P @ ( nth_nat @ Xs2 @ N ) ) ) ) ).

% list_ball_nth
thf(fact_8308_list__ball__nth,axiom,
    ! [N: nat,Xs2: list_int,P: int > $o] :
      ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs2 ) )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ ( set_int2 @ Xs2 ) )
           => ( P @ X3 ) )
       => ( P @ ( nth_int @ Xs2 @ N ) ) ) ) ).

% list_ball_nth
thf(fact_8309_in__set__conv__nth,axiom,
    ! [X4: extended_enat,Xs2: list_Extended_enat] :
      ( ( member_Extended_enat @ X4 @ ( set_Extended_enat2 @ Xs2 ) )
      = ( ? [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s3941691890525107288d_enat @ Xs2 ) )
            & ( ( nth_Extended_enat @ Xs2 @ I2 )
              = X4 ) ) ) ) ).

% in_set_conv_nth
thf(fact_8310_in__set__conv__nth,axiom,
    ! [X4: complex,Xs2: list_complex] :
      ( ( member_complex @ X4 @ ( set_complex2 @ Xs2 ) )
      = ( ? [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s3451745648224563538omplex @ Xs2 ) )
            & ( ( nth_complex @ Xs2 @ I2 )
              = X4 ) ) ) ) ).

% in_set_conv_nth
thf(fact_8311_in__set__conv__nth,axiom,
    ! [X4: real,Xs2: list_real] :
      ( ( member_real @ X4 @ ( set_real2 @ Xs2 ) )
      = ( ? [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_real @ Xs2 ) )
            & ( ( nth_real @ Xs2 @ I2 )
              = X4 ) ) ) ) ).

% in_set_conv_nth
thf(fact_8312_in__set__conv__nth,axiom,
    ! [X4: vEBT_VEBT,Xs2: list_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ Xs2 ) )
      = ( ? [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
            & ( ( nth_VEBT_VEBT @ Xs2 @ I2 )
              = X4 ) ) ) ) ).

% in_set_conv_nth
thf(fact_8313_in__set__conv__nth,axiom,
    ! [X4: $o,Xs2: list_o] :
      ( ( member_o @ X4 @ ( set_o2 @ Xs2 ) )
      = ( ? [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs2 ) )
            & ( ( nth_o @ Xs2 @ I2 )
              = X4 ) ) ) ) ).

% in_set_conv_nth
thf(fact_8314_in__set__conv__nth,axiom,
    ! [X4: nat,Xs2: list_nat] :
      ( ( member_nat @ X4 @ ( set_nat2 @ Xs2 ) )
      = ( ? [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs2 ) )
            & ( ( nth_nat @ Xs2 @ I2 )
              = X4 ) ) ) ) ).

% in_set_conv_nth
thf(fact_8315_in__set__conv__nth,axiom,
    ! [X4: int,Xs2: list_int] :
      ( ( member_int @ X4 @ ( set_int2 @ Xs2 ) )
      = ( ? [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs2 ) )
            & ( ( nth_int @ Xs2 @ I2 )
              = X4 ) ) ) ) ).

% in_set_conv_nth
thf(fact_8316_all__nth__imp__all__set,axiom,
    ! [Xs2: list_Extended_enat,P: extended_enat > $o,X4: extended_enat] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ ( size_s3941691890525107288d_enat @ Xs2 ) )
         => ( P @ ( nth_Extended_enat @ Xs2 @ I3 ) ) )
     => ( ( member_Extended_enat @ X4 @ ( set_Extended_enat2 @ Xs2 ) )
       => ( P @ X4 ) ) ) ).

% all_nth_imp_all_set
thf(fact_8317_all__nth__imp__all__set,axiom,
    ! [Xs2: list_complex,P: complex > $o,X4: complex] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ ( size_s3451745648224563538omplex @ Xs2 ) )
         => ( P @ ( nth_complex @ Xs2 @ I3 ) ) )
     => ( ( member_complex @ X4 @ ( set_complex2 @ Xs2 ) )
       => ( P @ X4 ) ) ) ).

% all_nth_imp_all_set
thf(fact_8318_all__nth__imp__all__set,axiom,
    ! [Xs2: list_real,P: real > $o,X4: real] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ ( size_size_list_real @ Xs2 ) )
         => ( P @ ( nth_real @ Xs2 @ I3 ) ) )
     => ( ( member_real @ X4 @ ( set_real2 @ Xs2 ) )
       => ( P @ X4 ) ) ) ).

% all_nth_imp_all_set
thf(fact_8319_all__nth__imp__all__set,axiom,
    ! [Xs2: list_VEBT_VEBT,P: vEBT_VEBT > $o,X4: vEBT_VEBT] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
         => ( P @ ( nth_VEBT_VEBT @ Xs2 @ I3 ) ) )
     => ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ Xs2 ) )
       => ( P @ X4 ) ) ) ).

% all_nth_imp_all_set
thf(fact_8320_all__nth__imp__all__set,axiom,
    ! [Xs2: list_o,P: $o > $o,X4: $o] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ ( size_size_list_o @ Xs2 ) )
         => ( P @ ( nth_o @ Xs2 @ I3 ) ) )
     => ( ( member_o @ X4 @ ( set_o2 @ Xs2 ) )
       => ( P @ X4 ) ) ) ).

% all_nth_imp_all_set
thf(fact_8321_all__nth__imp__all__set,axiom,
    ! [Xs2: list_nat,P: nat > $o,X4: nat] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs2 ) )
         => ( P @ ( nth_nat @ Xs2 @ I3 ) ) )
     => ( ( member_nat @ X4 @ ( set_nat2 @ Xs2 ) )
       => ( P @ X4 ) ) ) ).

% all_nth_imp_all_set
thf(fact_8322_all__nth__imp__all__set,axiom,
    ! [Xs2: list_int,P: int > $o,X4: int] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs2 ) )
         => ( P @ ( nth_int @ Xs2 @ I3 ) ) )
     => ( ( member_int @ X4 @ ( set_int2 @ Xs2 ) )
       => ( P @ X4 ) ) ) ).

% all_nth_imp_all_set
thf(fact_8323_all__set__conv__all__nth,axiom,
    ! [Xs2: list_VEBT_VEBT,P: vEBT_VEBT > $o] :
      ( ( ! [X: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs2 ) )
           => ( P @ X ) ) )
      = ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
           => ( P @ ( nth_VEBT_VEBT @ Xs2 @ I2 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_8324_all__set__conv__all__nth,axiom,
    ! [Xs2: list_o,P: $o > $o] :
      ( ( ! [X: $o] :
            ( ( member_o @ X @ ( set_o2 @ Xs2 ) )
           => ( P @ X ) ) )
      = ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs2 ) )
           => ( P @ ( nth_o @ Xs2 @ I2 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_8325_all__set__conv__all__nth,axiom,
    ! [Xs2: list_nat,P: nat > $o] :
      ( ( ! [X: nat] :
            ( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
           => ( P @ X ) ) )
      = ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs2 ) )
           => ( P @ ( nth_nat @ Xs2 @ I2 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_8326_all__set__conv__all__nth,axiom,
    ! [Xs2: list_int,P: int > $o] :
      ( ( ! [X: int] :
            ( ( member_int @ X @ ( set_int2 @ Xs2 ) )
           => ( P @ X ) ) )
      = ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs2 ) )
           => ( P @ ( nth_int @ Xs2 @ I2 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_8327_is__succ__in__set__def,axiom,
    ( vEBT_is_succ_in_set
    = ( ^ [Xs: set_nat,X: nat,Y: nat] :
          ( ( member_nat @ Y @ Xs )
          & ( ord_less_nat @ X @ Y )
          & ! [Z5: nat] :
              ( ( member_nat @ Z5 @ Xs )
             => ( ( ord_less_nat @ X @ Z5 )
               => ( ord_less_eq_nat @ Y @ Z5 ) ) ) ) ) ) ).

% is_succ_in_set_def
thf(fact_8328_inrange,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ord_less_eq_set_nat @ ( vEBT_VEBT_set_vebt @ T ) @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ).

% inrange
thf(fact_8329_fgh,axiom,
    ( ( vEBT_VEBT_set_vebt @ ( nth_VEBT_VEBT @ treeList @ sc ) )
   != bot_bot_set_nat ) ).

% fgh
thf(fact_8330__092_060open_062vebt__member_A_INode_A_ISome_A_Imi_M_Ama_J_J_Adeg_AtreeList_Asummary_J_Az_A_092_060and_062_Ax_A_060_Az_092_060close_062,axiom,
    ( ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ za )
    & ( ord_less_nat @ xa @ za ) ) ).

% \<open>vebt_member (Node (Some (mi, ma)) deg treeList summary) z \<and> x < z\<close>
thf(fact_8331__092_060open_062vebt__succ_Asummary_A_Ihigh_Ax_A_Ideg_Adiv_A2_J_J_A_061_ASome_Asc_092_060close_062,axiom,
    ( ( vEBT_vebt_succ @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
    = ( some_nat @ sc ) ) ).

% \<open>vebt_succ summary (high x (deg div 2)) = Some sc\<close>
thf(fact_8332_buildup__gives__empty,axiom,
    ! [N: nat] :
      ( ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_buildup @ N ) )
      = bot_bot_set_nat ) ).

% buildup_gives_empty
thf(fact_8333_power__shift,axiom,
    ! [X4: nat,Y3: nat,Z: nat] :
      ( ( ( power_power_nat @ X4 @ Y3 )
        = Z )
      = ( ( vEBT_VEBT_power @ ( some_nat @ X4 ) @ ( some_nat @ Y3 ) )
        = ( some_nat @ Z ) ) ) ).

% power_shift
thf(fact_8334__C5_Ohyps_C_I1_J,axiom,
    ! [X5: vEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ treeList ) )
     => ( ( vEBT_invar_vebt @ X5 @ na )
        & ! [Xa2: nat,Xb2: nat] :
            ( ( ( vEBT_vebt_succ @ X5 @ Xa2 )
              = ( some_nat @ Xb2 ) )
            = ( vEBT_is_succ_in_set @ ( vEBT_VEBT_set_vebt @ X5 ) @ Xa2 @ Xb2 ) ) ) ) ).

% "5.hyps"(1)
thf(fact_8335_mi__eq__ma__no__ch,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ Deg )
     => ( ( Mi = Ma )
       => ( ! [X5: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
             => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) )
          & ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_12 ) ) ) ) ).

% mi_eq_ma_no_ch
thf(fact_8336__C5_Ohyps_C_I3_J,axiom,
    ! [X4: nat,Sx: nat] :
      ( ( ( vEBT_vebt_succ @ summary @ X4 )
        = ( some_nat @ Sx ) )
      = ( vEBT_is_succ_in_set @ ( vEBT_VEBT_set_vebt @ summary ) @ X4 @ Sx ) ) ).

% "5.hyps"(3)
thf(fact_8337_atLeastAtMost__iff,axiom,
    ! [I: extended_enat,L2: extended_enat,U: extended_enat] :
      ( ( member_Extended_enat @ I @ ( set_or5403411693681687835d_enat @ L2 @ U ) )
      = ( ( ord_le2932123472753598470d_enat @ L2 @ I )
        & ( ord_le2932123472753598470d_enat @ I @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_8338_atLeastAtMost__iff,axiom,
    ! [I: set_nat,L2: set_nat,U: set_nat] :
      ( ( member_set_nat @ I @ ( set_or4548717258645045905et_nat @ L2 @ U ) )
      = ( ( ord_less_eq_set_nat @ L2 @ I )
        & ( ord_less_eq_set_nat @ I @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_8339_atLeastAtMost__iff,axiom,
    ! [I: rat,L2: rat,U: rat] :
      ( ( member_rat @ I @ ( set_or633870826150836451st_rat @ L2 @ U ) )
      = ( ( ord_less_eq_rat @ L2 @ I )
        & ( ord_less_eq_rat @ I @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_8340_atLeastAtMost__iff,axiom,
    ! [I: num,L2: num,U: num] :
      ( ( member_num @ I @ ( set_or7049704709247886629st_num @ L2 @ U ) )
      = ( ( ord_less_eq_num @ L2 @ I )
        & ( ord_less_eq_num @ I @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_8341_atLeastAtMost__iff,axiom,
    ! [I: nat,L2: nat,U: nat] :
      ( ( member_nat @ I @ ( set_or1269000886237332187st_nat @ L2 @ U ) )
      = ( ( ord_less_eq_nat @ L2 @ I )
        & ( ord_less_eq_nat @ I @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_8342_atLeastAtMost__iff,axiom,
    ! [I: int,L2: int,U: int] :
      ( ( member_int @ I @ ( set_or1266510415728281911st_int @ L2 @ U ) )
      = ( ( ord_less_eq_int @ L2 @ I )
        & ( ord_less_eq_int @ I @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_8343_atLeastAtMost__iff,axiom,
    ! [I: real,L2: real,U: real] :
      ( ( member_real @ I @ ( set_or1222579329274155063t_real @ L2 @ U ) )
      = ( ( ord_less_eq_real @ L2 @ I )
        & ( ord_less_eq_real @ I @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_8344_Icc__eq__Icc,axiom,
    ! [L2: set_nat,H2: set_nat,L4: set_nat,H3: set_nat] :
      ( ( ( set_or4548717258645045905et_nat @ L2 @ H2 )
        = ( set_or4548717258645045905et_nat @ L4 @ H3 ) )
      = ( ( ( L2 = L4 )
          & ( H2 = H3 ) )
        | ( ~ ( ord_less_eq_set_nat @ L2 @ H2 )
          & ~ ( ord_less_eq_set_nat @ L4 @ H3 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_8345_Icc__eq__Icc,axiom,
    ! [L2: rat,H2: rat,L4: rat,H3: rat] :
      ( ( ( set_or633870826150836451st_rat @ L2 @ H2 )
        = ( set_or633870826150836451st_rat @ L4 @ H3 ) )
      = ( ( ( L2 = L4 )
          & ( H2 = H3 ) )
        | ( ~ ( ord_less_eq_rat @ L2 @ H2 )
          & ~ ( ord_less_eq_rat @ L4 @ H3 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_8346_Icc__eq__Icc,axiom,
    ! [L2: num,H2: num,L4: num,H3: num] :
      ( ( ( set_or7049704709247886629st_num @ L2 @ H2 )
        = ( set_or7049704709247886629st_num @ L4 @ H3 ) )
      = ( ( ( L2 = L4 )
          & ( H2 = H3 ) )
        | ( ~ ( ord_less_eq_num @ L2 @ H2 )
          & ~ ( ord_less_eq_num @ L4 @ H3 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_8347_Icc__eq__Icc,axiom,
    ! [L2: nat,H2: nat,L4: nat,H3: nat] :
      ( ( ( set_or1269000886237332187st_nat @ L2 @ H2 )
        = ( set_or1269000886237332187st_nat @ L4 @ H3 ) )
      = ( ( ( L2 = L4 )
          & ( H2 = H3 ) )
        | ( ~ ( ord_less_eq_nat @ L2 @ H2 )
          & ~ ( ord_less_eq_nat @ L4 @ H3 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_8348_Icc__eq__Icc,axiom,
    ! [L2: int,H2: int,L4: int,H3: int] :
      ( ( ( set_or1266510415728281911st_int @ L2 @ H2 )
        = ( set_or1266510415728281911st_int @ L4 @ H3 ) )
      = ( ( ( L2 = L4 )
          & ( H2 = H3 ) )
        | ( ~ ( ord_less_eq_int @ L2 @ H2 )
          & ~ ( ord_less_eq_int @ L4 @ H3 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_8349_Icc__eq__Icc,axiom,
    ! [L2: real,H2: real,L4: real,H3: real] :
      ( ( ( set_or1222579329274155063t_real @ L2 @ H2 )
        = ( set_or1222579329274155063t_real @ L4 @ H3 ) )
      = ( ( ( L2 = L4 )
          & ( H2 = H3 ) )
        | ( ~ ( ord_less_eq_real @ L2 @ H2 )
          & ~ ( ord_less_eq_real @ L4 @ H3 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_8350_insert__simp__mima,axiom,
    ! [X4: nat,Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( X4 = Mi )
        | ( X4 = Ma ) )
     => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
       => ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X4 )
          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) ) ) ) ).

% insert_simp_mima
thf(fact_8351_Diff__cancel,axiom,
    ! [A3: set_Extended_enat] :
      ( ( minus_925952699566721837d_enat @ A3 @ A3 )
      = bot_bo7653980558646680370d_enat ) ).

% Diff_cancel
thf(fact_8352_Diff__cancel,axiom,
    ! [A3: set_real] :
      ( ( minus_minus_set_real @ A3 @ A3 )
      = bot_bot_set_real ) ).

% Diff_cancel
thf(fact_8353_Diff__cancel,axiom,
    ! [A3: set_int] :
      ( ( minus_minus_set_int @ A3 @ A3 )
      = bot_bot_set_int ) ).

% Diff_cancel
thf(fact_8354_Diff__cancel,axiom,
    ! [A3: set_nat] :
      ( ( minus_minus_set_nat @ A3 @ A3 )
      = bot_bot_set_nat ) ).

% Diff_cancel
thf(fact_8355_empty__Diff,axiom,
    ! [A3: set_Extended_enat] :
      ( ( minus_925952699566721837d_enat @ bot_bo7653980558646680370d_enat @ A3 )
      = bot_bo7653980558646680370d_enat ) ).

% empty_Diff
thf(fact_8356_empty__Diff,axiom,
    ! [A3: set_real] :
      ( ( minus_minus_set_real @ bot_bot_set_real @ A3 )
      = bot_bot_set_real ) ).

% empty_Diff
thf(fact_8357_empty__Diff,axiom,
    ! [A3: set_int] :
      ( ( minus_minus_set_int @ bot_bot_set_int @ A3 )
      = bot_bot_set_int ) ).

% empty_Diff
thf(fact_8358_empty__Diff,axiom,
    ! [A3: set_nat] :
      ( ( minus_minus_set_nat @ bot_bot_set_nat @ A3 )
      = bot_bot_set_nat ) ).

% empty_Diff
thf(fact_8359_Diff__empty,axiom,
    ! [A3: set_Extended_enat] :
      ( ( minus_925952699566721837d_enat @ A3 @ bot_bo7653980558646680370d_enat )
      = A3 ) ).

% Diff_empty
thf(fact_8360_Diff__empty,axiom,
    ! [A3: set_real] :
      ( ( minus_minus_set_real @ A3 @ bot_bot_set_real )
      = A3 ) ).

% Diff_empty
thf(fact_8361_Diff__empty,axiom,
    ! [A3: set_int] :
      ( ( minus_minus_set_int @ A3 @ bot_bot_set_int )
      = A3 ) ).

% Diff_empty
thf(fact_8362_Diff__empty,axiom,
    ! [A3: set_nat] :
      ( ( minus_minus_set_nat @ A3 @ bot_bot_set_nat )
      = A3 ) ).

% Diff_empty
thf(fact_8363_mi__ma__2__deg,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ N )
     => ( ( ord_less_eq_nat @ Mi @ Ma )
        & ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) ) ) ) ).

% mi_ma_2_deg
thf(fact_8364_succ__min,axiom,
    ! [Deg: nat,X4: nat,Mi: nat,Ma: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ( ord_less_nat @ X4 @ Mi )
       => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X4 )
          = ( some_nat @ Mi ) ) ) ) ).

% succ_min
thf(fact_8365_atLeastatMost__empty__iff,axiom,
    ! [A: extended_enat,B2: extended_enat] :
      ( ( ( set_or5403411693681687835d_enat @ A @ B2 )
        = bot_bo7653980558646680370d_enat )
      = ( ~ ( ord_le2932123472753598470d_enat @ A @ B2 ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_8366_atLeastatMost__empty__iff,axiom,
    ! [A: set_nat,B2: set_nat] :
      ( ( ( set_or4548717258645045905et_nat @ A @ B2 )
        = bot_bot_set_set_nat )
      = ( ~ ( ord_less_eq_set_nat @ A @ B2 ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_8367_atLeastatMost__empty__iff,axiom,
    ! [A: rat,B2: rat] :
      ( ( ( set_or633870826150836451st_rat @ A @ B2 )
        = bot_bot_set_rat )
      = ( ~ ( ord_less_eq_rat @ A @ B2 ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_8368_atLeastatMost__empty__iff,axiom,
    ! [A: num,B2: num] :
      ( ( ( set_or7049704709247886629st_num @ A @ B2 )
        = bot_bot_set_num )
      = ( ~ ( ord_less_eq_num @ A @ B2 ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_8369_atLeastatMost__empty__iff,axiom,
    ! [A: nat,B2: nat] :
      ( ( ( set_or1269000886237332187st_nat @ A @ B2 )
        = bot_bot_set_nat )
      = ( ~ ( ord_less_eq_nat @ A @ B2 ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_8370_atLeastatMost__empty__iff,axiom,
    ! [A: int,B2: int] :
      ( ( ( set_or1266510415728281911st_int @ A @ B2 )
        = bot_bot_set_int )
      = ( ~ ( ord_less_eq_int @ A @ B2 ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_8371_atLeastatMost__empty__iff,axiom,
    ! [A: real,B2: real] :
      ( ( ( set_or1222579329274155063t_real @ A @ B2 )
        = bot_bot_set_real )
      = ( ~ ( ord_less_eq_real @ A @ B2 ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_8372_atLeastatMost__empty__iff2,axiom,
    ! [A: extended_enat,B2: extended_enat] :
      ( ( bot_bo7653980558646680370d_enat
        = ( set_or5403411693681687835d_enat @ A @ B2 ) )
      = ( ~ ( ord_le2932123472753598470d_enat @ A @ B2 ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_8373_atLeastatMost__empty__iff2,axiom,
    ! [A: set_nat,B2: set_nat] :
      ( ( bot_bot_set_set_nat
        = ( set_or4548717258645045905et_nat @ A @ B2 ) )
      = ( ~ ( ord_less_eq_set_nat @ A @ B2 ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_8374_atLeastatMost__empty__iff2,axiom,
    ! [A: rat,B2: rat] :
      ( ( bot_bot_set_rat
        = ( set_or633870826150836451st_rat @ A @ B2 ) )
      = ( ~ ( ord_less_eq_rat @ A @ B2 ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_8375_atLeastatMost__empty__iff2,axiom,
    ! [A: num,B2: num] :
      ( ( bot_bot_set_num
        = ( set_or7049704709247886629st_num @ A @ B2 ) )
      = ( ~ ( ord_less_eq_num @ A @ B2 ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_8376_atLeastatMost__empty__iff2,axiom,
    ! [A: nat,B2: nat] :
      ( ( bot_bot_set_nat
        = ( set_or1269000886237332187st_nat @ A @ B2 ) )
      = ( ~ ( ord_less_eq_nat @ A @ B2 ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_8377_atLeastatMost__empty__iff2,axiom,
    ! [A: int,B2: int] :
      ( ( bot_bot_set_int
        = ( set_or1266510415728281911st_int @ A @ B2 ) )
      = ( ~ ( ord_less_eq_int @ A @ B2 ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_8378_atLeastatMost__empty__iff2,axiom,
    ! [A: real,B2: real] :
      ( ( bot_bot_set_real
        = ( set_or1222579329274155063t_real @ A @ B2 ) )
      = ( ~ ( ord_less_eq_real @ A @ B2 ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_8379_atLeastatMost__subset__iff,axiom,
    ! [A: set_nat,B2: set_nat,C: set_nat,D: set_nat] :
      ( ( ord_le6893508408891458716et_nat @ ( set_or4548717258645045905et_nat @ A @ B2 ) @ ( set_or4548717258645045905et_nat @ C @ D ) )
      = ( ~ ( ord_less_eq_set_nat @ A @ B2 )
        | ( ( ord_less_eq_set_nat @ C @ A )
          & ( ord_less_eq_set_nat @ B2 @ D ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_8380_atLeastatMost__subset__iff,axiom,
    ! [A: rat,B2: rat,C: rat,D: rat] :
      ( ( ord_less_eq_set_rat @ ( set_or633870826150836451st_rat @ A @ B2 ) @ ( set_or633870826150836451st_rat @ C @ D ) )
      = ( ~ ( ord_less_eq_rat @ A @ B2 )
        | ( ( ord_less_eq_rat @ C @ A )
          & ( ord_less_eq_rat @ B2 @ D ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_8381_atLeastatMost__subset__iff,axiom,
    ! [A: num,B2: num,C: num,D: num] :
      ( ( ord_less_eq_set_num @ ( set_or7049704709247886629st_num @ A @ B2 ) @ ( set_or7049704709247886629st_num @ C @ D ) )
      = ( ~ ( ord_less_eq_num @ A @ B2 )
        | ( ( ord_less_eq_num @ C @ A )
          & ( ord_less_eq_num @ B2 @ D ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_8382_atLeastatMost__subset__iff,axiom,
    ! [A: nat,B2: nat,C: nat,D: nat] :
      ( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ A @ B2 ) @ ( set_or1269000886237332187st_nat @ C @ D ) )
      = ( ~ ( ord_less_eq_nat @ A @ B2 )
        | ( ( ord_less_eq_nat @ C @ A )
          & ( ord_less_eq_nat @ B2 @ D ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_8383_atLeastatMost__subset__iff,axiom,
    ! [A: int,B2: int,C: int,D: int] :
      ( ( ord_less_eq_set_int @ ( set_or1266510415728281911st_int @ A @ B2 ) @ ( set_or1266510415728281911st_int @ C @ D ) )
      = ( ~ ( ord_less_eq_int @ A @ B2 )
        | ( ( ord_less_eq_int @ C @ A )
          & ( ord_less_eq_int @ B2 @ D ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_8384_atLeastatMost__subset__iff,axiom,
    ! [A: real,B2: real,C: real,D: real] :
      ( ( ord_less_eq_set_real @ ( set_or1222579329274155063t_real @ A @ B2 ) @ ( set_or1222579329274155063t_real @ C @ D ) )
      = ( ~ ( ord_less_eq_real @ A @ B2 )
        | ( ( ord_less_eq_real @ C @ A )
          & ( ord_less_eq_real @ B2 @ D ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_8385_atLeastatMost__empty,axiom,
    ! [B2: extended_enat,A: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ B2 @ A )
     => ( ( set_or5403411693681687835d_enat @ A @ B2 )
        = bot_bo7653980558646680370d_enat ) ) ).

% atLeastatMost_empty
thf(fact_8386_atLeastatMost__empty,axiom,
    ! [B2: rat,A: rat] :
      ( ( ord_less_rat @ B2 @ A )
     => ( ( set_or633870826150836451st_rat @ A @ B2 )
        = bot_bot_set_rat ) ) ).

% atLeastatMost_empty
thf(fact_8387_atLeastatMost__empty,axiom,
    ! [B2: num,A: num] :
      ( ( ord_less_num @ B2 @ A )
     => ( ( set_or7049704709247886629st_num @ A @ B2 )
        = bot_bot_set_num ) ) ).

% atLeastatMost_empty
thf(fact_8388_atLeastatMost__empty,axiom,
    ! [B2: nat,A: nat] :
      ( ( ord_less_nat @ B2 @ A )
     => ( ( set_or1269000886237332187st_nat @ A @ B2 )
        = bot_bot_set_nat ) ) ).

% atLeastatMost_empty
thf(fact_8389_atLeastatMost__empty,axiom,
    ! [B2: int,A: int] :
      ( ( ord_less_int @ B2 @ A )
     => ( ( set_or1266510415728281911st_int @ A @ B2 )
        = bot_bot_set_int ) ) ).

% atLeastatMost_empty
thf(fact_8390_atLeastatMost__empty,axiom,
    ! [B2: real,A: real] :
      ( ( ord_less_real @ B2 @ A )
     => ( ( set_or1222579329274155063t_real @ A @ B2 )
        = bot_bot_set_real ) ) ).

% atLeastatMost_empty
thf(fact_8391_Diff__eq__empty__iff,axiom,
    ! [A3: set_Extended_enat,B4: set_Extended_enat] :
      ( ( ( minus_925952699566721837d_enat @ A3 @ B4 )
        = bot_bo7653980558646680370d_enat )
      = ( ord_le7203529160286727270d_enat @ A3 @ B4 ) ) ).

% Diff_eq_empty_iff
thf(fact_8392_Diff__eq__empty__iff,axiom,
    ! [A3: set_real,B4: set_real] :
      ( ( ( minus_minus_set_real @ A3 @ B4 )
        = bot_bot_set_real )
      = ( ord_less_eq_set_real @ A3 @ B4 ) ) ).

% Diff_eq_empty_iff
thf(fact_8393_Diff__eq__empty__iff,axiom,
    ! [A3: set_int,B4: set_int] :
      ( ( ( minus_minus_set_int @ A3 @ B4 )
        = bot_bot_set_int )
      = ( ord_less_eq_set_int @ A3 @ B4 ) ) ).

% Diff_eq_empty_iff
thf(fact_8394_Diff__eq__empty__iff,axiom,
    ! [A3: set_nat,B4: set_nat] :
      ( ( ( minus_minus_set_nat @ A3 @ B4 )
        = bot_bot_set_nat )
      = ( ord_less_eq_set_nat @ A3 @ B4 ) ) ).

% Diff_eq_empty_iff
thf(fact_8395__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062sc_O_Avebt__succ_Asummary_A_Ihigh_Ax_A_Ideg_Adiv_A2_J_J_A_061_ASome_Asc_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [Sc: nat] :
        ( ( vEBT_vebt_succ @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
       != ( some_nat @ Sc ) ) ).

% \<open>\<And>thesis. (\<And>sc. vebt_succ summary (high x (deg div 2)) = Some sc \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_8396_both__member__options__from__complete__tree__to__child,axiom,
    ! [Deg: nat,Mi: nat,Ma: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X4: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ Deg )
     => ( ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X4 )
       => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
          | ( X4 = Mi )
          | ( X4 = Ma ) ) ) ) ).

% both_member_options_from_complete_tree_to_child
thf(fact_8397_member__inv,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X4: nat] :
      ( ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X4 )
     => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
        & ( ( X4 = Mi )
          | ( X4 = Ma )
          | ( ( ord_less_nat @ X4 @ Ma )
            & ( ord_less_nat @ Mi @ X4 )
            & ( ord_less_nat @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
            & ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% member_inv
thf(fact_8398_both__member__options__from__chilf__to__complete__tree,axiom,
    ! [X4: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
      ( ( ord_less_nat @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
     => ( ( ord_less_eq_nat @ one_one_nat @ Deg )
       => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
         => ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X4 ) ) ) ) ).

% both_member_options_from_chilf_to_complete_tree
thf(fact_8399__092_060open_062Some_Aminy_A_061_Avebt__mint_A_ItreeList_A_B_Asc_J_092_060close_062,axiom,
    ( ( some_nat @ miny )
    = ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ sc ) ) ) ).

% \<open>Some miny = vebt_mint (treeList ! sc)\<close>
thf(fact_8400_greater__shift,axiom,
    ( ord_less_nat
    = ( ^ [Y: nat,X: nat] : ( vEBT_VEBT_greater @ ( some_nat @ X ) @ ( some_nat @ Y ) ) ) ) ).

% greater_shift
thf(fact_8401_lesseq__shift,axiom,
    ( ord_less_eq_nat
    = ( ^ [X: nat,Y: nat] : ( vEBT_VEBT_lesseq @ ( some_nat @ X ) @ ( some_nat @ Y ) ) ) ) ).

% lesseq_shift
thf(fact_8402_False,axiom,
    ( ( vEBT_vebt_succ @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
   != none_nat ) ).

% False
thf(fact_8403_not__psubset__empty,axiom,
    ! [A3: set_Extended_enat] :
      ~ ( ord_le2529575680413868914d_enat @ A3 @ bot_bo7653980558646680370d_enat ) ).

% not_psubset_empty
thf(fact_8404_not__psubset__empty,axiom,
    ! [A3: set_real] :
      ~ ( ord_less_set_real @ A3 @ bot_bot_set_real ) ).

% not_psubset_empty
thf(fact_8405_not__psubset__empty,axiom,
    ! [A3: set_nat] :
      ~ ( ord_less_set_nat @ A3 @ bot_bot_set_nat ) ).

% not_psubset_empty
thf(fact_8406_not__psubset__empty,axiom,
    ! [A3: set_int] :
      ~ ( ord_less_set_int @ A3 @ bot_bot_set_int ) ).

% not_psubset_empty
thf(fact_8407_bot_Onot__eq__extremum,axiom,
    ! [A: set_Extended_enat] :
      ( ( A != bot_bo7653980558646680370d_enat )
      = ( ord_le2529575680413868914d_enat @ bot_bo7653980558646680370d_enat @ A ) ) ).

% bot.not_eq_extremum
thf(fact_8408_bot_Onot__eq__extremum,axiom,
    ! [A: set_real] :
      ( ( A != bot_bot_set_real )
      = ( ord_less_set_real @ bot_bot_set_real @ A ) ) ).

% bot.not_eq_extremum
thf(fact_8409_bot_Onot__eq__extremum,axiom,
    ! [A: set_nat] :
      ( ( A != bot_bot_set_nat )
      = ( ord_less_set_nat @ bot_bot_set_nat @ A ) ) ).

% bot.not_eq_extremum
thf(fact_8410_bot_Onot__eq__extremum,axiom,
    ! [A: set_int] :
      ( ( A != bot_bot_set_int )
      = ( ord_less_set_int @ bot_bot_set_int @ A ) ) ).

% bot.not_eq_extremum
thf(fact_8411_bot_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != bot_bot_nat )
      = ( ord_less_nat @ bot_bot_nat @ A ) ) ).

% bot.not_eq_extremum
thf(fact_8412_bot_Oextremum__strict,axiom,
    ! [A: set_Extended_enat] :
      ~ ( ord_le2529575680413868914d_enat @ A @ bot_bo7653980558646680370d_enat ) ).

% bot.extremum_strict
thf(fact_8413_bot_Oextremum__strict,axiom,
    ! [A: set_real] :
      ~ ( ord_less_set_real @ A @ bot_bot_set_real ) ).

% bot.extremum_strict
thf(fact_8414_bot_Oextremum__strict,axiom,
    ! [A: set_nat] :
      ~ ( ord_less_set_nat @ A @ bot_bot_set_nat ) ).

% bot.extremum_strict
thf(fact_8415_bot_Oextremum__strict,axiom,
    ! [A: set_int] :
      ~ ( ord_less_set_int @ A @ bot_bot_set_int ) ).

% bot.extremum_strict
thf(fact_8416_bot_Oextremum__strict,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ bot_bot_nat ) ).

% bot.extremum_strict
thf(fact_8417_bot_Oextremum,axiom,
    ! [A: set_Extended_enat] : ( ord_le7203529160286727270d_enat @ bot_bo7653980558646680370d_enat @ A ) ).

% bot.extremum
thf(fact_8418_bot_Oextremum,axiom,
    ! [A: set_real] : ( ord_less_eq_set_real @ bot_bot_set_real @ A ) ).

% bot.extremum
thf(fact_8419_bot_Oextremum,axiom,
    ! [A: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A ) ).

% bot.extremum
thf(fact_8420_bot_Oextremum,axiom,
    ! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).

% bot.extremum
thf(fact_8421_bot_Oextremum,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).

% bot.extremum
thf(fact_8422_bot_Oextremum__unique,axiom,
    ! [A: set_Extended_enat] :
      ( ( ord_le7203529160286727270d_enat @ A @ bot_bo7653980558646680370d_enat )
      = ( A = bot_bo7653980558646680370d_enat ) ) ).

% bot.extremum_unique
thf(fact_8423_bot_Oextremum__unique,axiom,
    ! [A: set_real] :
      ( ( ord_less_eq_set_real @ A @ bot_bot_set_real )
      = ( A = bot_bot_set_real ) ) ).

% bot.extremum_unique
thf(fact_8424_bot_Oextremum__unique,axiom,
    ! [A: set_int] :
      ( ( ord_less_eq_set_int @ A @ bot_bot_set_int )
      = ( A = bot_bot_set_int ) ) ).

% bot.extremum_unique
thf(fact_8425_bot_Oextremum__unique,axiom,
    ! [A: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
      = ( A = bot_bot_set_nat ) ) ).

% bot.extremum_unique
thf(fact_8426_bot_Oextremum__unique,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ bot_bot_nat )
      = ( A = bot_bot_nat ) ) ).

% bot.extremum_unique
thf(fact_8427_bot_Oextremum__uniqueI,axiom,
    ! [A: set_Extended_enat] :
      ( ( ord_le7203529160286727270d_enat @ A @ bot_bo7653980558646680370d_enat )
     => ( A = bot_bo7653980558646680370d_enat ) ) ).

% bot.extremum_uniqueI
thf(fact_8428_bot_Oextremum__uniqueI,axiom,
    ! [A: set_real] :
      ( ( ord_less_eq_set_real @ A @ bot_bot_set_real )
     => ( A = bot_bot_set_real ) ) ).

% bot.extremum_uniqueI
thf(fact_8429_bot_Oextremum__uniqueI,axiom,
    ! [A: set_int] :
      ( ( ord_less_eq_set_int @ A @ bot_bot_set_int )
     => ( A = bot_bot_set_int ) ) ).

% bot.extremum_uniqueI
thf(fact_8430_bot_Oextremum__uniqueI,axiom,
    ! [A: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
     => ( A = bot_bot_set_nat ) ) ).

% bot.extremum_uniqueI
thf(fact_8431_bot_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ bot_bot_nat )
     => ( A = bot_bot_nat ) ) ).

% bot.extremum_uniqueI
thf(fact_8432_diff__shunt__var,axiom,
    ! [X4: set_Extended_enat,Y3: set_Extended_enat] :
      ( ( ( minus_925952699566721837d_enat @ X4 @ Y3 )
        = bot_bo7653980558646680370d_enat )
      = ( ord_le7203529160286727270d_enat @ X4 @ Y3 ) ) ).

% diff_shunt_var
thf(fact_8433_diff__shunt__var,axiom,
    ! [X4: set_real,Y3: set_real] :
      ( ( ( minus_minus_set_real @ X4 @ Y3 )
        = bot_bot_set_real )
      = ( ord_less_eq_set_real @ X4 @ Y3 ) ) ).

% diff_shunt_var
thf(fact_8434_diff__shunt__var,axiom,
    ! [X4: set_int,Y3: set_int] :
      ( ( ( minus_minus_set_int @ X4 @ Y3 )
        = bot_bot_set_int )
      = ( ord_less_eq_set_int @ X4 @ Y3 ) ) ).

% diff_shunt_var
thf(fact_8435_diff__shunt__var,axiom,
    ! [X4: set_nat,Y3: set_nat] :
      ( ( ( minus_minus_set_nat @ X4 @ Y3 )
        = bot_bot_set_nat )
      = ( ord_less_eq_set_nat @ X4 @ Y3 ) ) ).

% diff_shunt_var
thf(fact_8436_ex__nat__less,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ? [M6: nat] :
            ( ( ord_less_eq_nat @ M6 @ N )
            & ( P @ M6 ) ) )
      = ( ? [X: nat] :
            ( ( member_nat @ X @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
            & ( P @ X ) ) ) ) ).

% ex_nat_less
thf(fact_8437_all__nat__less,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ! [M6: nat] :
            ( ( ord_less_eq_nat @ M6 @ N )
           => ( P @ M6 ) ) )
      = ( ! [X: nat] :
            ( ( member_nat @ X @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
           => ( P @ X ) ) ) ) ).

% all_nat_less
thf(fact_8438_atLeastatMost__psubset__iff,axiom,
    ! [A: set_nat,B2: set_nat,C: set_nat,D: set_nat] :
      ( ( ord_less_set_set_nat @ ( set_or4548717258645045905et_nat @ A @ B2 ) @ ( set_or4548717258645045905et_nat @ C @ D ) )
      = ( ( ~ ( ord_less_eq_set_nat @ A @ B2 )
          | ( ( ord_less_eq_set_nat @ C @ A )
            & ( ord_less_eq_set_nat @ B2 @ D )
            & ( ( ord_less_set_nat @ C @ A )
              | ( ord_less_set_nat @ B2 @ D ) ) ) )
        & ( ord_less_eq_set_nat @ C @ D ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_8439_atLeastatMost__psubset__iff,axiom,
    ! [A: rat,B2: rat,C: rat,D: rat] :
      ( ( ord_less_set_rat @ ( set_or633870826150836451st_rat @ A @ B2 ) @ ( set_or633870826150836451st_rat @ C @ D ) )
      = ( ( ~ ( ord_less_eq_rat @ A @ B2 )
          | ( ( ord_less_eq_rat @ C @ A )
            & ( ord_less_eq_rat @ B2 @ D )
            & ( ( ord_less_rat @ C @ A )
              | ( ord_less_rat @ B2 @ D ) ) ) )
        & ( ord_less_eq_rat @ C @ D ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_8440_atLeastatMost__psubset__iff,axiom,
    ! [A: num,B2: num,C: num,D: num] :
      ( ( ord_less_set_num @ ( set_or7049704709247886629st_num @ A @ B2 ) @ ( set_or7049704709247886629st_num @ C @ D ) )
      = ( ( ~ ( ord_less_eq_num @ A @ B2 )
          | ( ( ord_less_eq_num @ C @ A )
            & ( ord_less_eq_num @ B2 @ D )
            & ( ( ord_less_num @ C @ A )
              | ( ord_less_num @ B2 @ D ) ) ) )
        & ( ord_less_eq_num @ C @ D ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_8441_atLeastatMost__psubset__iff,axiom,
    ! [A: nat,B2: nat,C: nat,D: nat] :
      ( ( ord_less_set_nat @ ( set_or1269000886237332187st_nat @ A @ B2 ) @ ( set_or1269000886237332187st_nat @ C @ D ) )
      = ( ( ~ ( ord_less_eq_nat @ A @ B2 )
          | ( ( ord_less_eq_nat @ C @ A )
            & ( ord_less_eq_nat @ B2 @ D )
            & ( ( ord_less_nat @ C @ A )
              | ( ord_less_nat @ B2 @ D ) ) ) )
        & ( ord_less_eq_nat @ C @ D ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_8442_atLeastatMost__psubset__iff,axiom,
    ! [A: int,B2: int,C: int,D: int] :
      ( ( ord_less_set_int @ ( set_or1266510415728281911st_int @ A @ B2 ) @ ( set_or1266510415728281911st_int @ C @ D ) )
      = ( ( ~ ( ord_less_eq_int @ A @ B2 )
          | ( ( ord_less_eq_int @ C @ A )
            & ( ord_less_eq_int @ B2 @ D )
            & ( ( ord_less_int @ C @ A )
              | ( ord_less_int @ B2 @ D ) ) ) )
        & ( ord_less_eq_int @ C @ D ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_8443_atLeastatMost__psubset__iff,axiom,
    ! [A: real,B2: real,C: real,D: real] :
      ( ( ord_less_set_real @ ( set_or1222579329274155063t_real @ A @ B2 ) @ ( set_or1222579329274155063t_real @ C @ D ) )
      = ( ( ~ ( ord_less_eq_real @ A @ B2 )
          | ( ( ord_less_eq_real @ C @ A )
            & ( ord_less_eq_real @ B2 @ D )
            & ( ( ord_less_real @ C @ A )
              | ( ord_less_real @ B2 @ D ) ) ) )
        & ( ord_less_eq_real @ C @ D ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_8444_invar__vebt_Ointros_I4_J,axiom,
    ! [TreeList2: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
         => ( vEBT_invar_vebt @ X3 @ N ) )
     => ( ( vEBT_invar_vebt @ Summary @ M )
       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
         => ( ( M = N )
           => ( ( Deg
                = ( plus_plus_nat @ N @ M ) )
             => ( ! [I3: nat] :
                    ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
                   => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ X6 ) )
                      = ( vEBT_V8194947554948674370ptions @ Summary @ I3 ) ) )
               => ( ( ( Mi = Ma )
                   => ! [X3: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                       => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) ) )
                 => ( ( ord_less_eq_nat @ Mi @ Ma )
                   => ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
                     => ( ( ( Mi != Ma )
                         => ! [I3: nat] :
                              ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
                             => ( ( ( ( vEBT_VEBT_high @ Ma @ N )
                                    = I3 )
                                 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ ( vEBT_VEBT_low @ Ma @ N ) ) )
                                & ! [X3: nat] :
                                    ( ( ( ( vEBT_VEBT_high @ X3 @ N )
                                        = I3 )
                                      & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ ( vEBT_VEBT_low @ X3 @ N ) ) )
                                   => ( ( ord_less_nat @ Mi @ X3 )
                                      & ( ord_less_eq_nat @ X3 @ Ma ) ) ) ) ) )
                       => ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(4)
thf(fact_8445_invar__vebt_Ointros_I5_J,axiom,
    ! [TreeList2: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
         => ( vEBT_invar_vebt @ X3 @ N ) )
     => ( ( vEBT_invar_vebt @ Summary @ M )
       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
         => ( ( M
              = ( suc @ N ) )
           => ( ( Deg
                = ( plus_plus_nat @ N @ M ) )
             => ( ! [I3: nat] :
                    ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
                   => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ X6 ) )
                      = ( vEBT_V8194947554948674370ptions @ Summary @ I3 ) ) )
               => ( ( ( Mi = Ma )
                   => ! [X3: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                       => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) ) )
                 => ( ( ord_less_eq_nat @ Mi @ Ma )
                   => ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
                     => ( ( ( Mi != Ma )
                         => ! [I3: nat] :
                              ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
                             => ( ( ( ( vEBT_VEBT_high @ Ma @ N )
                                    = I3 )
                                 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ ( vEBT_VEBT_low @ Ma @ N ) ) )
                                & ! [X3: nat] :
                                    ( ( ( ( vEBT_VEBT_high @ X3 @ N )
                                        = I3 )
                                      & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ ( vEBT_VEBT_low @ X3 @ N ) ) )
                                   => ( ( ord_less_nat @ Mi @ X3 )
                                      & ( ord_less_eq_nat @ X3 @ Ma ) ) ) ) ) )
                       => ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(5)
thf(fact_8446_mintlistlength,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ N )
     => ( ( Mi != Ma )
       => ( ( ord_less_nat @ Mi @ Ma )
          & ? [M4: nat] :
              ( ( ( some_nat @ M4 )
                = ( vEBT_vebt_mint @ Summary ) )
              & ( ord_less_nat @ M4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% mintlistlength
thf(fact_8447__092_060open_062min__in__set_A_Iset__vebt_H_A_ItreeList_A_B_Athe_A_Ivebt__succ_Asummary_A_Ihigh_Ax_A_Ideg_Adiv_A2_J_J_J_J_J_Aminy_092_060close_062,axiom,
    vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ ( nth_VEBT_VEBT @ treeList @ ( the_nat @ ( vEBT_vebt_succ @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) @ miny ).

% \<open>min_in_set (set_vebt' (treeList ! the (vebt_succ summary (high x (deg div 2))))) miny\<close>
thf(fact_8448_scmem,axiom,
    vEBT_vebt_member @ ( nth_VEBT_VEBT @ treeList @ ( the_nat @ ( vEBT_vebt_succ @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ miny ).

% scmem
thf(fact_8449__092_060open_062invar__vebt_A_ItreeList_A_B_Athe_A_Ivebt__succ_Asummary_A_Ihigh_Ax_A_Ideg_Adiv_A2_J_J_J_J_An_092_060close_062,axiom,
    vEBT_invar_vebt @ ( nth_VEBT_VEBT @ treeList @ ( the_nat @ ( vEBT_vebt_succ @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ na ).

% \<open>invar_vebt (treeList ! the (vebt_succ summary (high x (deg div 2)))) n\<close>
thf(fact_8450_mint__member,axiom,
    ! [T: vEBT_VEBT,N: nat,Maxi: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_mint @ T )
          = ( some_nat @ Maxi ) )
       => ( vEBT_vebt_member @ T @ Maxi ) ) ) ).

% mint_member
thf(fact_8451_mint__corr__help,axiom,
    ! [T: vEBT_VEBT,N: nat,Mini: nat,X4: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_mint @ T )
          = ( some_nat @ Mini ) )
       => ( ( vEBT_vebt_member @ T @ X4 )
         => ( ord_less_eq_nat @ Mini @ X4 ) ) ) ) ).

% mint_corr_help
thf(fact_8452_mint__corr__help__empty,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_mint @ T )
          = none_nat )
       => ( ( vEBT_VEBT_set_vebt @ T )
          = bot_bot_set_nat ) ) ) ).

% mint_corr_help_empty
thf(fact_8453_mint__sound,axiom,
    ! [T: vEBT_VEBT,N: nat,X4: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X4 )
       => ( ( vEBT_vebt_mint @ T )
          = ( some_nat @ X4 ) ) ) ) ).

% mint_sound
thf(fact_8454_mint__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X4: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_mint @ T )
          = ( some_nat @ X4 ) )
       => ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X4 ) ) ) ).

% mint_corr
thf(fact_8455__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062miny_O_ASome_Aminy_A_061_Avebt__mint_A_ItreeList_A_B_Asc_J_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [Miny: nat] :
        ( ( some_nat @ Miny )
       != ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ sc ) ) ) ).

% \<open>\<And>thesis. (\<And>miny. Some miny = vebt_mint (treeList ! sc) \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_8456_misiz,axiom,
    ! [T: vEBT_VEBT,N: nat,M: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( some_nat @ M )
          = ( vEBT_vebt_mint @ T ) )
       => ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% misiz
thf(fact_8457_bit__0__eq,axiom,
    ( ( bit_se1146084159140164899it_int @ zero_zero_int )
    = bot_bot_nat_o ) ).

% bit_0_eq
thf(fact_8458_bit__0__eq,axiom,
    ( ( bit_se1148574629649215175it_nat @ zero_zero_nat )
    = bot_bot_nat_o ) ).

% bit_0_eq
thf(fact_8459_succ__list__to__short,axiom,
    ! [Deg: nat,Mi: nat,X4: nat,TreeList2: list_VEBT_VEBT,Ma: nat,Summary: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ( ord_less_eq_nat @ Mi @ X4 )
       => ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
         => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X4 )
            = none_nat ) ) ) ) ).

% succ_list_to_short
thf(fact_8460_nested__mint,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat,Va2: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ N )
     => ( ( N
          = ( suc @ ( suc @ Va2 ) ) )
       => ( ~ ( ord_less_nat @ Ma @ Mi )
         => ( ( Ma != Mi )
           => ( ord_less_nat @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Va2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( suc @ ( divide_divide_nat @ Va2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ).

% nested_mint
thf(fact_8461__092_060open_062Some_Aminy_A_061_Avebt__mint_A_ItreeList_A_B_Athe_A_Ivebt__succ_Asummary_A_Ihigh_Ax_A_Ideg_Adiv_A2_J_J_J_J_092_060close_062,axiom,
    ( ( some_nat @ miny )
    = ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( the_nat @ ( vEBT_vebt_succ @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% \<open>Some miny = vebt_mint (treeList ! the (vebt_succ summary (high x (deg div 2))))\<close>
thf(fact_8462_local_Opower__def,axiom,
    ( vEBT_VEBT_power
    = ( vEBT_V4262088993061758097ft_nat @ power_power_nat ) ) ).

% local.power_def
thf(fact_8463_bot__enat__def,axiom,
    bot_bo4199563552545308370d_enat = zero_z5237406670263579293d_enat ).

% bot_enat_def
thf(fact_8464_bot__nat__def,axiom,
    bot_bot_nat = zero_zero_nat ).

% bot_nat_def
thf(fact_8465_aset_I2_J,axiom,
    ! [D4: int,A3: set_int,P: int > $o,Q: int > $o] :
      ( ! [X3: int] :
          ( ! [Xa2: int] :
              ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb2: int] :
                  ( ( member_int @ Xb2 @ A3 )
                 => ( X3
                   != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
         => ( ( P @ X3 )
           => ( P @ ( plus_plus_int @ X3 @ D4 ) ) ) )
     => ( ! [X3: int] :
            ( ! [Xa2: int] :
                ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ A3 )
                   => ( X3
                     != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
           => ( ( Q @ X3 )
             => ( Q @ ( plus_plus_int @ X3 @ D4 ) ) ) )
       => ! [X5: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ A3 )
                   => ( X5
                     != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( ( P @ X5 )
                | ( Q @ X5 ) )
             => ( ( P @ ( plus_plus_int @ X5 @ D4 ) )
                | ( Q @ ( plus_plus_int @ X5 @ D4 ) ) ) ) ) ) ) ).

% aset(2)
thf(fact_8466_aset_I1_J,axiom,
    ! [D4: int,A3: set_int,P: int > $o,Q: int > $o] :
      ( ! [X3: int] :
          ( ! [Xa2: int] :
              ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb2: int] :
                  ( ( member_int @ Xb2 @ A3 )
                 => ( X3
                   != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
         => ( ( P @ X3 )
           => ( P @ ( plus_plus_int @ X3 @ D4 ) ) ) )
     => ( ! [X3: int] :
            ( ! [Xa2: int] :
                ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ A3 )
                   => ( X3
                     != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
           => ( ( Q @ X3 )
             => ( Q @ ( plus_plus_int @ X3 @ D4 ) ) ) )
       => ! [X5: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ A3 )
                   => ( X5
                     != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( ( P @ X5 )
                & ( Q @ X5 ) )
             => ( ( P @ ( plus_plus_int @ X5 @ D4 ) )
                & ( Q @ ( plus_plus_int @ X5 @ D4 ) ) ) ) ) ) ) ).

% aset(1)
thf(fact_8467_bset_I2_J,axiom,
    ! [D4: int,B4: set_int,P: int > $o,Q: int > $o] :
      ( ! [X3: int] :
          ( ! [Xa2: int] :
              ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb2: int] :
                  ( ( member_int @ Xb2 @ B4 )
                 => ( X3
                   != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
         => ( ( P @ X3 )
           => ( P @ ( minus_minus_int @ X3 @ D4 ) ) ) )
     => ( ! [X3: int] :
            ( ! [Xa2: int] :
                ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ B4 )
                   => ( X3
                     != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
           => ( ( Q @ X3 )
             => ( Q @ ( minus_minus_int @ X3 @ D4 ) ) ) )
       => ! [X5: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ B4 )
                   => ( X5
                     != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( ( P @ X5 )
                | ( Q @ X5 ) )
             => ( ( P @ ( minus_minus_int @ X5 @ D4 ) )
                | ( Q @ ( minus_minus_int @ X5 @ D4 ) ) ) ) ) ) ) ).

% bset(2)
thf(fact_8468_bset_I1_J,axiom,
    ! [D4: int,B4: set_int,P: int > $o,Q: int > $o] :
      ( ! [X3: int] :
          ( ! [Xa2: int] :
              ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb2: int] :
                  ( ( member_int @ Xb2 @ B4 )
                 => ( X3
                   != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
         => ( ( P @ X3 )
           => ( P @ ( minus_minus_int @ X3 @ D4 ) ) ) )
     => ( ! [X3: int] :
            ( ! [Xa2: int] :
                ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ B4 )
                   => ( X3
                     != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
           => ( ( Q @ X3 )
             => ( Q @ ( minus_minus_int @ X3 @ D4 ) ) ) )
       => ! [X5: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ B4 )
                   => ( X5
                     != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( ( P @ X5 )
                & ( Q @ X5 ) )
             => ( ( P @ ( minus_minus_int @ X5 @ D4 ) )
                & ( Q @ ( minus_minus_int @ X5 @ D4 ) ) ) ) ) ) ) ).

% bset(1)
thf(fact_8469_bset_I9_J,axiom,
    ! [D: int,D4: int,B4: set_int,T: int] :
      ( ( dvd_dvd_int @ D @ D4 )
     => ! [X5: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb3: int] :
                  ( ( member_int @ Xb3 @ B4 )
                 => ( X5
                   != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
         => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ T ) )
           => ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X5 @ D4 ) @ T ) ) ) ) ) ).

% bset(9)
thf(fact_8470_bset_I10_J,axiom,
    ! [D: int,D4: int,B4: set_int,T: int] :
      ( ( dvd_dvd_int @ D @ D4 )
     => ! [X5: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb3: int] :
                  ( ( member_int @ Xb3 @ B4 )
                 => ( X5
                   != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
         => ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ T ) )
           => ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X5 @ D4 ) @ T ) ) ) ) ) ).

% bset(10)
thf(fact_8471_aset_I9_J,axiom,
    ! [D: int,D4: int,A3: set_int,T: int] :
      ( ( dvd_dvd_int @ D @ D4 )
     => ! [X5: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb3: int] :
                  ( ( member_int @ Xb3 @ A3 )
                 => ( X5
                   != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
         => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ T ) )
           => ( dvd_dvd_int @ D @ ( plus_plus_int @ ( plus_plus_int @ X5 @ D4 ) @ T ) ) ) ) ) ).

% aset(9)
thf(fact_8472_aset_I10_J,axiom,
    ! [D: int,D4: int,A3: set_int,T: int] :
      ( ( dvd_dvd_int @ D @ D4 )
     => ! [X5: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb3: int] :
                  ( ( member_int @ Xb3 @ A3 )
                 => ( X5
                   != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
         => ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ T ) )
           => ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( plus_plus_int @ X5 @ D4 ) @ T ) ) ) ) ) ).

% aset(10)
thf(fact_8473_vebt__succ_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vc: list_VEBT_VEBT,Vd: vEBT_VEBT,Ve: nat] :
      ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vc @ Vd ) @ Ve )
      = none_nat ) ).

% vebt_succ.simps(4)
thf(fact_8474_periodic__finite__ex,axiom,
    ! [D: int,P: int > $o] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ! [X3: int,K2: int] :
            ( ( P @ X3 )
            = ( P @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D ) ) ) )
       => ( ( ? [X6: int] : ( P @ X6 ) )
          = ( ? [X: int] :
                ( ( member_int @ X @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
                & ( P @ X ) ) ) ) ) ) ).

% periodic_finite_ex
thf(fact_8475_aset_I7_J,axiom,
    ! [D4: int,A3: set_int,T: int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ! [X5: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb3: int] :
                  ( ( member_int @ Xb3 @ A3 )
                 => ( X5
                   != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
         => ( ( ord_less_int @ T @ X5 )
           => ( ord_less_int @ T @ ( plus_plus_int @ X5 @ D4 ) ) ) ) ) ).

% aset(7)
thf(fact_8476_aset_I5_J,axiom,
    ! [D4: int,T: int,A3: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ T @ A3 )
       => ! [X5: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ A3 )
                   => ( X5
                     != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( ord_less_int @ X5 @ T )
             => ( ord_less_int @ ( plus_plus_int @ X5 @ D4 ) @ T ) ) ) ) ) ).

% aset(5)
thf(fact_8477_aset_I4_J,axiom,
    ! [D4: int,T: int,A3: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ T @ A3 )
       => ! [X5: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ A3 )
                   => ( X5
                     != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( X5 != T )
             => ( ( plus_plus_int @ X5 @ D4 )
               != T ) ) ) ) ) ).

% aset(4)
thf(fact_8478_aset_I3_J,axiom,
    ! [D4: int,T: int,A3: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A3 )
       => ! [X5: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ A3 )
                   => ( X5
                     != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( X5 = T )
             => ( ( plus_plus_int @ X5 @ D4 )
                = T ) ) ) ) ) ).

% aset(3)
thf(fact_8479_bset_I7_J,axiom,
    ! [D4: int,T: int,B4: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ T @ B4 )
       => ! [X5: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ B4 )
                   => ( X5
                     != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( ord_less_int @ T @ X5 )
             => ( ord_less_int @ T @ ( minus_minus_int @ X5 @ D4 ) ) ) ) ) ) ).

% bset(7)
thf(fact_8480_bset_I5_J,axiom,
    ! [D4: int,B4: set_int,T: int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ! [X5: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb3: int] :
                  ( ( member_int @ Xb3 @ B4 )
                 => ( X5
                   != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
         => ( ( ord_less_int @ X5 @ T )
           => ( ord_less_int @ ( minus_minus_int @ X5 @ D4 ) @ T ) ) ) ) ).

% bset(5)
thf(fact_8481_bset_I4_J,axiom,
    ! [D4: int,T: int,B4: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ T @ B4 )
       => ! [X5: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ B4 )
                   => ( X5
                     != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( X5 != T )
             => ( ( minus_minus_int @ X5 @ D4 )
               != T ) ) ) ) ) ).

% bset(4)
thf(fact_8482_bset_I3_J,axiom,
    ! [D4: int,T: int,B4: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B4 )
       => ! [X5: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ B4 )
                   => ( X5
                     != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( X5 = T )
             => ( ( minus_minus_int @ X5 @ D4 )
                = T ) ) ) ) ) ).

% bset(3)
thf(fact_8483_vebt__succ_Osimps_I5_J,axiom,
    ! [V: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT,Vi: nat] :
      ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vg @ Vh ) @ Vi )
      = none_nat ) ).

% vebt_succ.simps(5)
thf(fact_8484_aset_I8_J,axiom,
    ! [D4: int,A3: set_int,T: int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ! [X5: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb3: int] :
                  ( ( member_int @ Xb3 @ A3 )
                 => ( X5
                   != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
         => ( ( ord_less_eq_int @ T @ X5 )
           => ( ord_less_eq_int @ T @ ( plus_plus_int @ X5 @ D4 ) ) ) ) ) ).

% aset(8)
thf(fact_8485_aset_I6_J,axiom,
    ! [D4: int,T: int,A3: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A3 )
       => ! [X5: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ A3 )
                   => ( X5
                     != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( ord_less_eq_int @ X5 @ T )
             => ( ord_less_eq_int @ ( plus_plus_int @ X5 @ D4 ) @ T ) ) ) ) ) ).

% aset(6)
thf(fact_8486_bset_I8_J,axiom,
    ! [D4: int,T: int,B4: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B4 )
       => ! [X5: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ B4 )
                   => ( X5
                     != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( ord_less_eq_int @ T @ X5 )
             => ( ord_less_eq_int @ T @ ( minus_minus_int @ X5 @ D4 ) ) ) ) ) ) ).

% bset(8)
thf(fact_8487_bset_I6_J,axiom,
    ! [D4: int,B4: set_int,T: int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ! [X5: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb3: int] :
                  ( ( member_int @ Xb3 @ B4 )
                 => ( X5
                   != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
         => ( ( ord_less_eq_int @ X5 @ T )
           => ( ord_less_eq_int @ ( minus_minus_int @ X5 @ D4 ) @ T ) ) ) ) ).

% bset(6)
thf(fact_8488_cpmi,axiom,
    ! [D4: int,P: int > $o,P4: int > $o,B4: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ? [Z3: int] :
          ! [X3: int] :
            ( ( ord_less_int @ X3 @ Z3 )
           => ( ( P @ X3 )
              = ( P4 @ X3 ) ) )
       => ( ! [X3: int] :
              ( ! [Xa2: int] :
                  ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
                 => ! [Xb2: int] :
                      ( ( member_int @ Xb2 @ B4 )
                     => ( X3
                       != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
             => ( ( P @ X3 )
               => ( P @ ( minus_minus_int @ X3 @ D4 ) ) ) )
         => ( ! [X3: int,K2: int] :
                ( ( P4 @ X3 )
                = ( P4 @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D4 ) ) ) )
           => ( ( ? [X6: int] : ( P @ X6 ) )
              = ( ? [X: int] :
                    ( ( member_int @ X @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
                    & ( P4 @ X ) )
                | ? [X: int] :
                    ( ( member_int @ X @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
                    & ? [Y: int] :
                        ( ( member_int @ Y @ B4 )
                        & ( P @ ( plus_plus_int @ Y @ X ) ) ) ) ) ) ) ) ) ) ).

% cpmi
thf(fact_8489_cppi,axiom,
    ! [D4: int,P: int > $o,P4: int > $o,A3: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ? [Z3: int] :
          ! [X3: int] :
            ( ( ord_less_int @ Z3 @ X3 )
           => ( ( P @ X3 )
              = ( P4 @ X3 ) ) )
       => ( ! [X3: int] :
              ( ! [Xa2: int] :
                  ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
                 => ! [Xb2: int] :
                      ( ( member_int @ Xb2 @ A3 )
                     => ( X3
                       != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
             => ( ( P @ X3 )
               => ( P @ ( plus_plus_int @ X3 @ D4 ) ) ) )
         => ( ! [X3: int,K2: int] :
                ( ( P4 @ X3 )
                = ( P4 @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D4 ) ) ) )
           => ( ( ? [X6: int] : ( P @ X6 ) )
              = ( ? [X: int] :
                    ( ( member_int @ X @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
                    & ( P4 @ X ) )
                | ? [X: int] :
                    ( ( member_int @ X @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
                    & ? [Y: int] :
                        ( ( member_int @ Y @ A3 )
                        & ( P @ ( minus_minus_int @ Y @ X ) ) ) ) ) ) ) ) ) ) ).

% cppi
thf(fact_8490__092_060open_062res_A_061_Athe_A_ISome_A_I2_A_094_A_Ideg_Adiv_A2_J_J_A_K_092_060_094sub_062o_Avebt__succ_Asummary_A_Ihigh_Ax_A_Ideg_Adiv_A2_J_J_A_L_092_060_094sub_062o_Avebt__mint_A_ItreeList_A_B_Athe_A_Ivebt__succ_Asummary_A_Ihigh_Ax_A_Ideg_Adiv_A2_J_J_J_J_J_092_060close_062,axiom,
    ( res
    = ( the_nat @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( the_nat @ ( vEBT_vebt_succ @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).

% \<open>res = the (Some (2 ^ (deg div 2)) *\<^sub>o vebt_succ summary (high x (deg div 2)) +\<^sub>o vebt_mint (treeList ! the (vebt_succ summary (high x (deg div 2)))))\<close>
thf(fact_8491_summaxma,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ Deg )
     => ( ( Mi != Ma )
       => ( ( the_nat @ ( vEBT_vebt_maxt @ Summary ) )
          = ( vEBT_VEBT_high @ Ma @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% summaxma
thf(fact_8492_maxt__sound,axiom,
    ! [T: vEBT_VEBT,N: nat,X4: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_VEBT_max_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X4 )
       => ( ( vEBT_vebt_maxt @ T )
          = ( some_nat @ X4 ) ) ) ) ).

% maxt_sound
thf(fact_8493_maxt__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X4: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_maxt @ T )
          = ( some_nat @ X4 ) )
       => ( vEBT_VEBT_max_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X4 ) ) ) ).

% maxt_corr
thf(fact_8494_add__shift,axiom,
    ! [X4: nat,Y3: nat,Z: nat] :
      ( ( ( plus_plus_nat @ X4 @ Y3 )
        = Z )
      = ( ( vEBT_VEBT_add @ ( some_nat @ X4 ) @ ( some_nat @ Y3 ) )
        = ( some_nat @ Z ) ) ) ).

% add_shift
thf(fact_8495_mul__shift,axiom,
    ! [X4: nat,Y3: nat,Z: nat] :
      ( ( ( times_times_nat @ X4 @ Y3 )
        = Z )
      = ( ( vEBT_VEBT_mul @ ( some_nat @ X4 ) @ ( some_nat @ Y3 ) )
        = ( some_nat @ Z ) ) ) ).

% mul_shift
thf(fact_8496_maxbmo,axiom,
    ! [T: vEBT_VEBT,X4: nat] :
      ( ( ( vEBT_vebt_maxt @ T )
        = ( some_nat @ X4 ) )
     => ( vEBT_V8194947554948674370ptions @ T @ X4 ) ) ).

% maxbmo
thf(fact_8497_add__def,axiom,
    ( vEBT_VEBT_add
    = ( vEBT_V4262088993061758097ft_nat @ plus_plus_nat ) ) ).

% add_def
thf(fact_8498_mul__def,axiom,
    ( vEBT_VEBT_mul
    = ( vEBT_V4262088993061758097ft_nat @ times_times_nat ) ) ).

% mul_def
thf(fact_8499_maxt__member,axiom,
    ! [T: vEBT_VEBT,N: nat,Maxi: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_maxt @ T )
          = ( some_nat @ Maxi ) )
       => ( vEBT_vebt_member @ T @ Maxi ) ) ) ).

% maxt_member
thf(fact_8500_maxt__corr__help,axiom,
    ! [T: vEBT_VEBT,N: nat,Maxi: nat,X4: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_maxt @ T )
          = ( some_nat @ Maxi ) )
       => ( ( vEBT_vebt_member @ T @ X4 )
         => ( ord_less_eq_nat @ X4 @ Maxi ) ) ) ) ).

% maxt_corr_help
thf(fact_8501_maxt__corr__help__empty,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_maxt @ T )
          = none_nat )
       => ( ( vEBT_VEBT_set_vebt @ T )
          = bot_bot_set_nat ) ) ) ).

% maxt_corr_help_empty
thf(fact_8502__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_J_A_K_092_060_094sub_062o_Avebt__succ_Asummary_A_Ihigh_Ax_A_Ideg_Adiv_A2_J_J_A_L_092_060_094sub_062o_Avebt__mint_A_ItreeList_A_B_Athe_A_Ivebt__succ_Asummary_A_Ihigh_Ax_A_Ideg_Adiv_A2_J_J_J_J_092_060close_062,axiom,
    ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ xa )
    = ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( the_nat @ ( vEBT_vebt_succ @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% \<open>vebt_succ (Node (Some (mi, ma)) deg treeList summary) x = Some (2 ^ (deg div 2)) *\<^sub>o vebt_succ summary (high x (deg div 2)) +\<^sub>o vebt_mint (treeList ! the (vebt_succ summary (high x (deg div 2))))\<close>
thf(fact_8503__C2_C,axiom,
    ( ( ( ( vEBT_vebt_succ @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
        = none_nat )
     => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ xa )
        = none_nat ) )
    & ( ( ( vEBT_vebt_succ @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
       != none_nat )
     => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ xa )
        = ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( the_nat @ ( vEBT_vebt_succ @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ).

% "2"
thf(fact_8504__092_060open_062vebt__member_A_INode_A_ISome_A_Imi_M_Ama_J_J_Adeg_AtreeList_Asummary_J_A_Ithe_A_ISome_A_I2_A_094_A_Ideg_Adiv_A2_J_J_A_K_092_060_094sub_062o_Avebt__succ_Asummary_A_Ihigh_Ax_A_Ideg_Adiv_A2_J_J_A_L_092_060_094sub_062o_Avebt__mint_A_ItreeList_A_B_Athe_A_Ivebt__succ_Asummary_A_Ihigh_Ax_A_Ideg_Adiv_A2_J_J_J_J_J_J_092_060close_062,axiom,
    vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ ( the_nat @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( the_nat @ ( vEBT_vebt_succ @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% \<open>vebt_member (Node (Some (mi, ma)) deg treeList summary) (the (Some (2 ^ (deg div 2)) *\<^sub>o vebt_succ summary (high x (deg div 2)) +\<^sub>o vebt_mint (treeList ! the (vebt_succ summary (high x (deg div 2))))))\<close>
thf(fact_8505__C1_C,axiom,
    ( ( ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
         != none_nat )
        & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
     => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ xa )
        = ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
    & ( ~ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
           != none_nat )
          & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
     => ( ( ( ( vEBT_vebt_succ @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
            = none_nat )
         => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ xa )
            = none_nat ) )
        & ( ( ( vEBT_vebt_succ @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
           != none_nat )
         => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ xa )
            = ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( the_nat @ ( vEBT_vebt_succ @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% "1"
thf(fact_8506_i1,axiom,
    ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
      = none_nat )
    | ~ ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% i1
thf(fact_8507_divmod__step__eq,axiom,
    ! [L2: num,R2: nat,Q2: nat] :
      ( ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ L2 ) @ R2 )
       => ( ( unique5026877609467782581ep_nat @ L2 @ ( product_Pair_nat_nat @ Q2 @ R2 ) )
          = ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q2 ) @ one_one_nat ) @ ( minus_minus_nat @ R2 @ ( numeral_numeral_nat @ L2 ) ) ) ) )
      & ( ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L2 ) @ R2 )
       => ( ( unique5026877609467782581ep_nat @ L2 @ ( product_Pair_nat_nat @ Q2 @ R2 ) )
          = ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q2 ) @ R2 ) ) ) ) ).

% divmod_step_eq
thf(fact_8508_divmod__step__eq,axiom,
    ! [L2: num,R2: int,Q2: int] :
      ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ L2 ) @ R2 )
       => ( ( unique5024387138958732305ep_int @ L2 @ ( product_Pair_int_int @ Q2 @ R2 ) )
          = ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q2 ) @ one_one_int ) @ ( minus_minus_int @ R2 @ ( numeral_numeral_int @ L2 ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ L2 ) @ R2 )
       => ( ( unique5024387138958732305ep_int @ L2 @ ( product_Pair_int_int @ Q2 @ R2 ) )
          = ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q2 ) @ R2 ) ) ) ) ).

% divmod_step_eq
thf(fact_8509_divmod__step__eq,axiom,
    ! [L2: num,R2: code_integer,Q2: code_integer] :
      ( ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L2 ) @ R2 )
       => ( ( unique4921790084139445826nteger @ L2 @ ( produc1086072967326762835nteger @ Q2 @ R2 ) )
          = ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q2 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R2 @ ( numera6620942414471956472nteger @ L2 ) ) ) ) )
      & ( ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L2 ) @ R2 )
       => ( ( unique4921790084139445826nteger @ L2 @ ( produc1086072967326762835nteger @ Q2 @ R2 ) )
          = ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q2 ) @ R2 ) ) ) ) ).

% divmod_step_eq
thf(fact_8510_less__shift,axiom,
    ( ord_less_nat
    = ( ^ [X: nat,Y: nat] : ( vEBT_VEBT_less @ ( some_nat @ X ) @ ( some_nat @ Y ) ) ) ) ).

% less_shift
thf(fact_8511_fold__atLeastAtMost__nat_Ocases,axiom,
    ! [X4: produc3368934014287244435at_num] :
      ~ ! [F3: nat > num > num,A5: nat,B5: nat,Acc2: num] :
          ( X4
         != ( produc851828971589881931at_num @ F3 @ ( produc1195630363706982562at_num @ A5 @ ( product_Pair_nat_num @ B5 @ Acc2 ) ) ) ) ).

% fold_atLeastAtMost_nat.cases
thf(fact_8512_fold__atLeastAtMost__nat_Ocases,axiom,
    ! [X4: produc4471711990508489141at_nat] :
      ~ ! [F3: nat > nat > nat,A5: nat,B5: nat,Acc2: nat] :
          ( X4
         != ( produc3209952032786966637at_nat @ F3 @ ( produc487386426758144856at_nat @ A5 @ ( product_Pair_nat_nat @ B5 @ Acc2 ) ) ) ) ).

% fold_atLeastAtMost_nat.cases
thf(fact_8513_xor__num_Ocases,axiom,
    ! [X4: product_prod_num_num] :
      ( ( X4
       != ( product_Pair_num_num @ one @ one ) )
     => ( ! [N4: num] :
            ( X4
           != ( product_Pair_num_num @ one @ ( bit0 @ N4 ) ) )
       => ( ! [N4: num] :
              ( X4
             != ( product_Pair_num_num @ one @ ( bit1 @ N4 ) ) )
         => ( ! [M4: num] :
                ( X4
               != ( product_Pair_num_num @ ( bit0 @ M4 ) @ one ) )
           => ( ! [M4: num,N4: num] :
                  ( X4
                 != ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit0 @ N4 ) ) )
             => ( ! [M4: num,N4: num] :
                    ( X4
                   != ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit1 @ N4 ) ) )
               => ( ! [M4: num] :
                      ( X4
                     != ( product_Pair_num_num @ ( bit1 @ M4 ) @ one ) )
                 => ( ! [M4: num,N4: num] :
                        ( X4
                       != ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit0 @ N4 ) ) )
                   => ~ ! [M4: num,N4: num] :
                          ( X4
                         != ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit1 @ N4 ) ) ) ) ) ) ) ) ) ) ) ).

% xor_num.cases
thf(fact_8514_minNullmin,axiom,
    ! [T: vEBT_VEBT] :
      ( ( vEBT_VEBT_minNull @ T )
     => ( ( vEBT_vebt_mint @ T )
        = none_nat ) ) ).

% minNullmin
thf(fact_8515_minminNull,axiom,
    ! [T: vEBT_VEBT] :
      ( ( ( vEBT_vebt_mint @ T )
        = none_nat )
     => ( vEBT_VEBT_minNull @ T ) ) ).

% minminNull
thf(fact_8516_vebt__member_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT,X4: nat] :
      ~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vb @ Vc ) @ X4 ) ).

% vebt_member.simps(4)
thf(fact_8517_succ__less__length__list,axiom,
    ! [Deg: nat,Mi: nat,X4: nat,TreeList2: list_VEBT_VEBT,Ma: nat,Summary: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ( ord_less_eq_nat @ Mi @ X4 )
       => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
         => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X4 )
            = ( if_option_nat
              @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
              @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
              @ ( if_option_nat
                @ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                  = none_nat )
                @ none_nat
                @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% succ_less_length_list
thf(fact_8518_not__min__Null__member,axiom,
    ! [T: vEBT_VEBT] :
      ( ~ ( vEBT_VEBT_minNull @ T )
     => ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ T @ X_1 ) ) ).

% not_min_Null_member
thf(fact_8519_min__Null__member,axiom,
    ! [T: vEBT_VEBT,X4: nat] :
      ( ( vEBT_VEBT_minNull @ T )
     => ~ ( vEBT_vebt_member @ T @ X4 ) ) ).

% min_Null_member
thf(fact_8520_set__vebt_H__def,axiom,
    ( vEBT_VEBT_set_vebt
    = ( ^ [T3: vEBT_VEBT] : ( collect_nat @ ( vEBT_vebt_member @ T3 ) ) ) ) ).

% set_vebt'_def
thf(fact_8521_succ__greatereq__min,axiom,
    ! [Deg: nat,Mi: nat,X4: nat,Ma: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ( ord_less_eq_nat @ Mi @ X4 )
       => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X4 )
          = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
            @ ( if_option_nat
              @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
              @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
              @ ( if_option_nat
                @ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                  = none_nat )
                @ none_nat
                @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
            @ none_nat ) ) ) ) ).

% succ_greatereq_min
thf(fact_8522_nat__less__as__int,axiom,
    ( ord_less_nat
    = ( ^ [A4: nat,B3: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).

% nat_less_as_int
thf(fact_8523_less__set__def,axiom,
    ( ord_le2529575680413868914d_enat
    = ( ^ [A6: set_Extended_enat,B7: set_Extended_enat] :
          ( ord_le8499522857272258027enat_o
          @ ^ [X: extended_enat] : ( member_Extended_enat @ X @ A6 )
          @ ^ [X: extended_enat] : ( member_Extended_enat @ X @ B7 ) ) ) ) ).

% less_set_def
thf(fact_8524_less__set__def,axiom,
    ( ord_less_set_complex
    = ( ^ [A6: set_complex,B7: set_complex] :
          ( ord_less_complex_o
          @ ^ [X: complex] : ( member_complex @ X @ A6 )
          @ ^ [X: complex] : ( member_complex @ X @ B7 ) ) ) ) ).

% less_set_def
thf(fact_8525_less__set__def,axiom,
    ( ord_less_set_real
    = ( ^ [A6: set_real,B7: set_real] :
          ( ord_less_real_o
          @ ^ [X: real] : ( member_real @ X @ A6 )
          @ ^ [X: real] : ( member_real @ X @ B7 ) ) ) ) ).

% less_set_def
thf(fact_8526_less__set__def,axiom,
    ( ord_less_set_nat
    = ( ^ [A6: set_nat,B7: set_nat] :
          ( ord_less_nat_o
          @ ^ [X: nat] : ( member_nat @ X @ A6 )
          @ ^ [X: nat] : ( member_nat @ X @ B7 ) ) ) ) ).

% less_set_def
thf(fact_8527_less__set__def,axiom,
    ( ord_less_set_int
    = ( ^ [A6: set_int,B7: set_int] :
          ( ord_less_int_o
          @ ^ [X: int] : ( member_int @ X @ A6 )
          @ ^ [X: int] : ( member_int @ X @ B7 ) ) ) ) ).

% less_set_def
thf(fact_8528_strict__subset__divisors__dvd,axiom,
    ! [A: complex,B2: complex] :
      ( ( ord_less_set_complex
        @ ( collect_complex
          @ ^ [C2: complex] : ( dvd_dvd_complex @ C2 @ A ) )
        @ ( collect_complex
          @ ^ [C2: complex] : ( dvd_dvd_complex @ C2 @ B2 ) ) )
      = ( ( dvd_dvd_complex @ A @ B2 )
        & ~ ( dvd_dvd_complex @ B2 @ A ) ) ) ).

% strict_subset_divisors_dvd
thf(fact_8529_strict__subset__divisors__dvd,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_set_real
        @ ( collect_real
          @ ^ [C2: real] : ( dvd_dvd_real @ C2 @ A ) )
        @ ( collect_real
          @ ^ [C2: real] : ( dvd_dvd_real @ C2 @ B2 ) ) )
      = ( ( dvd_dvd_real @ A @ B2 )
        & ~ ( dvd_dvd_real @ B2 @ A ) ) ) ).

% strict_subset_divisors_dvd
thf(fact_8530_strict__subset__divisors__dvd,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_set_nat
        @ ( collect_nat
          @ ^ [C2: nat] : ( dvd_dvd_nat @ C2 @ A ) )
        @ ( collect_nat
          @ ^ [C2: nat] : ( dvd_dvd_nat @ C2 @ B2 ) ) )
      = ( ( dvd_dvd_nat @ A @ B2 )
        & ~ ( dvd_dvd_nat @ B2 @ A ) ) ) ).

% strict_subset_divisors_dvd
thf(fact_8531_strict__subset__divisors__dvd,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_set_int
        @ ( collect_int
          @ ^ [C2: int] : ( dvd_dvd_int @ C2 @ A ) )
        @ ( collect_int
          @ ^ [C2: int] : ( dvd_dvd_int @ C2 @ B2 ) ) )
      = ( ( dvd_dvd_int @ A @ B2 )
        & ~ ( dvd_dvd_int @ B2 @ A ) ) ) ).

% strict_subset_divisors_dvd
thf(fact_8532_strict__subset__divisors__dvd,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( ord_le1307284697595431911nteger
        @ ( collect_Code_integer
          @ ^ [C2: code_integer] : ( dvd_dvd_Code_integer @ C2 @ A ) )
        @ ( collect_Code_integer
          @ ^ [C2: code_integer] : ( dvd_dvd_Code_integer @ C2 @ B2 ) ) )
      = ( ( dvd_dvd_Code_integer @ A @ B2 )
        & ~ ( dvd_dvd_Code_integer @ B2 @ A ) ) ) ).

% strict_subset_divisors_dvd
thf(fact_8533_set__diff__eq,axiom,
    ( minus_925952699566721837d_enat
    = ( ^ [A6: set_Extended_enat,B7: set_Extended_enat] :
          ( collec4429806609662206161d_enat
          @ ^ [X: extended_enat] :
              ( ( member_Extended_enat @ X @ A6 )
              & ~ ( member_Extended_enat @ X @ B7 ) ) ) ) ) ).

% set_diff_eq
thf(fact_8534_set__diff__eq,axiom,
    ( minus_811609699411566653omplex
    = ( ^ [A6: set_complex,B7: set_complex] :
          ( collect_complex
          @ ^ [X: complex] :
              ( ( member_complex @ X @ A6 )
              & ~ ( member_complex @ X @ B7 ) ) ) ) ) ).

% set_diff_eq
thf(fact_8535_set__diff__eq,axiom,
    ( minus_minus_set_real
    = ( ^ [A6: set_real,B7: set_real] :
          ( collect_real
          @ ^ [X: real] :
              ( ( member_real @ X @ A6 )
              & ~ ( member_real @ X @ B7 ) ) ) ) ) ).

% set_diff_eq
thf(fact_8536_set__diff__eq,axiom,
    ( minus_7954133019191499631st_nat
    = ( ^ [A6: set_list_nat,B7: set_list_nat] :
          ( collect_list_nat
          @ ^ [X: list_nat] :
              ( ( member_list_nat @ X @ A6 )
              & ~ ( member_list_nat @ X @ B7 ) ) ) ) ) ).

% set_diff_eq
thf(fact_8537_set__diff__eq,axiom,
    ( minus_minus_set_int
    = ( ^ [A6: set_int,B7: set_int] :
          ( collect_int
          @ ^ [X: int] :
              ( ( member_int @ X @ A6 )
              & ~ ( member_int @ X @ B7 ) ) ) ) ) ).

% set_diff_eq
thf(fact_8538_set__diff__eq,axiom,
    ( minus_minus_set_nat
    = ( ^ [A6: set_nat,B7: set_nat] :
          ( collect_nat
          @ ^ [X: nat] :
              ( ( member_nat @ X @ A6 )
              & ~ ( member_nat @ X @ B7 ) ) ) ) ) ).

% set_diff_eq
thf(fact_8539_minus__set__def,axiom,
    ( minus_925952699566721837d_enat
    = ( ^ [A6: set_Extended_enat,B7: set_Extended_enat] :
          ( collec4429806609662206161d_enat
          @ ( minus_2020553357622893040enat_o
            @ ^ [X: extended_enat] : ( member_Extended_enat @ X @ A6 )
            @ ^ [X: extended_enat] : ( member_Extended_enat @ X @ B7 ) ) ) ) ) ).

% minus_set_def
thf(fact_8540_minus__set__def,axiom,
    ( minus_811609699411566653omplex
    = ( ^ [A6: set_complex,B7: set_complex] :
          ( collect_complex
          @ ( minus_8727706125548526216plex_o
            @ ^ [X: complex] : ( member_complex @ X @ A6 )
            @ ^ [X: complex] : ( member_complex @ X @ B7 ) ) ) ) ) ).

% minus_set_def
thf(fact_8541_minus__set__def,axiom,
    ( minus_minus_set_real
    = ( ^ [A6: set_real,B7: set_real] :
          ( collect_real
          @ ( minus_minus_real_o
            @ ^ [X: real] : ( member_real @ X @ A6 )
            @ ^ [X: real] : ( member_real @ X @ B7 ) ) ) ) ) ).

% minus_set_def
thf(fact_8542_minus__set__def,axiom,
    ( minus_7954133019191499631st_nat
    = ( ^ [A6: set_list_nat,B7: set_list_nat] :
          ( collect_list_nat
          @ ( minus_1139252259498527702_nat_o
            @ ^ [X: list_nat] : ( member_list_nat @ X @ A6 )
            @ ^ [X: list_nat] : ( member_list_nat @ X @ B7 ) ) ) ) ) ).

% minus_set_def
thf(fact_8543_minus__set__def,axiom,
    ( minus_minus_set_int
    = ( ^ [A6: set_int,B7: set_int] :
          ( collect_int
          @ ( minus_minus_int_o
            @ ^ [X: int] : ( member_int @ X @ A6 )
            @ ^ [X: int] : ( member_int @ X @ B7 ) ) ) ) ) ).

% minus_set_def
thf(fact_8544_minus__set__def,axiom,
    ( minus_minus_set_nat
    = ( ^ [A6: set_nat,B7: set_nat] :
          ( collect_nat
          @ ( minus_minus_nat_o
            @ ^ [X: nat] : ( member_nat @ X @ A6 )
            @ ^ [X: nat] : ( member_nat @ X @ B7 ) ) ) ) ) ).

% minus_set_def
thf(fact_8545_numeral__code_I2_J,axiom,
    ! [N: num] :
      ( ( numera6690914467698888265omplex @ ( bit0 @ N ) )
      = ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ ( numera6690914467698888265omplex @ N ) ) ) ).

% numeral_code(2)
thf(fact_8546_numeral__code_I2_J,axiom,
    ! [N: num] :
      ( ( numeral_numeral_real @ ( bit0 @ N ) )
      = ( plus_plus_real @ ( numeral_numeral_real @ N ) @ ( numeral_numeral_real @ N ) ) ) ).

% numeral_code(2)
thf(fact_8547_numeral__code_I2_J,axiom,
    ! [N: num] :
      ( ( numeral_numeral_rat @ ( bit0 @ N ) )
      = ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ ( numeral_numeral_rat @ N ) ) ) ).

% numeral_code(2)
thf(fact_8548_numeral__code_I2_J,axiom,
    ! [N: num] :
      ( ( numeral_numeral_nat @ ( bit0 @ N ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) ) ).

% numeral_code(2)
thf(fact_8549_numeral__code_I2_J,axiom,
    ! [N: num] :
      ( ( numeral_numeral_int @ ( bit0 @ N ) )
      = ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) ) ).

% numeral_code(2)
thf(fact_8550_lambda__zero,axiom,
    ( ( ^ [H: real] : zero_zero_real )
    = ( times_times_real @ zero_zero_real ) ) ).

% lambda_zero
thf(fact_8551_lambda__zero,axiom,
    ( ( ^ [H: rat] : zero_zero_rat )
    = ( times_times_rat @ zero_zero_rat ) ) ).

% lambda_zero
thf(fact_8552_lambda__zero,axiom,
    ( ( ^ [H: nat] : zero_zero_nat )
    = ( times_times_nat @ zero_zero_nat ) ) ).

% lambda_zero
thf(fact_8553_lambda__zero,axiom,
    ( ( ^ [H: int] : zero_zero_int )
    = ( times_times_int @ zero_zero_int ) ) ).

% lambda_zero
thf(fact_8554_mult__commute__abs,axiom,
    ! [C: real] :
      ( ( ^ [X: real] : ( times_times_real @ X @ C ) )
      = ( times_times_real @ C ) ) ).

% mult_commute_abs
thf(fact_8555_mult__commute__abs,axiom,
    ! [C: rat] :
      ( ( ^ [X: rat] : ( times_times_rat @ X @ C ) )
      = ( times_times_rat @ C ) ) ).

% mult_commute_abs
thf(fact_8556_mult__commute__abs,axiom,
    ! [C: nat] :
      ( ( ^ [X: nat] : ( times_times_nat @ X @ C ) )
      = ( times_times_nat @ C ) ) ).

% mult_commute_abs
thf(fact_8557_mult__commute__abs,axiom,
    ! [C: int] :
      ( ( ^ [X: int] : ( times_times_int @ X @ C ) )
      = ( times_times_int @ C ) ) ).

% mult_commute_abs
thf(fact_8558_lambda__one,axiom,
    ( ( ^ [X: complex] : X )
    = ( times_times_complex @ one_one_complex ) ) ).

% lambda_one
thf(fact_8559_lambda__one,axiom,
    ( ( ^ [X: real] : X )
    = ( times_times_real @ one_one_real ) ) ).

% lambda_one
thf(fact_8560_lambda__one,axiom,
    ( ( ^ [X: rat] : X )
    = ( times_times_rat @ one_one_rat ) ) ).

% lambda_one
thf(fact_8561_lambda__one,axiom,
    ( ( ^ [X: nat] : X )
    = ( times_times_nat @ one_one_nat ) ) ).

% lambda_one
thf(fact_8562_lambda__one,axiom,
    ( ( ^ [X: int] : X )
    = ( times_times_int @ one_one_int ) ) ).

% lambda_one
thf(fact_8563_subset__divisors__dvd,axiom,
    ! [A: complex,B2: complex] :
      ( ( ord_le211207098394363844omplex
        @ ( collect_complex
          @ ^ [C2: complex] : ( dvd_dvd_complex @ C2 @ A ) )
        @ ( collect_complex
          @ ^ [C2: complex] : ( dvd_dvd_complex @ C2 @ B2 ) ) )
      = ( dvd_dvd_complex @ A @ B2 ) ) ).

% subset_divisors_dvd
thf(fact_8564_subset__divisors__dvd,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_set_real
        @ ( collect_real
          @ ^ [C2: real] : ( dvd_dvd_real @ C2 @ A ) )
        @ ( collect_real
          @ ^ [C2: real] : ( dvd_dvd_real @ C2 @ B2 ) ) )
      = ( dvd_dvd_real @ A @ B2 ) ) ).

% subset_divisors_dvd
thf(fact_8565_subset__divisors__dvd,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_set_int
        @ ( collect_int
          @ ^ [C2: int] : ( dvd_dvd_int @ C2 @ A ) )
        @ ( collect_int
          @ ^ [C2: int] : ( dvd_dvd_int @ C2 @ B2 ) ) )
      = ( dvd_dvd_int @ A @ B2 ) ) ).

% subset_divisors_dvd
thf(fact_8566_subset__divisors__dvd,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( ord_le7084787975880047091nteger
        @ ( collect_Code_integer
          @ ^ [C2: code_integer] : ( dvd_dvd_Code_integer @ C2 @ A ) )
        @ ( collect_Code_integer
          @ ^ [C2: code_integer] : ( dvd_dvd_Code_integer @ C2 @ B2 ) ) )
      = ( dvd_dvd_Code_integer @ A @ B2 ) ) ).

% subset_divisors_dvd
thf(fact_8567_subset__divisors__dvd,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_eq_set_nat
        @ ( collect_nat
          @ ^ [C2: nat] : ( dvd_dvd_nat @ C2 @ A ) )
        @ ( collect_nat
          @ ^ [C2: nat] : ( dvd_dvd_nat @ C2 @ B2 ) ) )
      = ( dvd_dvd_nat @ A @ B2 ) ) ).

% subset_divisors_dvd
thf(fact_8568_nat__leq__as__int,axiom,
    ( ord_less_eq_nat
    = ( ^ [A4: nat,B3: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).

% nat_leq_as_int
thf(fact_8569_power__numeral__even,axiom,
    ! [Z: complex,W: num] :
      ( ( power_power_complex @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
      = ( times_times_complex @ ( power_power_complex @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_complex @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_even
thf(fact_8570_power__numeral__even,axiom,
    ! [Z: real,W: num] :
      ( ( power_power_real @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
      = ( times_times_real @ ( power_power_real @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_real @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_even
thf(fact_8571_power__numeral__even,axiom,
    ! [Z: rat,W: num] :
      ( ( power_power_rat @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
      = ( times_times_rat @ ( power_power_rat @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_rat @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_even
thf(fact_8572_power__numeral__even,axiom,
    ! [Z: nat,W: num] :
      ( ( power_power_nat @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
      = ( times_times_nat @ ( power_power_nat @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_nat @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_even
thf(fact_8573_power__numeral__even,axiom,
    ! [Z: int,W: num] :
      ( ( power_power_int @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
      = ( times_times_int @ ( power_power_int @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_int @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_even
thf(fact_8574_numeral__code_I3_J,axiom,
    ! [N: num] :
      ( ( numera6690914467698888265omplex @ ( bit1 @ N ) )
      = ( plus_plus_complex @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ ( numera6690914467698888265omplex @ N ) ) @ one_one_complex ) ) ).

% numeral_code(3)
thf(fact_8575_numeral__code_I3_J,axiom,
    ! [N: num] :
      ( ( numeral_numeral_real @ ( bit1 @ N ) )
      = ( plus_plus_real @ ( plus_plus_real @ ( numeral_numeral_real @ N ) @ ( numeral_numeral_real @ N ) ) @ one_one_real ) ) ).

% numeral_code(3)
thf(fact_8576_numeral__code_I3_J,axiom,
    ! [N: num] :
      ( ( numeral_numeral_rat @ ( bit1 @ N ) )
      = ( plus_plus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ ( numeral_numeral_rat @ N ) ) @ one_one_rat ) ) ).

% numeral_code(3)
thf(fact_8577_numeral__code_I3_J,axiom,
    ! [N: num] :
      ( ( numeral_numeral_nat @ ( bit1 @ N ) )
      = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) @ one_one_nat ) ) ).

% numeral_code(3)
thf(fact_8578_numeral__code_I3_J,axiom,
    ! [N: num] :
      ( ( numeral_numeral_int @ ( bit1 @ N ) )
      = ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) @ one_one_int ) ) ).

% numeral_code(3)
thf(fact_8579_power__numeral__odd,axiom,
    ! [Z: complex,W: num] :
      ( ( power_power_complex @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
      = ( times_times_complex @ ( times_times_complex @ Z @ ( power_power_complex @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_complex @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_odd
thf(fact_8580_power__numeral__odd,axiom,
    ! [Z: real,W: num] :
      ( ( power_power_real @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
      = ( times_times_real @ ( times_times_real @ Z @ ( power_power_real @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_real @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_odd
thf(fact_8581_power__numeral__odd,axiom,
    ! [Z: rat,W: num] :
      ( ( power_power_rat @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
      = ( times_times_rat @ ( times_times_rat @ Z @ ( power_power_rat @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_rat @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_odd
thf(fact_8582_power__numeral__odd,axiom,
    ! [Z: nat,W: num] :
      ( ( power_power_nat @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
      = ( times_times_nat @ ( times_times_nat @ Z @ ( power_power_nat @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_nat @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_odd
thf(fact_8583_power__numeral__odd,axiom,
    ! [Z: int,W: num] :
      ( ( power_power_int @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
      = ( times_times_int @ ( times_times_int @ Z @ ( power_power_int @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_int @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_odd
thf(fact_8584_nat__plus__as__int,axiom,
    ( plus_plus_nat
    = ( ^ [A4: nat,B3: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ) ).

% nat_plus_as_int
thf(fact_8585_nat__times__as__int,axiom,
    ( times_times_nat
    = ( ^ [A4: nat,B3: nat] : ( nat2 @ ( times_times_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ) ).

% nat_times_as_int
thf(fact_8586_nat__minus__as__int,axiom,
    ( minus_minus_nat
    = ( ^ [A4: nat,B3: nat] : ( nat2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ) ).

% nat_minus_as_int
thf(fact_8587_nat__div__as__int,axiom,
    ( divide_divide_nat
    = ( ^ [A4: nat,B3: nat] : ( nat2 @ ( divide_divide_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ) ).

% nat_div_as_int
thf(fact_8588_nat__mod__as__int,axiom,
    ( modulo_modulo_nat
    = ( ^ [A4: nat,B3: nat] : ( nat2 @ ( modulo_modulo_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ) ).

% nat_mod_as_int
thf(fact_8589_set__conv__nth,axiom,
    ( set_complex2
    = ( ^ [Xs: list_complex] :
          ( collect_complex
          @ ^ [Uu: complex] :
            ? [I2: nat] :
              ( ( Uu
                = ( nth_complex @ Xs @ I2 ) )
              & ( ord_less_nat @ I2 @ ( size_s3451745648224563538omplex @ Xs ) ) ) ) ) ) ).

% set_conv_nth
thf(fact_8590_set__conv__nth,axiom,
    ( set_real2
    = ( ^ [Xs: list_real] :
          ( collect_real
          @ ^ [Uu: real] :
            ? [I2: nat] :
              ( ( Uu
                = ( nth_real @ Xs @ I2 ) )
              & ( ord_less_nat @ I2 @ ( size_size_list_real @ Xs ) ) ) ) ) ) ).

% set_conv_nth
thf(fact_8591_set__conv__nth,axiom,
    ( set_list_nat2
    = ( ^ [Xs: list_list_nat] :
          ( collect_list_nat
          @ ^ [Uu: list_nat] :
            ? [I2: nat] :
              ( ( Uu
                = ( nth_list_nat @ Xs @ I2 ) )
              & ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Xs ) ) ) ) ) ) ).

% set_conv_nth
thf(fact_8592_set__conv__nth,axiom,
    ( set_VEBT_VEBT2
    = ( ^ [Xs: list_VEBT_VEBT] :
          ( collect_VEBT_VEBT
          @ ^ [Uu: vEBT_VEBT] :
            ? [I2: nat] :
              ( ( Uu
                = ( nth_VEBT_VEBT @ Xs @ I2 ) )
              & ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs ) ) ) ) ) ) ).

% set_conv_nth
thf(fact_8593_set__conv__nth,axiom,
    ( set_o2
    = ( ^ [Xs: list_o] :
          ( collect_o
          @ ^ [Uu: $o] :
            ? [I2: nat] :
              ( ( Uu
                = ( nth_o @ Xs @ I2 ) )
              & ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs ) ) ) ) ) ) ).

% set_conv_nth
thf(fact_8594_set__conv__nth,axiom,
    ( set_nat2
    = ( ^ [Xs: list_nat] :
          ( collect_nat
          @ ^ [Uu: nat] :
            ? [I2: nat] :
              ( ( Uu
                = ( nth_nat @ Xs @ I2 ) )
              & ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) ) ) ) ) ) ).

% set_conv_nth
thf(fact_8595_set__conv__nth,axiom,
    ( set_int2
    = ( ^ [Xs: list_int] :
          ( collect_int
          @ ^ [Uu: int] :
            ? [I2: nat] :
              ( ( Uu
                = ( nth_int @ Xs @ I2 ) )
              & ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs ) ) ) ) ) ) ).

% set_conv_nth
thf(fact_8596_diff__nat__eq__if,axiom,
    ! [Z6: int,Z: int] :
      ( ( ( ord_less_int @ Z6 @ zero_zero_int )
       => ( ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z6 ) )
          = ( nat2 @ Z ) ) )
      & ( ~ ( ord_less_int @ Z6 @ zero_zero_int )
       => ( ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z6 ) )
          = ( if_nat @ ( ord_less_int @ ( minus_minus_int @ Z @ Z6 ) @ zero_zero_int ) @ zero_zero_nat @ ( nat2 @ ( minus_minus_int @ Z @ Z6 ) ) ) ) ) ) ).

% diff_nat_eq_if
thf(fact_8597_set__decode__def,axiom,
    ( nat_set_decode
    = ( ^ [X: nat] :
          ( collect_nat
          @ ^ [N2: nat] :
              ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ) ).

% set_decode_def
thf(fact_8598_signed__take__bit__code,axiom,
    ( bit_ri6519982836138164636nteger
    = ( ^ [N2: nat,A4: code_integer] : ( if_Code_integer @ ( bit_se9216721137139052372nteger @ ( bit_se1745604003318907178nteger @ ( suc @ N2 ) @ A4 ) @ N2 ) @ ( plus_p5714425477246183910nteger @ ( bit_se1745604003318907178nteger @ ( suc @ N2 ) @ A4 ) @ ( bit_se7788150548672797655nteger @ ( suc @ N2 ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) @ ( bit_se1745604003318907178nteger @ ( suc @ N2 ) @ A4 ) ) ) ) ).

% signed_take_bit_code
thf(fact_8599_signed__take__bit__code,axiom,
    ( bit_ri631733984087533419it_int
    = ( ^ [N2: nat,A4: int] : ( if_int @ ( bit_se1146084159140164899it_int @ ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ A4 ) @ N2 ) @ ( plus_plus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ A4 ) @ ( bit_se545348938243370406it_int @ ( suc @ N2 ) @ ( uminus_uminus_int @ one_one_int ) ) ) @ ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ A4 ) ) ) ) ).

% signed_take_bit_code
thf(fact_8600_pochhammer__code,axiom,
    ( comm_s2602460028002588243omplex
    = ( ^ [A4: complex,N2: nat] :
          ( if_complex @ ( N2 = zero_zero_nat ) @ one_one_complex
          @ ( set_fo1517530859248394432omplex
            @ ^ [O: nat] : ( times_times_complex @ ( plus_plus_complex @ A4 @ ( semiri8010041392384452111omplex @ O ) ) )
            @ zero_zero_nat
            @ ( minus_minus_nat @ N2 @ one_one_nat )
            @ one_one_complex ) ) ) ) ).

% pochhammer_code
thf(fact_8601_pochhammer__code,axiom,
    ( comm_s7457072308508201937r_real
    = ( ^ [A4: real,N2: nat] :
          ( if_real @ ( N2 = zero_zero_nat ) @ one_one_real
          @ ( set_fo3111899725591712190t_real
            @ ^ [O: nat] : ( times_times_real @ ( plus_plus_real @ A4 @ ( semiri5074537144036343181t_real @ O ) ) )
            @ zero_zero_nat
            @ ( minus_minus_nat @ N2 @ one_one_nat )
            @ one_one_real ) ) ) ) ).

% pochhammer_code
thf(fact_8602_pochhammer__code,axiom,
    ( comm_s4028243227959126397er_rat
    = ( ^ [A4: rat,N2: nat] :
          ( if_rat @ ( N2 = zero_zero_nat ) @ one_one_rat
          @ ( set_fo1949268297981939178at_rat
            @ ^ [O: nat] : ( times_times_rat @ ( plus_plus_rat @ A4 @ ( semiri681578069525770553at_rat @ O ) ) )
            @ zero_zero_nat
            @ ( minus_minus_nat @ N2 @ one_one_nat )
            @ one_one_rat ) ) ) ) ).

% pochhammer_code
thf(fact_8603_pochhammer__code,axiom,
    ( comm_s4660882817536571857er_int
    = ( ^ [A4: int,N2: nat] :
          ( if_int @ ( N2 = zero_zero_nat ) @ one_one_int
          @ ( set_fo2581907887559384638at_int
            @ ^ [O: nat] : ( times_times_int @ ( plus_plus_int @ A4 @ ( semiri1314217659103216013at_int @ O ) ) )
            @ zero_zero_nat
            @ ( minus_minus_nat @ N2 @ one_one_nat )
            @ one_one_int ) ) ) ) ).

% pochhammer_code
thf(fact_8604_pochhammer__code,axiom,
    ( comm_s4663373288045622133er_nat
    = ( ^ [A4: nat,N2: nat] :
          ( if_nat @ ( N2 = zero_zero_nat ) @ one_one_nat
          @ ( set_fo2584398358068434914at_nat
            @ ^ [O: nat] : ( times_times_nat @ ( plus_plus_nat @ A4 @ ( semiri1316708129612266289at_nat @ O ) ) )
            @ zero_zero_nat
            @ ( minus_minus_nat @ N2 @ one_one_nat )
            @ one_one_nat ) ) ) ) ).

% pochhammer_code
thf(fact_8605_gbinomial__code,axiom,
    ( gbinomial_complex
    = ( ^ [A4: complex,K3: nat] :
          ( if_complex @ ( K3 = zero_zero_nat ) @ one_one_complex
          @ ( divide1717551699836669952omplex
            @ ( set_fo1517530859248394432omplex
              @ ^ [L: nat] : ( times_times_complex @ ( minus_minus_complex @ A4 @ ( semiri8010041392384452111omplex @ L ) ) )
              @ zero_zero_nat
              @ ( minus_minus_nat @ K3 @ one_one_nat )
              @ one_one_complex )
            @ ( semiri5044797733671781792omplex @ K3 ) ) ) ) ) ).

% gbinomial_code
thf(fact_8606_gbinomial__code,axiom,
    ( gbinomial_rat
    = ( ^ [A4: rat,K3: nat] :
          ( if_rat @ ( K3 = zero_zero_nat ) @ one_one_rat
          @ ( divide_divide_rat
            @ ( set_fo1949268297981939178at_rat
              @ ^ [L: nat] : ( times_times_rat @ ( minus_minus_rat @ A4 @ ( semiri681578069525770553at_rat @ L ) ) )
              @ zero_zero_nat
              @ ( minus_minus_nat @ K3 @ one_one_nat )
              @ one_one_rat )
            @ ( semiri773545260158071498ct_rat @ K3 ) ) ) ) ) ).

% gbinomial_code
thf(fact_8607_gbinomial__code,axiom,
    ( gbinomial_real
    = ( ^ [A4: real,K3: nat] :
          ( if_real @ ( K3 = zero_zero_nat ) @ one_one_real
          @ ( divide_divide_real
            @ ( set_fo3111899725591712190t_real
              @ ^ [L: nat] : ( times_times_real @ ( minus_minus_real @ A4 @ ( semiri5074537144036343181t_real @ L ) ) )
              @ zero_zero_nat
              @ ( minus_minus_nat @ K3 @ one_one_nat )
              @ one_one_real )
            @ ( semiri2265585572941072030t_real @ K3 ) ) ) ) ) ).

% gbinomial_code
thf(fact_8608_vebt__member_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X4: nat] :
      ( ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary ) @ X4 )
      = ( ( X4 != Mi )
       => ( ( X4 != Ma )
         => ( ~ ( ord_less_nat @ X4 @ Mi )
            & ( ~ ( ord_less_nat @ X4 @ Mi )
             => ( ~ ( ord_less_nat @ Ma @ X4 )
                & ( ~ ( ord_less_nat @ Ma @ X4 )
                 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                     => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X4 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                    & ( ord_less_nat @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.simps(5)
thf(fact_8609_vebt__succ_Osimps_I6_J,axiom,
    ! [X4: nat,Mi: nat,Ma: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_nat @ X4 @ Mi )
       => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary ) @ X4 )
          = ( some_nat @ Mi ) ) )
      & ( ~ ( ord_less_nat @ X4 @ Mi )
       => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary ) @ X4 )
          = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
            @ ( if_option_nat
              @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ X4 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
              @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X4 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
              @ ( if_option_nat
                @ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                  = none_nat )
                @ none_nat
                @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
            @ none_nat ) ) ) ) ).

% vebt_succ.simps(6)
thf(fact_8610_of__int__code__if,axiom,
    ( ring_1_of_int_real
    = ( ^ [K3: int] :
          ( if_real @ ( K3 = zero_zero_int ) @ zero_zero_real
          @ ( if_real @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_real @ ( ring_1_of_int_real @ ( uminus_uminus_int @ K3 ) ) )
            @ ( if_real
              @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
                = zero_zero_int )
              @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
              @ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_real ) ) ) ) ) ) ).

% of_int_code_if
thf(fact_8611_of__int__code__if,axiom,
    ( ring_1_of_int_int
    = ( ^ [K3: int] :
          ( if_int @ ( K3 = zero_zero_int ) @ zero_zero_int
          @ ( if_int @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_int @ ( ring_1_of_int_int @ ( uminus_uminus_int @ K3 ) ) )
            @ ( if_int
              @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
                = zero_zero_int )
              @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( ring_1_of_int_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
              @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( ring_1_of_int_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_int ) ) ) ) ) ) ).

% of_int_code_if
thf(fact_8612_of__int__code__if,axiom,
    ( ring_17405671764205052669omplex
    = ( ^ [K3: int] :
          ( if_complex @ ( K3 = zero_zero_int ) @ zero_zero_complex
          @ ( if_complex @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus1482373934393186551omplex @ ( ring_17405671764205052669omplex @ ( uminus_uminus_int @ K3 ) ) )
            @ ( if_complex
              @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
                = zero_zero_int )
              @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( ring_17405671764205052669omplex @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
              @ ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( ring_17405671764205052669omplex @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_complex ) ) ) ) ) ) ).

% of_int_code_if
thf(fact_8613_of__int__code__if,axiom,
    ( ring_18347121197199848620nteger
    = ( ^ [K3: int] :
          ( if_Code_integer @ ( K3 = zero_zero_int ) @ zero_z3403309356797280102nteger
          @ ( if_Code_integer @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus1351360451143612070nteger @ ( ring_18347121197199848620nteger @ ( uminus_uminus_int @ K3 ) ) )
            @ ( if_Code_integer
              @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
                = zero_zero_int )
              @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( ring_18347121197199848620nteger @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
              @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( ring_18347121197199848620nteger @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_Code_integer ) ) ) ) ) ) ).

% of_int_code_if
thf(fact_8614_of__int__code__if,axiom,
    ( ring_1_of_int_rat
    = ( ^ [K3: int] :
          ( if_rat @ ( K3 = zero_zero_int ) @ zero_zero_rat
          @ ( if_rat @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_rat @ ( ring_1_of_int_rat @ ( uminus_uminus_int @ K3 ) ) )
            @ ( if_rat
              @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
                = zero_zero_int )
              @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( ring_1_of_int_rat @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
              @ ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( ring_1_of_int_rat @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_rat ) ) ) ) ) ) ).

% of_int_code_if
thf(fact_8615_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
    ! [Mi: nat,Ma: nat,V: nat,TreeList2: list_VEBT_VEBT,Vc: vEBT_VEBT,X4: nat] :
      ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ V ) @ TreeList2 @ Vc ) @ X4 )
      = ( ( X4 = Mi )
        | ( X4 = Ma )
        | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
           => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X4 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
          & ( ord_less_nat @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ).

% VEBT_internal.membermima.simps(4)
thf(fact_8616_monoseq__arctan__series,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X4 ) @ one_one_real )
     => ( topolo6980174941875973593q_real
        @ ^ [N2: nat] : ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X4 @ ( plus_plus_nat @ ( times_times_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ).

% monoseq_arctan_series
thf(fact_8617_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
    ! [Uy: option4927543243414619207at_nat,V: nat,TreeList2: list_VEBT_VEBT,S: vEBT_VEBT,X4: nat] :
      ( ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uy @ ( suc @ V ) @ TreeList2 @ S ) @ X4 )
      = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
         => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X4 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
        & ( ord_less_nat @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ).

% VEBT_internal.naive_member.simps(3)
thf(fact_8618_buildup__nothing__in__min__max,axiom,
    ! [N: nat,X4: nat] :
      ~ ( vEBT_VEBT_membermima @ ( vEBT_vebt_buildup @ N ) @ X4 ) ).

% buildup_nothing_in_min_max
thf(fact_8619_buildup__nothing__in__leaf,axiom,
    ! [N: nat,X4: nat] :
      ~ ( vEBT_V5719532721284313246member @ ( vEBT_vebt_buildup @ N ) @ X4 ) ).

% buildup_nothing_in_leaf
thf(fact_8620_both__member__options__def,axiom,
    ( vEBT_V8194947554948674370ptions
    = ( ^ [T3: vEBT_VEBT,X: nat] :
          ( ( vEBT_V5719532721284313246member @ T3 @ X )
          | ( vEBT_VEBT_membermima @ T3 @ X ) ) ) ) ).

% both_member_options_def
thf(fact_8621_member__valid__both__member__options,axiom,
    ! [Tree: vEBT_VEBT,N: nat,X4: nat] :
      ( ( vEBT_invar_vebt @ Tree @ N )
     => ( ( vEBT_vebt_member @ Tree @ X4 )
       => ( ( vEBT_V5719532721284313246member @ Tree @ X4 )
          | ( vEBT_VEBT_membermima @ Tree @ X4 ) ) ) ) ).

% member_valid_both_member_options
thf(fact_8622_monoseq__realpow,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_eq_real @ X4 @ one_one_real )
       => ( topolo6980174941875973593q_real @ ( power_power_real @ X4 ) ) ) ) ).

% monoseq_realpow
thf(fact_8623_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
    ! [X4: vEBT_VEBT,Xa: nat] :
      ( ( vEBT_VEBT_membermima @ X4 @ Xa )
     => ( ! [Mi2: nat,Ma2: nat] :
            ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                ( X4
                = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
           => ~ ( ( Xa = Mi2 )
                | ( Xa = Ma2 ) ) )
       => ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList3: list_VEBT_VEBT] :
              ( ? [Vc2: vEBT_VEBT] :
                  ( X4
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ Vc2 ) )
             => ~ ( ( Xa = Mi2 )
                  | ( Xa = Ma2 )
                  | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                     => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                    & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) )
         => ~ ! [V3: nat,TreeList3: list_VEBT_VEBT] :
                ( ? [Vd2: vEBT_VEBT] :
                    ( X4
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V3 ) @ TreeList3 @ Vd2 ) )
               => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                     => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                    & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(2)
thf(fact_8624_vebt__member_Oelims_I2_J,axiom,
    ! [X4: vEBT_VEBT,Xa: nat] :
      ( ( vEBT_vebt_member @ X4 @ Xa )
     => ( ! [A5: $o,B5: $o] :
            ( ( X4
              = ( vEBT_Leaf @ A5 @ B5 ) )
           => ~ ( ( ( Xa = zero_zero_nat )
                 => A5 )
                & ( ( Xa != zero_zero_nat )
                 => ( ( ( Xa = one_one_nat )
                     => B5 )
                    & ( Xa = one_one_nat ) ) ) ) )
       => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList3: list_VEBT_VEBT] :
              ( ? [Summary2: vEBT_VEBT] :
                  ( X4
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) )
             => ~ ( ( Xa != Mi2 )
                 => ( ( Xa != Ma2 )
                   => ( ~ ( ord_less_nat @ Xa @ Mi2 )
                      & ( ~ ( ord_less_nat @ Xa @ Mi2 )
                       => ( ~ ( ord_less_nat @ Ma2 @ Xa )
                          & ( ~ ( ord_less_nat @ Ma2 @ Xa )
                           => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                               => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                              & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(2)
thf(fact_8625_vebt__insert_Osimps_I3_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT,X4: nat] :
      ( ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts @ S ) @ X4 )
      = ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts @ S ) ) ).

% vebt_insert.simps(3)
thf(fact_8626_pochhammer__times__pochhammer__half,axiom,
    ! [Z: complex,N: nat] :
      ( ( times_times_complex @ ( comm_s2602460028002588243omplex @ Z @ ( suc @ N ) ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z @ ( divide1717551699836669952omplex @ one_one_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) @ ( suc @ N ) ) )
      = ( groups6464643781859351333omplex
        @ ^ [K3: nat] : ( plus_plus_complex @ Z @ ( divide1717551699836669952omplex @ ( semiri8010041392384452111omplex @ K3 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ).

% pochhammer_times_pochhammer_half
thf(fact_8627_pochhammer__times__pochhammer__half,axiom,
    ! [Z: real,N: nat] :
      ( ( times_times_real @ ( comm_s7457072308508201937r_real @ Z @ ( suc @ N ) ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( suc @ N ) ) )
      = ( groups129246275422532515t_real
        @ ^ [K3: nat] : ( plus_plus_real @ Z @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ K3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ).

% pochhammer_times_pochhammer_half
thf(fact_8628_pochhammer__times__pochhammer__half,axiom,
    ! [Z: rat,N: nat] :
      ( ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z @ ( suc @ N ) ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( suc @ N ) ) )
      = ( groups73079841787564623at_rat
        @ ^ [K3: nat] : ( plus_plus_rat @ Z @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ K3 ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ).

% pochhammer_times_pochhammer_half
thf(fact_8629_Leaf__0__not,axiom,
    ! [A: $o,B2: $o] :
      ~ ( vEBT_invar_vebt @ ( vEBT_Leaf @ A @ B2 ) @ zero_zero_nat ) ).

% Leaf_0_not
thf(fact_8630_deg__1__Leafy,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( N = one_one_nat )
       => ? [A5: $o,B5: $o] :
            ( T
            = ( vEBT_Leaf @ A5 @ B5 ) ) ) ) ).

% deg_1_Leafy
thf(fact_8631_deg__1__Leaf,axiom,
    ! [T: vEBT_VEBT] :
      ( ( vEBT_invar_vebt @ T @ one_one_nat )
     => ? [A5: $o,B5: $o] :
          ( T
          = ( vEBT_Leaf @ A5 @ B5 ) ) ) ).

% deg_1_Leaf
thf(fact_8632_deg1Leaf,axiom,
    ! [T: vEBT_VEBT] :
      ( ( vEBT_invar_vebt @ T @ one_one_nat )
      = ( ? [A4: $o,B3: $o] :
            ( T
            = ( vEBT_Leaf @ A4 @ B3 ) ) ) ) ).

% deg1Leaf
thf(fact_8633_prod_Ocl__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: nat > complex] :
      ( ( ( ord_less_nat @ ( suc @ N ) @ M )
       => ( ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
          = one_one_complex ) )
      & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
       => ( ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
          = ( times_times_complex @ ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).

% prod.cl_ivl_Suc
thf(fact_8634_prod_Ocl__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: nat > real] :
      ( ( ( ord_less_nat @ ( suc @ N ) @ M )
       => ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
          = one_one_real ) )
      & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
       => ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
          = ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).

% prod.cl_ivl_Suc
thf(fact_8635_prod_Ocl__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: nat > rat] :
      ( ( ( ord_less_nat @ ( suc @ N ) @ M )
       => ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
          = one_one_rat ) )
      & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
       => ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
          = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).

% prod.cl_ivl_Suc
thf(fact_8636_prod_Ocl__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: nat > nat] :
      ( ( ( ord_less_nat @ ( suc @ N ) @ M )
       => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
          = one_one_nat ) )
      & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
       => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
          = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).

% prod.cl_ivl_Suc
thf(fact_8637_prod_Ocl__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: nat > int] :
      ( ( ( ord_less_nat @ ( suc @ N ) @ M )
       => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
          = one_one_int ) )
      & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
       => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
          = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).

% prod.cl_ivl_Suc
thf(fact_8638_mod__prod__eq,axiom,
    ! [F: nat > nat,A: nat,A3: set_nat] :
      ( ( modulo_modulo_nat
        @ ( groups708209901874060359at_nat
          @ ^ [I2: nat] : ( modulo_modulo_nat @ ( F @ I2 ) @ A )
          @ A3 )
        @ A )
      = ( modulo_modulo_nat @ ( groups708209901874060359at_nat @ F @ A3 ) @ A ) ) ).

% mod_prod_eq
thf(fact_8639_mod__prod__eq,axiom,
    ! [F: nat > int,A: int,A3: set_nat] :
      ( ( modulo_modulo_int
        @ ( groups705719431365010083at_int
          @ ^ [I2: nat] : ( modulo_modulo_int @ ( F @ I2 ) @ A )
          @ A3 )
        @ A )
      = ( modulo_modulo_int @ ( groups705719431365010083at_int @ F @ A3 ) @ A ) ) ).

% mod_prod_eq
thf(fact_8640_mod__prod__eq,axiom,
    ! [F: int > int,A: int,A3: set_int] :
      ( ( modulo_modulo_int
        @ ( groups1705073143266064639nt_int
          @ ^ [I2: int] : ( modulo_modulo_int @ ( F @ I2 ) @ A )
          @ A3 )
        @ A )
      = ( modulo_modulo_int @ ( groups1705073143266064639nt_int @ F @ A3 ) @ A ) ) ).

% mod_prod_eq
thf(fact_8641_vebt__insert_Ocases,axiom,
    ! [X4: produc9072475918466114483BT_nat] :
      ( ! [A5: $o,B5: $o,X3: nat] :
          ( X4
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B5 ) @ X3 ) )
     => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT,X3: nat] :
            ( X4
           != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S3 ) @ X3 ) )
       => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT,X3: nat] :
              ( X4
             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S3 ) @ X3 ) )
         => ( ! [V3: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
                ( X4
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V3 ) ) @ TreeList3 @ Summary2 ) @ X3 ) )
           => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
                  ( X4
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) @ X3 ) ) ) ) ) ) ).

% vebt_insert.cases
thf(fact_8642_vebt__insert_Osimps_I1_J,axiom,
    ! [X4: nat,A: $o,B2: $o] :
      ( ( ( X4 = zero_zero_nat )
       => ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A @ B2 ) @ X4 )
          = ( vEBT_Leaf @ $true @ B2 ) ) )
      & ( ( X4 != zero_zero_nat )
       => ( ( ( X4 = one_one_nat )
           => ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A @ B2 ) @ X4 )
              = ( vEBT_Leaf @ A @ $true ) ) )
          & ( ( X4 != one_one_nat )
           => ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A @ B2 ) @ X4 )
              = ( vEBT_Leaf @ A @ B2 ) ) ) ) ) ) ).

% vebt_insert.simps(1)
thf(fact_8643_prod_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: nat > nat,M: nat,N: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( suc @ N ) ) )
      = ( groups708209901874060359at_nat
        @ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
        @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).

% prod.shift_bounds_cl_Suc_ivl
thf(fact_8644_prod_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: nat > int,M: nat,N: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( suc @ N ) ) )
      = ( groups705719431365010083at_int
        @ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
        @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).

% prod.shift_bounds_cl_Suc_ivl
thf(fact_8645_prod_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: nat > nat,M: nat,K: nat,N: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) ) )
      = ( groups708209901874060359at_nat
        @ ^ [I2: nat] : ( G @ ( plus_plus_nat @ I2 @ K ) )
        @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).

% prod.shift_bounds_cl_nat_ivl
thf(fact_8646_prod_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: nat > int,M: nat,K: nat,N: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) ) )
      = ( groups705719431365010083at_int
        @ ^ [I2: nat] : ( G @ ( plus_plus_nat @ I2 @ K ) )
        @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).

% prod.shift_bounds_cl_nat_ivl
thf(fact_8647_vebt__member_Ocases,axiom,
    ! [X4: produc9072475918466114483BT_nat] :
      ( ! [A5: $o,B5: $o,X3: nat] :
          ( X4
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B5 ) @ X3 ) )
     => ( ! [Uu2: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,X3: nat] :
            ( X4
           != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv @ Uw ) @ X3 ) )
       => ( ! [V3: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz: vEBT_VEBT,X3: nat] :
              ( X4
             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V3 ) @ zero_zero_nat @ Uy2 @ Uz ) @ X3 ) )
         => ( ! [V3: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT,X3: nat] :
                ( X4
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V3 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ X3 ) )
           => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
                  ( X4
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) @ X3 ) ) ) ) ) ) ).

% vebt_member.cases
thf(fact_8648_vebt__succ_Ocases,axiom,
    ! [X4: produc9072475918466114483BT_nat] :
      ( ! [Uu2: $o,B5: $o] :
          ( X4
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ B5 ) @ zero_zero_nat ) )
     => ( ! [Uv: $o,Uw: $o,N4: nat] :
            ( X4
           != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uv @ Uw ) @ ( suc @ N4 ) ) )
       => ( ! [Ux: nat,Uy2: list_VEBT_VEBT,Uz: vEBT_VEBT,Va3: nat] :
              ( X4
             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux @ Uy2 @ Uz ) @ Va3 ) )
         => ( ! [V3: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT,Ve2: nat] :
                ( X4
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V3 ) @ zero_zero_nat @ Vc2 @ Vd2 ) @ Ve2 ) )
           => ( ! [V3: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT,Vi2: nat] :
                  ( X4
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V3 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) @ Vi2 ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
                    ( X4
                   != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) @ X3 ) ) ) ) ) ) ) ).

% vebt_succ.cases
thf(fact_8649_VEBT__internal_Omembermima_Ocases,axiom,
    ! [X4: produc9072475918466114483BT_nat] :
      ( ! [Uu2: $o,Uv: $o,Uw: nat] :
          ( X4
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv ) @ Uw ) )
     => ( ! [Ux: list_VEBT_VEBT,Uy2: vEBT_VEBT,Uz: nat] :
            ( X4
           != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux @ Uy2 ) @ Uz ) )
       => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT,X3: nat] :
              ( X4
             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ X3 ) )
         => ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList3: list_VEBT_VEBT,Vc2: vEBT_VEBT,X3: nat] :
                ( X4
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ Vc2 ) @ X3 ) )
           => ~ ! [V3: nat,TreeList3: list_VEBT_VEBT,Vd2: vEBT_VEBT,X3: nat] :
                  ( X4
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V3 ) @ TreeList3 @ Vd2 ) @ X3 ) ) ) ) ) ) ).

% VEBT_internal.membermima.cases
thf(fact_8650_VEBT__internal_Onaive__member_Ocases,axiom,
    ! [X4: produc9072475918466114483BT_nat] :
      ( ! [A5: $o,B5: $o,X3: nat] :
          ( X4
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B5 ) @ X3 ) )
     => ( ! [Uu2: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,Ux: nat] :
            ( X4
           != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv @ Uw ) @ Ux ) )
       => ~ ! [Uy2: option4927543243414619207at_nat,V3: nat,TreeList3: list_VEBT_VEBT,S3: vEBT_VEBT,X3: nat] :
              ( X4
             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S3 ) @ X3 ) ) ) ) ).

% VEBT_internal.naive_member.cases
thf(fact_8651_invar__vebt_Ointros_I1_J,axiom,
    ! [A: $o,B2: $o] : ( vEBT_invar_vebt @ ( vEBT_Leaf @ A @ B2 ) @ ( suc @ zero_zero_nat ) ) ).

% invar_vebt.intros(1)
thf(fact_8652_vebt__member_Osimps_I1_J,axiom,
    ! [A: $o,B2: $o,X4: nat] :
      ( ( vEBT_vebt_member @ ( vEBT_Leaf @ A @ B2 ) @ X4 )
      = ( ( ( X4 = zero_zero_nat )
         => A )
        & ( ( X4 != zero_zero_nat )
         => ( ( ( X4 = one_one_nat )
             => B2 )
            & ( X4 = one_one_nat ) ) ) ) ) ).

% vebt_member.simps(1)
thf(fact_8653_vebt__buildup_Osimps_I2_J,axiom,
    ( ( vEBT_vebt_buildup @ ( suc @ zero_zero_nat ) )
    = ( vEBT_Leaf @ $false @ $false ) ) ).

% vebt_buildup.simps(2)
thf(fact_8654_vebt__succ_Osimps_I2_J,axiom,
    ! [Uv2: $o,Uw2: $o,N: nat] :
      ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uv2 @ Uw2 ) @ ( suc @ N ) )
      = none_nat ) ).

% vebt_succ.simps(2)
thf(fact_8655_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
    ! [A: $o,B2: $o,X4: nat] :
      ( ( vEBT_V5719532721284313246member @ ( vEBT_Leaf @ A @ B2 ) @ X4 )
      = ( ( ( X4 = zero_zero_nat )
         => A )
        & ( ( X4 != zero_zero_nat )
         => ( ( ( X4 = one_one_nat )
             => B2 )
            & ( X4 = one_one_nat ) ) ) ) ) ).

% VEBT_internal.naive_member.simps(1)
thf(fact_8656_prod_OatLeastAtMost__rev,axiom,
    ! [G: nat > nat,N: nat,M: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ N @ M ) )
      = ( groups708209901874060359at_nat
        @ ^ [I2: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ I2 ) )
        @ ( set_or1269000886237332187st_nat @ N @ M ) ) ) ).

% prod.atLeastAtMost_rev
thf(fact_8657_prod_OatLeastAtMost__rev,axiom,
    ! [G: nat > int,N: nat,M: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ N @ M ) )
      = ( groups705719431365010083at_int
        @ ^ [I2: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ I2 ) )
        @ ( set_or1269000886237332187st_nat @ N @ M ) ) ) ).

% prod.atLeastAtMost_rev
thf(fact_8658_vebt__succ_Osimps_I3_J,axiom,
    ! [Ux2: nat,Uy: list_VEBT_VEBT,Uz2: vEBT_VEBT,Va2: nat] :
      ( ( vEBT_vebt_succ @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy @ Uz2 ) @ Va2 )
      = none_nat ) ).

% vebt_succ.simps(3)
thf(fact_8659_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: nat > real,N: nat] :
      ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
      = ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).

% prod.atLeast0_atMost_Suc
thf(fact_8660_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: nat > rat,N: nat] :
      ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
      = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).

% prod.atLeast0_atMost_Suc
thf(fact_8661_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: nat > nat,N: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
      = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).

% prod.atLeast0_atMost_Suc
thf(fact_8662_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: nat > int,N: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
      = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).

% prod.atLeast0_atMost_Suc
thf(fact_8663_prod_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N: nat,G: nat > real] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
        = ( times_times_real @ ( G @ M ) @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).

% prod.atLeast_Suc_atMost
thf(fact_8664_prod_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N: nat,G: nat > rat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
        = ( times_times_rat @ ( G @ M ) @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).

% prod.atLeast_Suc_atMost
thf(fact_8665_prod_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
        = ( times_times_nat @ ( G @ M ) @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).

% prod.atLeast_Suc_atMost
thf(fact_8666_prod_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
        = ( times_times_int @ ( G @ M ) @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).

% prod.atLeast_Suc_atMost
thf(fact_8667_prod_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N: nat,G: nat > real] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
     => ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
        = ( times_times_real @ ( G @ ( suc @ N ) ) @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).

% prod.nat_ivl_Suc'
thf(fact_8668_prod_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N: nat,G: nat > rat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
     => ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
        = ( times_times_rat @ ( G @ ( suc @ N ) ) @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).

% prod.nat_ivl_Suc'
thf(fact_8669_prod_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
     => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
        = ( times_times_nat @ ( G @ ( suc @ N ) ) @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).

% prod.nat_ivl_Suc'
thf(fact_8670_prod_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
     => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
        = ( times_times_int @ ( G @ ( suc @ N ) ) @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).

% prod.nat_ivl_Suc'
thf(fact_8671_vebt__insert_Osimps_I4_J,axiom,
    ! [V: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X4: nat] :
      ( ( vEBT_vebt_insert @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V ) ) @ TreeList2 @ Summary ) @ X4 )
      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ X4 @ X4 ) ) @ ( suc @ ( suc @ V ) ) @ TreeList2 @ Summary ) ) ).

% vebt_insert.simps(4)
thf(fact_8672_prod_OSuc__reindex__ivl,axiom,
    ! [M: nat,N: nat,G: nat > real] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
        = ( times_times_real @ ( G @ M )
          @ ( groups129246275422532515t_real
            @ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
            @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).

% prod.Suc_reindex_ivl
thf(fact_8673_prod_OSuc__reindex__ivl,axiom,
    ! [M: nat,N: nat,G: nat > rat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
        = ( times_times_rat @ ( G @ M )
          @ ( groups73079841787564623at_rat
            @ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
            @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).

% prod.Suc_reindex_ivl
thf(fact_8674_prod_OSuc__reindex__ivl,axiom,
    ! [M: nat,N: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
        = ( times_times_nat @ ( G @ M )
          @ ( groups708209901874060359at_nat
            @ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
            @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).

% prod.Suc_reindex_ivl
thf(fact_8675_prod_OSuc__reindex__ivl,axiom,
    ! [M: nat,N: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
        = ( times_times_int @ ( G @ M )
          @ ( groups705719431365010083at_int
            @ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
            @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).

% prod.Suc_reindex_ivl
thf(fact_8676_fact__prod,axiom,
    ( semiri5044797733671781792omplex
    = ( ^ [N2: nat] :
          ( semiri8010041392384452111omplex
          @ ( groups708209901874060359at_nat
            @ ^ [X: nat] : X
            @ ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) ) ) ) ) ).

% fact_prod
thf(fact_8677_fact__prod,axiom,
    ( semiri773545260158071498ct_rat
    = ( ^ [N2: nat] :
          ( semiri681578069525770553at_rat
          @ ( groups708209901874060359at_nat
            @ ^ [X: nat] : X
            @ ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) ) ) ) ) ).

% fact_prod
thf(fact_8678_fact__prod,axiom,
    ( semiri1406184849735516958ct_int
    = ( ^ [N2: nat] :
          ( semiri1314217659103216013at_int
          @ ( groups708209901874060359at_nat
            @ ^ [X: nat] : X
            @ ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) ) ) ) ) ).

% fact_prod
thf(fact_8679_fact__prod,axiom,
    ( semiri1408675320244567234ct_nat
    = ( ^ [N2: nat] :
          ( semiri1316708129612266289at_nat
          @ ( groups708209901874060359at_nat
            @ ^ [X: nat] : X
            @ ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) ) ) ) ) ).

% fact_prod
thf(fact_8680_fact__prod,axiom,
    ( semiri2265585572941072030t_real
    = ( ^ [N2: nat] :
          ( semiri5074537144036343181t_real
          @ ( groups708209901874060359at_nat
            @ ^ [X: nat] : X
            @ ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) ) ) ) ) ).

% fact_prod
thf(fact_8681_prod__atLeastAtMost__code,axiom,
    ! [F: nat > complex,A: nat,B2: nat] :
      ( ( groups6464643781859351333omplex @ F @ ( set_or1269000886237332187st_nat @ A @ B2 ) )
      = ( set_fo1517530859248394432omplex
        @ ^ [A4: nat] : ( times_times_complex @ ( F @ A4 ) )
        @ A
        @ B2
        @ one_one_complex ) ) ).

% prod_atLeastAtMost_code
thf(fact_8682_prod__atLeastAtMost__code,axiom,
    ! [F: nat > real,A: nat,B2: nat] :
      ( ( groups129246275422532515t_real @ F @ ( set_or1269000886237332187st_nat @ A @ B2 ) )
      = ( set_fo3111899725591712190t_real
        @ ^ [A4: nat] : ( times_times_real @ ( F @ A4 ) )
        @ A
        @ B2
        @ one_one_real ) ) ).

% prod_atLeastAtMost_code
thf(fact_8683_prod__atLeastAtMost__code,axiom,
    ! [F: nat > rat,A: nat,B2: nat] :
      ( ( groups73079841787564623at_rat @ F @ ( set_or1269000886237332187st_nat @ A @ B2 ) )
      = ( set_fo1949268297981939178at_rat
        @ ^ [A4: nat] : ( times_times_rat @ ( F @ A4 ) )
        @ A
        @ B2
        @ one_one_rat ) ) ).

% prod_atLeastAtMost_code
thf(fact_8684_prod__atLeastAtMost__code,axiom,
    ! [F: nat > nat,A: nat,B2: nat] :
      ( ( groups708209901874060359at_nat @ F @ ( set_or1269000886237332187st_nat @ A @ B2 ) )
      = ( set_fo2584398358068434914at_nat
        @ ^ [A4: nat] : ( times_times_nat @ ( F @ A4 ) )
        @ A
        @ B2
        @ one_one_nat ) ) ).

% prod_atLeastAtMost_code
thf(fact_8685_prod__atLeastAtMost__code,axiom,
    ! [F: nat > int,A: nat,B2: nat] :
      ( ( groups705719431365010083at_int @ F @ ( set_or1269000886237332187st_nat @ A @ B2 ) )
      = ( set_fo2581907887559384638at_int
        @ ^ [A4: nat] : ( times_times_int @ ( F @ A4 ) )
        @ A
        @ B2
        @ one_one_int ) ) ).

% prod_atLeastAtMost_code
thf(fact_8686_prod_Oub__add__nat,axiom,
    ! [M: nat,N: nat,G: nat > real,P2: nat] :
      ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
     => ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P2 ) ) )
        = ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P2 ) ) ) ) ) ) ).

% prod.ub_add_nat
thf(fact_8687_prod_Oub__add__nat,axiom,
    ! [M: nat,N: nat,G: nat > rat,P2: nat] :
      ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
     => ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P2 ) ) )
        = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P2 ) ) ) ) ) ) ).

% prod.ub_add_nat
thf(fact_8688_prod_Oub__add__nat,axiom,
    ! [M: nat,N: nat,G: nat > nat,P2: nat] :
      ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
     => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P2 ) ) )
        = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P2 ) ) ) ) ) ) ).

% prod.ub_add_nat
thf(fact_8689_prod_Oub__add__nat,axiom,
    ! [M: nat,N: nat,G: nat > int,P2: nat] :
      ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
     => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P2 ) ) )
        = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P2 ) ) ) ) ) ) ).

% prod.ub_add_nat
thf(fact_8690_vebt__succ_Osimps_I1_J,axiom,
    ! [B2: $o,Uu3: $o] :
      ( ( B2
       => ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uu3 @ B2 ) @ zero_zero_nat )
          = ( some_nat @ one_one_nat ) ) )
      & ( ~ B2
       => ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uu3 @ B2 ) @ zero_zero_nat )
          = none_nat ) ) ) ).

% vebt_succ.simps(1)
thf(fact_8691_fact__eq__fact__times,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( semiri1408675320244567234ct_nat @ M )
        = ( times_times_nat @ ( semiri1408675320244567234ct_nat @ N )
          @ ( groups708209901874060359at_nat
            @ ^ [X: nat] : X
            @ ( set_or1269000886237332187st_nat @ ( suc @ N ) @ M ) ) ) ) ) ).

% fact_eq_fact_times
thf(fact_8692_pochhammer__Suc__prod,axiom,
    ! [A: complex,N: nat] :
      ( ( comm_s2602460028002588243omplex @ A @ ( suc @ N ) )
      = ( groups6464643781859351333omplex
        @ ^ [I2: nat] : ( plus_plus_complex @ A @ ( semiri8010041392384452111omplex @ I2 ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).

% pochhammer_Suc_prod
thf(fact_8693_pochhammer__Suc__prod,axiom,
    ! [A: real,N: nat] :
      ( ( comm_s7457072308508201937r_real @ A @ ( suc @ N ) )
      = ( groups129246275422532515t_real
        @ ^ [I2: nat] : ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ I2 ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).

% pochhammer_Suc_prod
thf(fact_8694_pochhammer__Suc__prod,axiom,
    ! [A: rat,N: nat] :
      ( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N ) )
      = ( groups73079841787564623at_rat
        @ ^ [I2: nat] : ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ I2 ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).

% pochhammer_Suc_prod
thf(fact_8695_pochhammer__Suc__prod,axiom,
    ! [A: nat,N: nat] :
      ( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N ) )
      = ( groups708209901874060359at_nat
        @ ^ [I2: nat] : ( plus_plus_nat @ A @ ( semiri1316708129612266289at_nat @ I2 ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).

% pochhammer_Suc_prod
thf(fact_8696_pochhammer__Suc__prod,axiom,
    ! [A: int,N: nat] :
      ( ( comm_s4660882817536571857er_int @ A @ ( suc @ N ) )
      = ( groups705719431365010083at_int
        @ ^ [I2: nat] : ( plus_plus_int @ A @ ( semiri1314217659103216013at_int @ I2 ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).

% pochhammer_Suc_prod
thf(fact_8697_pochhammer__prod__rev,axiom,
    ( comm_s2602460028002588243omplex
    = ( ^ [A4: complex,N2: nat] :
          ( groups6464643781859351333omplex
          @ ^ [I2: nat] : ( plus_plus_complex @ A4 @ ( semiri8010041392384452111omplex @ ( minus_minus_nat @ N2 @ I2 ) ) )
          @ ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) ) ) ) ).

% pochhammer_prod_rev
thf(fact_8698_pochhammer__prod__rev,axiom,
    ( comm_s7457072308508201937r_real
    = ( ^ [A4: real,N2: nat] :
          ( groups129246275422532515t_real
          @ ^ [I2: nat] : ( plus_plus_real @ A4 @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N2 @ I2 ) ) )
          @ ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) ) ) ) ).

% pochhammer_prod_rev
thf(fact_8699_pochhammer__prod__rev,axiom,
    ( comm_s4028243227959126397er_rat
    = ( ^ [A4: rat,N2: nat] :
          ( groups73079841787564623at_rat
          @ ^ [I2: nat] : ( plus_plus_rat @ A4 @ ( semiri681578069525770553at_rat @ ( minus_minus_nat @ N2 @ I2 ) ) )
          @ ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) ) ) ) ).

% pochhammer_prod_rev
thf(fact_8700_pochhammer__prod__rev,axiom,
    ( comm_s4663373288045622133er_nat
    = ( ^ [A4: nat,N2: nat] :
          ( groups708209901874060359at_nat
          @ ^ [I2: nat] : ( plus_plus_nat @ A4 @ ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ N2 @ I2 ) ) )
          @ ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) ) ) ) ).

% pochhammer_prod_rev
thf(fact_8701_pochhammer__prod__rev,axiom,
    ( comm_s4660882817536571857er_int
    = ( ^ [A4: int,N2: nat] :
          ( groups705719431365010083at_int
          @ ^ [I2: nat] : ( plus_plus_int @ A4 @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ N2 @ I2 ) ) )
          @ ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) ) ) ) ).

% pochhammer_prod_rev
thf(fact_8702_fact__div__fact,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) )
        = ( groups708209901874060359at_nat
          @ ^ [X: nat] : X
          @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M ) ) ) ) ).

% fact_div_fact
thf(fact_8703_prod_Oin__pairs,axiom,
    ! [G: nat > real,M: nat,N: nat] :
      ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
      = ( groups129246275422532515t_real
        @ ^ [I2: nat] : ( times_times_real @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).

% prod.in_pairs
thf(fact_8704_prod_Oin__pairs,axiom,
    ! [G: nat > rat,M: nat,N: nat] :
      ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
      = ( groups73079841787564623at_rat
        @ ^ [I2: nat] : ( times_times_rat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).

% prod.in_pairs
thf(fact_8705_prod_Oin__pairs,axiom,
    ! [G: nat > nat,M: nat,N: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
      = ( groups708209901874060359at_nat
        @ ^ [I2: nat] : ( times_times_nat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).

% prod.in_pairs
thf(fact_8706_prod_Oin__pairs,axiom,
    ! [G: nat > int,M: nat,N: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
      = ( groups705719431365010083at_int
        @ ^ [I2: nat] : ( times_times_int @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).

% prod.in_pairs
thf(fact_8707_pochhammer__Suc__prod__rev,axiom,
    ! [A: complex,N: nat] :
      ( ( comm_s2602460028002588243omplex @ A @ ( suc @ N ) )
      = ( groups6464643781859351333omplex
        @ ^ [I2: nat] : ( plus_plus_complex @ A @ ( semiri8010041392384452111omplex @ ( minus_minus_nat @ N @ I2 ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).

% pochhammer_Suc_prod_rev
thf(fact_8708_pochhammer__Suc__prod__rev,axiom,
    ! [A: real,N: nat] :
      ( ( comm_s7457072308508201937r_real @ A @ ( suc @ N ) )
      = ( groups129246275422532515t_real
        @ ^ [I2: nat] : ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N @ I2 ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).

% pochhammer_Suc_prod_rev
thf(fact_8709_pochhammer__Suc__prod__rev,axiom,
    ! [A: rat,N: nat] :
      ( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N ) )
      = ( groups73079841787564623at_rat
        @ ^ [I2: nat] : ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ ( minus_minus_nat @ N @ I2 ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).

% pochhammer_Suc_prod_rev
thf(fact_8710_pochhammer__Suc__prod__rev,axiom,
    ! [A: nat,N: nat] :
      ( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N ) )
      = ( groups708209901874060359at_nat
        @ ^ [I2: nat] : ( plus_plus_nat @ A @ ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ N @ I2 ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).

% pochhammer_Suc_prod_rev
thf(fact_8711_pochhammer__Suc__prod__rev,axiom,
    ! [A: int,N: nat] :
      ( ( comm_s4660882817536571857er_int @ A @ ( suc @ N ) )
      = ( groups705719431365010083at_int
        @ ^ [I2: nat] : ( plus_plus_int @ A @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ N @ I2 ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).

% pochhammer_Suc_prod_rev
thf(fact_8712_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
    ! [X4: vEBT_VEBT,Xa: nat,Y3: $o] :
      ( ( ( vEBT_VEBT_membermima @ X4 @ Xa )
        = Y3 )
     => ( ( ? [Uu2: $o,Uv: $o] :
              ( X4
              = ( vEBT_Leaf @ Uu2 @ Uv ) )
         => Y3 )
       => ( ( ? [Ux: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( X4
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux @ Uy2 ) )
           => Y3 )
         => ( ! [Mi2: nat,Ma2: nat] :
                ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                    ( X4
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
               => ( Y3
                  = ( ~ ( ( Xa = Mi2 )
                        | ( Xa = Ma2 ) ) ) ) )
           => ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList3: list_VEBT_VEBT] :
                  ( ? [Vc2: vEBT_VEBT] :
                      ( X4
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ Vc2 ) )
                 => ( Y3
                    = ( ~ ( ( Xa = Mi2 )
                          | ( Xa = Ma2 )
                          | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                             => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                            & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) )
             => ~ ! [V3: nat,TreeList3: list_VEBT_VEBT] :
                    ( ? [Vd2: vEBT_VEBT] :
                        ( X4
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V3 ) @ TreeList3 @ Vd2 ) )
                   => ( Y3
                      = ( ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                             => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                            & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(1)
thf(fact_8713_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
    ! [X4: vEBT_VEBT,Xa: nat] :
      ( ~ ( vEBT_VEBT_membermima @ X4 @ Xa )
     => ( ! [Uu2: $o,Uv: $o] :
            ( X4
           != ( vEBT_Leaf @ Uu2 @ Uv ) )
       => ( ! [Ux: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
              ( X4
             != ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux @ Uy2 ) )
         => ( ! [Mi2: nat,Ma2: nat] :
                ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                    ( X4
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
               => ( ( Xa = Mi2 )
                  | ( Xa = Ma2 ) ) )
           => ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList3: list_VEBT_VEBT] :
                  ( ? [Vc2: vEBT_VEBT] :
                      ( X4
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ Vc2 ) )
                 => ( ( Xa = Mi2 )
                    | ( Xa = Ma2 )
                    | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                       => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) )
             => ~ ! [V3: nat,TreeList3: list_VEBT_VEBT] :
                    ( ? [Vd2: vEBT_VEBT] :
                        ( X4
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V3 ) @ TreeList3 @ Vd2 ) )
                   => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                       => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(3)
thf(fact_8714_gbinomial__Suc,axiom,
    ! [A: complex,K: nat] :
      ( ( gbinomial_complex @ A @ ( suc @ K ) )
      = ( divide1717551699836669952omplex
        @ ( groups6464643781859351333omplex
          @ ^ [I2: nat] : ( minus_minus_complex @ A @ ( semiri8010041392384452111omplex @ I2 ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) )
        @ ( semiri5044797733671781792omplex @ ( suc @ K ) ) ) ) ).

% gbinomial_Suc
thf(fact_8715_gbinomial__Suc,axiom,
    ! [A: rat,K: nat] :
      ( ( gbinomial_rat @ A @ ( suc @ K ) )
      = ( divide_divide_rat
        @ ( groups73079841787564623at_rat
          @ ^ [I2: nat] : ( minus_minus_rat @ A @ ( semiri681578069525770553at_rat @ I2 ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) )
        @ ( semiri773545260158071498ct_rat @ ( suc @ K ) ) ) ) ).

% gbinomial_Suc
thf(fact_8716_gbinomial__Suc,axiom,
    ! [A: real,K: nat] :
      ( ( gbinomial_real @ A @ ( suc @ K ) )
      = ( divide_divide_real
        @ ( groups129246275422532515t_real
          @ ^ [I2: nat] : ( minus_minus_real @ A @ ( semiri5074537144036343181t_real @ I2 ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) )
        @ ( semiri2265585572941072030t_real @ ( suc @ K ) ) ) ) ).

% gbinomial_Suc
thf(fact_8717_gbinomial__Suc,axiom,
    ! [A: nat,K: nat] :
      ( ( gbinomial_nat @ A @ ( suc @ K ) )
      = ( divide_divide_nat
        @ ( groups708209901874060359at_nat
          @ ^ [I2: nat] : ( minus_minus_nat @ A @ ( semiri1316708129612266289at_nat @ I2 ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) )
        @ ( semiri1408675320244567234ct_nat @ ( suc @ K ) ) ) ) ).

% gbinomial_Suc
thf(fact_8718_gbinomial__Suc,axiom,
    ! [A: int,K: nat] :
      ( ( gbinomial_int @ A @ ( suc @ K ) )
      = ( divide_divide_int
        @ ( groups705719431365010083at_int
          @ ^ [I2: nat] : ( minus_minus_int @ A @ ( semiri1314217659103216013at_int @ I2 ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) )
        @ ( semiri1406184849735516958ct_int @ ( suc @ K ) ) ) ) ).

% gbinomial_Suc
thf(fact_8719_vebt__member_Oelims_I1_J,axiom,
    ! [X4: vEBT_VEBT,Xa: nat,Y3: $o] :
      ( ( ( vEBT_vebt_member @ X4 @ Xa )
        = Y3 )
     => ( ! [A5: $o,B5: $o] :
            ( ( X4
              = ( vEBT_Leaf @ A5 @ B5 ) )
           => ( Y3
              = ( ~ ( ( ( Xa = zero_zero_nat )
                     => A5 )
                    & ( ( Xa != zero_zero_nat )
                     => ( ( ( Xa = one_one_nat )
                         => B5 )
                        & ( Xa = one_one_nat ) ) ) ) ) ) )
       => ( ( ? [Uu2: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( X4
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv @ Uw ) )
           => Y3 )
         => ( ( ? [V3: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz: vEBT_VEBT] :
                  ( X4
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V3 ) @ zero_zero_nat @ Uy2 @ Uz ) )
             => Y3 )
           => ( ( ? [V3: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( X4
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V3 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
               => Y3 )
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList3: list_VEBT_VEBT] :
                    ( ? [Summary2: vEBT_VEBT] :
                        ( X4
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) )
                   => ( Y3
                      = ( ~ ( ( Xa != Mi2 )
                           => ( ( Xa != Ma2 )
                             => ( ~ ( ord_less_nat @ Xa @ Mi2 )
                                & ( ~ ( ord_less_nat @ Xa @ Mi2 )
                                 => ( ~ ( ord_less_nat @ Ma2 @ Xa )
                                    & ( ~ ( ord_less_nat @ Ma2 @ Xa )
                                     => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                                         => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(1)
thf(fact_8720_vebt__member_Oelims_I3_J,axiom,
    ! [X4: vEBT_VEBT,Xa: nat] :
      ( ~ ( vEBT_vebt_member @ X4 @ Xa )
     => ( ! [A5: $o,B5: $o] :
            ( ( X4
              = ( vEBT_Leaf @ A5 @ B5 ) )
           => ( ( ( Xa = zero_zero_nat )
               => A5 )
              & ( ( Xa != zero_zero_nat )
               => ( ( ( Xa = one_one_nat )
                   => B5 )
                  & ( Xa = one_one_nat ) ) ) ) )
       => ( ! [Uu2: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
              ( X4
             != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv @ Uw ) )
         => ( ! [V3: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz: vEBT_VEBT] :
                ( X4
               != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V3 ) @ zero_zero_nat @ Uy2 @ Uz ) )
           => ( ! [V3: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                  ( X4
                 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V3 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList3: list_VEBT_VEBT] :
                    ( ? [Summary2: vEBT_VEBT] :
                        ( X4
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) )
                   => ( ( Xa != Mi2 )
                     => ( ( Xa != Ma2 )
                       => ( ~ ( ord_less_nat @ Xa @ Mi2 )
                          & ( ~ ( ord_less_nat @ Xa @ Mi2 )
                           => ( ~ ( ord_less_nat @ Ma2 @ Xa )
                              & ( ~ ( ord_less_nat @ Ma2 @ Xa )
                               => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                                   => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                  & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(3)
thf(fact_8721_invar__vebt_Osimps,axiom,
    ( vEBT_invar_vebt
    = ( ^ [A1: vEBT_VEBT,A22: nat] :
          ( ( ? [A4: $o,B3: $o] :
                ( A1
                = ( vEBT_Leaf @ A4 @ B3 ) )
            & ( A22
              = ( suc @ zero_zero_nat ) ) )
          | ? [TreeList: list_VEBT_VEBT,N2: nat,Summary3: vEBT_VEBT] :
              ( ( A1
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ A22 @ TreeList @ Summary3 ) )
              & ! [X: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
                 => ( vEBT_invar_vebt @ X @ N2 ) )
              & ( vEBT_invar_vebt @ Summary3 @ N2 )
              & ( ( size_s6755466524823107622T_VEBT @ TreeList )
                = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
              & ( A22
                = ( plus_plus_nat @ N2 @ N2 ) )
              & ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X6 )
              & ! [X: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
                 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) )
          | ? [TreeList: list_VEBT_VEBT,N2: nat,Summary3: vEBT_VEBT] :
              ( ( A1
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ A22 @ TreeList @ Summary3 ) )
              & ! [X: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
                 => ( vEBT_invar_vebt @ X @ N2 ) )
              & ( vEBT_invar_vebt @ Summary3 @ ( suc @ N2 ) )
              & ( ( size_s6755466524823107622T_VEBT @ TreeList )
                = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) )
              & ( A22
                = ( plus_plus_nat @ N2 @ ( suc @ N2 ) ) )
              & ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X6 )
              & ! [X: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
                 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) )
          | ? [TreeList: list_VEBT_VEBT,N2: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
              ( ( A1
                = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ A22 @ TreeList @ Summary3 ) )
              & ! [X: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
                 => ( vEBT_invar_vebt @ X @ N2 ) )
              & ( vEBT_invar_vebt @ Summary3 @ N2 )
              & ( ( size_s6755466524823107622T_VEBT @ TreeList )
                = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
              & ( A22
                = ( plus_plus_nat @ N2 @ N2 ) )
              & ! [I2: nat] :
                  ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
                 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ X6 ) )
                    = ( vEBT_V8194947554948674370ptions @ Summary3 @ I2 ) ) )
              & ( ( Mi3 = Ma3 )
               => ! [X: vEBT_VEBT] :
                    ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
                   => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) )
              & ( ord_less_eq_nat @ Mi3 @ Ma3 )
              & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A22 ) )
              & ( ( Mi3 != Ma3 )
               => ! [I2: nat] :
                    ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
                   => ( ( ( ( vEBT_VEBT_high @ Ma3 @ N2 )
                          = I2 )
                       => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ ( vEBT_VEBT_low @ Ma3 @ N2 ) ) )
                      & ! [X: nat] :
                          ( ( ( ( vEBT_VEBT_high @ X @ N2 )
                              = I2 )
                            & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ ( vEBT_VEBT_low @ X @ N2 ) ) )
                         => ( ( ord_less_nat @ Mi3 @ X )
                            & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) )
          | ? [TreeList: list_VEBT_VEBT,N2: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
              ( ( A1
                = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ A22 @ TreeList @ Summary3 ) )
              & ! [X: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
                 => ( vEBT_invar_vebt @ X @ N2 ) )
              & ( vEBT_invar_vebt @ Summary3 @ ( suc @ N2 ) )
              & ( ( size_s6755466524823107622T_VEBT @ TreeList )
                = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) )
              & ( A22
                = ( plus_plus_nat @ N2 @ ( suc @ N2 ) ) )
              & ! [I2: nat] :
                  ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) )
                 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ X6 ) )
                    = ( vEBT_V8194947554948674370ptions @ Summary3 @ I2 ) ) )
              & ( ( Mi3 = Ma3 )
               => ! [X: vEBT_VEBT] :
                    ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
                   => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) )
              & ( ord_less_eq_nat @ Mi3 @ Ma3 )
              & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A22 ) )
              & ( ( Mi3 != Ma3 )
               => ! [I2: nat] :
                    ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) )
                   => ( ( ( ( vEBT_VEBT_high @ Ma3 @ N2 )
                          = I2 )
                       => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ ( vEBT_VEBT_low @ Ma3 @ N2 ) ) )
                      & ! [X: nat] :
                          ( ( ( ( vEBT_VEBT_high @ X @ N2 )
                              = I2 )
                            & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ ( vEBT_VEBT_low @ X @ N2 ) ) )
                         => ( ( ord_less_nat @ Mi3 @ X )
                            & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.simps
thf(fact_8722_invar__vebt_Ocases,axiom,
    ! [A12: vEBT_VEBT,A23: nat] :
      ( ( vEBT_invar_vebt @ A12 @ A23 )
     => ( ( ? [A5: $o,B5: $o] :
              ( A12
              = ( vEBT_Leaf @ A5 @ B5 ) )
         => ( A23
           != ( suc @ zero_zero_nat ) ) )
       => ( ! [TreeList3: list_VEBT_VEBT,N4: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat] :
              ( ( A12
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList3 @ Summary2 ) )
             => ( ( A23 = Deg2 )
               => ( ! [X5: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                     => ( vEBT_invar_vebt @ X5 @ N4 ) )
                 => ( ( vEBT_invar_vebt @ Summary2 @ M4 )
                   => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
                        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
                     => ( ( M4 = N4 )
                       => ( ( Deg2
                            = ( plus_plus_nat @ N4 @ M4 ) )
                         => ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_12 )
                           => ~ ! [X5: vEBT_VEBT] :
                                  ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                                 => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) ) ) ) ) ) ) ) ) )
         => ( ! [TreeList3: list_VEBT_VEBT,N4: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat] :
                ( ( A12
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList3 @ Summary2 ) )
               => ( ( A23 = Deg2 )
                 => ( ! [X5: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                       => ( vEBT_invar_vebt @ X5 @ N4 ) )
                   => ( ( vEBT_invar_vebt @ Summary2 @ M4 )
                     => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
                          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
                       => ( ( M4
                            = ( suc @ N4 ) )
                         => ( ( Deg2
                              = ( plus_plus_nat @ N4 @ M4 ) )
                           => ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_12 )
                             => ~ ! [X5: vEBT_VEBT] :
                                    ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                                   => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) ) ) ) ) ) ) ) ) )
           => ( ! [TreeList3: list_VEBT_VEBT,N4: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
                  ( ( A12
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList3 @ Summary2 ) )
                 => ( ( A23 = Deg2 )
                   => ( ! [X5: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                         => ( vEBT_invar_vebt @ X5 @ N4 ) )
                     => ( ( vEBT_invar_vebt @ Summary2 @ M4 )
                       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
                            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
                         => ( ( M4 = N4 )
                           => ( ( Deg2
                                = ( plus_plus_nat @ N4 @ M4 ) )
                             => ( ! [I4: nat] :
                                    ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
                                   => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ X6 ) )
                                      = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
                               => ( ( ( Mi2 = Ma2 )
                                   => ! [X5: vEBT_VEBT] :
                                        ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                                       => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) ) )
                                 => ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
                                   => ( ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                     => ~ ( ( Mi2 != Ma2 )
                                         => ! [I4: nat] :
                                              ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
                                             => ( ( ( ( vEBT_VEBT_high @ Ma2 @ N4 )
                                                    = I4 )
                                                 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ ( vEBT_VEBT_low @ Ma2 @ N4 ) ) )
                                                & ! [X5: nat] :
                                                    ( ( ( ( vEBT_VEBT_high @ X5 @ N4 )
                                                        = I4 )
                                                      & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ ( vEBT_VEBT_low @ X5 @ N4 ) ) )
                                                   => ( ( ord_less_nat @ Mi2 @ X5 )
                                                      & ( ord_less_eq_nat @ X5 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
             => ~ ! [TreeList3: list_VEBT_VEBT,N4: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
                    ( ( A12
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList3 @ Summary2 ) )
                   => ( ( A23 = Deg2 )
                     => ( ! [X5: vEBT_VEBT] :
                            ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                           => ( vEBT_invar_vebt @ X5 @ N4 ) )
                       => ( ( vEBT_invar_vebt @ Summary2 @ M4 )
                         => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
                              = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
                           => ( ( M4
                                = ( suc @ N4 ) )
                             => ( ( Deg2
                                  = ( plus_plus_nat @ N4 @ M4 ) )
                               => ( ! [I4: nat] :
                                      ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
                                     => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ X6 ) )
                                        = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
                                 => ( ( ( Mi2 = Ma2 )
                                     => ! [X5: vEBT_VEBT] :
                                          ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                                         => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) ) )
                                   => ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
                                     => ( ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                       => ~ ( ( Mi2 != Ma2 )
                                           => ! [I4: nat] :
                                                ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
                                               => ( ( ( ( vEBT_VEBT_high @ Ma2 @ N4 )
                                                      = I4 )
                                                   => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ ( vEBT_VEBT_low @ Ma2 @ N4 ) ) )
                                                  & ! [X5: nat] :
                                                      ( ( ( ( vEBT_VEBT_high @ X5 @ N4 )
                                                          = I4 )
                                                        & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ ( vEBT_VEBT_low @ X5 @ N4 ) ) )
                                                     => ( ( ord_less_nat @ Mi2 @ X5 )
                                                        & ( ord_less_eq_nat @ X5 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.cases
thf(fact_8723_invar__vebt_Ointros_I2_J,axiom,
    ! [TreeList2: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
         => ( vEBT_invar_vebt @ X3 @ N ) )
     => ( ( vEBT_invar_vebt @ Summary @ M )
       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
         => ( ( M = N )
           => ( ( Deg
                = ( plus_plus_nat @ N @ M ) )
             => ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_1 )
               => ( ! [X3: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                     => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) )
                 => ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(2)
thf(fact_8724_invar__vebt_Ointros_I3_J,axiom,
    ! [TreeList2: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
         => ( vEBT_invar_vebt @ X3 @ N ) )
     => ( ( vEBT_invar_vebt @ Summary @ M )
       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
         => ( ( M
              = ( suc @ N ) )
           => ( ( Deg
                = ( plus_plus_nat @ N @ M ) )
             => ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_1 )
               => ( ! [X3: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                     => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) )
                 => ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(3)
thf(fact_8725_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
    ! [V: nat,TreeList2: list_VEBT_VEBT,Vd: vEBT_VEBT,X4: nat] :
      ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V ) @ TreeList2 @ Vd ) @ X4 )
      = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
         => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X4 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
        & ( ord_less_nat @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ).

% VEBT_internal.membermima.simps(5)
thf(fact_8726_vebt__succ_Oelims,axiom,
    ! [X4: vEBT_VEBT,Xa: nat,Y3: option_nat] :
      ( ( ( vEBT_vebt_succ @ X4 @ Xa )
        = Y3 )
     => ( ! [Uu2: $o,B5: $o] :
            ( ( X4
              = ( vEBT_Leaf @ Uu2 @ B5 ) )
           => ( ( Xa = zero_zero_nat )
             => ~ ( ( B5
                   => ( Y3
                      = ( some_nat @ one_one_nat ) ) )
                  & ( ~ B5
                   => ( Y3 = none_nat ) ) ) ) )
       => ( ( ? [Uv: $o,Uw: $o] :
                ( X4
                = ( vEBT_Leaf @ Uv @ Uw ) )
           => ( ? [N4: nat] :
                  ( Xa
                  = ( suc @ N4 ) )
             => ( Y3 != none_nat ) ) )
         => ( ( ? [Ux: nat,Uy2: list_VEBT_VEBT,Uz: vEBT_VEBT] :
                  ( X4
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux @ Uy2 @ Uz ) )
             => ( Y3 != none_nat ) )
           => ( ( ? [V3: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                    ( X4
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V3 ) @ zero_zero_nat @ Vc2 @ Vd2 ) )
               => ( Y3 != none_nat ) )
             => ( ( ? [V3: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] :
                      ( X4
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V3 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) )
                 => ( Y3 != none_nat ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X4
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) )
                     => ~ ( ( ( ord_less_nat @ Xa @ Mi2 )
                           => ( Y3
                              = ( some_nat @ Mi2 ) ) )
                          & ( ~ ( ord_less_nat @ Xa @ Mi2 )
                           => ( Y3
                              = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                                @ ( if_option_nat
                                  @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                     != none_nat )
                                    & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                  @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                  @ ( if_option_nat
                                    @ ( ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                                      = none_nat )
                                    @ none_nat
                                    @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
                                @ none_nat ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_succ.elims
thf(fact_8727_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
    ! [X4: vEBT_VEBT,Xa: nat] :
      ( ~ ( vEBT_V5719532721284313246member @ X4 @ Xa )
     => ( ! [A5: $o,B5: $o] :
            ( ( X4
              = ( vEBT_Leaf @ A5 @ B5 ) )
           => ( ( ( Xa = zero_zero_nat )
               => A5 )
              & ( ( Xa != zero_zero_nat )
               => ( ( ( Xa = one_one_nat )
                   => B5 )
                  & ( Xa = one_one_nat ) ) ) ) )
       => ( ! [Uu2: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
              ( X4
             != ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv @ Uw ) )
         => ~ ! [Uy2: option4927543243414619207at_nat,V3: nat,TreeList3: list_VEBT_VEBT] :
                ( ? [S3: vEBT_VEBT] :
                    ( X4
                    = ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S3 ) )
               => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                   => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(3)
thf(fact_8728_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
    ! [X4: vEBT_VEBT,Xa: nat] :
      ( ( vEBT_V5719532721284313246member @ X4 @ Xa )
     => ( ! [A5: $o,B5: $o] :
            ( ( X4
              = ( vEBT_Leaf @ A5 @ B5 ) )
           => ~ ( ( ( Xa = zero_zero_nat )
                 => A5 )
                & ( ( Xa != zero_zero_nat )
                 => ( ( ( Xa = one_one_nat )
                     => B5 )
                    & ( Xa = one_one_nat ) ) ) ) )
       => ~ ! [Uy2: option4927543243414619207at_nat,V3: nat,TreeList3: list_VEBT_VEBT] :
              ( ? [S3: vEBT_VEBT] :
                  ( X4
                  = ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S3 ) )
             => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                   => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(2)
thf(fact_8729_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
    ! [X4: vEBT_VEBT,Xa: nat,Y3: $o] :
      ( ( ( vEBT_V5719532721284313246member @ X4 @ Xa )
        = Y3 )
     => ( ! [A5: $o,B5: $o] :
            ( ( X4
              = ( vEBT_Leaf @ A5 @ B5 ) )
           => ( Y3
              = ( ~ ( ( ( Xa = zero_zero_nat )
                     => A5 )
                    & ( ( Xa != zero_zero_nat )
                     => ( ( ( Xa = one_one_nat )
                         => B5 )
                        & ( Xa = one_one_nat ) ) ) ) ) ) )
       => ( ( ? [Uu2: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( X4
                = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv @ Uw ) )
           => Y3 )
         => ~ ! [Uy2: option4927543243414619207at_nat,V3: nat,TreeList3: list_VEBT_VEBT] :
                ( ? [S3: vEBT_VEBT] :
                    ( X4
                    = ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S3 ) )
               => ( Y3
                  = ( ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                         => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(1)
thf(fact_8730_vebt__maxt_Oelims,axiom,
    ! [X4: vEBT_VEBT,Y3: option_nat] :
      ( ( ( vEBT_vebt_maxt @ X4 )
        = Y3 )
     => ( ! [A5: $o,B5: $o] :
            ( ( X4
              = ( vEBT_Leaf @ A5 @ B5 ) )
           => ~ ( ( B5
                 => ( Y3
                    = ( some_nat @ one_one_nat ) ) )
                & ( ~ B5
                 => ( ( A5
                     => ( Y3
                        = ( some_nat @ zero_zero_nat ) ) )
                    & ( ~ A5
                     => ( Y3 = none_nat ) ) ) ) ) )
       => ( ( ? [Uu2: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( X4
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv @ Uw ) )
           => ( Y3 != none_nat ) )
         => ~ ! [Mi2: nat,Ma2: nat] :
                ( ? [Ux: nat,Uy2: list_VEBT_VEBT,Uz: vEBT_VEBT] :
                    ( X4
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy2 @ Uz ) )
               => ( Y3
                 != ( some_nat @ Ma2 ) ) ) ) ) ) ).

% vebt_maxt.elims
thf(fact_8731_vebt__mint_Oelims,axiom,
    ! [X4: vEBT_VEBT,Y3: option_nat] :
      ( ( ( vEBT_vebt_mint @ X4 )
        = Y3 )
     => ( ! [A5: $o,B5: $o] :
            ( ( X4
              = ( vEBT_Leaf @ A5 @ B5 ) )
           => ~ ( ( A5
                 => ( Y3
                    = ( some_nat @ zero_zero_nat ) ) )
                & ( ~ A5
                 => ( ( B5
                     => ( Y3
                        = ( some_nat @ one_one_nat ) ) )
                    & ( ~ B5
                     => ( Y3 = none_nat ) ) ) ) ) )
       => ( ( ? [Uu2: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( X4
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv @ Uw ) )
           => ( Y3 != none_nat ) )
         => ~ ! [Mi2: nat] :
                ( ? [Ma2: nat,Ux: nat,Uy2: list_VEBT_VEBT,Uz: vEBT_VEBT] :
                    ( X4
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy2 @ Uz ) )
               => ( Y3
                 != ( some_nat @ Mi2 ) ) ) ) ) ) ).

% vebt_mint.elims
thf(fact_8732_prod_Oempty,axiom,
    ! [G: extended_enat > complex] :
      ( ( groups4622424608036095791omplex @ G @ bot_bo7653980558646680370d_enat )
      = one_one_complex ) ).

% prod.empty
thf(fact_8733_prod_Oempty,axiom,
    ! [G: extended_enat > real] :
      ( ( groups97031904164794029t_real @ G @ bot_bo7653980558646680370d_enat )
      = one_one_real ) ).

% prod.empty
thf(fact_8734_prod_Oempty,axiom,
    ! [G: extended_enat > rat] :
      ( ( groups2245840878043517529at_rat @ G @ bot_bo7653980558646680370d_enat )
      = one_one_rat ) ).

% prod.empty
thf(fact_8735_prod_Oempty,axiom,
    ! [G: extended_enat > nat] :
      ( ( groups2880970938130013265at_nat @ G @ bot_bo7653980558646680370d_enat )
      = one_one_nat ) ).

% prod.empty
thf(fact_8736_prod_Oempty,axiom,
    ! [G: extended_enat > int] :
      ( ( groups2878480467620962989at_int @ G @ bot_bo7653980558646680370d_enat )
      = one_one_int ) ).

% prod.empty
thf(fact_8737_prod_Oempty,axiom,
    ! [G: real > complex] :
      ( ( groups713298508707869441omplex @ G @ bot_bot_set_real )
      = one_one_complex ) ).

% prod.empty
thf(fact_8738_prod_Oempty,axiom,
    ! [G: real > real] :
      ( ( groups1681761925125756287l_real @ G @ bot_bot_set_real )
      = one_one_real ) ).

% prod.empty
thf(fact_8739_prod_Oempty,axiom,
    ! [G: real > rat] :
      ( ( groups4061424788464935467al_rat @ G @ bot_bot_set_real )
      = one_one_rat ) ).

% prod.empty
thf(fact_8740_prod_Oempty,axiom,
    ! [G: real > nat] :
      ( ( groups4696554848551431203al_nat @ G @ bot_bot_set_real )
      = one_one_nat ) ).

% prod.empty
thf(fact_8741_prod_Oempty,axiom,
    ! [G: real > int] :
      ( ( groups4694064378042380927al_int @ G @ bot_bot_set_real )
      = one_one_int ) ).

% prod.empty
thf(fact_8742_vebt__maxt_Osimps_I1_J,axiom,
    ! [B2: $o,A: $o] :
      ( ( B2
       => ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A @ B2 ) )
          = ( some_nat @ one_one_nat ) ) )
      & ( ~ B2
       => ( ( A
           => ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A @ B2 ) )
              = ( some_nat @ zero_zero_nat ) ) )
          & ( ~ A
           => ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A @ B2 ) )
              = none_nat ) ) ) ) ) ).

% vebt_maxt.simps(1)
thf(fact_8743_prod_Oneutral__const,axiom,
    ! [A3: set_nat] :
      ( ( groups708209901874060359at_nat
        @ ^ [Uu: nat] : one_one_nat
        @ A3 )
      = one_one_nat ) ).

% prod.neutral_const
thf(fact_8744_prod_Oneutral__const,axiom,
    ! [A3: set_nat] :
      ( ( groups705719431365010083at_int
        @ ^ [Uu: nat] : one_one_int
        @ A3 )
      = one_one_int ) ).

% prod.neutral_const
thf(fact_8745_prod_Oneutral__const,axiom,
    ! [A3: set_int] :
      ( ( groups1705073143266064639nt_int
        @ ^ [Uu: int] : one_one_int
        @ A3 )
      = one_one_int ) ).

% prod.neutral_const
thf(fact_8746_prod__int__eq,axiom,
    ! [I: nat,J: nat] :
      ( ( groups705719431365010083at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ I @ J ) )
      = ( groups1705073143266064639nt_int
        @ ^ [X: int] : X
        @ ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J ) ) ) ) ).

% prod_int_eq
thf(fact_8747_prod__int__plus__eq,axiom,
    ! [I: nat,J: nat] :
      ( ( groups705719431365010083at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ I @ ( plus_plus_nat @ I @ J ) ) )
      = ( groups1705073143266064639nt_int
        @ ^ [X: int] : X
        @ ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ I @ J ) ) ) ) ) ).

% prod_int_plus_eq
thf(fact_8748_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: extended_enat > complex,A3: set_Extended_enat] :
      ( ( ( groups4622424608036095791omplex @ G @ A3 )
       != one_one_complex )
     => ~ ! [A5: extended_enat] :
            ( ( member_Extended_enat @ A5 @ A3 )
           => ( ( G @ A5 )
              = one_one_complex ) ) ) ).

% prod.not_neutral_contains_not_neutral
thf(fact_8749_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: complex > complex,A3: set_complex] :
      ( ( ( groups3708469109370488835omplex @ G @ A3 )
       != one_one_complex )
     => ~ ! [A5: complex] :
            ( ( member_complex @ A5 @ A3 )
           => ( ( G @ A5 )
              = one_one_complex ) ) ) ).

% prod.not_neutral_contains_not_neutral
thf(fact_8750_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: real > complex,A3: set_real] :
      ( ( ( groups713298508707869441omplex @ G @ A3 )
       != one_one_complex )
     => ~ ! [A5: real] :
            ( ( member_real @ A5 @ A3 )
           => ( ( G @ A5 )
              = one_one_complex ) ) ) ).

% prod.not_neutral_contains_not_neutral
thf(fact_8751_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: nat > complex,A3: set_nat] :
      ( ( ( groups6464643781859351333omplex @ G @ A3 )
       != one_one_complex )
     => ~ ! [A5: nat] :
            ( ( member_nat @ A5 @ A3 )
           => ( ( G @ A5 )
              = one_one_complex ) ) ) ).

% prod.not_neutral_contains_not_neutral
thf(fact_8752_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: int > complex,A3: set_int] :
      ( ( ( groups7440179247065528705omplex @ G @ A3 )
       != one_one_complex )
     => ~ ! [A5: int] :
            ( ( member_int @ A5 @ A3 )
           => ( ( G @ A5 )
              = one_one_complex ) ) ) ).

% prod.not_neutral_contains_not_neutral
thf(fact_8753_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: extended_enat > real,A3: set_Extended_enat] :
      ( ( ( groups97031904164794029t_real @ G @ A3 )
       != one_one_real )
     => ~ ! [A5: extended_enat] :
            ( ( member_Extended_enat @ A5 @ A3 )
           => ( ( G @ A5 )
              = one_one_real ) ) ) ).

% prod.not_neutral_contains_not_neutral
thf(fact_8754_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: complex > real,A3: set_complex] :
      ( ( ( groups766887009212190081x_real @ G @ A3 )
       != one_one_real )
     => ~ ! [A5: complex] :
            ( ( member_complex @ A5 @ A3 )
           => ( ( G @ A5 )
              = one_one_real ) ) ) ).

% prod.not_neutral_contains_not_neutral
thf(fact_8755_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: real > real,A3: set_real] :
      ( ( ( groups1681761925125756287l_real @ G @ A3 )
       != one_one_real )
     => ~ ! [A5: real] :
            ( ( member_real @ A5 @ A3 )
           => ( ( G @ A5 )
              = one_one_real ) ) ) ).

% prod.not_neutral_contains_not_neutral
thf(fact_8756_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: nat > real,A3: set_nat] :
      ( ( ( groups129246275422532515t_real @ G @ A3 )
       != one_one_real )
     => ~ ! [A5: nat] :
            ( ( member_nat @ A5 @ A3 )
           => ( ( G @ A5 )
              = one_one_real ) ) ) ).

% prod.not_neutral_contains_not_neutral
thf(fact_8757_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: int > real,A3: set_int] :
      ( ( ( groups2316167850115554303t_real @ G @ A3 )
       != one_one_real )
     => ~ ! [A5: int] :
            ( ( member_int @ A5 @ A3 )
           => ( ( G @ A5 )
              = one_one_real ) ) ) ).

% prod.not_neutral_contains_not_neutral
thf(fact_8758_prod_Oneutral,axiom,
    ! [A3: set_nat,G: nat > nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A3 )
         => ( ( G @ X3 )
            = one_one_nat ) )
     => ( ( groups708209901874060359at_nat @ G @ A3 )
        = one_one_nat ) ) ).

% prod.neutral
thf(fact_8759_prod_Oneutral,axiom,
    ! [A3: set_nat,G: nat > int] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A3 )
         => ( ( G @ X3 )
            = one_one_int ) )
     => ( ( groups705719431365010083at_int @ G @ A3 )
        = one_one_int ) ) ).

% prod.neutral
thf(fact_8760_prod_Oneutral,axiom,
    ! [A3: set_int,G: int > int] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A3 )
         => ( ( G @ X3 )
            = one_one_int ) )
     => ( ( groups1705073143266064639nt_int @ G @ A3 )
        = one_one_int ) ) ).

% prod.neutral
thf(fact_8761_prod_Odistrib,axiom,
    ! [G: nat > nat,H2: nat > nat,A3: set_nat] :
      ( ( groups708209901874060359at_nat
        @ ^ [X: nat] : ( times_times_nat @ ( G @ X ) @ ( H2 @ X ) )
        @ A3 )
      = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ A3 ) @ ( groups708209901874060359at_nat @ H2 @ A3 ) ) ) ).

% prod.distrib
thf(fact_8762_prod_Odistrib,axiom,
    ! [G: nat > int,H2: nat > int,A3: set_nat] :
      ( ( groups705719431365010083at_int
        @ ^ [X: nat] : ( times_times_int @ ( G @ X ) @ ( H2 @ X ) )
        @ A3 )
      = ( times_times_int @ ( groups705719431365010083at_int @ G @ A3 ) @ ( groups705719431365010083at_int @ H2 @ A3 ) ) ) ).

% prod.distrib
thf(fact_8763_prod_Odistrib,axiom,
    ! [G: int > int,H2: int > int,A3: set_int] :
      ( ( groups1705073143266064639nt_int
        @ ^ [X: int] : ( times_times_int @ ( G @ X ) @ ( H2 @ X ) )
        @ A3 )
      = ( times_times_int @ ( groups1705073143266064639nt_int @ G @ A3 ) @ ( groups1705073143266064639nt_int @ H2 @ A3 ) ) ) ).

% prod.distrib
thf(fact_8764_prod__power__distrib,axiom,
    ! [F: nat > nat,A3: set_nat,N: nat] :
      ( ( power_power_nat @ ( groups708209901874060359at_nat @ F @ A3 ) @ N )
      = ( groups708209901874060359at_nat
        @ ^ [X: nat] : ( power_power_nat @ ( F @ X ) @ N )
        @ A3 ) ) ).

% prod_power_distrib
thf(fact_8765_prod__power__distrib,axiom,
    ! [F: nat > int,A3: set_nat,N: nat] :
      ( ( power_power_int @ ( groups705719431365010083at_int @ F @ A3 ) @ N )
      = ( groups705719431365010083at_int
        @ ^ [X: nat] : ( power_power_int @ ( F @ X ) @ N )
        @ A3 ) ) ).

% prod_power_distrib
thf(fact_8766_prod__power__distrib,axiom,
    ! [F: int > int,A3: set_int,N: nat] :
      ( ( power_power_int @ ( groups1705073143266064639nt_int @ F @ A3 ) @ N )
      = ( groups1705073143266064639nt_int
        @ ^ [X: int] : ( power_power_int @ ( F @ X ) @ N )
        @ A3 ) ) ).

% prod_power_distrib
thf(fact_8767_prod__nonneg,axiom,
    ! [A3: set_nat,F: nat > nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A3 )
         => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( groups708209901874060359at_nat @ F @ A3 ) ) ) ).

% prod_nonneg
thf(fact_8768_prod__nonneg,axiom,
    ! [A3: set_nat,F: nat > int] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A3 )
         => ( ord_less_eq_int @ zero_zero_int @ ( F @ X3 ) ) )
     => ( ord_less_eq_int @ zero_zero_int @ ( groups705719431365010083at_int @ F @ A3 ) ) ) ).

% prod_nonneg
thf(fact_8769_prod__nonneg,axiom,
    ! [A3: set_int,F: int > int] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A3 )
         => ( ord_less_eq_int @ zero_zero_int @ ( F @ X3 ) ) )
     => ( ord_less_eq_int @ zero_zero_int @ ( groups1705073143266064639nt_int @ F @ A3 ) ) ) ).

% prod_nonneg
thf(fact_8770_prod__mono,axiom,
    ! [A3: set_Extended_enat,F: extended_enat > real,G: extended_enat > real] :
      ( ! [I3: extended_enat] :
          ( ( member_Extended_enat @ I3 @ A3 )
         => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) )
            & ( ord_less_eq_real @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
     => ( ord_less_eq_real @ ( groups97031904164794029t_real @ F @ A3 ) @ ( groups97031904164794029t_real @ G @ A3 ) ) ) ).

% prod_mono
thf(fact_8771_prod__mono,axiom,
    ! [A3: set_complex,F: complex > real,G: complex > real] :
      ( ! [I3: complex] :
          ( ( member_complex @ I3 @ A3 )
         => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) )
            & ( ord_less_eq_real @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
     => ( ord_less_eq_real @ ( groups766887009212190081x_real @ F @ A3 ) @ ( groups766887009212190081x_real @ G @ A3 ) ) ) ).

% prod_mono
thf(fact_8772_prod__mono,axiom,
    ! [A3: set_real,F: real > real,G: real > real] :
      ( ! [I3: real] :
          ( ( member_real @ I3 @ A3 )
         => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) )
            & ( ord_less_eq_real @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
     => ( ord_less_eq_real @ ( groups1681761925125756287l_real @ F @ A3 ) @ ( groups1681761925125756287l_real @ G @ A3 ) ) ) ).

% prod_mono
thf(fact_8773_prod__mono,axiom,
    ! [A3: set_nat,F: nat > real,G: nat > real] :
      ( ! [I3: nat] :
          ( ( member_nat @ I3 @ A3 )
         => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) )
            & ( ord_less_eq_real @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
     => ( ord_less_eq_real @ ( groups129246275422532515t_real @ F @ A3 ) @ ( groups129246275422532515t_real @ G @ A3 ) ) ) ).

% prod_mono
thf(fact_8774_prod__mono,axiom,
    ! [A3: set_int,F: int > real,G: int > real] :
      ( ! [I3: int] :
          ( ( member_int @ I3 @ A3 )
         => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) )
            & ( ord_less_eq_real @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
     => ( ord_less_eq_real @ ( groups2316167850115554303t_real @ F @ A3 ) @ ( groups2316167850115554303t_real @ G @ A3 ) ) ) ).

% prod_mono
thf(fact_8775_prod__mono,axiom,
    ! [A3: set_Extended_enat,F: extended_enat > rat,G: extended_enat > rat] :
      ( ! [I3: extended_enat] :
          ( ( member_Extended_enat @ I3 @ A3 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) )
            & ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
     => ( ord_less_eq_rat @ ( groups2245840878043517529at_rat @ F @ A3 ) @ ( groups2245840878043517529at_rat @ G @ A3 ) ) ) ).

% prod_mono
thf(fact_8776_prod__mono,axiom,
    ! [A3: set_complex,F: complex > rat,G: complex > rat] :
      ( ! [I3: complex] :
          ( ( member_complex @ I3 @ A3 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) )
            & ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
     => ( ord_less_eq_rat @ ( groups225925009352817453ex_rat @ F @ A3 ) @ ( groups225925009352817453ex_rat @ G @ A3 ) ) ) ).

% prod_mono
thf(fact_8777_prod__mono,axiom,
    ! [A3: set_real,F: real > rat,G: real > rat] :
      ( ! [I3: real] :
          ( ( member_real @ I3 @ A3 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) )
            & ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
     => ( ord_less_eq_rat @ ( groups4061424788464935467al_rat @ F @ A3 ) @ ( groups4061424788464935467al_rat @ G @ A3 ) ) ) ).

% prod_mono
thf(fact_8778_prod__mono,axiom,
    ! [A3: set_nat,F: nat > rat,G: nat > rat] :
      ( ! [I3: nat] :
          ( ( member_nat @ I3 @ A3 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) )
            & ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
     => ( ord_less_eq_rat @ ( groups73079841787564623at_rat @ F @ A3 ) @ ( groups73079841787564623at_rat @ G @ A3 ) ) ) ).

% prod_mono
thf(fact_8779_prod__mono,axiom,
    ! [A3: set_int,F: int > rat,G: int > rat] :
      ( ! [I3: int] :
          ( ( member_int @ I3 @ A3 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) )
            & ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
     => ( ord_less_eq_rat @ ( groups1072433553688619179nt_rat @ F @ A3 ) @ ( groups1072433553688619179nt_rat @ G @ A3 ) ) ) ).

% prod_mono
thf(fact_8780_prod__pos,axiom,
    ! [A3: set_nat,F: nat > nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A3 )
         => ( ord_less_nat @ zero_zero_nat @ ( F @ X3 ) ) )
     => ( ord_less_nat @ zero_zero_nat @ ( groups708209901874060359at_nat @ F @ A3 ) ) ) ).

% prod_pos
thf(fact_8781_prod__pos,axiom,
    ! [A3: set_nat,F: nat > int] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A3 )
         => ( ord_less_int @ zero_zero_int @ ( F @ X3 ) ) )
     => ( ord_less_int @ zero_zero_int @ ( groups705719431365010083at_int @ F @ A3 ) ) ) ).

% prod_pos
thf(fact_8782_prod__pos,axiom,
    ! [A3: set_int,F: int > int] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A3 )
         => ( ord_less_int @ zero_zero_int @ ( F @ X3 ) ) )
     => ( ord_less_int @ zero_zero_int @ ( groups1705073143266064639nt_int @ F @ A3 ) ) ) ).

% prod_pos
thf(fact_8783_prod__ge__1,axiom,
    ! [A3: set_Extended_enat,F: extended_enat > real] :
      ( ! [X3: extended_enat] :
          ( ( member_Extended_enat @ X3 @ A3 )
         => ( ord_less_eq_real @ one_one_real @ ( F @ X3 ) ) )
     => ( ord_less_eq_real @ one_one_real @ ( groups97031904164794029t_real @ F @ A3 ) ) ) ).

% prod_ge_1
thf(fact_8784_prod__ge__1,axiom,
    ! [A3: set_complex,F: complex > real] :
      ( ! [X3: complex] :
          ( ( member_complex @ X3 @ A3 )
         => ( ord_less_eq_real @ one_one_real @ ( F @ X3 ) ) )
     => ( ord_less_eq_real @ one_one_real @ ( groups766887009212190081x_real @ F @ A3 ) ) ) ).

% prod_ge_1
thf(fact_8785_prod__ge__1,axiom,
    ! [A3: set_real,F: real > real] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A3 )
         => ( ord_less_eq_real @ one_one_real @ ( F @ X3 ) ) )
     => ( ord_less_eq_real @ one_one_real @ ( groups1681761925125756287l_real @ F @ A3 ) ) ) ).

% prod_ge_1
thf(fact_8786_prod__ge__1,axiom,
    ! [A3: set_nat,F: nat > real] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A3 )
         => ( ord_less_eq_real @ one_one_real @ ( F @ X3 ) ) )
     => ( ord_less_eq_real @ one_one_real @ ( groups129246275422532515t_real @ F @ A3 ) ) ) ).

% prod_ge_1
thf(fact_8787_prod__ge__1,axiom,
    ! [A3: set_int,F: int > real] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A3 )
         => ( ord_less_eq_real @ one_one_real @ ( F @ X3 ) ) )
     => ( ord_less_eq_real @ one_one_real @ ( groups2316167850115554303t_real @ F @ A3 ) ) ) ).

% prod_ge_1
thf(fact_8788_prod__ge__1,axiom,
    ! [A3: set_Extended_enat,F: extended_enat > rat] :
      ( ! [X3: extended_enat] :
          ( ( member_Extended_enat @ X3 @ A3 )
         => ( ord_less_eq_rat @ one_one_rat @ ( F @ X3 ) ) )
     => ( ord_less_eq_rat @ one_one_rat @ ( groups2245840878043517529at_rat @ F @ A3 ) ) ) ).

% prod_ge_1
thf(fact_8789_prod__ge__1,axiom,
    ! [A3: set_complex,F: complex > rat] :
      ( ! [X3: complex] :
          ( ( member_complex @ X3 @ A3 )
         => ( ord_less_eq_rat @ one_one_rat @ ( F @ X3 ) ) )
     => ( ord_less_eq_rat @ one_one_rat @ ( groups225925009352817453ex_rat @ F @ A3 ) ) ) ).

% prod_ge_1
thf(fact_8790_prod__ge__1,axiom,
    ! [A3: set_real,F: real > rat] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A3 )
         => ( ord_less_eq_rat @ one_one_rat @ ( F @ X3 ) ) )
     => ( ord_less_eq_rat @ one_one_rat @ ( groups4061424788464935467al_rat @ F @ A3 ) ) ) ).

% prod_ge_1
thf(fact_8791_prod__ge__1,axiom,
    ! [A3: set_nat,F: nat > rat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A3 )
         => ( ord_less_eq_rat @ one_one_rat @ ( F @ X3 ) ) )
     => ( ord_less_eq_rat @ one_one_rat @ ( groups73079841787564623at_rat @ F @ A3 ) ) ) ).

% prod_ge_1
thf(fact_8792_prod__ge__1,axiom,
    ! [A3: set_int,F: int > rat] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A3 )
         => ( ord_less_eq_rat @ one_one_rat @ ( F @ X3 ) ) )
     => ( ord_less_eq_rat @ one_one_rat @ ( groups1072433553688619179nt_rat @ F @ A3 ) ) ) ).

% prod_ge_1
thf(fact_8793_prod__le__1,axiom,
    ! [A3: set_Extended_enat,F: extended_enat > real] :
      ( ! [X3: extended_enat] :
          ( ( member_Extended_enat @ X3 @ A3 )
         => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) )
            & ( ord_less_eq_real @ ( F @ X3 ) @ one_one_real ) ) )
     => ( ord_less_eq_real @ ( groups97031904164794029t_real @ F @ A3 ) @ one_one_real ) ) ).

% prod_le_1
thf(fact_8794_prod__le__1,axiom,
    ! [A3: set_complex,F: complex > real] :
      ( ! [X3: complex] :
          ( ( member_complex @ X3 @ A3 )
         => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) )
            & ( ord_less_eq_real @ ( F @ X3 ) @ one_one_real ) ) )
     => ( ord_less_eq_real @ ( groups766887009212190081x_real @ F @ A3 ) @ one_one_real ) ) ).

% prod_le_1
thf(fact_8795_prod__le__1,axiom,
    ! [A3: set_real,F: real > real] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A3 )
         => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) )
            & ( ord_less_eq_real @ ( F @ X3 ) @ one_one_real ) ) )
     => ( ord_less_eq_real @ ( groups1681761925125756287l_real @ F @ A3 ) @ one_one_real ) ) ).

% prod_le_1
thf(fact_8796_prod__le__1,axiom,
    ! [A3: set_nat,F: nat > real] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A3 )
         => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) )
            & ( ord_less_eq_real @ ( F @ X3 ) @ one_one_real ) ) )
     => ( ord_less_eq_real @ ( groups129246275422532515t_real @ F @ A3 ) @ one_one_real ) ) ).

% prod_le_1
thf(fact_8797_prod__le__1,axiom,
    ! [A3: set_int,F: int > real] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A3 )
         => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) )
            & ( ord_less_eq_real @ ( F @ X3 ) @ one_one_real ) ) )
     => ( ord_less_eq_real @ ( groups2316167850115554303t_real @ F @ A3 ) @ one_one_real ) ) ).

% prod_le_1
thf(fact_8798_prod__le__1,axiom,
    ! [A3: set_Extended_enat,F: extended_enat > rat] :
      ( ! [X3: extended_enat] :
          ( ( member_Extended_enat @ X3 @ A3 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) )
            & ( ord_less_eq_rat @ ( F @ X3 ) @ one_one_rat ) ) )
     => ( ord_less_eq_rat @ ( groups2245840878043517529at_rat @ F @ A3 ) @ one_one_rat ) ) ).

% prod_le_1
thf(fact_8799_prod__le__1,axiom,
    ! [A3: set_complex,F: complex > rat] :
      ( ! [X3: complex] :
          ( ( member_complex @ X3 @ A3 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) )
            & ( ord_less_eq_rat @ ( F @ X3 ) @ one_one_rat ) ) )
     => ( ord_less_eq_rat @ ( groups225925009352817453ex_rat @ F @ A3 ) @ one_one_rat ) ) ).

% prod_le_1
thf(fact_8800_prod__le__1,axiom,
    ! [A3: set_real,F: real > rat] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A3 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) )
            & ( ord_less_eq_rat @ ( F @ X3 ) @ one_one_rat ) ) )
     => ( ord_less_eq_rat @ ( groups4061424788464935467al_rat @ F @ A3 ) @ one_one_rat ) ) ).

% prod_le_1
thf(fact_8801_prod__le__1,axiom,
    ! [A3: set_nat,F: nat > rat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A3 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) )
            & ( ord_less_eq_rat @ ( F @ X3 ) @ one_one_rat ) ) )
     => ( ord_less_eq_rat @ ( groups73079841787564623at_rat @ F @ A3 ) @ one_one_rat ) ) ).

% prod_le_1
thf(fact_8802_prod__le__1,axiom,
    ! [A3: set_int,F: int > rat] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A3 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) )
            & ( ord_less_eq_rat @ ( F @ X3 ) @ one_one_rat ) ) )
     => ( ord_less_eq_rat @ ( groups1072433553688619179nt_rat @ F @ A3 ) @ one_one_rat ) ) ).

% prod_le_1
thf(fact_8803_vebt__mint_Osimps_I1_J,axiom,
    ! [A: $o,B2: $o] :
      ( ( A
       => ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A @ B2 ) )
          = ( some_nat @ zero_zero_nat ) ) )
      & ( ~ A
       => ( ( B2
           => ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A @ B2 ) )
              = ( some_nat @ one_one_nat ) ) )
          & ( ~ B2
           => ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A @ B2 ) )
              = none_nat ) ) ) ) ) ).

% vebt_mint.simps(1)
thf(fact_8804_vebt__succ_Opelims,axiom,
    ! [X4: vEBT_VEBT,Xa: nat,Y3: option_nat] :
      ( ( ( vEBT_vebt_succ @ X4 @ Xa )
        = Y3 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ X4 @ Xa ) )
       => ( ! [Uu2: $o,B5: $o] :
              ( ( X4
                = ( vEBT_Leaf @ Uu2 @ B5 ) )
             => ( ( Xa = zero_zero_nat )
               => ( ( ( B5
                     => ( Y3
                        = ( some_nat @ one_one_nat ) ) )
                    & ( ~ B5
                     => ( Y3 = none_nat ) ) )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ B5 ) @ zero_zero_nat ) ) ) ) )
         => ( ! [Uv: $o,Uw: $o] :
                ( ( X4
                  = ( vEBT_Leaf @ Uv @ Uw ) )
               => ! [N4: nat] :
                    ( ( Xa
                      = ( suc @ N4 ) )
                   => ( ( Y3 = none_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uv @ Uw ) @ ( suc @ N4 ) ) ) ) ) )
           => ( ! [Ux: nat,Uy2: list_VEBT_VEBT,Uz: vEBT_VEBT] :
                  ( ( X4
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux @ Uy2 @ Uz ) )
                 => ( ( Y3 = none_nat )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux @ Uy2 @ Uz ) @ Xa ) ) ) )
             => ( ! [V3: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                    ( ( X4
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V3 ) @ zero_zero_nat @ Vc2 @ Vd2 ) )
                   => ( ( Y3 = none_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V3 ) @ zero_zero_nat @ Vc2 @ Vd2 ) @ Xa ) ) ) )
               => ( ! [V3: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] :
                      ( ( X4
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V3 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) )
                     => ( ( Y3 = none_nat )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V3 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) @ Xa ) ) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X4
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) )
                       => ( ( ( ( ord_less_nat @ Xa @ Mi2 )
                             => ( Y3
                                = ( some_nat @ Mi2 ) ) )
                            & ( ~ ( ord_less_nat @ Xa @ Mi2 )
                             => ( Y3
                                = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                                  @ ( if_option_nat
                                    @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                       != none_nat )
                                      & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                    @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                    @ ( if_option_nat
                                      @ ( ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                                        = none_nat )
                                      @ none_nat
                                      @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
                                  @ none_nat ) ) ) )
                         => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_succ.pelims
thf(fact_8805_ln__series,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( ord_less_real @ X4 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
       => ( ( ln_ln_real @ X4 )
          = ( suminf_real
            @ ^ [N2: nat] : ( times_times_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) @ ( power_power_real @ ( minus_minus_real @ X4 @ one_one_real ) @ ( suc @ N2 ) ) ) ) ) ) ) ).

% ln_series
thf(fact_8806_arctan__series,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X4 ) @ one_one_real )
     => ( ( arctan @ X4 )
        = ( suminf_real
          @ ^ [K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X4 @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ) ) ).

% arctan_series
thf(fact_8807_divmod__algorithm__code_I7_J,axiom,
    ! [M: num,N: num] :
      ( ( ( ord_less_eq_num @ M @ N )
       => ( ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit1 @ N ) )
          = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) ) ) )
      & ( ~ ( ord_less_eq_num @ M @ N )
       => ( ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit1 @ N ) )
          = ( unique5026877609467782581ep_nat @ ( bit1 @ N ) @ ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).

% divmod_algorithm_code(7)
thf(fact_8808_divmod__algorithm__code_I7_J,axiom,
    ! [M: num,N: num] :
      ( ( ( ord_less_eq_num @ M @ N )
       => ( ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit1 @ N ) )
          = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) ) ) )
      & ( ~ ( ord_less_eq_num @ M @ N )
       => ( ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit1 @ N ) )
          = ( unique5024387138958732305ep_int @ ( bit1 @ N ) @ ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).

% divmod_algorithm_code(7)
thf(fact_8809_divmod__algorithm__code_I7_J,axiom,
    ! [M: num,N: num] :
      ( ( ( ord_less_eq_num @ M @ N )
       => ( ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit1 @ N ) )
          = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) ) ) )
      & ( ~ ( ord_less_eq_num @ M @ N )
       => ( ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit1 @ N ) )
          = ( unique4921790084139445826nteger @ ( bit1 @ N ) @ ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).

% divmod_algorithm_code(7)
thf(fact_8810_divmod__algorithm__code_I2_J,axiom,
    ! [M: num] :
      ( ( unique5052692396658037445od_int @ M @ one )
      = ( product_Pair_int_int @ ( numeral_numeral_int @ M ) @ zero_zero_int ) ) ).

% divmod_algorithm_code(2)
thf(fact_8811_divmod__algorithm__code_I2_J,axiom,
    ! [M: num] :
      ( ( unique5055182867167087721od_nat @ M @ one )
      = ( product_Pair_nat_nat @ ( numeral_numeral_nat @ M ) @ zero_zero_nat ) ) ).

% divmod_algorithm_code(2)
thf(fact_8812_divmod__algorithm__code_I2_J,axiom,
    ! [M: num] :
      ( ( unique3479559517661332726nteger @ M @ one )
      = ( produc1086072967326762835nteger @ ( numera6620942414471956472nteger @ M ) @ zero_z3403309356797280102nteger ) ) ).

% divmod_algorithm_code(2)
thf(fact_8813_powser__zero,axiom,
    ! [F: nat > complex] :
      ( ( suminf_complex
        @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ zero_zero_complex @ N2 ) ) )
      = ( F @ zero_zero_nat ) ) ).

% powser_zero
thf(fact_8814_powser__zero,axiom,
    ! [F: nat > real] :
      ( ( suminf_real
        @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ zero_zero_real @ N2 ) ) )
      = ( F @ zero_zero_nat ) ) ).

% powser_zero
thf(fact_8815_divmod__algorithm__code_I3_J,axiom,
    ! [N: num] :
      ( ( unique5052692396658037445od_int @ one @ ( bit0 @ N ) )
      = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ one ) ) ) ).

% divmod_algorithm_code(3)
thf(fact_8816_divmod__algorithm__code_I3_J,axiom,
    ! [N: num] :
      ( ( unique5055182867167087721od_nat @ one @ ( bit0 @ N ) )
      = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ one ) ) ) ).

% divmod_algorithm_code(3)
thf(fact_8817_divmod__algorithm__code_I3_J,axiom,
    ! [N: num] :
      ( ( unique3479559517661332726nteger @ one @ ( bit0 @ N ) )
      = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ one ) ) ) ).

% divmod_algorithm_code(3)
thf(fact_8818_divmod__algorithm__code_I4_J,axiom,
    ! [N: num] :
      ( ( unique5052692396658037445od_int @ one @ ( bit1 @ N ) )
      = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ one ) ) ) ).

% divmod_algorithm_code(4)
thf(fact_8819_divmod__algorithm__code_I4_J,axiom,
    ! [N: num] :
      ( ( unique5055182867167087721od_nat @ one @ ( bit1 @ N ) )
      = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ one ) ) ) ).

% divmod_algorithm_code(4)
thf(fact_8820_divmod__algorithm__code_I4_J,axiom,
    ! [N: num] :
      ( ( unique3479559517661332726nteger @ one @ ( bit1 @ N ) )
      = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ one ) ) ) ).

% divmod_algorithm_code(4)
thf(fact_8821_divmod__algorithm__code_I8_J,axiom,
    ! [M: num,N: num] :
      ( ( ( ord_less_num @ M @ N )
       => ( ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit1 @ N ) )
          = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) ) ) )
      & ( ~ ( ord_less_num @ M @ N )
       => ( ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit1 @ N ) )
          = ( unique5026877609467782581ep_nat @ ( bit1 @ N ) @ ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).

% divmod_algorithm_code(8)
thf(fact_8822_divmod__algorithm__code_I8_J,axiom,
    ! [M: num,N: num] :
      ( ( ( ord_less_num @ M @ N )
       => ( ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit1 @ N ) )
          = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) ) ) )
      & ( ~ ( ord_less_num @ M @ N )
       => ( ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit1 @ N ) )
          = ( unique5024387138958732305ep_int @ ( bit1 @ N ) @ ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).

% divmod_algorithm_code(8)
thf(fact_8823_divmod__algorithm__code_I8_J,axiom,
    ! [M: num,N: num] :
      ( ( ( ord_less_num @ M @ N )
       => ( ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit1 @ N ) )
          = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) ) ) )
      & ( ~ ( ord_less_num @ M @ N )
       => ( ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit1 @ N ) )
          = ( unique4921790084139445826nteger @ ( bit1 @ N ) @ ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).

% divmod_algorithm_code(8)
thf(fact_8824_divmod__int__def,axiom,
    ( unique5052692396658037445od_int
    = ( ^ [M6: num,N2: num] : ( product_Pair_int_int @ ( divide_divide_int @ ( numeral_numeral_int @ M6 ) @ ( numeral_numeral_int @ N2 ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ M6 ) @ ( numeral_numeral_int @ N2 ) ) ) ) ) ).

% divmod_int_def
thf(fact_8825_divmod__def,axiom,
    ( unique5052692396658037445od_int
    = ( ^ [M6: num,N2: num] : ( product_Pair_int_int @ ( divide_divide_int @ ( numeral_numeral_int @ M6 ) @ ( numeral_numeral_int @ N2 ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ M6 ) @ ( numeral_numeral_int @ N2 ) ) ) ) ) ).

% divmod_def
thf(fact_8826_divmod__def,axiom,
    ( unique5055182867167087721od_nat
    = ( ^ [M6: num,N2: num] : ( product_Pair_nat_nat @ ( divide_divide_nat @ ( numeral_numeral_nat @ M6 ) @ ( numeral_numeral_nat @ N2 ) ) @ ( modulo_modulo_nat @ ( numeral_numeral_nat @ M6 ) @ ( numeral_numeral_nat @ N2 ) ) ) ) ) ).

% divmod_def
thf(fact_8827_divmod__def,axiom,
    ( unique3479559517661332726nteger
    = ( ^ [M6: num,N2: num] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( numera6620942414471956472nteger @ M6 ) @ ( numera6620942414471956472nteger @ N2 ) ) @ ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M6 ) @ ( numera6620942414471956472nteger @ N2 ) ) ) ) ) ).

% divmod_def
thf(fact_8828_divmod_H__nat__def,axiom,
    ( unique5055182867167087721od_nat
    = ( ^ [M6: num,N2: num] : ( product_Pair_nat_nat @ ( divide_divide_nat @ ( numeral_numeral_nat @ M6 ) @ ( numeral_numeral_nat @ N2 ) ) @ ( modulo_modulo_nat @ ( numeral_numeral_nat @ M6 ) @ ( numeral_numeral_nat @ N2 ) ) ) ) ) ).

% divmod'_nat_def
thf(fact_8829_divmod__divmod__step,axiom,
    ( unique5055182867167087721od_nat
    = ( ^ [M6: num,N2: num] : ( if_Pro6206227464963214023at_nat @ ( ord_less_num @ M6 @ N2 ) @ ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ M6 ) ) @ ( unique5026877609467782581ep_nat @ N2 @ ( unique5055182867167087721od_nat @ M6 @ ( bit0 @ N2 ) ) ) ) ) ) ).

% divmod_divmod_step
thf(fact_8830_divmod__divmod__step,axiom,
    ( unique5052692396658037445od_int
    = ( ^ [M6: num,N2: num] : ( if_Pro3027730157355071871nt_int @ ( ord_less_num @ M6 @ N2 ) @ ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ M6 ) ) @ ( unique5024387138958732305ep_int @ N2 @ ( unique5052692396658037445od_int @ M6 @ ( bit0 @ N2 ) ) ) ) ) ) ).

% divmod_divmod_step
thf(fact_8831_divmod__divmod__step,axiom,
    ( unique3479559517661332726nteger
    = ( ^ [M6: num,N2: num] : ( if_Pro6119634080678213985nteger @ ( ord_less_num @ M6 @ N2 ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ M6 ) ) @ ( unique4921790084139445826nteger @ N2 @ ( unique3479559517661332726nteger @ M6 @ ( bit0 @ N2 ) ) ) ) ) ) ).

% divmod_divmod_step
thf(fact_8832_pi__series,axiom,
    ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) )
    = ( suminf_real
      @ ^ [K3: nat] : ( divide_divide_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ).

% pi_series
thf(fact_8833_vebt__member_Opelims_I3_J,axiom,
    ! [X4: vEBT_VEBT,Xa: nat] :
      ( ~ ( vEBT_vebt_member @ X4 @ Xa )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X4 @ Xa ) )
       => ( ! [A5: $o,B5: $o] :
              ( ( X4
                = ( vEBT_Leaf @ A5 @ B5 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B5 ) @ Xa ) )
               => ( ( ( Xa = zero_zero_nat )
                   => A5 )
                  & ( ( Xa != zero_zero_nat )
                   => ( ( ( Xa = one_one_nat )
                       => B5 )
                      & ( Xa = one_one_nat ) ) ) ) ) )
         => ( ! [Uu2: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( X4
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv @ Uw ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv @ Uw ) @ Xa ) ) )
           => ( ! [V3: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz: vEBT_VEBT] :
                  ( ( X4
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V3 ) @ zero_zero_nat @ Uy2 @ Uz ) )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V3 ) @ zero_zero_nat @ Uy2 @ Uz ) @ Xa ) ) )
             => ( ! [V3: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X4
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V3 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V3 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ Xa ) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X4
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) )
                     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) @ Xa ) )
                       => ( ( Xa != Mi2 )
                         => ( ( Xa != Ma2 )
                           => ( ~ ( ord_less_nat @ Xa @ Mi2 )
                              & ( ~ ( ord_less_nat @ Xa @ Mi2 )
                               => ( ~ ( ord_less_nat @ Ma2 @ Xa )
                                  & ( ~ ( ord_less_nat @ Ma2 @ Xa )
                                   => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                                       => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                      & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(3)
thf(fact_8834_vebt__member_Opelims_I1_J,axiom,
    ! [X4: vEBT_VEBT,Xa: nat,Y3: $o] :
      ( ( ( vEBT_vebt_member @ X4 @ Xa )
        = Y3 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X4 @ Xa ) )
       => ( ! [A5: $o,B5: $o] :
              ( ( X4
                = ( vEBT_Leaf @ A5 @ B5 ) )
             => ( ( Y3
                  = ( ( ( Xa = zero_zero_nat )
                     => A5 )
                    & ( ( Xa != zero_zero_nat )
                     => ( ( ( Xa = one_one_nat )
                         => B5 )
                        & ( Xa = one_one_nat ) ) ) ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B5 ) @ Xa ) ) ) )
         => ( ! [Uu2: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( X4
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv @ Uw ) )
               => ( ~ Y3
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv @ Uw ) @ Xa ) ) ) )
           => ( ! [V3: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz: vEBT_VEBT] :
                  ( ( X4
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V3 ) @ zero_zero_nat @ Uy2 @ Uz ) )
                 => ( ~ Y3
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V3 ) @ zero_zero_nat @ Uy2 @ Uz ) @ Xa ) ) ) )
             => ( ! [V3: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X4
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V3 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
                   => ( ~ Y3
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V3 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ Xa ) ) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X4
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) )
                     => ( ( Y3
                          = ( ( Xa != Mi2 )
                           => ( ( Xa != Ma2 )
                             => ( ~ ( ord_less_nat @ Xa @ Mi2 )
                                & ( ~ ( ord_less_nat @ Xa @ Mi2 )
                                 => ( ~ ( ord_less_nat @ Ma2 @ Xa )
                                    & ( ~ ( ord_less_nat @ Ma2 @ Xa )
                                     => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                                         => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(1)
thf(fact_8835_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
    ! [X4: vEBT_VEBT,Xa: nat,Y3: $o] :
      ( ( ( vEBT_V5719532721284313246member @ X4 @ Xa )
        = Y3 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X4 @ Xa ) )
       => ( ! [A5: $o,B5: $o] :
              ( ( X4
                = ( vEBT_Leaf @ A5 @ B5 ) )
             => ( ( Y3
                  = ( ( ( Xa = zero_zero_nat )
                     => A5 )
                    & ( ( Xa != zero_zero_nat )
                     => ( ( ( Xa = one_one_nat )
                         => B5 )
                        & ( Xa = one_one_nat ) ) ) ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B5 ) @ Xa ) ) ) )
         => ( ! [Uu2: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( X4
                  = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv @ Uw ) )
               => ( ~ Y3
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv @ Uw ) @ Xa ) ) ) )
           => ~ ! [Uy2: option4927543243414619207at_nat,V3: nat,TreeList3: list_VEBT_VEBT,S3: vEBT_VEBT] :
                  ( ( X4
                    = ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S3 ) )
                 => ( ( Y3
                      = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                         => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S3 ) @ Xa ) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(1)
thf(fact_8836_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
    ! [X4: vEBT_VEBT,Xa: nat] :
      ( ( vEBT_V5719532721284313246member @ X4 @ Xa )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X4 @ Xa ) )
       => ( ! [A5: $o,B5: $o] :
              ( ( X4
                = ( vEBT_Leaf @ A5 @ B5 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B5 ) @ Xa ) )
               => ~ ( ( ( Xa = zero_zero_nat )
                     => A5 )
                    & ( ( Xa != zero_zero_nat )
                     => ( ( ( Xa = one_one_nat )
                         => B5 )
                        & ( Xa = one_one_nat ) ) ) ) ) )
         => ~ ! [Uy2: option4927543243414619207at_nat,V3: nat,TreeList3: list_VEBT_VEBT,S3: vEBT_VEBT] :
                ( ( X4
                  = ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S3 ) )
               => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S3 ) @ Xa ) )
                 => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                       => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(2)
thf(fact_8837_vebt__member_Opelims_I2_J,axiom,
    ! [X4: vEBT_VEBT,Xa: nat] :
      ( ( vEBT_vebt_member @ X4 @ Xa )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X4 @ Xa ) )
       => ( ! [A5: $o,B5: $o] :
              ( ( X4
                = ( vEBT_Leaf @ A5 @ B5 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B5 ) @ Xa ) )
               => ~ ( ( ( Xa = zero_zero_nat )
                     => A5 )
                    & ( ( Xa != zero_zero_nat )
                     => ( ( ( Xa = one_one_nat )
                         => B5 )
                        & ( Xa = one_one_nat ) ) ) ) ) )
         => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X4
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) )
               => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) @ Xa ) )
                 => ~ ( ( Xa != Mi2 )
                     => ( ( Xa != Ma2 )
                       => ( ~ ( ord_less_nat @ Xa @ Mi2 )
                          & ( ~ ( ord_less_nat @ Xa @ Mi2 )
                           => ( ~ ( ord_less_nat @ Ma2 @ Xa )
                              & ( ~ ( ord_less_nat @ Ma2 @ Xa )
                               => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                                   => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                  & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(2)
thf(fact_8838_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
    ! [X4: vEBT_VEBT,Xa: nat] :
      ( ~ ( vEBT_V5719532721284313246member @ X4 @ Xa )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X4 @ Xa ) )
       => ( ! [A5: $o,B5: $o] :
              ( ( X4
                = ( vEBT_Leaf @ A5 @ B5 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B5 ) @ Xa ) )
               => ( ( ( Xa = zero_zero_nat )
                   => A5 )
                  & ( ( Xa != zero_zero_nat )
                   => ( ( ( Xa = one_one_nat )
                       => B5 )
                      & ( Xa = one_one_nat ) ) ) ) ) )
         => ( ! [Uu2: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( X4
                  = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv @ Uw ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv @ Uw ) @ Xa ) ) )
           => ~ ! [Uy2: option4927543243414619207at_nat,V3: nat,TreeList3: list_VEBT_VEBT,S3: vEBT_VEBT] :
                  ( ( X4
                    = ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S3 ) )
                 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S3 ) @ Xa ) )
                   => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                       => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(3)
thf(fact_8839_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
    ! [X4: vEBT_VEBT,Xa: nat,Y3: $o] :
      ( ( ( vEBT_VEBT_membermima @ X4 @ Xa )
        = Y3 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X4 @ Xa ) )
       => ( ! [Uu2: $o,Uv: $o] :
              ( ( X4
                = ( vEBT_Leaf @ Uu2 @ Uv ) )
             => ( ~ Y3
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv ) @ Xa ) ) ) )
         => ( ! [Ux: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( X4
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux @ Uy2 ) )
               => ( ~ Y3
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux @ Uy2 ) @ Xa ) ) ) )
           => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                  ( ( X4
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
                 => ( ( Y3
                      = ( ( Xa = Mi2 )
                        | ( Xa = Ma2 ) ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ Xa ) ) ) )
             => ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList3: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X4
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ Vc2 ) )
                   => ( ( Y3
                        = ( ( Xa = Mi2 )
                          | ( Xa = Ma2 )
                          | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                             => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                            & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ Vc2 ) @ Xa ) ) ) )
               => ~ ! [V3: nat,TreeList3: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                      ( ( X4
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V3 ) @ TreeList3 @ Vd2 ) )
                     => ( ( Y3
                          = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                             => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                            & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V3 ) @ TreeList3 @ Vd2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(1)
thf(fact_8840_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
    ! [X4: vEBT_VEBT,Xa: nat] :
      ( ~ ( vEBT_VEBT_membermima @ X4 @ Xa )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X4 @ Xa ) )
       => ( ! [Uu2: $o,Uv: $o] :
              ( ( X4
                = ( vEBT_Leaf @ Uu2 @ Uv ) )
             => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv ) @ Xa ) ) )
         => ( ! [Ux: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( X4
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux @ Uy2 ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux @ Uy2 ) @ Xa ) ) )
           => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                  ( ( X4
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
                 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ Xa ) )
                   => ( ( Xa = Mi2 )
                      | ( Xa = Ma2 ) ) ) )
             => ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList3: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X4
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ Vc2 ) )
                   => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ Vc2 ) @ Xa ) )
                     => ( ( Xa = Mi2 )
                        | ( Xa = Ma2 )
                        | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                           => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                          & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) )
               => ~ ! [V3: nat,TreeList3: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                      ( ( X4
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V3 ) @ TreeList3 @ Vd2 ) )
                     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V3 ) @ TreeList3 @ Vd2 ) @ Xa ) )
                       => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                           => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                          & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(3)
thf(fact_8841_suminf__geometric,axiom,
    ! [C: real] :
      ( ( ord_less_real @ ( real_V7735802525324610683m_real @ C ) @ one_one_real )
     => ( ( suminf_real @ ( power_power_real @ C ) )
        = ( divide_divide_real @ one_one_real @ ( minus_minus_real @ one_one_real @ C ) ) ) ) ).

% suminf_geometric
thf(fact_8842_suminf__geometric,axiom,
    ! [C: complex] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real )
     => ( ( suminf_complex @ ( power_power_complex @ C ) )
        = ( divide1717551699836669952omplex @ one_one_complex @ ( minus_minus_complex @ one_one_complex @ C ) ) ) ) ).

% suminf_geometric
thf(fact_8843_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
    ! [X4: vEBT_VEBT,Xa: nat] :
      ( ( vEBT_VEBT_membermima @ X4 @ Xa )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X4 @ Xa ) )
       => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
              ( ( X4
                = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ Xa ) )
               => ~ ( ( Xa = Mi2 )
                    | ( Xa = Ma2 ) ) ) )
         => ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList3: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                ( ( X4
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ Vc2 ) )
               => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ Vc2 ) @ Xa ) )
                 => ~ ( ( Xa = Mi2 )
                      | ( Xa = Ma2 )
                      | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                         => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) )
           => ~ ! [V3: nat,TreeList3: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                  ( ( X4
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V3 ) @ TreeList3 @ Vd2 ) )
                 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V3 ) @ TreeList3 @ Vd2 ) @ Xa ) )
                   => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                         => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(2)
thf(fact_8844_minus__one__div__numeral,axiom,
    ! [N: num] :
      ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N ) )
      = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ one @ N ) ) ) ) ).

% minus_one_div_numeral
thf(fact_8845_one__div__minus__numeral,axiom,
    ! [N: num] :
      ( ( divide_divide_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ one @ N ) ) ) ) ).

% one_div_minus_numeral
thf(fact_8846_dvd__numeral__simp,axiom,
    ! [M: num,N: num] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
      = ( unique6319869463603278526ux_int @ ( unique5052692396658037445od_int @ N @ M ) ) ) ).

% dvd_numeral_simp
thf(fact_8847_dvd__numeral__simp,axiom,
    ! [M: num,N: num] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
      = ( unique6322359934112328802ux_nat @ ( unique5055182867167087721od_nat @ N @ M ) ) ) ).

% dvd_numeral_simp
thf(fact_8848_dvd__numeral__simp,axiom,
    ! [M: num,N: num] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ N ) )
      = ( unique5706413561485394159nteger @ ( unique3479559517661332726nteger @ N @ M ) ) ) ).

% dvd_numeral_simp
thf(fact_8849_minus__numeral__div__numeral,axiom,
    ! [M: num,N: num] :
      ( ( divide_divide_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
      = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ M @ N ) ) ) ) ).

% minus_numeral_div_numeral
thf(fact_8850_divides__aux__eq,axiom,
    ! [Q2: nat,R2: nat] :
      ( ( unique6322359934112328802ux_nat @ ( product_Pair_nat_nat @ Q2 @ R2 ) )
      = ( R2 = zero_zero_nat ) ) ).

% divides_aux_eq
thf(fact_8851_divides__aux__eq,axiom,
    ! [Q2: int,R2: int] :
      ( ( unique6319869463603278526ux_int @ ( product_Pair_int_int @ Q2 @ R2 ) )
      = ( R2 = zero_zero_int ) ) ).

% divides_aux_eq
thf(fact_8852_Divides_Oadjust__div__eq,axiom,
    ! [Q2: int,R2: int] :
      ( ( adjust_div @ ( product_Pair_int_int @ Q2 @ R2 ) )
      = ( plus_plus_int @ Q2 @ ( zero_n2684676970156552555ol_int @ ( R2 != zero_zero_int ) ) ) ) ).

% Divides.adjust_div_eq
thf(fact_8853_numeral__div__minus__numeral,axiom,
    ! [M: num,N: num] :
      ( ( divide_divide_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ M @ N ) ) ) ) ).

% numeral_div_minus_numeral
thf(fact_8854_neg__eucl__rel__int__mult__2,axiom,
    ! [B2: int,A: int,Q2: int,R2: int] :
      ( ( ord_less_eq_int @ B2 @ zero_zero_int )
     => ( ( eucl_rel_int @ ( plus_plus_int @ A @ one_one_int ) @ B2 @ ( product_Pair_int_int @ Q2 @ R2 ) )
       => ( eucl_rel_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) @ ( product_Pair_int_int @ Q2 @ ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R2 ) @ one_one_int ) ) ) ) ) ).

% neg_eucl_rel_int_mult_2
thf(fact_8855_summable__arctan__series,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X4 ) @ one_one_real )
     => ( summable_real
        @ ^ [K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X4 @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ) ).

% summable_arctan_series
thf(fact_8856_pos__eucl__rel__int__mult__2,axiom,
    ! [B2: int,A: int,Q2: int,R2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ B2 )
     => ( ( eucl_rel_int @ A @ B2 @ ( product_Pair_int_int @ Q2 @ R2 ) )
       => ( eucl_rel_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) @ ( product_Pair_int_int @ Q2 @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R2 ) ) ) ) ) ) ).

% pos_eucl_rel_int_mult_2
thf(fact_8857_product__nth,axiom,
    ! [N: nat,Xs2: list_num,Ys2: list_num] :
      ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_num @ Xs2 ) @ ( size_size_list_num @ Ys2 ) ) )
     => ( ( nth_Pr6456567536196504476um_num @ ( product_num_num @ Xs2 @ Ys2 ) @ N )
        = ( product_Pair_num_num @ ( nth_num @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_num @ Ys2 ) ) ) @ ( nth_num @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_size_list_num @ Ys2 ) ) ) ) ) ) ).

% product_nth
thf(fact_8858_product__nth,axiom,
    ! [N: nat,Xs2: list_VEBT_VEBT,Ys2: list_VEBT_VEBT] :
      ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) )
     => ( ( nth_Pr4953567300277697838T_VEBT @ ( produc4743750530478302277T_VEBT @ Xs2 @ Ys2 ) @ N )
        = ( produc537772716801021591T_VEBT @ ( nth_VEBT_VEBT @ Xs2 @ ( divide_divide_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) ) @ ( nth_VEBT_VEBT @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) ) ) ) ) ).

% product_nth
thf(fact_8859_product__nth,axiom,
    ! [N: nat,Xs2: list_VEBT_VEBT,Ys2: list_o] :
      ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_o @ Ys2 ) ) )
     => ( ( nth_Pr4606735188037164562VEBT_o @ ( product_VEBT_VEBT_o @ Xs2 @ Ys2 ) @ N )
        = ( produc8721562602347293563VEBT_o @ ( nth_VEBT_VEBT @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_o @ Ys2 ) ) ) @ ( nth_o @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_size_list_o @ Ys2 ) ) ) ) ) ) ).

% product_nth
thf(fact_8860_product__nth,axiom,
    ! [N: nat,Xs2: list_VEBT_VEBT,Ys2: list_nat] :
      ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_nat @ Ys2 ) ) )
     => ( ( nth_Pr1791586995822124652BT_nat @ ( produc7295137177222721919BT_nat @ Xs2 @ Ys2 ) @ N )
        = ( produc738532404422230701BT_nat @ ( nth_VEBT_VEBT @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_nat @ Ys2 ) ) ) @ ( nth_nat @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_size_list_nat @ Ys2 ) ) ) ) ) ) ).

% product_nth
thf(fact_8861_product__nth,axiom,
    ! [N: nat,Xs2: list_VEBT_VEBT,Ys2: list_int] :
      ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_int @ Ys2 ) ) )
     => ( ( nth_Pr6837108013167703752BT_int @ ( produc7292646706713671643BT_int @ Xs2 @ Ys2 ) @ N )
        = ( produc736041933913180425BT_int @ ( nth_VEBT_VEBT @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_int @ Ys2 ) ) ) @ ( nth_int @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_size_list_int @ Ys2 ) ) ) ) ) ) ).

% product_nth
thf(fact_8862_product__nth,axiom,
    ! [N: nat,Xs2: list_o,Ys2: list_VEBT_VEBT] :
      ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) )
     => ( ( nth_Pr6777367263587873994T_VEBT @ ( product_o_VEBT_VEBT @ Xs2 @ Ys2 ) @ N )
        = ( produc2982872950893828659T_VEBT @ ( nth_o @ Xs2 @ ( divide_divide_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) ) @ ( nth_VEBT_VEBT @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) ) ) ) ) ).

% product_nth
thf(fact_8863_product__nth,axiom,
    ! [N: nat,Xs2: list_o,Ys2: list_o] :
      ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_size_list_o @ Ys2 ) ) )
     => ( ( nth_Product_prod_o_o @ ( product_o_o @ Xs2 @ Ys2 ) @ N )
        = ( product_Pair_o_o @ ( nth_o @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_o @ Ys2 ) ) ) @ ( nth_o @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_size_list_o @ Ys2 ) ) ) ) ) ) ).

% product_nth
thf(fact_8864_product__nth,axiom,
    ! [N: nat,Xs2: list_o,Ys2: list_nat] :
      ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_size_list_nat @ Ys2 ) ) )
     => ( ( nth_Pr5826913651314560976_o_nat @ ( product_o_nat @ Xs2 @ Ys2 ) @ N )
        = ( product_Pair_o_nat @ ( nth_o @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_nat @ Ys2 ) ) ) @ ( nth_nat @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_size_list_nat @ Ys2 ) ) ) ) ) ) ).

% product_nth
thf(fact_8865_product__nth,axiom,
    ! [N: nat,Xs2: list_o,Ys2: list_int] :
      ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_size_list_int @ Ys2 ) ) )
     => ( ( nth_Pr1649062631805364268_o_int @ ( product_o_int @ Xs2 @ Ys2 ) @ N )
        = ( product_Pair_o_int @ ( nth_o @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_int @ Ys2 ) ) ) @ ( nth_int @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_size_list_int @ Ys2 ) ) ) ) ) ) ).

% product_nth
thf(fact_8866_product__nth,axiom,
    ! [N: nat,Xs2: list_nat,Ys2: list_num] :
      ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_nat @ Xs2 ) @ ( size_size_list_num @ Ys2 ) ) )
     => ( ( nth_Pr8326237132889035090at_num @ ( product_nat_num @ Xs2 @ Ys2 ) @ N )
        = ( product_Pair_nat_num @ ( nth_nat @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_num @ Ys2 ) ) ) @ ( nth_num @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_size_list_num @ Ys2 ) ) ) ) ) ) ).

% product_nth
thf(fact_8867_summable__iff__shift,axiom,
    ! [F: nat > real,K: nat] :
      ( ( summable_real
        @ ^ [N2: nat] : ( F @ ( plus_plus_nat @ N2 @ K ) ) )
      = ( summable_real @ F ) ) ).

% summable_iff_shift
thf(fact_8868_summable__cmult__iff,axiom,
    ! [C: real,F: nat > real] :
      ( ( summable_real
        @ ^ [N2: nat] : ( times_times_real @ C @ ( F @ N2 ) ) )
      = ( ( C = zero_zero_real )
        | ( summable_real @ F ) ) ) ).

% summable_cmult_iff
thf(fact_8869_summable__divide__iff,axiom,
    ! [F: nat > complex,C: complex] :
      ( ( summable_complex
        @ ^ [N2: nat] : ( divide1717551699836669952omplex @ ( F @ N2 ) @ C ) )
      = ( ( C = zero_zero_complex )
        | ( summable_complex @ F ) ) ) ).

% summable_divide_iff
thf(fact_8870_summable__divide__iff,axiom,
    ! [F: nat > real,C: real] :
      ( ( summable_real
        @ ^ [N2: nat] : ( divide_divide_real @ ( F @ N2 ) @ C ) )
      = ( ( C = zero_zero_real )
        | ( summable_real @ F ) ) ) ).

% summable_divide_iff
thf(fact_8871_length__product,axiom,
    ! [Xs2: list_VEBT_VEBT,Ys2: list_VEBT_VEBT] :
      ( ( size_s7466405169056248089T_VEBT @ ( produc4743750530478302277T_VEBT @ Xs2 @ Ys2 ) )
      = ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) ) ).

% length_product
thf(fact_8872_length__product,axiom,
    ! [Xs2: list_VEBT_VEBT,Ys2: list_o] :
      ( ( size_s9168528473962070013VEBT_o @ ( product_VEBT_VEBT_o @ Xs2 @ Ys2 ) )
      = ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_o @ Ys2 ) ) ) ).

% length_product
thf(fact_8873_length__product,axiom,
    ! [Xs2: list_VEBT_VEBT,Ys2: list_nat] :
      ( ( size_s6152045936467909847BT_nat @ ( produc7295137177222721919BT_nat @ Xs2 @ Ys2 ) )
      = ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_nat @ Ys2 ) ) ) ).

% length_product
thf(fact_8874_length__product,axiom,
    ! [Xs2: list_VEBT_VEBT,Ys2: list_int] :
      ( ( size_s3661962791536183091BT_int @ ( produc7292646706713671643BT_int @ Xs2 @ Ys2 ) )
      = ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_int @ Ys2 ) ) ) ).

% length_product
thf(fact_8875_length__product,axiom,
    ! [Xs2: list_o,Ys2: list_VEBT_VEBT] :
      ( ( size_s4313452262239582901T_VEBT @ ( product_o_VEBT_VEBT @ Xs2 @ Ys2 ) )
      = ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) ) ).

% length_product
thf(fact_8876_length__product,axiom,
    ! [Xs2: list_o,Ys2: list_o] :
      ( ( size_s1515746228057227161od_o_o @ ( product_o_o @ Xs2 @ Ys2 ) )
      = ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_size_list_o @ Ys2 ) ) ) ).

% length_product
thf(fact_8877_length__product,axiom,
    ! [Xs2: list_o,Ys2: list_nat] :
      ( ( size_s5443766701097040955_o_nat @ ( product_o_nat @ Xs2 @ Ys2 ) )
      = ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_size_list_nat @ Ys2 ) ) ) ).

% length_product
thf(fact_8878_length__product,axiom,
    ! [Xs2: list_o,Ys2: list_int] :
      ( ( size_s2953683556165314199_o_int @ ( product_o_int @ Xs2 @ Ys2 ) )
      = ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_size_list_int @ Ys2 ) ) ) ).

% length_product
thf(fact_8879_length__product,axiom,
    ! [Xs2: list_nat,Ys2: list_VEBT_VEBT] :
      ( ( size_s4762443039079500285T_VEBT @ ( produc7156399406898700509T_VEBT @ Xs2 @ Ys2 ) )
      = ( times_times_nat @ ( size_size_list_nat @ Xs2 ) @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) ) ).

% length_product
thf(fact_8880_length__product,axiom,
    ! [Xs2: list_nat,Ys2: list_o] :
      ( ( size_s6491369823275344609_nat_o @ ( product_nat_o @ Xs2 @ Ys2 ) )
      = ( times_times_nat @ ( size_size_list_nat @ Xs2 ) @ ( size_size_list_o @ Ys2 ) ) ) ).

% length_product
thf(fact_8881_summable__geometric__iff,axiom,
    ! [C: complex] :
      ( ( summable_complex @ ( power_power_complex @ C ) )
      = ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real ) ) ).

% summable_geometric_iff
thf(fact_8882_unique__quotient,axiom,
    ! [A: int,B2: int,Q2: int,R2: int,Q5: int,R4: int] :
      ( ( eucl_rel_int @ A @ B2 @ ( product_Pair_int_int @ Q2 @ R2 ) )
     => ( ( eucl_rel_int @ A @ B2 @ ( product_Pair_int_int @ Q5 @ R4 ) )
       => ( Q2 = Q5 ) ) ) ).

% unique_quotient
thf(fact_8883_unique__remainder,axiom,
    ! [A: int,B2: int,Q2: int,R2: int,Q5: int,R4: int] :
      ( ( eucl_rel_int @ A @ B2 @ ( product_Pair_int_int @ Q2 @ R2 ) )
     => ( ( eucl_rel_int @ A @ B2 @ ( product_Pair_int_int @ Q5 @ R4 ) )
       => ( R2 = R4 ) ) ) ).

% unique_remainder
thf(fact_8884_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
thf(fact_8885_summable__rabs__comparison__test,axiom,
    ! [F: nat > real,G: nat > real] :
      ( ? [N7: nat] :
        ! [N4: nat] :
          ( ( ord_less_eq_nat @ N7 @ N4 )
         => ( ord_less_eq_real @ ( abs_abs_real @ ( F @ N4 ) ) @ ( G @ N4 ) ) )
     => ( ( summable_real @ G )
       => ( summable_real
          @ ^ [N2: nat] : ( abs_abs_real @ ( F @ N2 ) ) ) ) ) ).

% summable_rabs_comparison_test
thf(fact_8886_div__int__unique,axiom,
    ! [K: int,L2: int,Q2: int,R2: int] :
      ( ( eucl_rel_int @ K @ L2 @ ( product_Pair_int_int @ Q2 @ R2 ) )
     => ( ( divide_divide_int @ K @ L2 )
        = Q2 ) ) ).

% div_int_unique
thf(fact_8887_mod__int__unique,axiom,
    ! [K: int,L2: int,Q2: int,R2: int] :
      ( ( eucl_rel_int @ K @ L2 @ ( product_Pair_int_int @ Q2 @ R2 ) )
     => ( ( modulo_modulo_int @ K @ L2 )
        = R2 ) ) ).

% mod_int_unique
thf(fact_8888_summable__rabs,axiom,
    ! [F: nat > real] :
      ( ( summable_real
        @ ^ [N2: nat] : ( abs_abs_real @ ( F @ N2 ) ) )
     => ( ord_less_eq_real @ ( abs_abs_real @ ( suminf_real @ F ) )
        @ ( suminf_real
          @ ^ [N2: nat] : ( abs_abs_real @ ( F @ N2 ) ) ) ) ) ).

% summable_rabs
thf(fact_8889_eucl__rel__int__dividesI,axiom,
    ! [L2: int,K: int,Q2: int] :
      ( ( L2 != zero_zero_int )
     => ( ( K
          = ( times_times_int @ Q2 @ L2 ) )
       => ( eucl_rel_int @ K @ L2 @ ( product_Pair_int_int @ Q2 @ zero_zero_int ) ) ) ) ).

% eucl_rel_int_dividesI
thf(fact_8890_eucl__rel__int,axiom,
    ! [K: int,L2: int] : ( eucl_rel_int @ K @ L2 @ ( product_Pair_int_int @ ( divide_divide_int @ K @ L2 ) @ ( modulo_modulo_int @ K @ L2 ) ) ) ).

% eucl_rel_int
thf(fact_8891_summable__power__series,axiom,
    ! [F: nat > real,Z: real] :
      ( ! [I3: nat] : ( ord_less_eq_real @ ( F @ I3 ) @ one_one_real )
     => ( ! [I3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) )
       => ( ( ord_less_eq_real @ zero_zero_real @ Z )
         => ( ( ord_less_real @ Z @ one_one_real )
           => ( summable_real
              @ ^ [I2: nat] : ( times_times_real @ ( F @ I2 ) @ ( power_power_real @ Z @ I2 ) ) ) ) ) ) ) ).

% summable_power_series
thf(fact_8892_zminus1__lemma,axiom,
    ! [A: int,B2: int,Q2: int,R2: int] :
      ( ( eucl_rel_int @ A @ B2 @ ( product_Pair_int_int @ Q2 @ R2 ) )
     => ( ( B2 != zero_zero_int )
       => ( eucl_rel_int @ ( uminus_uminus_int @ A ) @ B2 @ ( product_Pair_int_int @ ( if_int @ ( R2 = zero_zero_int ) @ ( uminus_uminus_int @ Q2 ) @ ( minus_minus_int @ ( uminus_uminus_int @ Q2 ) @ one_one_int ) ) @ ( if_int @ ( R2 = zero_zero_int ) @ zero_zero_int @ ( minus_minus_int @ B2 @ R2 ) ) ) ) ) ) ).

% zminus1_lemma
thf(fact_8893_eucl__rel__int__iff,axiom,
    ! [K: int,L2: int,Q2: int,R2: int] :
      ( ( eucl_rel_int @ K @ L2 @ ( product_Pair_int_int @ Q2 @ R2 ) )
      = ( ( K
          = ( plus_plus_int @ ( times_times_int @ L2 @ Q2 ) @ R2 ) )
        & ( ( ord_less_int @ zero_zero_int @ L2 )
         => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
            & ( ord_less_int @ R2 @ L2 ) ) )
        & ( ~ ( ord_less_int @ zero_zero_int @ L2 )
         => ( ( ( ord_less_int @ L2 @ zero_zero_int )
             => ( ( ord_less_int @ L2 @ R2 )
                & ( ord_less_eq_int @ R2 @ zero_zero_int ) ) )
            & ( ~ ( ord_less_int @ L2 @ zero_zero_int )
             => ( Q2 = zero_zero_int ) ) ) ) ) ) ).

% eucl_rel_int_iff
thf(fact_8894_eucl__rel__int__remainderI,axiom,
    ! [R2: int,L2: int,K: int,Q2: int] :
      ( ( ( sgn_sgn_int @ R2 )
        = ( sgn_sgn_int @ L2 ) )
     => ( ( ord_less_int @ ( abs_abs_int @ R2 ) @ ( abs_abs_int @ L2 ) )
       => ( ( K
            = ( plus_plus_int @ ( times_times_int @ Q2 @ L2 ) @ R2 ) )
         => ( eucl_rel_int @ K @ L2 @ ( product_Pair_int_int @ Q2 @ R2 ) ) ) ) ) ).

% eucl_rel_int_remainderI
thf(fact_8895_eucl__rel__int_Osimps,axiom,
    ( eucl_rel_int
    = ( ^ [A1: int,A22: int,A32: product_prod_int_int] :
          ( ? [K3: int] :
              ( ( A1 = K3 )
              & ( A22 = zero_zero_int )
              & ( A32
                = ( product_Pair_int_int @ zero_zero_int @ K3 ) ) )
          | ? [L: int,K3: int,Q4: int] :
              ( ( A1 = K3 )
              & ( A22 = L )
              & ( A32
                = ( product_Pair_int_int @ Q4 @ zero_zero_int ) )
              & ( L != zero_zero_int )
              & ( K3
                = ( times_times_int @ Q4 @ L ) ) )
          | ? [R5: int,L: int,K3: int,Q4: int] :
              ( ( A1 = K3 )
              & ( A22 = L )
              & ( A32
                = ( product_Pair_int_int @ Q4 @ R5 ) )
              & ( ( sgn_sgn_int @ R5 )
                = ( sgn_sgn_int @ L ) )
              & ( ord_less_int @ ( abs_abs_int @ R5 ) @ ( abs_abs_int @ L ) )
              & ( K3
                = ( plus_plus_int @ ( times_times_int @ Q4 @ L ) @ R5 ) ) ) ) ) ) ).

% eucl_rel_int.simps
thf(fact_8896_eucl__rel__int_Ocases,axiom,
    ! [A12: int,A23: int,A33: product_prod_int_int] :
      ( ( eucl_rel_int @ A12 @ A23 @ A33 )
     => ( ( ( A23 = zero_zero_int )
         => ( A33
           != ( product_Pair_int_int @ zero_zero_int @ A12 ) ) )
       => ( ! [Q3: int] :
              ( ( A33
                = ( product_Pair_int_int @ Q3 @ zero_zero_int ) )
             => ( ( A23 != zero_zero_int )
               => ( A12
                 != ( times_times_int @ Q3 @ A23 ) ) ) )
         => ~ ! [R3: int,Q3: int] :
                ( ( A33
                  = ( product_Pair_int_int @ Q3 @ R3 ) )
               => ( ( ( sgn_sgn_int @ R3 )
                    = ( sgn_sgn_int @ A23 ) )
                 => ( ( ord_less_int @ ( abs_abs_int @ R3 ) @ ( abs_abs_int @ A23 ) )
                   => ( A12
                     != ( plus_plus_int @ ( times_times_int @ Q3 @ A23 ) @ R3 ) ) ) ) ) ) ) ) ).

% eucl_rel_int.cases
thf(fact_8897_and__int_Oelims,axiom,
    ! [X4: int,Xa: int,Y3: int] :
      ( ( ( bit_se725231765392027082nd_int @ X4 @ Xa )
        = Y3 )
     => ( ( ( ( member_int @ X4 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
            & ( member_int @ Xa @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
         => ( Y3
            = ( uminus_uminus_int
              @ ( zero_n2684676970156552555ol_int
                @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X4 )
                  & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa ) ) ) ) ) )
        & ( ~ ( ( member_int @ X4 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
              & ( member_int @ Xa @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
         => ( Y3
            = ( plus_plus_int
              @ ( zero_n2684676970156552555ol_int
                @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X4 )
                  & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa ) ) )
              @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ X4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ Xa @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% and_int.elims
thf(fact_8898_and__int_Osimps,axiom,
    ( bit_se725231765392027082nd_int
    = ( ^ [K3: int,L: int] :
          ( if_int
          @ ( ( member_int @ K3 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
            & ( member_int @ L @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
          @ ( uminus_uminus_int
            @ ( zero_n2684676970156552555ol_int
              @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 )
                & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) ) )
          @ ( plus_plus_int
            @ ( zero_n2684676970156552555ol_int
              @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 )
                & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) )
            @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% and_int.simps
thf(fact_8899_vebt__buildup_Oelims,axiom,
    ! [X4: nat,Y3: vEBT_VEBT] :
      ( ( ( vEBT_vebt_buildup @ X4 )
        = Y3 )
     => ( ( ( X4 = zero_zero_nat )
         => ( Y3
           != ( vEBT_Leaf @ $false @ $false ) ) )
       => ( ( ( X4
              = ( suc @ zero_zero_nat ) )
           => ( Y3
             != ( vEBT_Leaf @ $false @ $false ) ) )
         => ~ ! [Va: nat] :
                ( ( X4
                  = ( suc @ ( suc @ Va ) ) )
               => ~ ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
                     => ( Y3
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
                    & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
                     => ( Y3
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_buildup.elims
thf(fact_8900_sin__paired,axiom,
    ! [X4: real] :
      ( sums_real
      @ ^ [N2: nat] : ( times_times_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( semiri2265585572941072030t_real @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) ) @ ( power_power_real @ X4 @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) )
      @ ( sin_real @ X4 ) ) ).

% sin_paired
thf(fact_8901_atLeastAtMostPlus1__int__conv,axiom,
    ! [M: int,N: int] :
      ( ( ord_less_eq_int @ M @ ( plus_plus_int @ one_one_int @ N ) )
     => ( ( set_or1266510415728281911st_int @ M @ ( plus_plus_int @ one_one_int @ N ) )
        = ( insert_int @ ( plus_plus_int @ one_one_int @ N ) @ ( set_or1266510415728281911st_int @ M @ N ) ) ) ) ).

% atLeastAtMostPlus1_int_conv
thf(fact_8902_simp__from__to,axiom,
    ( set_or1266510415728281911st_int
    = ( ^ [I2: int,J3: int] : ( if_set_int @ ( ord_less_int @ J3 @ I2 ) @ bot_bot_set_int @ ( insert_int @ I2 @ ( set_or1266510415728281911st_int @ ( plus_plus_int @ I2 @ one_one_int ) @ J3 ) ) ) ) ) ).

% simp_from_to
thf(fact_8903_power__half__series,axiom,
    ( sums_real
    @ ^ [N2: nat] : ( power_power_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( suc @ N2 ) )
    @ one_one_real ) ).

% power_half_series
thf(fact_8904_sums__if_H,axiom,
    ! [G: nat > real,X4: real] :
      ( ( sums_real @ G @ X4 )
     => ( sums_real
        @ ^ [N2: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ zero_zero_real @ ( G @ ( divide_divide_nat @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
        @ X4 ) ) ).

% sums_if'
thf(fact_8905_sums__if,axiom,
    ! [G: nat > real,X4: real,F: nat > real,Y3: real] :
      ( ( sums_real @ G @ X4 )
     => ( ( sums_real @ F @ Y3 )
       => ( sums_real
          @ ^ [N2: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ ( F @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( G @ ( divide_divide_nat @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
          @ ( plus_plus_real @ X4 @ Y3 ) ) ) ) ).

% sums_if
thf(fact_8906_cos__paired,axiom,
    ! [X4: real] :
      ( sums_real
      @ ^ [N2: nat] : ( times_times_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( semiri2265585572941072030t_real @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) @ ( power_power_real @ X4 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
      @ ( cos_real @ X4 ) ) ).

% cos_paired
thf(fact_8907_vebt__buildup_Osimps_I3_J,axiom,
    ! [Va2: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
       => ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va2 ) ) )
          = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va2 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
       => ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va2 ) ) )
          = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va2 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% vebt_buildup.simps(3)
thf(fact_8908_and__int_Opelims,axiom,
    ! [X4: int,Xa: int,Y3: int] :
      ( ( ( bit_se725231765392027082nd_int @ X4 @ Xa )
        = Y3 )
     => ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ X4 @ Xa ) )
       => ~ ( ( ( ( ( member_int @ X4 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
                  & ( member_int @ Xa @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
               => ( Y3
                  = ( uminus_uminus_int
                    @ ( zero_n2684676970156552555ol_int
                      @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X4 )
                        & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa ) ) ) ) ) )
              & ( ~ ( ( member_int @ X4 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
                    & ( member_int @ Xa @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
               => ( Y3
                  = ( plus_plus_int
                    @ ( zero_n2684676970156552555ol_int
                      @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X4 )
                        & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa ) ) )
                    @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ X4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ Xa @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) )
           => ~ ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ X4 @ Xa ) ) ) ) ) ).

% and_int.pelims
thf(fact_8909_and__int_Opsimps,axiom,
    ! [K: int,L2: int] :
      ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K @ L2 ) )
     => ( ( ( ( member_int @ K @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
            & ( member_int @ L2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
         => ( ( bit_se725231765392027082nd_int @ K @ L2 )
            = ( uminus_uminus_int
              @ ( zero_n2684676970156552555ol_int
                @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
                  & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) ) ) ) )
        & ( ~ ( ( member_int @ K @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
              & ( member_int @ L2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
         => ( ( bit_se725231765392027082nd_int @ K @ L2 )
            = ( plus_plus_int
              @ ( zero_n2684676970156552555ol_int
                @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
                  & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) )
              @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% and_int.psimps
thf(fact_8910_obtain__set__succ,axiom,
    ! [X4: nat,Z: nat,A3: set_nat,B4: set_nat] :
      ( ( ord_less_nat @ X4 @ Z )
     => ( ( vEBT_VEBT_max_in_set @ A3 @ Z )
       => ( ( finite_finite_nat @ B4 )
         => ( ( A3 = B4 )
           => ? [X_1: nat] : ( vEBT_is_succ_in_set @ A3 @ X4 @ X_1 ) ) ) ) ) ).

% obtain_set_succ
thf(fact_8911_set__vebt__finite,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( finite_finite_nat @ ( vEBT_VEBT_set_vebt @ T ) ) ) ).

% set_vebt_finite
thf(fact_8912_succ__none__empty,axiom,
    ! [Xs2: set_nat,A: nat] :
      ( ~ ? [X_1: nat] : ( vEBT_is_succ_in_set @ Xs2 @ A @ X_1 )
     => ( ( finite_finite_nat @ Xs2 )
       => ~ ? [X5: nat] :
              ( ( member_nat @ X5 @ Xs2 )
              & ( ord_less_nat @ A @ X5 ) ) ) ) ).

% succ_none_empty
thf(fact_8913_finite__atLeastAtMost,axiom,
    ! [L2: nat,U: nat] : ( finite_finite_nat @ ( set_or1269000886237332187st_nat @ L2 @ U ) ) ).

% finite_atLeastAtMost
thf(fact_8914_finite__nat__set__iff__bounded,axiom,
    ( finite_finite_nat
    = ( ^ [N8: set_nat] :
        ? [M6: nat] :
        ! [X: nat] :
          ( ( member_nat @ X @ N8 )
         => ( ord_less_nat @ X @ M6 ) ) ) ) ).

% finite_nat_set_iff_bounded
thf(fact_8915_bounded__nat__set__is__finite,axiom,
    ! [N3: set_nat,N: nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ N3 )
         => ( ord_less_nat @ X3 @ N ) )
     => ( finite_finite_nat @ N3 ) ) ).

% bounded_nat_set_is_finite
thf(fact_8916_finite__nat__set__iff__bounded__le,axiom,
    ( finite_finite_nat
    = ( ^ [N8: set_nat] :
        ? [M6: nat] :
        ! [X: nat] :
          ( ( member_nat @ X @ N8 )
         => ( ord_less_eq_nat @ X @ M6 ) ) ) ) ).

% finite_nat_set_iff_bounded_le
thf(fact_8917_finite__M__bounded__by__nat,axiom,
    ! [P: nat > $o,I: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [K3: nat] :
            ( ( P @ K3 )
            & ( ord_less_nat @ K3 @ I ) ) ) ) ).

% finite_M_bounded_by_nat
thf(fact_8918_finite__less__ub,axiom,
    ! [F: nat > nat,U: nat] :
      ( ! [N4: nat] : ( ord_less_eq_nat @ N4 @ ( F @ N4 ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ U ) ) ) ) ).

% finite_less_ub
thf(fact_8919_atLeast0__atMost__Suc,axiom,
    ! [N: nat] :
      ( ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) )
      = ( insert_nat @ ( suc @ N ) @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).

% atLeast0_atMost_Suc
thf(fact_8920_Icc__eq__insert__lb__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( set_or1269000886237332187st_nat @ M @ N )
        = ( insert_nat @ M @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ).

% Icc_eq_insert_lb_nat
thf(fact_8921_atLeastAtMostSuc__conv,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
     => ( ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) )
        = ( insert_nat @ ( suc @ N ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ).

% atLeastAtMostSuc_conv
thf(fact_8922_atLeastAtMost__insertL,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( insert_nat @ M @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) )
        = ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).

% atLeastAtMost_insertL
thf(fact_8923_finite__divisors__nat,axiom,
    ! [M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [D2: nat] : ( dvd_dvd_nat @ D2 @ M ) ) ) ) ).

% finite_divisors_nat
thf(fact_8924_subset__eq__atLeast0__atMost__finite,axiom,
    ! [N3: set_nat,N: nat] :
      ( ( ord_less_eq_set_nat @ N3 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
     => ( finite_finite_nat @ N3 ) ) ).

% subset_eq_atLeast0_atMost_finite
thf(fact_8925_set__decode__plus__power__2,axiom,
    ! [N: nat,Z: nat] :
      ( ~ ( member_nat @ N @ ( nat_set_decode @ Z ) )
     => ( ( nat_set_decode @ ( plus_plus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ Z ) )
        = ( insert_nat @ N @ ( nat_set_decode @ Z ) ) ) ) ).

% set_decode_plus_power_2
thf(fact_8926_and__int_Opinduct,axiom,
    ! [A0: int,A12: int,P: int > int > $o] :
      ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ A0 @ A12 ) )
     => ( ! [K2: int,L3: int] :
            ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K2 @ L3 ) )
           => ( ( ~ ( ( member_int @ K2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
                    & ( member_int @ L3 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
               => ( P @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
             => ( P @ K2 @ L3 ) ) )
       => ( P @ A0 @ A12 ) ) ) ).

% and_int.pinduct
thf(fact_8927_finite__Collect__le__nat,axiom,
    ! [K: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [N2: nat] : ( ord_less_eq_nat @ N2 @ K ) ) ) ).

% finite_Collect_le_nat
thf(fact_8928_finite__Collect__less__nat,axiom,
    ! [K: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [N2: nat] : ( ord_less_nat @ N2 @ K ) ) ) ).

% finite_Collect_less_nat
thf(fact_8929_finite__atLeastAtMost__int,axiom,
    ! [L2: int,U: int] : ( finite_finite_int @ ( set_or1266510415728281911st_int @ L2 @ U ) ) ).

% finite_atLeastAtMost_int
thf(fact_8930_finite__interval__int4,axiom,
    ! [A: int,B2: int] :
      ( finite_finite_int
      @ ( collect_int
        @ ^ [I2: int] :
            ( ( ord_less_int @ A @ I2 )
            & ( ord_less_int @ I2 @ B2 ) ) ) ) ).

% finite_interval_int4
thf(fact_8931_finite__interval__int3,axiom,
    ! [A: int,B2: int] :
      ( finite_finite_int
      @ ( collect_int
        @ ^ [I2: int] :
            ( ( ord_less_int @ A @ I2 )
            & ( ord_less_eq_int @ I2 @ B2 ) ) ) ) ).

% finite_interval_int3
thf(fact_8932_finite__interval__int2,axiom,
    ! [A: int,B2: int] :
      ( finite_finite_int
      @ ( collect_int
        @ ^ [I2: int] :
            ( ( ord_less_eq_int @ A @ I2 )
            & ( ord_less_int @ I2 @ B2 ) ) ) ) ).

% finite_interval_int2
thf(fact_8933_finite__nth__roots,axiom,
    ! [N: nat,C: complex] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [Z5: complex] :
              ( ( power_power_complex @ Z5 @ N )
              = C ) ) ) ) ).

% finite_nth_roots
thf(fact_8934_set__encode__insert,axiom,
    ! [A3: set_nat,N: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ~ ( member_nat @ N @ A3 )
       => ( ( nat_set_encode @ ( insert_nat @ N @ A3 ) )
          = ( plus_plus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ ( nat_set_encode @ A3 ) ) ) ) ) ).

% set_encode_insert
thf(fact_8935_upto_Opinduct,axiom,
    ! [A0: int,A12: int,P: int > int > $o] :
      ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ A0 @ A12 ) )
     => ( ! [I3: int,J2: int] :
            ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I3 @ J2 ) )
           => ( ( ( ord_less_eq_int @ I3 @ J2 )
               => ( P @ ( plus_plus_int @ I3 @ one_one_int ) @ J2 ) )
             => ( P @ I3 @ J2 ) ) )
       => ( P @ A0 @ A12 ) ) ) ).

% upto.pinduct
thf(fact_8936_set__encode__def,axiom,
    ( nat_set_encode
    = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% set_encode_def
thf(fact_8937_mask__eq__sum__exp__nat,axiom,
    ! [N: nat] :
      ( ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ ( suc @ zero_zero_nat ) )
      = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        @ ( collect_nat
          @ ^ [Q4: nat] : ( ord_less_nat @ Q4 @ N ) ) ) ) ).

% mask_eq_sum_exp_nat
thf(fact_8938_gauss__sum__nat,axiom,
    ! [N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [X: nat] : X
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
      = ( divide_divide_nat @ ( times_times_nat @ N @ ( suc @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% gauss_sum_nat
thf(fact_8939_arith__series__nat,axiom,
    ! [A: nat,D: nat,N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [I2: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ I2 @ D ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
      = ( divide_divide_nat @ ( times_times_nat @ ( suc @ N ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( times_times_nat @ N @ D ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% arith_series_nat
thf(fact_8940_Sum__Icc__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [X: nat] : X
        @ ( set_or1269000886237332187st_nat @ M @ N ) )
      = ( divide_divide_nat @ ( minus_minus_nat @ ( times_times_nat @ N @ ( plus_plus_nat @ N @ one_one_nat ) ) @ ( times_times_nat @ M @ ( minus_minus_nat @ M @ one_one_nat ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% Sum_Icc_nat
thf(fact_8941_even__set__encode__iff,axiom,
    ! [A3: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( nat_set_encode @ A3 ) )
        = ( ~ ( member_nat @ zero_zero_nat @ A3 ) ) ) ) ).

% even_set_encode_iff
thf(fact_8942_Maclaurin__minus__cos__expansion,axiom,
    ! [N: nat,X4: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ X4 @ zero_zero_real )
       => ? [T2: real] :
            ( ( ord_less_real @ X4 @ T2 )
            & ( ord_less_real @ T2 @ zero_zero_real )
            & ( ( cos_real @ X4 )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M6: nat] : ( times_times_real @ ( cos_coeff @ M6 ) @ ( power_power_real @ X4 @ M6 ) )
                  @ ( set_ord_lessThan_nat @ N ) )
                @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T2 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X4 @ N ) ) ) ) ) ) ) ).

% Maclaurin_minus_cos_expansion
thf(fact_8943_Maclaurin__cos__expansion2,axiom,
    ! [X4: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ? [T2: real] :
            ( ( ord_less_real @ zero_zero_real @ T2 )
            & ( ord_less_real @ T2 @ X4 )
            & ( ( cos_real @ X4 )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M6: nat] : ( times_times_real @ ( cos_coeff @ M6 ) @ ( power_power_real @ X4 @ M6 ) )
                  @ ( set_ord_lessThan_nat @ N ) )
                @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T2 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X4 @ N ) ) ) ) ) ) ) ).

% Maclaurin_cos_expansion2
thf(fact_8944_Maclaurin__sin__expansion3,axiom,
    ! [N: nat,X4: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ X4 )
       => ? [T2: real] :
            ( ( ord_less_real @ zero_zero_real @ T2 )
            & ( ord_less_real @ T2 @ X4 )
            & ( ( sin_real @ X4 )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M6: nat] : ( times_times_real @ ( sin_coeff @ M6 ) @ ( power_power_real @ X4 @ M6 ) )
                  @ ( set_ord_lessThan_nat @ N ) )
                @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T2 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X4 @ N ) ) ) ) ) ) ) ).

% Maclaurin_sin_expansion3
thf(fact_8945_Maclaurin__sin__expansion4,axiom,
    ! [X4: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ? [T2: real] :
          ( ( ord_less_real @ zero_zero_real @ T2 )
          & ( ord_less_eq_real @ T2 @ X4 )
          & ( ( sin_real @ X4 )
            = ( plus_plus_real
              @ ( groups6591440286371151544t_real
                @ ^ [M6: nat] : ( times_times_real @ ( sin_coeff @ M6 ) @ ( power_power_real @ X4 @ M6 ) )
                @ ( set_ord_lessThan_nat @ N ) )
              @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T2 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X4 @ N ) ) ) ) ) ) ).

% Maclaurin_sin_expansion4
thf(fact_8946_finite__lessThan,axiom,
    ! [K: nat] : ( finite_finite_nat @ ( set_ord_lessThan_nat @ K ) ) ).

% finite_lessThan
thf(fact_8947_lessThan__0,axiom,
    ( ( set_ord_lessThan_nat @ zero_zero_nat )
    = bot_bot_set_nat ) ).

% lessThan_0
thf(fact_8948_sumr__cos__zero__one,axiom,
    ! [N: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [M6: nat] : ( times_times_real @ ( cos_coeff @ M6 ) @ ( power_power_real @ zero_zero_real @ M6 ) )
        @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
      = one_one_real ) ).

% sumr_cos_zero_one
thf(fact_8949_lessThan__Suc,axiom,
    ! [K: nat] :
      ( ( set_ord_lessThan_nat @ ( suc @ K ) )
      = ( insert_nat @ K @ ( set_ord_lessThan_nat @ K ) ) ) ).

% lessThan_Suc
thf(fact_8950_lessThan__empty__iff,axiom,
    ! [N: nat] :
      ( ( ( set_ord_lessThan_nat @ N )
        = bot_bot_set_nat )
      = ( N = zero_zero_nat ) ) ).

% lessThan_empty_iff
thf(fact_8951_lessThan__nat__numeral,axiom,
    ! [K: num] :
      ( ( set_ord_lessThan_nat @ ( numeral_numeral_nat @ K ) )
      = ( insert_nat @ ( pred_numeral @ K ) @ ( set_ord_lessThan_nat @ ( pred_numeral @ K ) ) ) ) ).

% lessThan_nat_numeral
thf(fact_8952_sum__nth__roots,axiom,
    ! [N: nat,C: complex] :
      ( ( ord_less_nat @ one_one_nat @ N )
     => ( ( groups7754918857620584856omplex
          @ ^ [X: complex] : X
          @ ( collect_complex
            @ ^ [Z5: complex] :
                ( ( power_power_complex @ Z5 @ N )
                = C ) ) )
        = zero_zero_complex ) ) ).

% sum_nth_roots
thf(fact_8953_sum__roots__unity,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ one_one_nat @ N )
     => ( ( groups7754918857620584856omplex
          @ ^ [X: complex] : X
          @ ( collect_complex
            @ ^ [Z5: complex] :
                ( ( power_power_complex @ Z5 @ N )
                = one_one_complex ) ) )
        = zero_zero_complex ) ) ).

% sum_roots_unity
thf(fact_8954_Maclaurin__lemma,axiom,
    ! [H2: real,F: real > real,J: nat > real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ H2 )
     => ? [B8: real] :
          ( ( F @ H2 )
          = ( plus_plus_real
            @ ( groups6591440286371151544t_real
              @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( J @ M6 ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ H2 @ M6 ) )
              @ ( set_ord_lessThan_nat @ N ) )
            @ ( times_times_real @ B8 @ ( divide_divide_real @ ( power_power_real @ H2 @ N ) @ ( semiri2265585572941072030t_real @ N ) ) ) ) ) ) ).

% Maclaurin_lemma
thf(fact_8955_sum__split__even__odd,axiom,
    ! [F: nat > real,G: nat > real,N: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [I2: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) @ ( F @ I2 ) @ ( G @ I2 ) )
        @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( plus_plus_real
        @ ( groups6591440286371151544t_real
          @ ^ [I2: nat] : ( F @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) )
          @ ( set_ord_lessThan_nat @ N ) )
        @ ( groups6591440286371151544t_real
          @ ^ [I2: nat] : ( G @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) @ one_one_nat ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% sum_split_even_odd
thf(fact_8956_Maclaurin__exp__le,axiom,
    ! [X4: real,N: nat] :
    ? [T2: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ T2 ) @ ( abs_abs_real @ X4 ) )
      & ( ( exp_real @ X4 )
        = ( plus_plus_real
          @ ( groups6591440286371151544t_real
            @ ^ [M6: nat] : ( divide_divide_real @ ( power_power_real @ X4 @ M6 ) @ ( semiri2265585572941072030t_real @ M6 ) )
            @ ( set_ord_lessThan_nat @ N ) )
          @ ( times_times_real @ ( divide_divide_real @ ( exp_real @ T2 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X4 @ N ) ) ) ) ) ).

% Maclaurin_exp_le
thf(fact_8957_Maclaurin__sin__bound,axiom,
    ! [X4: real,N: nat] :
      ( ord_less_eq_real
      @ ( abs_abs_real
        @ ( minus_minus_real @ ( sin_real @ X4 )
          @ ( groups6591440286371151544t_real
            @ ^ [M6: nat] : ( times_times_real @ ( sin_coeff @ M6 ) @ ( power_power_real @ X4 @ M6 ) )
            @ ( set_ord_lessThan_nat @ N ) ) ) )
      @ ( times_times_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( abs_abs_real @ X4 ) @ N ) ) ) ).

% Maclaurin_sin_bound
thf(fact_8958_Sum__Icc__int,axiom,
    ! [M: int,N: int] :
      ( ( ord_less_eq_int @ M @ N )
     => ( ( groups4538972089207619220nt_int
          @ ^ [X: int] : X
          @ ( set_or1266510415728281911st_int @ M @ N ) )
        = ( divide_divide_int @ ( minus_minus_int @ ( times_times_int @ N @ ( plus_plus_int @ N @ one_one_int ) ) @ ( times_times_int @ M @ ( minus_minus_int @ M @ one_one_int ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).

% Sum_Icc_int
thf(fact_8959_sum__pos__lt__pair,axiom,
    ! [F: nat > real,K: nat] :
      ( ( summable_real @ F )
     => ( ! [D3: nat] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( F @ ( plus_plus_nat @ K @ ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ D3 ) ) ) @ ( F @ ( plus_plus_nat @ K @ ( plus_plus_nat @ ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ D3 ) @ one_one_nat ) ) ) ) )
       => ( ord_less_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ K ) ) @ ( suminf_real @ F ) ) ) ) ).

% sum_pos_lt_pair
thf(fact_8960_Maclaurin__exp__lt,axiom,
    ! [X4: real,N: nat] :
      ( ( X4 != zero_zero_real )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ? [T2: real] :
            ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ T2 ) )
            & ( ord_less_real @ ( abs_abs_real @ T2 ) @ ( abs_abs_real @ X4 ) )
            & ( ( exp_real @ X4 )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M6: nat] : ( divide_divide_real @ ( power_power_real @ X4 @ M6 ) @ ( semiri2265585572941072030t_real @ M6 ) )
                  @ ( set_ord_lessThan_nat @ N ) )
                @ ( times_times_real @ ( divide_divide_real @ ( exp_real @ T2 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X4 @ N ) ) ) ) ) ) ) ).

% Maclaurin_exp_lt
thf(fact_8961_Maclaurin__sin__expansion,axiom,
    ! [X4: real,N: nat] :
    ? [T2: real] :
      ( ( sin_real @ X4 )
      = ( plus_plus_real
        @ ( groups6591440286371151544t_real
          @ ^ [M6: nat] : ( times_times_real @ ( sin_coeff @ M6 ) @ ( power_power_real @ X4 @ M6 ) )
          @ ( set_ord_lessThan_nat @ N ) )
        @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T2 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X4 @ N ) ) ) ) ).

% Maclaurin_sin_expansion
thf(fact_8962_Maclaurin__sin__expansion2,axiom,
    ! [X4: real,N: nat] :
    ? [T2: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ T2 ) @ ( abs_abs_real @ X4 ) )
      & ( ( sin_real @ X4 )
        = ( plus_plus_real
          @ ( groups6591440286371151544t_real
            @ ^ [M6: nat] : ( times_times_real @ ( sin_coeff @ M6 ) @ ( power_power_real @ X4 @ M6 ) )
            @ ( set_ord_lessThan_nat @ N ) )
          @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T2 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X4 @ N ) ) ) ) ) ).

% Maclaurin_sin_expansion2
thf(fact_8963_Maclaurin__cos__expansion,axiom,
    ! [X4: real,N: nat] :
    ? [T2: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ T2 ) @ ( abs_abs_real @ X4 ) )
      & ( ( cos_real @ X4 )
        = ( plus_plus_real
          @ ( groups6591440286371151544t_real
            @ ^ [M6: nat] : ( times_times_real @ ( cos_coeff @ M6 ) @ ( power_power_real @ X4 @ M6 ) )
            @ ( set_ord_lessThan_nat @ N ) )
          @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T2 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X4 @ N ) ) ) ) ) ).

% Maclaurin_cos_expansion
thf(fact_8964_bij__betw__roots__unity,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( bij_betw_nat_complex
        @ ^ [K3: nat] : ( cis @ ( divide_divide_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( semiri5074537144036343181t_real @ K3 ) ) @ ( semiri5074537144036343181t_real @ N ) ) )
        @ ( set_ord_lessThan_nat @ N )
        @ ( collect_complex
          @ ^ [Z5: complex] :
              ( ( power_power_complex @ Z5 @ N )
              = one_one_complex ) ) ) ) ).

% bij_betw_roots_unity
thf(fact_8965_finite__atMost,axiom,
    ! [K: nat] : ( finite_finite_nat @ ( set_ord_atMost_nat @ K ) ) ).

% finite_atMost
thf(fact_8966_atMost__0,axiom,
    ( ( set_ord_atMost_nat @ zero_zero_nat )
    = ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ).

% atMost_0
thf(fact_8967_atMost__atLeast0,axiom,
    ( set_ord_atMost_nat
    = ( set_or1269000886237332187st_nat @ zero_zero_nat ) ) ).

% atMost_atLeast0
thf(fact_8968_lessThan__Suc__atMost,axiom,
    ! [K: nat] :
      ( ( set_ord_lessThan_nat @ ( suc @ K ) )
      = ( set_ord_atMost_nat @ K ) ) ).

% lessThan_Suc_atMost
thf(fact_8969_atMost__Suc,axiom,
    ! [K: nat] :
      ( ( set_ord_atMost_nat @ ( suc @ K ) )
      = ( insert_nat @ ( suc @ K ) @ ( set_ord_atMost_nat @ K ) ) ) ).

% atMost_Suc
thf(fact_8970_atMost__nat__numeral,axiom,
    ! [K: num] :
      ( ( set_ord_atMost_nat @ ( numeral_numeral_nat @ K ) )
      = ( insert_nat @ ( numeral_numeral_nat @ K ) @ ( set_ord_atMost_nat @ ( pred_numeral @ K ) ) ) ) ).

% atMost_nat_numeral
thf(fact_8971_sum__choose__upper,axiom,
    ! [M: nat,N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [K3: nat] : ( binomial @ K3 @ M )
        @ ( set_ord_atMost_nat @ N ) )
      = ( binomial @ ( suc @ N ) @ ( suc @ M ) ) ) ).

% sum_choose_upper
thf(fact_8972_sum__choose__lower,axiom,
    ! [R2: nat,N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [K3: nat] : ( binomial @ ( plus_plus_nat @ R2 @ K3 ) @ K3 )
        @ ( set_ord_atMost_nat @ N ) )
      = ( binomial @ ( suc @ ( plus_plus_nat @ R2 @ N ) ) @ N ) ) ).

% sum_choose_lower
thf(fact_8973_choose__rising__sum_I2_J,axiom,
    ! [N: nat,M: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [J3: nat] : ( binomial @ ( plus_plus_nat @ N @ J3 ) @ N )
        @ ( set_ord_atMost_nat @ M ) )
      = ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ N @ M ) @ one_one_nat ) @ M ) ) ).

% choose_rising_sum(2)
thf(fact_8974_choose__rising__sum_I1_J,axiom,
    ! [N: nat,M: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [J3: nat] : ( binomial @ ( plus_plus_nat @ N @ J3 ) @ N )
        @ ( set_ord_atMost_nat @ M ) )
      = ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ N @ M ) @ one_one_nat ) @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ).

% choose_rising_sum(1)
thf(fact_8975_atLeast1__atMost__eq__remove0,axiom,
    ! [N: nat] :
      ( ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N )
      = ( minus_minus_set_nat @ ( set_ord_atMost_nat @ N ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).

% atLeast1_atMost_eq_remove0
thf(fact_8976_sum__choose__diagonal,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( groups3542108847815614940at_nat
          @ ^ [K3: nat] : ( binomial @ ( minus_minus_nat @ N @ K3 ) @ ( minus_minus_nat @ M @ K3 ) )
          @ ( set_ord_atMost_nat @ M ) )
        = ( binomial @ ( suc @ N ) @ M ) ) ) ).

% sum_choose_diagonal
thf(fact_8977_vandermonde,axiom,
    ! [M: nat,N: nat,R2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [K3: nat] : ( times_times_nat @ ( binomial @ M @ K3 ) @ ( binomial @ N @ ( minus_minus_nat @ R2 @ K3 ) ) )
        @ ( set_ord_atMost_nat @ R2 ) )
      = ( binomial @ ( plus_plus_nat @ M @ N ) @ R2 ) ) ).

% vandermonde
thf(fact_8978_choose__row__sum,axiom,
    ! [N: nat] :
      ( ( groups3542108847815614940at_nat @ ( binomial @ N ) @ ( set_ord_atMost_nat @ N ) )
      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).

% choose_row_sum
thf(fact_8979_binomial,axiom,
    ! [A: nat,B2: nat,N: nat] :
      ( ( power_power_nat @ ( plus_plus_nat @ A @ B2 ) @ N )
      = ( groups3542108847815614940at_nat
        @ ^ [K3: nat] : ( times_times_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( binomial @ N @ K3 ) ) @ ( power_power_nat @ A @ K3 ) ) @ ( power_power_nat @ B2 @ ( minus_minus_nat @ N @ K3 ) ) )
        @ ( set_ord_atMost_nat @ N ) ) ) ).

% binomial
thf(fact_8980_polynomial__product__nat,axiom,
    ! [M: nat,A: nat > nat,N: nat,B2: nat > nat,X4: nat] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ M @ I3 )
         => ( ( A @ I3 )
            = zero_zero_nat ) )
     => ( ! [J2: nat] :
            ( ( ord_less_nat @ N @ J2 )
           => ( ( B2 @ J2 )
              = zero_zero_nat ) )
       => ( ( times_times_nat
            @ ( groups3542108847815614940at_nat
              @ ^ [I2: nat] : ( times_times_nat @ ( A @ I2 ) @ ( power_power_nat @ X4 @ I2 ) )
              @ ( set_ord_atMost_nat @ M ) )
            @ ( groups3542108847815614940at_nat
              @ ^ [J3: nat] : ( times_times_nat @ ( B2 @ J3 ) @ ( power_power_nat @ X4 @ J3 ) )
              @ ( set_ord_atMost_nat @ N ) ) )
          = ( groups3542108847815614940at_nat
            @ ^ [R5: nat] :
                ( times_times_nat
                @ ( groups3542108847815614940at_nat
                  @ ^ [K3: nat] : ( times_times_nat @ ( A @ K3 ) @ ( B2 @ ( minus_minus_nat @ R5 @ K3 ) ) )
                  @ ( set_ord_atMost_nat @ R5 ) )
                @ ( power_power_nat @ X4 @ R5 ) )
            @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N ) ) ) ) ) ) ).

% polynomial_product_nat
thf(fact_8981_choose__square__sum,axiom,
    ! [N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [K3: nat] : ( power_power_nat @ ( binomial @ N @ K3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        @ ( set_ord_atMost_nat @ N ) )
      = ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ N ) ) ).

% choose_square_sum
thf(fact_8982_binomial__r__part__sum,axiom,
    ! [M: nat] :
      ( ( groups3542108847815614940at_nat @ ( binomial @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ one_one_nat ) ) @ ( set_ord_atMost_nat @ M ) )
      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).

% binomial_r_part_sum
thf(fact_8983_choose__linear__sum,axiom,
    ! [N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [I2: nat] : ( times_times_nat @ I2 @ ( binomial @ N @ I2 ) )
        @ ( set_ord_atMost_nat @ N ) )
      = ( times_times_nat @ N @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).

% choose_linear_sum
thf(fact_8984_of__nat__id,axiom,
    ( semiri1316708129612266289at_nat
    = ( ^ [N2: nat] : N2 ) ) ).

% of_nat_id
thf(fact_8985_real__scaleR__def,axiom,
    real_V1485227260804924795R_real = times_times_real ).

% real_scaleR_def
thf(fact_8986_complex__scaleR,axiom,
    ! [R2: real,A: real,B2: real] :
      ( ( real_V2046097035970521341omplex @ R2 @ ( complex2 @ A @ B2 ) )
      = ( complex2 @ ( times_times_real @ R2 @ A ) @ ( times_times_real @ R2 @ B2 ) ) ) ).

% complex_scaleR
thf(fact_8987_bij__betw__nth__root__unity,axiom,
    ! [C: complex,N: nat] :
      ( ( C != zero_zero_complex )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( bij_be1856998921033663316omplex @ ( times_times_complex @ ( times_times_complex @ ( real_V4546457046886955230omplex @ ( root @ N @ ( real_V1022390504157884413omplex @ C ) ) ) @ ( cis @ ( divide_divide_real @ ( arg @ C ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) )
          @ ( collect_complex
            @ ^ [Z5: complex] :
                ( ( power_power_complex @ Z5 @ N )
                = one_one_complex ) )
          @ ( collect_complex
            @ ^ [Z5: complex] :
                ( ( power_power_complex @ Z5 @ N )
                = C ) ) ) ) ) ).

% bij_betw_nth_root_unity
thf(fact_8988_infinite__int__iff__unbounded,axiom,
    ! [S2: set_int] :
      ( ( ~ ( finite_finite_int @ S2 ) )
      = ( ! [M6: int] :
          ? [N2: int] :
            ( ( ord_less_int @ M6 @ ( abs_abs_int @ N2 ) )
            & ( member_int @ N2 @ S2 ) ) ) ) ).

% infinite_int_iff_unbounded
thf(fact_8989_Arg__def,axiom,
    ( arg
    = ( ^ [Z5: complex] :
          ( if_real @ ( Z5 = zero_zero_complex ) @ zero_zero_real
          @ ( fChoice_real
            @ ^ [A4: real] :
                ( ( ( sgn_sgn_complex @ Z5 )
                  = ( cis @ A4 ) )
                & ( ord_less_real @ ( uminus_uminus_real @ pi ) @ A4 )
                & ( ord_less_eq_real @ A4 @ pi ) ) ) ) ) ) ).

% Arg_def
thf(fact_8990_real__root__zero,axiom,
    ! [N: nat] :
      ( ( root @ N @ zero_zero_real )
      = zero_zero_real ) ).

% real_root_zero
thf(fact_8991_real__root__Suc__0,axiom,
    ! [X4: real] :
      ( ( root @ ( suc @ zero_zero_nat ) @ X4 )
      = X4 ) ).

% real_root_Suc_0
thf(fact_8992_real__root__eq__iff,axiom,
    ! [N: nat,X4: real,Y3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ( root @ N @ X4 )
          = ( root @ N @ Y3 ) )
        = ( X4 = Y3 ) ) ) ).

% real_root_eq_iff
thf(fact_8993_root__0,axiom,
    ! [X4: real] :
      ( ( root @ zero_zero_nat @ X4 )
      = zero_zero_real ) ).

% root_0
thf(fact_8994_real__root__eq__0__iff,axiom,
    ! [N: nat,X4: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ( root @ N @ X4 )
          = zero_zero_real )
        = ( X4 = zero_zero_real ) ) ) ).

% real_root_eq_0_iff
thf(fact_8995_real__root__less__iff,axiom,
    ! [N: nat,X4: real,Y3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ ( root @ N @ X4 ) @ ( root @ N @ Y3 ) )
        = ( ord_less_real @ X4 @ Y3 ) ) ) ).

% real_root_less_iff
thf(fact_8996_real__root__le__iff,axiom,
    ! [N: nat,X4: real,Y3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ ( root @ N @ X4 ) @ ( root @ N @ Y3 ) )
        = ( ord_less_eq_real @ X4 @ Y3 ) ) ) ).

% real_root_le_iff
thf(fact_8997_real__root__one,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( root @ N @ one_one_real )
        = one_one_real ) ) ).

% real_root_one
thf(fact_8998_real__root__eq__1__iff,axiom,
    ! [N: nat,X4: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ( root @ N @ X4 )
          = one_one_real )
        = ( X4 = one_one_real ) ) ) ).

% real_root_eq_1_iff
thf(fact_8999_real__root__gt__0__iff,axiom,
    ! [N: nat,Y3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ ( root @ N @ Y3 ) )
        = ( ord_less_real @ zero_zero_real @ Y3 ) ) ) ).

% real_root_gt_0_iff
thf(fact_9000_real__root__lt__0__iff,axiom,
    ! [N: nat,X4: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ ( root @ N @ X4 ) @ zero_zero_real )
        = ( ord_less_real @ X4 @ zero_zero_real ) ) ) ).

% real_root_lt_0_iff
thf(fact_9001_real__root__le__0__iff,axiom,
    ! [N: nat,X4: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ ( root @ N @ X4 ) @ zero_zero_real )
        = ( ord_less_eq_real @ X4 @ zero_zero_real ) ) ) ).

% real_root_le_0_iff
thf(fact_9002_real__root__ge__0__iff,axiom,
    ! [N: nat,Y3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ Y3 ) )
        = ( ord_less_eq_real @ zero_zero_real @ Y3 ) ) ) ).

% real_root_ge_0_iff
thf(fact_9003_real__root__gt__1__iff,axiom,
    ! [N: nat,Y3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ one_one_real @ ( root @ N @ Y3 ) )
        = ( ord_less_real @ one_one_real @ Y3 ) ) ) ).

% real_root_gt_1_iff
thf(fact_9004_real__root__lt__1__iff,axiom,
    ! [N: nat,X4: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ ( root @ N @ X4 ) @ one_one_real )
        = ( ord_less_real @ X4 @ one_one_real ) ) ) ).

% real_root_lt_1_iff
thf(fact_9005_real__root__ge__1__iff,axiom,
    ! [N: nat,Y3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ one_one_real @ ( root @ N @ Y3 ) )
        = ( ord_less_eq_real @ one_one_real @ Y3 ) ) ) ).

% real_root_ge_1_iff
thf(fact_9006_real__root__le__1__iff,axiom,
    ! [N: nat,X4: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ ( root @ N @ X4 ) @ one_one_real )
        = ( ord_less_eq_real @ X4 @ one_one_real ) ) ) ).

% real_root_le_1_iff
thf(fact_9007_real__root__pow__pos2,axiom,
    ! [N: nat,X4: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ zero_zero_real @ X4 )
       => ( ( power_power_real @ ( root @ N @ X4 ) @ N )
          = X4 ) ) ) ).

% real_root_pow_pos2
thf(fact_9008_real__root__minus,axiom,
    ! [N: nat,X4: real] :
      ( ( root @ N @ ( uminus_uminus_real @ X4 ) )
      = ( uminus_uminus_real @ ( root @ N @ X4 ) ) ) ).

% real_root_minus
thf(fact_9009_real__root__inverse,axiom,
    ! [N: nat,X4: real] :
      ( ( root @ N @ ( inverse_inverse_real @ X4 ) )
      = ( inverse_inverse_real @ ( root @ N @ X4 ) ) ) ).

% real_root_inverse
thf(fact_9010_real__root__commute,axiom,
    ! [M: nat,N: nat,X4: real] :
      ( ( root @ M @ ( root @ N @ X4 ) )
      = ( root @ N @ ( root @ M @ X4 ) ) ) ).

% real_root_commute
thf(fact_9011_real__root__mult,axiom,
    ! [N: nat,X4: real,Y3: real] :
      ( ( root @ N @ ( times_times_real @ X4 @ Y3 ) )
      = ( times_times_real @ ( root @ N @ X4 ) @ ( root @ N @ Y3 ) ) ) ).

% real_root_mult
thf(fact_9012_real__root__mult__exp,axiom,
    ! [M: nat,N: nat,X4: real] :
      ( ( root @ ( times_times_nat @ M @ N ) @ X4 )
      = ( root @ M @ ( root @ N @ X4 ) ) ) ).

% real_root_mult_exp
thf(fact_9013_real__root__divide,axiom,
    ! [N: nat,X4: real,Y3: real] :
      ( ( root @ N @ ( divide_divide_real @ X4 @ Y3 ) )
      = ( divide_divide_real @ ( root @ N @ X4 ) @ ( root @ N @ Y3 ) ) ) ).

% real_root_divide
thf(fact_9014_real__root__pos__pos__le,axiom,
    ! [X4: real,N: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ X4 ) ) ) ).

% real_root_pos_pos_le
thf(fact_9015_real__root__less__mono,axiom,
    ! [N: nat,X4: real,Y3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ X4 @ Y3 )
       => ( ord_less_real @ ( root @ N @ X4 ) @ ( root @ N @ Y3 ) ) ) ) ).

% real_root_less_mono
thf(fact_9016_real__root__le__mono,axiom,
    ! [N: nat,X4: real,Y3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ X4 @ Y3 )
       => ( ord_less_eq_real @ ( root @ N @ X4 ) @ ( root @ N @ Y3 ) ) ) ) ).

% real_root_le_mono
thf(fact_9017_real__root__power,axiom,
    ! [N: nat,X4: real,K: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( root @ N @ ( power_power_real @ X4 @ K ) )
        = ( power_power_real @ ( root @ N @ X4 ) @ K ) ) ) ).

% real_root_power
thf(fact_9018_real__root__abs,axiom,
    ! [N: nat,X4: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( root @ N @ ( abs_abs_real @ X4 ) )
        = ( abs_abs_real @ ( root @ N @ X4 ) ) ) ) ).

% real_root_abs
thf(fact_9019_sgn__root,axiom,
    ! [N: nat,X4: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( sgn_sgn_real @ ( root @ N @ X4 ) )
        = ( sgn_sgn_real @ X4 ) ) ) ).

% sgn_root
thf(fact_9020_real__root__gt__zero,axiom,
    ! [N: nat,X4: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ X4 )
       => ( ord_less_real @ zero_zero_real @ ( root @ N @ X4 ) ) ) ) ).

% real_root_gt_zero
thf(fact_9021_real__root__strict__decreasing,axiom,
    ! [N: nat,N3: nat,X4: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_nat @ N @ N3 )
       => ( ( ord_less_real @ one_one_real @ X4 )
         => ( ord_less_real @ ( root @ N3 @ X4 ) @ ( root @ N @ X4 ) ) ) ) ) ).

% real_root_strict_decreasing
thf(fact_9022_sqrt__def,axiom,
    ( sqrt
    = ( root @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% sqrt_def
thf(fact_9023_root__abs__power,axiom,
    ! [N: nat,Y3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( abs_abs_real @ ( root @ N @ ( power_power_real @ Y3 @ N ) ) )
        = ( abs_abs_real @ Y3 ) ) ) ).

% root_abs_power
thf(fact_9024_real__root__pos__pos,axiom,
    ! [N: nat,X4: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ X4 )
       => ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ X4 ) ) ) ) ).

% real_root_pos_pos
thf(fact_9025_real__root__strict__increasing,axiom,
    ! [N: nat,N3: nat,X4: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_nat @ N @ N3 )
       => ( ( ord_less_real @ zero_zero_real @ X4 )
         => ( ( ord_less_real @ X4 @ one_one_real )
           => ( ord_less_real @ ( root @ N @ X4 ) @ ( root @ N3 @ X4 ) ) ) ) ) ) ).

% real_root_strict_increasing
thf(fact_9026_real__root__decreasing,axiom,
    ! [N: nat,N3: nat,X4: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_nat @ N @ N3 )
       => ( ( ord_less_eq_real @ one_one_real @ X4 )
         => ( ord_less_eq_real @ ( root @ N3 @ X4 ) @ ( root @ N @ X4 ) ) ) ) ) ).

% real_root_decreasing
thf(fact_9027_real__root__pow__pos,axiom,
    ! [N: nat,X4: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ X4 )
       => ( ( power_power_real @ ( root @ N @ X4 ) @ N )
          = X4 ) ) ) ).

% real_root_pow_pos
thf(fact_9028_real__root__pos__unique,axiom,
    ! [N: nat,Y3: real,X4: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
       => ( ( ( power_power_real @ Y3 @ N )
            = X4 )
         => ( ( root @ N @ X4 )
            = Y3 ) ) ) ) ).

% real_root_pos_unique
thf(fact_9029_real__root__power__cancel,axiom,
    ! [N: nat,X4: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ zero_zero_real @ X4 )
       => ( ( root @ N @ ( power_power_real @ X4 @ N ) )
          = X4 ) ) ) ).

% real_root_power_cancel
thf(fact_9030_odd__real__root__pow,axiom,
    ! [N: nat,X4: real] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_real @ ( root @ N @ X4 ) @ N )
        = X4 ) ) ).

% odd_real_root_pow
thf(fact_9031_odd__real__root__unique,axiom,
    ! [N: nat,Y3: real,X4: real] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( ( power_power_real @ Y3 @ N )
          = X4 )
       => ( ( root @ N @ X4 )
          = Y3 ) ) ) ).

% odd_real_root_unique
thf(fact_9032_odd__real__root__power__cancel,axiom,
    ! [N: nat,X4: real] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( root @ N @ ( power_power_real @ X4 @ N ) )
        = X4 ) ) ).

% odd_real_root_power_cancel
thf(fact_9033_real__root__increasing,axiom,
    ! [N: nat,N3: nat,X4: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_nat @ N @ N3 )
       => ( ( ord_less_eq_real @ zero_zero_real @ X4 )
         => ( ( ord_less_eq_real @ X4 @ one_one_real )
           => ( ord_less_eq_real @ ( root @ N @ X4 ) @ ( root @ N3 @ X4 ) ) ) ) ) ) ).

% real_root_increasing
thf(fact_9034_sgn__power__root,axiom,
    ! [N: nat,X4: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( times_times_real @ ( sgn_sgn_real @ ( root @ N @ X4 ) ) @ ( power_power_real @ ( abs_abs_real @ ( root @ N @ X4 ) ) @ N ) )
        = X4 ) ) ).

% sgn_power_root
thf(fact_9035_root__sgn__power,axiom,
    ! [N: nat,Y3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( root @ N @ ( times_times_real @ ( sgn_sgn_real @ Y3 ) @ ( power_power_real @ ( abs_abs_real @ Y3 ) @ N ) ) )
        = Y3 ) ) ).

% root_sgn_power
thf(fact_9036_ln__root,axiom,
    ! [N: nat,B2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ B2 )
       => ( ( ln_ln_real @ ( root @ N @ B2 ) )
          = ( divide_divide_real @ ( ln_ln_real @ B2 ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).

% ln_root
thf(fact_9037_log__root,axiom,
    ! [N: nat,A: real,B2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ( ( log @ B2 @ ( root @ N @ A ) )
          = ( divide_divide_real @ ( log @ B2 @ A ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).

% log_root
thf(fact_9038_log__base__root,axiom,
    ! [N: nat,B2: real,X4: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ B2 )
       => ( ( log @ ( root @ N @ B2 ) @ X4 )
          = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B2 @ X4 ) ) ) ) ) ).

% log_base_root
thf(fact_9039_split__root,axiom,
    ! [P: real > $o,N: nat,X4: real] :
      ( ( P @ ( root @ N @ X4 ) )
      = ( ( ( N = zero_zero_nat )
         => ( P @ zero_zero_real ) )
        & ( ( ord_less_nat @ zero_zero_nat @ N )
         => ! [Y: real] :
              ( ( ( times_times_real @ ( sgn_sgn_real @ Y ) @ ( power_power_real @ ( abs_abs_real @ Y ) @ N ) )
                = X4 )
             => ( P @ Y ) ) ) ) ) ).

% split_root
thf(fact_9040_infinite__nat__iff__unbounded,axiom,
    ! [S2: set_nat] :
      ( ( ~ ( finite_finite_nat @ S2 ) )
      = ( ! [M6: nat] :
          ? [N2: nat] :
            ( ( ord_less_nat @ M6 @ N2 )
            & ( member_nat @ N2 @ S2 ) ) ) ) ).

% infinite_nat_iff_unbounded
thf(fact_9041_unbounded__k__infinite,axiom,
    ! [K: nat,S2: set_nat] :
      ( ! [M4: nat] :
          ( ( ord_less_nat @ K @ M4 )
         => ? [N6: nat] :
              ( ( ord_less_nat @ M4 @ N6 )
              & ( member_nat @ N6 @ S2 ) ) )
     => ~ ( finite_finite_nat @ S2 ) ) ).

% unbounded_k_infinite
thf(fact_9042_infinite__nat__iff__unbounded__le,axiom,
    ! [S2: set_nat] :
      ( ( ~ ( finite_finite_nat @ S2 ) )
      = ( ! [M6: nat] :
          ? [N2: nat] :
            ( ( ord_less_eq_nat @ M6 @ N2 )
            & ( member_nat @ N2 @ S2 ) ) ) ) ).

% infinite_nat_iff_unbounded_le
thf(fact_9043_root__powr__inverse,axiom,
    ! [N: nat,X4: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ X4 )
       => ( ( root @ N @ X4 )
          = ( powr_real @ X4 @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ) ).

% root_powr_inverse
thf(fact_9044_divmod__BitM__2__eq,axiom,
    ! [M: num] :
      ( ( unique5052692396658037445od_int @ ( bitM @ M ) @ ( bit0 @ one ) )
      = ( product_Pair_int_int @ ( minus_minus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ one_one_int ) ) ).

% divmod_BitM_2_eq
thf(fact_9045_insert__simp__excp,axiom,
    ! [Mi: nat,Deg: nat,TreeList2: list_VEBT_VEBT,X4: nat,Ma: nat,Summary: vEBT_VEBT] :
      ( ( ord_less_nat @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
     => ( ( ord_less_nat @ X4 @ Mi )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( X4 != Ma )
           => ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X4 )
              = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ X4 @ ( ord_max_nat @ Mi @ Ma ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary ) ) ) ) ) ) ) ).

% insert_simp_excp
thf(fact_9046_insert__simp__norm,axiom,
    ! [X4: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
      ( ( ord_less_nat @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
     => ( ( ord_less_nat @ Mi @ X4 )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( X4 != Ma )
           => ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X4 )
              = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ ( ord_max_nat @ X4 @ Ma ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary ) ) ) ) ) ) ) ).

% insert_simp_norm
thf(fact_9047_max__Suc__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_max_nat @ ( suc @ M ) @ ( suc @ N ) )
      = ( suc @ ( ord_max_nat @ M @ N ) ) ) ).

% max_Suc_Suc
thf(fact_9048_max__0R,axiom,
    ! [N: nat] :
      ( ( ord_max_nat @ N @ zero_zero_nat )
      = N ) ).

% max_0R
thf(fact_9049_max__0L,axiom,
    ! [N: nat] :
      ( ( ord_max_nat @ zero_zero_nat @ N )
      = N ) ).

% max_0L
thf(fact_9050_max__nat_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( ord_max_nat @ A @ zero_zero_nat )
      = A ) ).

% max_nat.right_neutral
thf(fact_9051_max__nat_Oneutr__eq__iff,axiom,
    ! [A: nat,B2: nat] :
      ( ( zero_zero_nat
        = ( ord_max_nat @ A @ B2 ) )
      = ( ( A = zero_zero_nat )
        & ( B2 = zero_zero_nat ) ) ) ).

% max_nat.neutr_eq_iff
thf(fact_9052_max__nat_Oleft__neutral,axiom,
    ! [A: nat] :
      ( ( ord_max_nat @ zero_zero_nat @ A )
      = A ) ).

% max_nat.left_neutral
thf(fact_9053_max__nat_Oeq__neutr__iff,axiom,
    ! [A: nat,B2: nat] :
      ( ( ( ord_max_nat @ A @ B2 )
        = zero_zero_nat )
      = ( ( A = zero_zero_nat )
        & ( B2 = zero_zero_nat ) ) ) ).

% max_nat.eq_neutr_iff
thf(fact_9054_max__numeral__Suc,axiom,
    ! [K: num,N: nat] :
      ( ( ord_max_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
      = ( suc @ ( ord_max_nat @ ( pred_numeral @ K ) @ N ) ) ) ).

% max_numeral_Suc
thf(fact_9055_max__Suc__numeral,axiom,
    ! [N: nat,K: num] :
      ( ( ord_max_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
      = ( suc @ ( ord_max_nat @ N @ ( pred_numeral @ K ) ) ) ) ).

% max_Suc_numeral
thf(fact_9056_pred__numeral__simps_I2_J,axiom,
    ! [K: num] :
      ( ( pred_numeral @ ( bit0 @ K ) )
      = ( numeral_numeral_nat @ ( bitM @ K ) ) ) ).

% pred_numeral_simps(2)
thf(fact_9057_nat__add__max__right,axiom,
    ! [M: nat,N: nat,Q2: nat] :
      ( ( plus_plus_nat @ M @ ( ord_max_nat @ N @ Q2 ) )
      = ( ord_max_nat @ ( plus_plus_nat @ M @ N ) @ ( plus_plus_nat @ M @ Q2 ) ) ) ).

% nat_add_max_right
thf(fact_9058_nat__add__max__left,axiom,
    ! [M: nat,N: nat,Q2: nat] :
      ( ( plus_plus_nat @ ( ord_max_nat @ M @ N ) @ Q2 )
      = ( ord_max_nat @ ( plus_plus_nat @ M @ Q2 ) @ ( plus_plus_nat @ N @ Q2 ) ) ) ).

% nat_add_max_left
thf(fact_9059_nat__mult__max__right,axiom,
    ! [M: nat,N: nat,Q2: nat] :
      ( ( times_times_nat @ M @ ( ord_max_nat @ N @ Q2 ) )
      = ( ord_max_nat @ ( times_times_nat @ M @ N ) @ ( times_times_nat @ M @ Q2 ) ) ) ).

% nat_mult_max_right
thf(fact_9060_nat__mult__max__left,axiom,
    ! [M: nat,N: nat,Q2: nat] :
      ( ( times_times_nat @ ( ord_max_nat @ M @ N ) @ Q2 )
      = ( ord_max_nat @ ( times_times_nat @ M @ Q2 ) @ ( times_times_nat @ N @ Q2 ) ) ) ).

% nat_mult_max_left
thf(fact_9061_semiring__norm_I26_J,axiom,
    ( ( bitM @ one )
    = one ) ).

% semiring_norm(26)
thf(fact_9062_nat__minus__add__max,axiom,
    ! [N: nat,M: nat] :
      ( ( plus_plus_nat @ ( minus_minus_nat @ N @ M ) @ M )
      = ( ord_max_nat @ N @ M ) ) ).

% nat_minus_add_max
thf(fact_9063_semiring__norm_I27_J,axiom,
    ! [N: num] :
      ( ( bitM @ ( bit0 @ N ) )
      = ( bit1 @ ( bitM @ N ) ) ) ).

% semiring_norm(27)
thf(fact_9064_semiring__norm_I28_J,axiom,
    ! [N: num] :
      ( ( bitM @ ( bit1 @ N ) )
      = ( bit1 @ ( bit0 @ N ) ) ) ).

% semiring_norm(28)
thf(fact_9065_inc__BitM__eq,axiom,
    ! [N: num] :
      ( ( inc @ ( bitM @ N ) )
      = ( bit0 @ N ) ) ).

% inc_BitM_eq
thf(fact_9066_BitM__inc__eq,axiom,
    ! [N: num] :
      ( ( bitM @ ( inc @ N ) )
      = ( bit1 @ N ) ) ).

% BitM_inc_eq
thf(fact_9067_eval__nat__numeral_I2_J,axiom,
    ! [N: num] :
      ( ( numeral_numeral_nat @ ( bit0 @ N ) )
      = ( suc @ ( numeral_numeral_nat @ ( bitM @ N ) ) ) ) ).

% eval_nat_numeral(2)
thf(fact_9068_one__plus__BitM,axiom,
    ! [N: num] :
      ( ( plus_plus_num @ one @ ( bitM @ N ) )
      = ( bit0 @ N ) ) ).

% one_plus_BitM
thf(fact_9069_BitM__plus__one,axiom,
    ! [N: num] :
      ( ( plus_plus_num @ ( bitM @ N ) @ one )
      = ( bit0 @ N ) ) ).

% BitM_plus_one
thf(fact_9070_vebt__insert_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X4: nat] :
      ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary ) @ X4 )
      = ( if_VEBT_VEBT
        @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X4 @ Mi ) @ Mi @ X4 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
          & ~ ( ( X4 = Mi )
              | ( X4 = Ma ) ) )
        @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ ( if_nat @ ( ord_less_nat @ X4 @ Mi ) @ X4 @ Mi ) @ ( ord_max_nat @ ( if_nat @ ( ord_less_nat @ X4 @ Mi ) @ Mi @ X4 ) @ Ma ) ) ) @ ( suc @ ( suc @ Va2 ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X4 @ Mi ) @ Mi @ X4 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X4 @ Mi ) @ Mi @ X4 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ X4 @ Mi ) @ Mi @ X4 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X4 @ Mi ) @ Mi @ X4 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X4 @ Mi ) @ Mi @ X4 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary ) )
        @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary ) ) ) ).

% vebt_insert.simps(5)
thf(fact_9071_vebt__insert_Oelims,axiom,
    ! [X4: vEBT_VEBT,Xa: nat,Y3: vEBT_VEBT] :
      ( ( ( vEBT_vebt_insert @ X4 @ Xa )
        = Y3 )
     => ( ! [A5: $o,B5: $o] :
            ( ( X4
              = ( vEBT_Leaf @ A5 @ B5 ) )
           => ~ ( ( ( Xa = zero_zero_nat )
                 => ( Y3
                    = ( vEBT_Leaf @ $true @ B5 ) ) )
                & ( ( Xa != zero_zero_nat )
                 => ( ( ( Xa = one_one_nat )
                     => ( Y3
                        = ( vEBT_Leaf @ A5 @ $true ) ) )
                    & ( ( Xa != one_one_nat )
                     => ( Y3
                        = ( vEBT_Leaf @ A5 @ B5 ) ) ) ) ) ) )
       => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT] :
              ( ( X4
                = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S3 ) )
             => ( Y3
               != ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S3 ) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT] :
                ( ( X4
                  = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S3 ) )
               => ( Y3
                 != ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S3 ) ) )
           => ( ! [V3: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                  ( ( X4
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V3 ) ) @ TreeList3 @ Summary2 ) )
                 => ( Y3
                   != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xa @ Xa ) ) @ ( suc @ ( suc @ V3 ) ) @ TreeList3 @ Summary2 ) ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X4
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) )
                   => ( Y3
                     != ( if_VEBT_VEBT
                        @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                          & ~ ( ( Xa = Mi2 )
                              | ( Xa = Ma2 ) ) )
                        @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Xa @ Mi2 ) @ ( ord_max_nat @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ Ma2 ) ) ) @ ( suc @ ( suc @ Va ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary2 ) )
                        @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) ) ) ) ) ) ) ) ) ).

% vebt_insert.elims
thf(fact_9072_vebt__insert_Opelims,axiom,
    ! [X4: vEBT_VEBT,Xa: nat,Y3: vEBT_VEBT] :
      ( ( ( vEBT_vebt_insert @ X4 @ Xa )
        = Y3 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ X4 @ Xa ) )
       => ( ! [A5: $o,B5: $o] :
              ( ( X4
                = ( vEBT_Leaf @ A5 @ B5 ) )
             => ( ( ( ( Xa = zero_zero_nat )
                   => ( Y3
                      = ( vEBT_Leaf @ $true @ B5 ) ) )
                  & ( ( Xa != zero_zero_nat )
                   => ( ( ( Xa = one_one_nat )
                       => ( Y3
                          = ( vEBT_Leaf @ A5 @ $true ) ) )
                      & ( ( Xa != one_one_nat )
                       => ( Y3
                          = ( vEBT_Leaf @ A5 @ B5 ) ) ) ) ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B5 ) @ Xa ) ) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT] :
                ( ( X4
                  = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S3 ) )
               => ( ( Y3
                    = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S3 ) )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S3 ) @ Xa ) ) ) )
           => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT] :
                  ( ( X4
                    = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S3 ) )
                 => ( ( Y3
                      = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S3 ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S3 ) @ Xa ) ) ) )
             => ( ! [V3: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X4
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V3 ) ) @ TreeList3 @ Summary2 ) )
                   => ( ( Y3
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xa @ Xa ) ) @ ( suc @ ( suc @ V3 ) ) @ TreeList3 @ Summary2 ) )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V3 ) ) @ TreeList3 @ Summary2 ) @ Xa ) ) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X4
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) )
                     => ( ( Y3
                          = ( if_VEBT_VEBT
                            @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                              & ~ ( ( Xa = Mi2 )
                                  | ( Xa = Ma2 ) ) )
                            @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Xa @ Mi2 ) @ ( ord_max_nat @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ Ma2 ) ) ) @ ( suc @ ( suc @ Va ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary2 ) )
                            @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ).

% vebt_insert.pelims
thf(fact_9073_vebt__buildup_Opelims,axiom,
    ! [X4: nat,Y3: vEBT_VEBT] :
      ( ( ( vEBT_vebt_buildup @ X4 )
        = Y3 )
     => ( ( accp_nat @ vEBT_v4011308405150292612up_rel @ X4 )
       => ( ( ( X4 = zero_zero_nat )
           => ( ( Y3
                = ( vEBT_Leaf @ $false @ $false ) )
             => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ zero_zero_nat ) ) )
         => ( ( ( X4
                = ( suc @ zero_zero_nat ) )
             => ( ( Y3
                  = ( vEBT_Leaf @ $false @ $false ) )
               => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ ( suc @ zero_zero_nat ) ) ) )
           => ~ ! [Va: nat] :
                  ( ( X4
                    = ( suc @ ( suc @ Va ) ) )
                 => ( ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
                       => ( Y3
                          = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
                      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
                       => ( Y3
                          = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
                   => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ ( suc @ ( suc @ Va ) ) ) ) ) ) ) ) ) ).

% vebt_buildup.pelims
thf(fact_9074_max__enat__simps_I2_J,axiom,
    ! [Q2: extended_enat] :
      ( ( ord_ma741700101516333627d_enat @ Q2 @ zero_z5237406670263579293d_enat )
      = Q2 ) ).

% max_enat_simps(2)
thf(fact_9075_max__enat__simps_I3_J,axiom,
    ! [Q2: extended_enat] :
      ( ( ord_ma741700101516333627d_enat @ zero_z5237406670263579293d_enat @ Q2 )
      = Q2 ) ).

% max_enat_simps(3)
thf(fact_9076_divmod__step__nat__def,axiom,
    ( unique5026877609467782581ep_nat
    = ( ^ [L: num] :
          ( produc2626176000494625587at_nat
          @ ^ [Q4: nat,R5: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L ) @ R5 ) @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q4 ) @ one_one_nat ) @ ( minus_minus_nat @ R5 @ ( numeral_numeral_nat @ L ) ) ) @ ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q4 ) @ R5 ) ) ) ) ) ).

% divmod_step_nat_def
thf(fact_9077_divmod__step__int__def,axiom,
    ( unique5024387138958732305ep_int
    = ( ^ [L: num] :
          ( produc4245557441103728435nt_int
          @ ^ [Q4: int,R5: int] : ( if_Pro3027730157355071871nt_int @ ( ord_less_eq_int @ ( numeral_numeral_int @ L ) @ R5 ) @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q4 ) @ one_one_int ) @ ( minus_minus_int @ R5 @ ( numeral_numeral_int @ L ) ) ) @ ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q4 ) @ R5 ) ) ) ) ) ).

% divmod_step_int_def
thf(fact_9078_divmod__nat__if,axiom,
    ( divmod_nat
    = ( ^ [M6: nat,N2: nat] :
          ( if_Pro6206227464963214023at_nat
          @ ( ( N2 = zero_zero_nat )
            | ( ord_less_nat @ M6 @ N2 ) )
          @ ( product_Pair_nat_nat @ zero_zero_nat @ M6 )
          @ ( produc2626176000494625587at_nat
            @ ^ [Q4: nat] : ( product_Pair_nat_nat @ ( suc @ Q4 ) )
            @ ( divmod_nat @ ( minus_minus_nat @ M6 @ N2 ) @ N2 ) ) ) ) ) ).

% divmod_nat_if
thf(fact_9079_arctan__def,axiom,
    ( arctan
    = ( ^ [Y: real] :
          ( the_real
          @ ^ [X: real] :
              ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
              & ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
              & ( ( tan_real @ X )
                = Y ) ) ) ) ) ).

% arctan_def
thf(fact_9080_ln__neg__is__const,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ X4 @ zero_zero_real )
     => ( ( ln_ln_real @ X4 )
        = ( the_real
          @ ^ [X: real] : $false ) ) ) ).

% ln_neg_is_const
thf(fact_9081_arccos__def,axiom,
    ( arccos
    = ( ^ [Y: real] :
          ( the_real
          @ ^ [X: real] :
              ( ( ord_less_eq_real @ zero_zero_real @ X )
              & ( ord_less_eq_real @ X @ pi )
              & ( ( cos_real @ X )
                = Y ) ) ) ) ) ).

% arccos_def
thf(fact_9082_divmod__nat__def,axiom,
    ( divmod_nat
    = ( ^ [M6: nat,N2: nat] : ( product_Pair_nat_nat @ ( divide_divide_nat @ M6 @ N2 ) @ ( modulo_modulo_nat @ M6 @ N2 ) ) ) ) ).

% divmod_nat_def
thf(fact_9083_pi__half,axiom,
    ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
    = ( the_real
      @ ^ [X: real] :
          ( ( ord_less_eq_real @ zero_zero_real @ X )
          & ( ord_less_eq_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
          & ( ( cos_real @ X )
            = zero_zero_real ) ) ) ) ).

% pi_half
thf(fact_9084_pi__def,axiom,
    ( pi
    = ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
      @ ( the_real
        @ ^ [X: real] :
            ( ( ord_less_eq_real @ zero_zero_real @ X )
            & ( ord_less_eq_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
            & ( ( cos_real @ X )
              = zero_zero_real ) ) ) ) ) ).

% pi_def
thf(fact_9085_arcsin__def,axiom,
    ( arcsin
    = ( ^ [Y: real] :
          ( the_real
          @ ^ [X: real] :
              ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
              & ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
              & ( ( sin_real @ X )
                = Y ) ) ) ) ) ).

% arcsin_def
thf(fact_9086_or__int__unfold,axiom,
    ( bit_se1409905431419307370or_int
    = ( ^ [K3: int,L: int] :
          ( if_int
          @ ( ( K3
              = ( uminus_uminus_int @ one_one_int ) )
            | ( L
              = ( uminus_uminus_int @ one_one_int ) ) )
          @ ( uminus_uminus_int @ one_one_int )
          @ ( if_int @ ( K3 = zero_zero_int ) @ L @ ( if_int @ ( L = zero_zero_int ) @ K3 @ ( plus_plus_int @ ( ord_max_int @ ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).

% or_int_unfold
thf(fact_9087_horner__sum__of__bool__2__less,axiom,
    ! [Bs: list_o] : ( ord_less_int @ ( groups9116527308978886569_o_int @ zero_n2684676970156552555ol_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Bs ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( size_size_list_o @ Bs ) ) ) ).

% horner_sum_of_bool_2_less
thf(fact_9088_or__nonnegative__int__iff,axiom,
    ! [K: int,L2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se1409905431419307370or_int @ K @ L2 ) )
      = ( ( ord_less_eq_int @ zero_zero_int @ K )
        & ( ord_less_eq_int @ zero_zero_int @ L2 ) ) ) ).

% or_nonnegative_int_iff
thf(fact_9089_or__negative__int__iff,axiom,
    ! [K: int,L2: int] :
      ( ( ord_less_int @ ( bit_se1409905431419307370or_int @ K @ L2 ) @ zero_zero_int )
      = ( ( ord_less_int @ K @ zero_zero_int )
        | ( ord_less_int @ L2 @ zero_zero_int ) ) ) ).

% or_negative_int_iff
thf(fact_9090_or__minus__numerals_I2_J,axiom,
    ! [N: num] :
      ( ( bit_se1409905431419307370or_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) ) ).

% or_minus_numerals(2)
thf(fact_9091_or__minus__numerals_I6_J,axiom,
    ! [N: num] :
      ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ one_one_int )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) ) ).

% or_minus_numerals(6)
thf(fact_9092_or__minus__minus__numerals,axiom,
    ! [M: num,N: num] :
      ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( bit_ri7919022796975470100ot_int @ ( bit_se725231765392027082nd_int @ ( minus_minus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( minus_minus_int @ ( numeral_numeral_int @ N ) @ one_one_int ) ) ) ) ).

% or_minus_minus_numerals
thf(fact_9093_and__minus__minus__numerals,axiom,
    ! [M: num,N: num] :
      ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( bit_ri7919022796975470100ot_int @ ( bit_se1409905431419307370or_int @ ( minus_minus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( minus_minus_int @ ( numeral_numeral_int @ N ) @ one_one_int ) ) ) ) ).

% and_minus_minus_numerals
thf(fact_9094_bit__or__int__iff,axiom,
    ! [K: int,L2: int,N: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_se1409905431419307370or_int @ K @ L2 ) @ N )
      = ( ( bit_se1146084159140164899it_int @ K @ N )
        | ( bit_se1146084159140164899it_int @ L2 @ N ) ) ) ).

% bit_or_int_iff
thf(fact_9095_OR__lower,axiom,
    ! [X4: int,Y3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X4 )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
       => ( ord_less_eq_int @ zero_zero_int @ ( bit_se1409905431419307370or_int @ X4 @ Y3 ) ) ) ) ).

% OR_lower
thf(fact_9096_or__greater__eq,axiom,
    ! [L2: int,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ L2 )
     => ( ord_less_eq_int @ K @ ( bit_se1409905431419307370or_int @ K @ L2 ) ) ) ).

% or_greater_eq
thf(fact_9097_plus__and__or,axiom,
    ! [X4: int,Y3: int] :
      ( ( plus_plus_int @ ( bit_se725231765392027082nd_int @ X4 @ Y3 ) @ ( bit_se1409905431419307370or_int @ X4 @ Y3 ) )
      = ( plus_plus_int @ X4 @ Y3 ) ) ).

% plus_and_or
thf(fact_9098_or__int__def,axiom,
    ( bit_se1409905431419307370or_int
    = ( ^ [K3: int,L: int] : ( bit_ri7919022796975470100ot_int @ ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ K3 ) @ ( bit_ri7919022796975470100ot_int @ L ) ) ) ) ) ).

% or_int_def
thf(fact_9099_or__not__numerals_I1_J,axiom,
    ( ( bit_se1409905431419307370or_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
    = ( bit_ri7919022796975470100ot_int @ zero_zero_int ) ) ).

% or_not_numerals(1)
thf(fact_9100_xor__int__def,axiom,
    ( bit_se6526347334894502574or_int
    = ( ^ [K3: int,L: int] : ( bit_se1409905431419307370or_int @ ( bit_se725231765392027082nd_int @ K3 @ ( bit_ri7919022796975470100ot_int @ L ) ) @ ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ K3 ) @ L ) ) ) ) ).

% xor_int_def
thf(fact_9101_concat__bit__def,axiom,
    ( bit_concat_bit
    = ( ^ [N2: nat,K3: int,L: int] : ( bit_se1409905431419307370or_int @ ( bit_se2923211474154528505it_int @ N2 @ K3 ) @ ( bit_se545348938243370406it_int @ N2 @ L ) ) ) ) ).

% concat_bit_def
thf(fact_9102_set__bit__int__def,axiom,
    ( bit_se7879613467334960850it_int
    = ( ^ [N2: nat,K3: int] : ( bit_se1409905431419307370or_int @ K3 @ ( bit_se545348938243370406it_int @ N2 @ one_one_int ) ) ) ) ).

% set_bit_int_def
thf(fact_9103_or__not__numerals_I4_J,axiom,
    ! [M: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
      = ( bit_ri7919022796975470100ot_int @ one_one_int ) ) ).

% or_not_numerals(4)
thf(fact_9104_or__not__numerals_I2_J,axiom,
    ! [N: num] :
      ( ( bit_se1409905431419307370or_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
      = ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) ) ).

% or_not_numerals(2)
thf(fact_9105_or__not__numerals_I3_J,axiom,
    ! [N: num] :
      ( ( bit_se1409905431419307370or_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
      = ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) ) ).

% or_not_numerals(3)
thf(fact_9106_or__not__numerals_I7_J,axiom,
    ! [M: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
      = ( bit_ri7919022796975470100ot_int @ zero_zero_int ) ) ).

% or_not_numerals(7)
thf(fact_9107_or__not__numerals_I6_J,axiom,
    ! [M: num,N: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).

% or_not_numerals(6)
thf(fact_9108_OR__upper,axiom,
    ! [X4: int,N: nat,Y3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X4 )
     => ( ( ord_less_int @ X4 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
       => ( ( ord_less_int @ Y3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
         => ( ord_less_int @ ( bit_se1409905431419307370or_int @ X4 @ Y3 ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).

% OR_upper
thf(fact_9109_or__not__numerals_I5_J,axiom,
    ! [M: num,N: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ) ).

% or_not_numerals(5)
thf(fact_9110_or__not__numerals_I9_J,axiom,
    ! [M: num,N: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ) ).

% or_not_numerals(9)
thf(fact_9111_or__not__numerals_I8_J,axiom,
    ! [M: num,N: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ) ).

% or_not_numerals(8)
thf(fact_9112_or__int__rec,axiom,
    ( bit_se1409905431419307370or_int
    = ( ^ [K3: int,L: int] :
          ( plus_plus_int
          @ ( zero_n2684676970156552555ol_int
            @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 )
              | ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) )
          @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% or_int_rec
thf(fact_9113_or__minus__numerals_I5_J,axiom,
    ! [N: num] :
      ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ one_one_int )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ one @ ( bitM @ N ) ) ) ) ) ).

% or_minus_numerals(5)
thf(fact_9114_or__minus__numerals_I1_J,axiom,
    ! [N: num] :
      ( ( bit_se1409905431419307370or_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ one @ ( bitM @ N ) ) ) ) ) ).

% or_minus_numerals(1)
thf(fact_9115_or__nat__numerals_I2_J,axiom,
    ! [Y3: num] :
      ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) )
      = ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) ) ).

% or_nat_numerals(2)
thf(fact_9116_or__nat__numerals_I4_J,axiom,
    ! [X4: num] :
      ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit1 @ X4 ) ) @ ( suc @ zero_zero_nat ) )
      = ( numeral_numeral_nat @ ( bit1 @ X4 ) ) ) ).

% or_nat_numerals(4)
thf(fact_9117_or__nat__numerals_I1_J,axiom,
    ! [Y3: num] :
      ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) )
      = ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) ) ).

% or_nat_numerals(1)
thf(fact_9118_or__nat__numerals_I3_J,axiom,
    ! [X4: num] :
      ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit0 @ X4 ) ) @ ( suc @ zero_zero_nat ) )
      = ( numeral_numeral_nat @ ( bit1 @ X4 ) ) ) ).

% or_nat_numerals(3)
thf(fact_9119_or__minus__numerals_I4_J,axiom,
    ! [M: num,N: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bit0 @ N ) ) ) ) ) ).

% or_minus_numerals(4)
thf(fact_9120_or__minus__numerals_I8_J,axiom,
    ! [N: num,M: num] :
      ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ ( numeral_numeral_int @ M ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bit0 @ N ) ) ) ) ) ).

% or_minus_numerals(8)
thf(fact_9121_or__minus__numerals_I7_J,axiom,
    ! [N: num,M: num] :
      ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ ( numeral_numeral_int @ M ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bitM @ N ) ) ) ) ) ).

% or_minus_numerals(7)
thf(fact_9122_or__minus__numerals_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bitM @ N ) ) ) ) ) ).

% or_minus_numerals(3)
thf(fact_9123_or__not__num__neg_Osimps_I1_J,axiom,
    ( ( bit_or_not_num_neg @ one @ one )
    = one ) ).

% or_not_num_neg.simps(1)
thf(fact_9124_set__bit__nat__def,axiom,
    ( bit_se7882103937844011126it_nat
    = ( ^ [M6: nat,N2: nat] : ( bit_se1412395901928357646or_nat @ N2 @ ( bit_se547839408752420682it_nat @ M6 @ one_one_nat ) ) ) ) ).

% set_bit_nat_def
thf(fact_9125_or__not__num__neg_Osimps_I4_J,axiom,
    ! [N: num] :
      ( ( bit_or_not_num_neg @ ( bit0 @ N ) @ one )
      = ( bit0 @ one ) ) ).

% or_not_num_neg.simps(4)
thf(fact_9126_or__not__num__neg_Osimps_I6_J,axiom,
    ! [N: num,M: num] :
      ( ( bit_or_not_num_neg @ ( bit0 @ N ) @ ( bit1 @ M ) )
      = ( bit0 @ ( bit_or_not_num_neg @ N @ M ) ) ) ).

% or_not_num_neg.simps(6)
thf(fact_9127_or__not__num__neg_Osimps_I3_J,axiom,
    ! [M: num] :
      ( ( bit_or_not_num_neg @ one @ ( bit1 @ M ) )
      = ( bit1 @ M ) ) ).

% or_not_num_neg.simps(3)
thf(fact_9128_or__not__num__neg_Osimps_I7_J,axiom,
    ! [N: num] :
      ( ( bit_or_not_num_neg @ ( bit1 @ N ) @ one )
      = one ) ).

% or_not_num_neg.simps(7)
thf(fact_9129_or__not__num__neg_Osimps_I5_J,axiom,
    ! [N: num,M: num] :
      ( ( bit_or_not_num_neg @ ( bit0 @ N ) @ ( bit0 @ M ) )
      = ( bitM @ ( bit_or_not_num_neg @ N @ M ) ) ) ).

% or_not_num_neg.simps(5)
thf(fact_9130_or__not__num__neg_Osimps_I9_J,axiom,
    ! [N: num,M: num] :
      ( ( bit_or_not_num_neg @ ( bit1 @ N ) @ ( bit1 @ M ) )
      = ( bitM @ ( bit_or_not_num_neg @ N @ M ) ) ) ).

% or_not_num_neg.simps(9)
thf(fact_9131_or__nat__def,axiom,
    ( bit_se1412395901928357646or_nat
    = ( ^ [M6: nat,N2: nat] : ( nat2 @ ( bit_se1409905431419307370or_int @ ( semiri1314217659103216013at_int @ M6 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).

% or_nat_def
thf(fact_9132_or__not__num__neg_Osimps_I2_J,axiom,
    ! [M: num] :
      ( ( bit_or_not_num_neg @ one @ ( bit0 @ M ) )
      = ( bit1 @ M ) ) ).

% or_not_num_neg.simps(2)
thf(fact_9133_or__not__num__neg_Osimps_I8_J,axiom,
    ! [N: num,M: num] :
      ( ( bit_or_not_num_neg @ ( bit1 @ N ) @ ( bit0 @ M ) )
      = ( bitM @ ( bit_or_not_num_neg @ N @ M ) ) ) ).

% or_not_num_neg.simps(8)
thf(fact_9134_or__not__num__neg_Oelims,axiom,
    ! [X4: num,Xa: num,Y3: num] :
      ( ( ( bit_or_not_num_neg @ X4 @ Xa )
        = Y3 )
     => ( ( ( X4 = one )
         => ( ( Xa = one )
           => ( Y3 != one ) ) )
       => ( ( ( X4 = one )
           => ! [M4: num] :
                ( ( Xa
                  = ( bit0 @ M4 ) )
               => ( Y3
                 != ( bit1 @ M4 ) ) ) )
         => ( ( ( X4 = one )
             => ! [M4: num] :
                  ( ( Xa
                    = ( bit1 @ M4 ) )
                 => ( Y3
                   != ( bit1 @ M4 ) ) ) )
           => ( ( ? [N4: num] :
                    ( X4
                    = ( bit0 @ N4 ) )
               => ( ( Xa = one )
                 => ( Y3
                   != ( bit0 @ one ) ) ) )
             => ( ! [N4: num] :
                    ( ( X4
                      = ( bit0 @ N4 ) )
                   => ! [M4: num] :
                        ( ( Xa
                          = ( bit0 @ M4 ) )
                       => ( Y3
                         != ( bitM @ ( bit_or_not_num_neg @ N4 @ M4 ) ) ) ) )
               => ( ! [N4: num] :
                      ( ( X4
                        = ( bit0 @ N4 ) )
                     => ! [M4: num] :
                          ( ( Xa
                            = ( bit1 @ M4 ) )
                         => ( Y3
                           != ( bit0 @ ( bit_or_not_num_neg @ N4 @ M4 ) ) ) ) )
                 => ( ( ? [N4: num] :
                          ( X4
                          = ( bit1 @ N4 ) )
                     => ( ( Xa = one )
                       => ( Y3 != one ) ) )
                   => ( ! [N4: num] :
                          ( ( X4
                            = ( bit1 @ N4 ) )
                         => ! [M4: num] :
                              ( ( Xa
                                = ( bit0 @ M4 ) )
                             => ( Y3
                               != ( bitM @ ( bit_or_not_num_neg @ N4 @ M4 ) ) ) ) )
                     => ~ ! [N4: num] :
                            ( ( X4
                              = ( bit1 @ N4 ) )
                           => ! [M4: num] :
                                ( ( Xa
                                  = ( bit1 @ M4 ) )
                               => ( Y3
                                 != ( bitM @ ( bit_or_not_num_neg @ N4 @ M4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% or_not_num_neg.elims
thf(fact_9135_numeral__or__not__num__eq,axiom,
    ! [M: num,N: num] :
      ( ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ N ) )
      = ( uminus_uminus_int @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).

% numeral_or_not_num_eq
thf(fact_9136_int__numeral__not__or__num__neg,axiom,
    ! [M: num,N: num] :
      ( ( bit_se1409905431419307370or_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ N @ M ) ) ) ) ).

% int_numeral_not_or_num_neg
thf(fact_9137_int__numeral__or__not__num__neg,axiom,
    ! [M: num,N: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ N ) ) ) ) ).

% int_numeral_or_not_num_neg
thf(fact_9138_or__Suc__0__eq,axiom,
    ! [N: nat] :
      ( ( bit_se1412395901928357646or_nat @ N @ ( suc @ zero_zero_nat ) )
      = ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% or_Suc_0_eq
thf(fact_9139_Suc__0__or__eq,axiom,
    ! [N: nat] :
      ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ N )
      = ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% Suc_0_or_eq
thf(fact_9140_or__nat__rec,axiom,
    ( bit_se1412395901928357646or_nat
    = ( ^ [M6: nat,N2: nat] :
          ( plus_plus_nat
          @ ( zero_n2687167440665602831ol_nat
            @ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M6 )
              | ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
          @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( divide_divide_nat @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% or_nat_rec
thf(fact_9141_or__nat__unfold,axiom,
    ( bit_se1412395901928357646or_nat
    = ( ^ [M6: nat,N2: nat] : ( if_nat @ ( M6 = zero_zero_nat ) @ N2 @ ( if_nat @ ( N2 = zero_zero_nat ) @ M6 @ ( plus_plus_nat @ ( ord_max_nat @ ( modulo_modulo_nat @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( divide_divide_nat @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% or_nat_unfold
thf(fact_9142_Sum__Ico__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [X: nat] : X
        @ ( set_or4665077453230672383an_nat @ M @ N ) )
      = ( divide_divide_nat @ ( minus_minus_nat @ ( times_times_nat @ N @ ( minus_minus_nat @ N @ one_one_nat ) ) @ ( times_times_nat @ M @ ( minus_minus_nat @ M @ one_one_nat ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% Sum_Ico_nat
thf(fact_9143_sum__power2,axiom,
    ! [K: nat] :
      ( ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K ) )
      = ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K ) @ one_one_nat ) ) ).

% sum_power2
thf(fact_9144_finite__atLeastLessThan,axiom,
    ! [L2: nat,U: nat] : ( finite_finite_nat @ ( set_or4665077453230672383an_nat @ L2 @ U ) ) ).

% finite_atLeastLessThan
thf(fact_9145_atLeastLessThan__singleton,axiom,
    ! [M: nat] :
      ( ( set_or4665077453230672383an_nat @ M @ ( suc @ M ) )
      = ( insert_nat @ M @ bot_bot_set_nat ) ) ).

% atLeastLessThan_singleton
thf(fact_9146_all__nat__less__eq,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ! [M6: nat] :
            ( ( ord_less_nat @ M6 @ N )
           => ( P @ M6 ) ) )
      = ( ! [X: nat] :
            ( ( member_nat @ X @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
           => ( P @ X ) ) ) ) ).

% all_nat_less_eq
thf(fact_9147_ex__nat__less__eq,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ? [M6: nat] :
            ( ( ord_less_nat @ M6 @ N )
            & ( P @ M6 ) ) )
      = ( ? [X: nat] :
            ( ( member_nat @ X @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
            & ( P @ X ) ) ) ) ).

% ex_nat_less_eq
thf(fact_9148_atLeastLessThanSuc__atLeastAtMost,axiom,
    ! [L2: nat,U: nat] :
      ( ( set_or4665077453230672383an_nat @ L2 @ ( suc @ U ) )
      = ( set_or1269000886237332187st_nat @ L2 @ U ) ) ).

% atLeastLessThanSuc_atLeastAtMost
thf(fact_9149_lessThan__atLeast0,axiom,
    ( set_ord_lessThan_nat
    = ( set_or4665077453230672383an_nat @ zero_zero_nat ) ) ).

% lessThan_atLeast0
thf(fact_9150_atLeastLessThan0,axiom,
    ! [M: nat] :
      ( ( set_or4665077453230672383an_nat @ M @ zero_zero_nat )
      = bot_bot_set_nat ) ).

% atLeastLessThan0
thf(fact_9151_atLeast0__lessThan__Suc,axiom,
    ! [N: nat] :
      ( ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) )
      = ( insert_nat @ N @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ).

% atLeast0_lessThan_Suc
thf(fact_9152_subset__eq__atLeast0__lessThan__finite,axiom,
    ! [N3: set_nat,N: nat] :
      ( ( ord_less_eq_set_nat @ N3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
     => ( finite_finite_nat @ N3 ) ) ).

% subset_eq_atLeast0_lessThan_finite
thf(fact_9153_atLeastLessThanSuc,axiom,
    ! [M: nat,N: nat] :
      ( ( ( ord_less_eq_nat @ M @ N )
       => ( ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) )
          = ( insert_nat @ N @ ( set_or4665077453230672383an_nat @ M @ N ) ) ) )
      & ( ~ ( ord_less_eq_nat @ M @ N )
       => ( ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) )
          = bot_bot_set_nat ) ) ) ).

% atLeastLessThanSuc
thf(fact_9154_prod__Suc__fact,axiom,
    ! [N: nat] :
      ( ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
      = ( semiri1408675320244567234ct_nat @ N ) ) ).

% prod_Suc_fact
thf(fact_9155_prod__Suc__Suc__fact,axiom,
    ! [N: nat] :
      ( ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ N ) )
      = ( semiri1408675320244567234ct_nat @ N ) ) ).

% prod_Suc_Suc_fact
thf(fact_9156_atLeastLessThan__nat__numeral,axiom,
    ! [M: nat,K: num] :
      ( ( ( ord_less_eq_nat @ M @ ( pred_numeral @ K ) )
       => ( ( set_or4665077453230672383an_nat @ M @ ( numeral_numeral_nat @ K ) )
          = ( insert_nat @ ( pred_numeral @ K ) @ ( set_or4665077453230672383an_nat @ M @ ( pred_numeral @ K ) ) ) ) )
      & ( ~ ( ord_less_eq_nat @ M @ ( pred_numeral @ K ) )
       => ( ( set_or4665077453230672383an_nat @ M @ ( numeral_numeral_nat @ K ) )
          = bot_bot_set_nat ) ) ) ).

% atLeastLessThan_nat_numeral
thf(fact_9157_atLeast1__lessThan__eq__remove0,axiom,
    ! [N: nat] :
      ( ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ N )
      = ( minus_minus_set_nat @ ( set_ord_lessThan_nat @ N ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).

% atLeast1_lessThan_eq_remove0
thf(fact_9158_Chebyshev__sum__upper__nat,axiom,
    ! [N: nat,A: nat > nat,B2: nat > nat] :
      ( ! [I3: nat,J2: nat] :
          ( ( ord_less_eq_nat @ I3 @ J2 )
         => ( ( ord_less_nat @ J2 @ N )
           => ( ord_less_eq_nat @ ( A @ I3 ) @ ( A @ J2 ) ) ) )
     => ( ! [I3: nat,J2: nat] :
            ( ( ord_less_eq_nat @ I3 @ J2 )
           => ( ( ord_less_nat @ J2 @ N )
             => ( ord_less_eq_nat @ ( B2 @ J2 ) @ ( B2 @ I3 ) ) ) )
       => ( ord_less_eq_nat
          @ ( times_times_nat @ N
            @ ( groups3542108847815614940at_nat
              @ ^ [I2: nat] : ( times_times_nat @ ( A @ I2 ) @ ( B2 @ I2 ) )
              @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) )
          @ ( times_times_nat @ ( groups3542108847815614940at_nat @ A @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) @ ( groups3542108847815614940at_nat @ B2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ) ) ).

% Chebyshev_sum_upper_nat
thf(fact_9159_finite__atLeastLessThan__int,axiom,
    ! [L2: int,U: int] : ( finite_finite_int @ ( set_or4662586982721622107an_int @ L2 @ U ) ) ).

% finite_atLeastLessThan_int
thf(fact_9160_finite__atLeastZeroLessThan__int,axiom,
    ! [U: int] : ( finite_finite_int @ ( set_or4662586982721622107an_int @ zero_zero_int @ U ) ) ).

% finite_atLeastZeroLessThan_int
thf(fact_9161_atLeastLessThanPlusOne__atLeastAtMost__int,axiom,
    ! [L2: int,U: int] :
      ( ( set_or4662586982721622107an_int @ L2 @ ( plus_plus_int @ U @ one_one_int ) )
      = ( set_or1266510415728281911st_int @ L2 @ U ) ) ).

% atLeastLessThanPlusOne_atLeastAtMost_int
thf(fact_9162_valid__eq,axiom,
    vEBT_VEBT_valid = vEBT_invar_vebt ).

% valid_eq
thf(fact_9163_valid__eq1,axiom,
    ! [T: vEBT_VEBT,D: nat] :
      ( ( vEBT_invar_vebt @ T @ D )
     => ( vEBT_VEBT_valid @ T @ D ) ) ).

% valid_eq1
thf(fact_9164_valid__eq2,axiom,
    ! [T: vEBT_VEBT,D: nat] :
      ( ( vEBT_VEBT_valid @ T @ D )
     => ( vEBT_invar_vebt @ T @ D ) ) ).

% valid_eq2
thf(fact_9165_VEBT__internal_Ovalid_H_Osimps_I1_J,axiom,
    ! [Uu3: $o,Uv2: $o,D: nat] :
      ( ( vEBT_VEBT_valid @ ( vEBT_Leaf @ Uu3 @ Uv2 ) @ D )
      = ( D = one_one_nat ) ) ).

% VEBT_internal.valid'.simps(1)
thf(fact_9166_divmod__step__integer__def,axiom,
    ( unique4921790084139445826nteger
    = ( ^ [L: num] :
          ( produc6916734918728496179nteger
          @ ^ [Q4: code_integer,R5: code_integer] : ( if_Pro6119634080678213985nteger @ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L ) @ R5 ) @ ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q4 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R5 @ ( numera6620942414471956472nteger @ L ) ) ) @ ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q4 ) @ R5 ) ) ) ) ) ).

% divmod_step_integer_def
thf(fact_9167_csqrt_Osimps_I1_J,axiom,
    ! [Z: complex] :
      ( ( re @ ( csqrt @ Z ) )
      = ( sqrt @ ( divide_divide_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Z ) @ ( re @ Z ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% csqrt.simps(1)
thf(fact_9168_complex__Re__numeral,axiom,
    ! [V: num] :
      ( ( re @ ( numera6690914467698888265omplex @ V ) )
      = ( numeral_numeral_real @ V ) ) ).

% complex_Re_numeral
thf(fact_9169_Re__divide__of__nat,axiom,
    ! [Z: complex,N: nat] :
      ( ( re @ ( divide1717551699836669952omplex @ Z @ ( semiri8010041392384452111omplex @ N ) ) )
      = ( divide_divide_real @ ( re @ Z ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).

% Re_divide_of_nat
thf(fact_9170_Re__divide__of__real,axiom,
    ! [Z: complex,R2: real] :
      ( ( re @ ( divide1717551699836669952omplex @ Z @ ( real_V4546457046886955230omplex @ R2 ) ) )
      = ( divide_divide_real @ ( re @ Z ) @ R2 ) ) ).

% Re_divide_of_real
thf(fact_9171_Re__sgn,axiom,
    ! [Z: complex] :
      ( ( re @ ( sgn_sgn_complex @ Z ) )
      = ( divide_divide_real @ ( re @ Z ) @ ( real_V1022390504157884413omplex @ Z ) ) ) ).

% Re_sgn
thf(fact_9172_Re__divide__numeral,axiom,
    ! [Z: complex,W: num] :
      ( ( re @ ( divide1717551699836669952omplex @ Z @ ( numera6690914467698888265omplex @ W ) ) )
      = ( divide_divide_real @ ( re @ Z ) @ ( numeral_numeral_real @ W ) ) ) ).

% Re_divide_numeral
thf(fact_9173_plus__integer__code_I1_J,axiom,
    ! [K: code_integer] :
      ( ( plus_p5714425477246183910nteger @ K @ zero_z3403309356797280102nteger )
      = K ) ).

% plus_integer_code(1)
thf(fact_9174_plus__integer__code_I2_J,axiom,
    ! [L2: code_integer] :
      ( ( plus_p5714425477246183910nteger @ zero_z3403309356797280102nteger @ L2 )
      = L2 ) ).

% plus_integer_code(2)
thf(fact_9175_times__integer__code_I1_J,axiom,
    ! [K: code_integer] :
      ( ( times_3573771949741848930nteger @ K @ zero_z3403309356797280102nteger )
      = zero_z3403309356797280102nteger ) ).

% times_integer_code(1)
thf(fact_9176_times__integer__code_I2_J,axiom,
    ! [L2: code_integer] :
      ( ( times_3573771949741848930nteger @ zero_z3403309356797280102nteger @ L2 )
      = zero_z3403309356797280102nteger ) ).

% times_integer_code(2)
thf(fact_9177_divmod__integer_H__def,axiom,
    ( unique3479559517661332726nteger
    = ( ^ [M6: num,N2: num] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( numera6620942414471956472nteger @ M6 ) @ ( numera6620942414471956472nteger @ N2 ) ) @ ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M6 ) @ ( numera6620942414471956472nteger @ N2 ) ) ) ) ) ).

% divmod_integer'_def
thf(fact_9178_sgn__integer__code,axiom,
    ( sgn_sgn_Code_integer
    = ( ^ [K3: code_integer] : ( if_Code_integer @ ( K3 = zero_z3403309356797280102nteger ) @ zero_z3403309356797280102nteger @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ K3 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ) ) ) ) ).

% sgn_integer_code
thf(fact_9179_minus__integer__code_I2_J,axiom,
    ! [L2: code_integer] :
      ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ L2 )
      = ( uminus1351360451143612070nteger @ L2 ) ) ).

% minus_integer_code(2)
thf(fact_9180_minus__integer__code_I1_J,axiom,
    ! [K: code_integer] :
      ( ( minus_8373710615458151222nteger @ K @ zero_z3403309356797280102nteger )
      = K ) ).

% minus_integer_code(1)
thf(fact_9181_complex__Re__le__cmod,axiom,
    ! [X4: complex] : ( ord_less_eq_real @ ( re @ X4 ) @ ( real_V1022390504157884413omplex @ X4 ) ) ).

% complex_Re_le_cmod
thf(fact_9182_one__complex_Osimps_I1_J,axiom,
    ( ( re @ one_one_complex )
    = one_one_real ) ).

% one_complex.simps(1)
thf(fact_9183_plus__complex_Osimps_I1_J,axiom,
    ! [X4: complex,Y3: complex] :
      ( ( re @ ( plus_plus_complex @ X4 @ Y3 ) )
      = ( plus_plus_real @ ( re @ X4 ) @ ( re @ Y3 ) ) ) ).

% plus_complex.simps(1)
thf(fact_9184_scaleR__complex_Osimps_I1_J,axiom,
    ! [R2: real,X4: complex] :
      ( ( re @ ( real_V2046097035970521341omplex @ R2 @ X4 ) )
      = ( times_times_real @ R2 @ ( re @ X4 ) ) ) ).

% scaleR_complex.simps(1)
thf(fact_9185_minus__complex_Osimps_I1_J,axiom,
    ! [X4: complex,Y3: complex] :
      ( ( re @ ( minus_minus_complex @ X4 @ Y3 ) )
      = ( minus_minus_real @ ( re @ X4 ) @ ( re @ Y3 ) ) ) ).

% minus_complex.simps(1)
thf(fact_9186_abs__Re__le__cmod,axiom,
    ! [X4: complex] : ( ord_less_eq_real @ ( abs_abs_real @ ( re @ X4 ) ) @ ( real_V1022390504157884413omplex @ X4 ) ) ).

% abs_Re_le_cmod
thf(fact_9187_Re__csqrt,axiom,
    ! [Z: complex] : ( ord_less_eq_real @ zero_zero_real @ ( re @ ( csqrt @ Z ) ) ) ).

% Re_csqrt
thf(fact_9188_one__natural_Orsp,axiom,
    one_one_nat = one_one_nat ).

% one_natural.rsp
thf(fact_9189_cmod__plus__Re__le__0__iff,axiom,
    ! [Z: complex] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Z ) @ ( re @ Z ) ) @ zero_zero_real )
      = ( ( re @ Z )
        = ( uminus_uminus_real @ ( real_V1022390504157884413omplex @ Z ) ) ) ) ).

% cmod_plus_Re_le_0_iff
thf(fact_9190_cos__n__Re__cis__pow__n,axiom,
    ! [N: nat,A: real] :
      ( ( cos_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ A ) )
      = ( re @ ( power_power_complex @ ( cis @ A ) @ N ) ) ) ).

% cos_n_Re_cis_pow_n
thf(fact_9191_csqrt_Ocode,axiom,
    ( csqrt
    = ( ^ [Z5: complex] :
          ( complex2 @ ( sqrt @ ( divide_divide_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Z5 ) @ ( re @ Z5 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
          @ ( times_times_real
            @ ( if_real
              @ ( ( im @ Z5 )
                = zero_zero_real )
              @ one_one_real
              @ ( sgn_sgn_real @ ( im @ Z5 ) ) )
            @ ( sqrt @ ( divide_divide_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ Z5 ) @ ( re @ Z5 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% csqrt.code
thf(fact_9192_csqrt_Osimps_I2_J,axiom,
    ! [Z: complex] :
      ( ( im @ ( csqrt @ Z ) )
      = ( times_times_real
        @ ( if_real
          @ ( ( im @ Z )
            = zero_zero_real )
          @ one_one_real
          @ ( sgn_sgn_real @ ( im @ Z ) ) )
        @ ( sqrt @ ( divide_divide_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ Z ) @ ( re @ Z ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% csqrt.simps(2)
thf(fact_9193_integer__of__int__code,axiom,
    ( code_integer_of_int
    = ( ^ [K3: int] :
          ( if_Code_integer @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus1351360451143612070nteger @ ( code_integer_of_int @ ( uminus_uminus_int @ K3 ) ) )
          @ ( if_Code_integer @ ( K3 = zero_zero_int ) @ zero_z3403309356797280102nteger
            @ ( if_Code_integer
              @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
                = zero_zero_int )
              @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( code_integer_of_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
              @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( code_integer_of_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_Code_integer ) ) ) ) ) ) ).

% integer_of_int_code
thf(fact_9194_Im__i__times,axiom,
    ! [Z: complex] :
      ( ( im @ ( times_times_complex @ imaginary_unit @ Z ) )
      = ( re @ Z ) ) ).

% Im_i_times
thf(fact_9195_Im__divide__of__real,axiom,
    ! [Z: complex,R2: real] :
      ( ( im @ ( divide1717551699836669952omplex @ Z @ ( real_V4546457046886955230omplex @ R2 ) ) )
      = ( divide_divide_real @ ( im @ Z ) @ R2 ) ) ).

% Im_divide_of_real
thf(fact_9196_Im__sgn,axiom,
    ! [Z: complex] :
      ( ( im @ ( sgn_sgn_complex @ Z ) )
      = ( divide_divide_real @ ( im @ Z ) @ ( real_V1022390504157884413omplex @ Z ) ) ) ).

% Im_sgn
thf(fact_9197_Re__power__real,axiom,
    ! [X4: complex,N: nat] :
      ( ( ( im @ X4 )
        = zero_zero_real )
     => ( ( re @ ( power_power_complex @ X4 @ N ) )
        = ( power_power_real @ ( re @ X4 ) @ N ) ) ) ).

% Re_power_real
thf(fact_9198_Re__i__times,axiom,
    ! [Z: complex] :
      ( ( re @ ( times_times_complex @ imaginary_unit @ Z ) )
      = ( uminus_uminus_real @ ( im @ Z ) ) ) ).

% Re_i_times
thf(fact_9199_Im__divide__numeral,axiom,
    ! [Z: complex,W: num] :
      ( ( im @ ( divide1717551699836669952omplex @ Z @ ( numera6690914467698888265omplex @ W ) ) )
      = ( divide_divide_real @ ( im @ Z ) @ ( numeral_numeral_real @ W ) ) ) ).

% Im_divide_numeral
thf(fact_9200_Im__divide__of__nat,axiom,
    ! [Z: complex,N: nat] :
      ( ( im @ ( divide1717551699836669952omplex @ Z @ ( semiri8010041392384452111omplex @ N ) ) )
      = ( divide_divide_real @ ( im @ Z ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).

% Im_divide_of_nat
thf(fact_9201_csqrt__of__real__nonneg,axiom,
    ! [X4: complex] :
      ( ( ( im @ X4 )
        = zero_zero_real )
     => ( ( ord_less_eq_real @ zero_zero_real @ ( re @ X4 ) )
       => ( ( csqrt @ X4 )
          = ( real_V4546457046886955230omplex @ ( sqrt @ ( re @ X4 ) ) ) ) ) ) ).

% csqrt_of_real_nonneg
thf(fact_9202_csqrt__minus,axiom,
    ! [X4: complex] :
      ( ( ( ord_less_real @ ( im @ X4 ) @ zero_zero_real )
        | ( ( ( im @ X4 )
            = zero_zero_real )
          & ( ord_less_eq_real @ zero_zero_real @ ( re @ X4 ) ) ) )
     => ( ( csqrt @ ( uminus1482373934393186551omplex @ X4 ) )
        = ( times_times_complex @ imaginary_unit @ ( csqrt @ X4 ) ) ) ) ).

% csqrt_minus
thf(fact_9203_csqrt__of__real__nonpos,axiom,
    ! [X4: complex] :
      ( ( ( im @ X4 )
        = zero_zero_real )
     => ( ( ord_less_eq_real @ ( re @ X4 ) @ zero_zero_real )
       => ( ( csqrt @ X4 )
          = ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sqrt @ ( abs_abs_real @ ( re @ X4 ) ) ) ) ) ) ) ) ).

% csqrt_of_real_nonpos
thf(fact_9204_divide__integer_Oabs__eq,axiom,
    ! [Xa: int,X4: int] :
      ( ( divide6298287555418463151nteger @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X4 ) )
      = ( code_integer_of_int @ ( divide_divide_int @ Xa @ X4 ) ) ) ).

% divide_integer.abs_eq
thf(fact_9205_modulo__integer_Oabs__eq,axiom,
    ! [Xa: int,X4: int] :
      ( ( modulo364778990260209775nteger @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X4 ) )
      = ( code_integer_of_int @ ( modulo_modulo_int @ Xa @ X4 ) ) ) ).

% modulo_integer.abs_eq
thf(fact_9206_less__integer__code_I1_J,axiom,
    ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger ) ).

% less_integer_code(1)
thf(fact_9207_less__integer_Oabs__eq,axiom,
    ! [Xa: int,X4: int] :
      ( ( ord_le6747313008572928689nteger @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X4 ) )
      = ( ord_less_int @ Xa @ X4 ) ) ).

% less_integer.abs_eq
thf(fact_9208_abs__integer__code,axiom,
    ( abs_abs_Code_integer
    = ( ^ [K3: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ K3 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ K3 ) @ K3 ) ) ) ).

% abs_integer_code
thf(fact_9209_imaginary__unit_Osimps_I2_J,axiom,
    ( ( im @ imaginary_unit )
    = one_one_real ) ).

% imaginary_unit.simps(2)
thf(fact_9210_plus__integer_Oabs__eq,axiom,
    ! [Xa: int,X4: int] :
      ( ( plus_p5714425477246183910nteger @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X4 ) )
      = ( code_integer_of_int @ ( plus_plus_int @ Xa @ X4 ) ) ) ).

% plus_integer.abs_eq
thf(fact_9211_times__integer_Oabs__eq,axiom,
    ! [Xa: int,X4: int] :
      ( ( times_3573771949741848930nteger @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X4 ) )
      = ( code_integer_of_int @ ( times_times_int @ Xa @ X4 ) ) ) ).

% times_integer.abs_eq
thf(fact_9212_minus__integer_Oabs__eq,axiom,
    ! [Xa: int,X4: int] :
      ( ( minus_8373710615458151222nteger @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X4 ) )
      = ( code_integer_of_int @ ( minus_minus_int @ Xa @ X4 ) ) ) ).

% minus_integer.abs_eq
thf(fact_9213_plus__complex_Osimps_I2_J,axiom,
    ! [X4: complex,Y3: complex] :
      ( ( im @ ( plus_plus_complex @ X4 @ Y3 ) )
      = ( plus_plus_real @ ( im @ X4 ) @ ( im @ Y3 ) ) ) ).

% plus_complex.simps(2)
thf(fact_9214_scaleR__complex_Osimps_I2_J,axiom,
    ! [R2: real,X4: complex] :
      ( ( im @ ( real_V2046097035970521341omplex @ R2 @ X4 ) )
      = ( times_times_real @ R2 @ ( im @ X4 ) ) ) ).

% scaleR_complex.simps(2)
thf(fact_9215_minus__complex_Osimps_I2_J,axiom,
    ! [X4: complex,Y3: complex] :
      ( ( im @ ( minus_minus_complex @ X4 @ Y3 ) )
      = ( minus_minus_real @ ( im @ X4 ) @ ( im @ Y3 ) ) ) ).

% minus_complex.simps(2)
thf(fact_9216_abs__Im__le__cmod,axiom,
    ! [X4: complex] : ( ord_less_eq_real @ ( abs_abs_real @ ( im @ X4 ) ) @ ( real_V1022390504157884413omplex @ X4 ) ) ).

% abs_Im_le_cmod
thf(fact_9217_times__complex_Osimps_I2_J,axiom,
    ! [X4: complex,Y3: complex] :
      ( ( im @ ( times_times_complex @ X4 @ Y3 ) )
      = ( plus_plus_real @ ( times_times_real @ ( re @ X4 ) @ ( im @ Y3 ) ) @ ( times_times_real @ ( im @ X4 ) @ ( re @ Y3 ) ) ) ) ).

% times_complex.simps(2)
thf(fact_9218_cmod__Im__le__iff,axiom,
    ! [X4: complex,Y3: complex] :
      ( ( ( re @ X4 )
        = ( re @ Y3 ) )
     => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X4 ) @ ( real_V1022390504157884413omplex @ Y3 ) )
        = ( ord_less_eq_real @ ( abs_abs_real @ ( im @ X4 ) ) @ ( abs_abs_real @ ( im @ Y3 ) ) ) ) ) ).

% cmod_Im_le_iff
thf(fact_9219_cmod__Re__le__iff,axiom,
    ! [X4: complex,Y3: complex] :
      ( ( ( im @ X4 )
        = ( im @ Y3 ) )
     => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X4 ) @ ( real_V1022390504157884413omplex @ Y3 ) )
        = ( ord_less_eq_real @ ( abs_abs_real @ ( re @ X4 ) ) @ ( abs_abs_real @ ( re @ Y3 ) ) ) ) ) ).

% cmod_Re_le_iff
thf(fact_9220_times__complex_Osimps_I1_J,axiom,
    ! [X4: complex,Y3: complex] :
      ( ( re @ ( times_times_complex @ X4 @ Y3 ) )
      = ( minus_minus_real @ ( times_times_real @ ( re @ X4 ) @ ( re @ Y3 ) ) @ ( times_times_real @ ( im @ X4 ) @ ( im @ Y3 ) ) ) ) ).

% times_complex.simps(1)
thf(fact_9221_plus__complex_Ocode,axiom,
    ( plus_plus_complex
    = ( ^ [X: complex,Y: complex] : ( complex2 @ ( plus_plus_real @ ( re @ X ) @ ( re @ Y ) ) @ ( plus_plus_real @ ( im @ X ) @ ( im @ Y ) ) ) ) ) ).

% plus_complex.code
thf(fact_9222_scaleR__complex_Ocode,axiom,
    ( real_V2046097035970521341omplex
    = ( ^ [R5: real,X: complex] : ( complex2 @ ( times_times_real @ R5 @ ( re @ X ) ) @ ( times_times_real @ R5 @ ( im @ X ) ) ) ) ) ).

% scaleR_complex.code
thf(fact_9223_minus__complex_Ocode,axiom,
    ( minus_minus_complex
    = ( ^ [X: complex,Y: complex] : ( complex2 @ ( minus_minus_real @ ( re @ X ) @ ( re @ Y ) ) @ ( minus_minus_real @ ( im @ X ) @ ( im @ Y ) ) ) ) ) ).

% minus_complex.code
thf(fact_9224_csqrt__principal,axiom,
    ! [Z: complex] :
      ( ( ord_less_real @ zero_zero_real @ ( re @ ( csqrt @ Z ) ) )
      | ( ( ( re @ ( csqrt @ Z ) )
          = zero_zero_real )
        & ( ord_less_eq_real @ zero_zero_real @ ( im @ ( csqrt @ Z ) ) ) ) ) ).

% csqrt_principal
thf(fact_9225_cmod__le,axiom,
    ! [Z: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ ( plus_plus_real @ ( abs_abs_real @ ( re @ Z ) ) @ ( abs_abs_real @ ( im @ Z ) ) ) ) ).

% cmod_le
thf(fact_9226_sin__n__Im__cis__pow__n,axiom,
    ! [N: nat,A: real] :
      ( ( sin_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ A ) )
      = ( im @ ( power_power_complex @ ( cis @ A ) @ N ) ) ) ).

% sin_n_Im_cis_pow_n
thf(fact_9227_Re__exp,axiom,
    ! [Z: complex] :
      ( ( re @ ( exp_complex @ Z ) )
      = ( times_times_real @ ( exp_real @ ( re @ Z ) ) @ ( cos_real @ ( im @ Z ) ) ) ) ).

% Re_exp
thf(fact_9228_Im__exp,axiom,
    ! [Z: complex] :
      ( ( im @ ( exp_complex @ Z ) )
      = ( times_times_real @ ( exp_real @ ( re @ Z ) ) @ ( sin_real @ ( im @ Z ) ) ) ) ).

% Im_exp
thf(fact_9229_complex__eq,axiom,
    ! [A: complex] :
      ( A
      = ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( re @ A ) ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( im @ A ) ) ) ) ) ).

% complex_eq
thf(fact_9230_times__complex_Ocode,axiom,
    ( times_times_complex
    = ( ^ [X: complex,Y: complex] : ( complex2 @ ( minus_minus_real @ ( times_times_real @ ( re @ X ) @ ( re @ Y ) ) @ ( times_times_real @ ( im @ X ) @ ( im @ Y ) ) ) @ ( plus_plus_real @ ( times_times_real @ ( re @ X ) @ ( im @ Y ) ) @ ( times_times_real @ ( im @ X ) @ ( re @ Y ) ) ) ) ) ) ).

% times_complex.code
thf(fact_9231_exp__eq__polar,axiom,
    ( exp_complex
    = ( ^ [Z5: complex] : ( times_times_complex @ ( real_V4546457046886955230omplex @ ( exp_real @ ( re @ Z5 ) ) ) @ ( cis @ ( im @ Z5 ) ) ) ) ) ).

% exp_eq_polar
thf(fact_9232_cmod__power2,axiom,
    ! [Z: complex] :
      ( ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_real @ ( power_power_real @ ( re @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% cmod_power2
thf(fact_9233_Im__power2,axiom,
    ! [X4: complex] :
      ( ( im @ ( power_power_complex @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( re @ X4 ) ) @ ( im @ X4 ) ) ) ).

% Im_power2
thf(fact_9234_Re__power2,axiom,
    ! [X4: complex] :
      ( ( re @ ( power_power_complex @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( minus_minus_real @ ( power_power_real @ ( re @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% Re_power2
thf(fact_9235_complex__eq__0,axiom,
    ! [Z: complex] :
      ( ( Z = zero_zero_complex )
      = ( ( plus_plus_real @ ( power_power_real @ ( re @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = zero_zero_real ) ) ).

% complex_eq_0
thf(fact_9236_norm__complex__def,axiom,
    ( real_V1022390504157884413omplex
    = ( ^ [Z5: complex] : ( sqrt @ ( plus_plus_real @ ( power_power_real @ ( re @ Z5 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z5 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% norm_complex_def
thf(fact_9237_inverse__complex_Osimps_I1_J,axiom,
    ! [X4: complex] :
      ( ( re @ ( invers8013647133539491842omplex @ X4 ) )
      = ( divide_divide_real @ ( re @ X4 ) @ ( plus_plus_real @ ( power_power_real @ ( re @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% inverse_complex.simps(1)
thf(fact_9238_complex__neq__0,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero_complex )
      = ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ ( re @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% complex_neq_0
thf(fact_9239_Re__divide,axiom,
    ! [X4: complex,Y3: complex] :
      ( ( re @ ( divide1717551699836669952omplex @ X4 @ Y3 ) )
      = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ ( re @ X4 ) @ ( re @ Y3 ) ) @ ( times_times_real @ ( im @ X4 ) @ ( im @ Y3 ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% Re_divide
thf(fact_9240_csqrt__unique,axiom,
    ! [W: complex,Z: complex] :
      ( ( ( power_power_complex @ W @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = Z )
     => ( ( ( ord_less_real @ zero_zero_real @ ( re @ W ) )
          | ( ( ( re @ W )
              = zero_zero_real )
            & ( ord_less_eq_real @ zero_zero_real @ ( im @ W ) ) ) )
       => ( ( csqrt @ Z )
          = W ) ) ) ).

% csqrt_unique
thf(fact_9241_csqrt__square,axiom,
    ! [B2: complex] :
      ( ( ( ord_less_real @ zero_zero_real @ ( re @ B2 ) )
        | ( ( ( re @ B2 )
            = zero_zero_real )
          & ( ord_less_eq_real @ zero_zero_real @ ( im @ B2 ) ) ) )
     => ( ( csqrt @ ( power_power_complex @ B2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = B2 ) ) ).

% csqrt_square
thf(fact_9242_inverse__complex_Osimps_I2_J,axiom,
    ! [X4: complex] :
      ( ( im @ ( invers8013647133539491842omplex @ X4 ) )
      = ( divide_divide_real @ ( uminus_uminus_real @ ( im @ X4 ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% inverse_complex.simps(2)
thf(fact_9243_Im__divide,axiom,
    ! [X4: complex,Y3: complex] :
      ( ( im @ ( divide1717551699836669952omplex @ X4 @ Y3 ) )
      = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ ( im @ X4 ) @ ( re @ Y3 ) ) @ ( times_times_real @ ( re @ X4 ) @ ( im @ Y3 ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% Im_divide
thf(fact_9244_complex__abs__le__norm,axiom,
    ! [Z: complex] : ( ord_less_eq_real @ ( plus_plus_real @ ( abs_abs_real @ ( re @ Z ) ) @ ( abs_abs_real @ ( im @ Z ) ) ) @ ( times_times_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( real_V1022390504157884413omplex @ Z ) ) ) ).

% complex_abs_le_norm
thf(fact_9245_complex__unit__circle,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero_complex )
     => ( ( plus_plus_real @ ( power_power_real @ ( divide_divide_real @ ( re @ Z ) @ ( real_V1022390504157884413omplex @ Z ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( divide_divide_real @ ( im @ Z ) @ ( real_V1022390504157884413omplex @ Z ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = one_one_real ) ) ).

% complex_unit_circle
thf(fact_9246_inverse__complex_Ocode,axiom,
    ( invers8013647133539491842omplex
    = ( ^ [X: complex] : ( complex2 @ ( divide_divide_real @ ( re @ X ) @ ( plus_plus_real @ ( power_power_real @ ( re @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( divide_divide_real @ ( uminus_uminus_real @ ( im @ X ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% inverse_complex.code
thf(fact_9247_Complex__divide,axiom,
    ( divide1717551699836669952omplex
    = ( ^ [X: complex,Y: complex] : ( complex2 @ ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ ( re @ X ) @ ( re @ Y ) ) @ ( times_times_real @ ( im @ X ) @ ( im @ Y ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ ( im @ X ) @ ( re @ Y ) ) @ ( times_times_real @ ( re @ X ) @ ( im @ Y ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% Complex_divide
thf(fact_9248_Im__Reals__divide,axiom,
    ! [R2: complex,Z: complex] :
      ( ( member_complex @ R2 @ real_V2521375963428798218omplex )
     => ( ( im @ ( divide1717551699836669952omplex @ R2 @ Z ) )
        = ( divide_divide_real @ ( times_times_real @ ( uminus_uminus_real @ ( re @ R2 ) ) @ ( im @ Z ) ) @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% Im_Reals_divide
thf(fact_9249_Re__Reals__divide,axiom,
    ! [R2: complex,Z: complex] :
      ( ( member_complex @ R2 @ real_V2521375963428798218omplex )
     => ( ( re @ ( divide1717551699836669952omplex @ R2 @ Z ) )
        = ( divide_divide_real @ ( times_times_real @ ( re @ R2 ) @ ( re @ Z ) ) @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% Re_Reals_divide
thf(fact_9250_Re__divide__Reals,axiom,
    ! [R2: complex,Z: complex] :
      ( ( member_complex @ R2 @ real_V2521375963428798218omplex )
     => ( ( re @ ( divide1717551699836669952omplex @ Z @ R2 ) )
        = ( divide_divide_real @ ( re @ Z ) @ ( re @ R2 ) ) ) ) ).

% Re_divide_Reals
thf(fact_9251_real__eq__imaginary__iff,axiom,
    ! [Y3: complex,X4: complex] :
      ( ( member_complex @ Y3 @ real_V2521375963428798218omplex )
     => ( ( member_complex @ X4 @ real_V2521375963428798218omplex )
       => ( ( X4
            = ( times_times_complex @ imaginary_unit @ Y3 ) )
          = ( ( X4 = zero_zero_complex )
            & ( Y3 = zero_zero_complex ) ) ) ) ) ).

% real_eq_imaginary_iff
thf(fact_9252_imaginary__eq__real__iff,axiom,
    ! [Y3: complex,X4: complex] :
      ( ( member_complex @ Y3 @ real_V2521375963428798218omplex )
     => ( ( member_complex @ X4 @ real_V2521375963428798218omplex )
       => ( ( ( times_times_complex @ imaginary_unit @ Y3 )
            = X4 )
          = ( ( X4 = zero_zero_complex )
            & ( Y3 = zero_zero_complex ) ) ) ) ) ).

% imaginary_eq_real_iff
thf(fact_9253_Im__divide__Reals,axiom,
    ! [R2: complex,Z: complex] :
      ( ( member_complex @ R2 @ real_V2521375963428798218omplex )
     => ( ( im @ ( divide1717551699836669952omplex @ Z @ R2 ) )
        = ( divide_divide_real @ ( im @ Z ) @ ( re @ R2 ) ) ) ) ).

% Im_divide_Reals
thf(fact_9254_complex__diff__cnj,axiom,
    ! [Z: complex] :
      ( ( minus_minus_complex @ Z @ ( cnj @ Z ) )
      = ( times_times_complex @ ( real_V4546457046886955230omplex @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( im @ Z ) ) ) @ imaginary_unit ) ) ).

% complex_diff_cnj
thf(fact_9255_complex__mult__cnj,axiom,
    ! [Z: complex] :
      ( ( times_times_complex @ Z @ ( cnj @ Z ) )
      = ( real_V4546457046886955230omplex @ ( plus_plus_real @ ( power_power_real @ ( re @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% complex_mult_cnj
thf(fact_9256_integer__of__num_I3_J,axiom,
    ! [N: num] :
      ( ( code_integer_of_num @ ( bit1 @ N ) )
      = ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( code_integer_of_num @ N ) @ ( code_integer_of_num @ N ) ) @ one_one_Code_integer ) ) ).

% integer_of_num(3)
thf(fact_9257_complex__cnj__mult,axiom,
    ! [X4: complex,Y3: complex] :
      ( ( cnj @ ( times_times_complex @ X4 @ Y3 ) )
      = ( times_times_complex @ ( cnj @ X4 ) @ ( cnj @ Y3 ) ) ) ).

% complex_cnj_mult
thf(fact_9258_complex__cnj__divide,axiom,
    ! [X4: complex,Y3: complex] :
      ( ( cnj @ ( divide1717551699836669952omplex @ X4 @ Y3 ) )
      = ( divide1717551699836669952omplex @ ( cnj @ X4 ) @ ( cnj @ Y3 ) ) ) ).

% complex_cnj_divide
thf(fact_9259_complex__cnj__add,axiom,
    ! [X4: complex,Y3: complex] :
      ( ( cnj @ ( plus_plus_complex @ X4 @ Y3 ) )
      = ( plus_plus_complex @ ( cnj @ X4 ) @ ( cnj @ Y3 ) ) ) ).

% complex_cnj_add
thf(fact_9260_complex__cnj__diff,axiom,
    ! [X4: complex,Y3: complex] :
      ( ( cnj @ ( minus_minus_complex @ X4 @ Y3 ) )
      = ( minus_minus_complex @ ( cnj @ X4 ) @ ( cnj @ Y3 ) ) ) ).

% complex_cnj_diff
thf(fact_9261_complex__In__mult__cnj__zero,axiom,
    ! [Z: complex] :
      ( ( im @ ( times_times_complex @ Z @ ( cnj @ Z ) ) )
      = zero_zero_real ) ).

% complex_In_mult_cnj_zero
thf(fact_9262_Re__complex__div__eq__0,axiom,
    ! [A: complex,B2: complex] :
      ( ( ( re @ ( divide1717551699836669952omplex @ A @ B2 ) )
        = zero_zero_real )
      = ( ( re @ ( times_times_complex @ A @ ( cnj @ B2 ) ) )
        = zero_zero_real ) ) ).

% Re_complex_div_eq_0
thf(fact_9263_Im__complex__div__eq__0,axiom,
    ! [A: complex,B2: complex] :
      ( ( ( im @ ( divide1717551699836669952omplex @ A @ B2 ) )
        = zero_zero_real )
      = ( ( im @ ( times_times_complex @ A @ ( cnj @ B2 ) ) )
        = zero_zero_real ) ) ).

% Im_complex_div_eq_0
thf(fact_9264_complex__mod__sqrt__Re__mult__cnj,axiom,
    ( real_V1022390504157884413omplex
    = ( ^ [Z5: complex] : ( sqrt @ ( re @ ( times_times_complex @ Z5 @ ( cnj @ Z5 ) ) ) ) ) ) ).

% complex_mod_sqrt_Re_mult_cnj
thf(fact_9265_integer__of__num__triv_I1_J,axiom,
    ( ( code_integer_of_num @ one )
    = one_one_Code_integer ) ).

% integer_of_num_triv(1)
thf(fact_9266_Re__complex__div__lt__0,axiom,
    ! [A: complex,B2: complex] :
      ( ( ord_less_real @ ( re @ ( divide1717551699836669952omplex @ A @ B2 ) ) @ zero_zero_real )
      = ( ord_less_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B2 ) ) ) @ zero_zero_real ) ) ).

% Re_complex_div_lt_0
thf(fact_9267_Re__complex__div__gt__0,axiom,
    ! [A: complex,B2: complex] :
      ( ( ord_less_real @ zero_zero_real @ ( re @ ( divide1717551699836669952omplex @ A @ B2 ) ) )
      = ( ord_less_real @ zero_zero_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B2 ) ) ) ) ) ).

% Re_complex_div_gt_0
thf(fact_9268_Re__complex__div__ge__0,axiom,
    ! [A: complex,B2: complex] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( re @ ( divide1717551699836669952omplex @ A @ B2 ) ) )
      = ( ord_less_eq_real @ zero_zero_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B2 ) ) ) ) ) ).

% Re_complex_div_ge_0
thf(fact_9269_Re__complex__div__le__0,axiom,
    ! [A: complex,B2: complex] :
      ( ( ord_less_eq_real @ ( re @ ( divide1717551699836669952omplex @ A @ B2 ) ) @ zero_zero_real )
      = ( ord_less_eq_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B2 ) ) ) @ zero_zero_real ) ) ).

% Re_complex_div_le_0
thf(fact_9270_Im__complex__div__lt__0,axiom,
    ! [A: complex,B2: complex] :
      ( ( ord_less_real @ ( im @ ( divide1717551699836669952omplex @ A @ B2 ) ) @ zero_zero_real )
      = ( ord_less_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B2 ) ) ) @ zero_zero_real ) ) ).

% Im_complex_div_lt_0
thf(fact_9271_Im__complex__div__gt__0,axiom,
    ! [A: complex,B2: complex] :
      ( ( ord_less_real @ zero_zero_real @ ( im @ ( divide1717551699836669952omplex @ A @ B2 ) ) )
      = ( ord_less_real @ zero_zero_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B2 ) ) ) ) ) ).

% Im_complex_div_gt_0
thf(fact_9272_Im__complex__div__ge__0,axiom,
    ! [A: complex,B2: complex] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( im @ ( divide1717551699836669952omplex @ A @ B2 ) ) )
      = ( ord_less_eq_real @ zero_zero_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B2 ) ) ) ) ) ).

% Im_complex_div_ge_0
thf(fact_9273_Im__complex__div__le__0,axiom,
    ! [A: complex,B2: complex] :
      ( ( ord_less_eq_real @ ( im @ ( divide1717551699836669952omplex @ A @ B2 ) ) @ zero_zero_real )
      = ( ord_less_eq_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B2 ) ) ) @ zero_zero_real ) ) ).

% Im_complex_div_le_0
thf(fact_9274_integer__of__num_I2_J,axiom,
    ! [N: num] :
      ( ( code_integer_of_num @ ( bit0 @ N ) )
      = ( plus_p5714425477246183910nteger @ ( code_integer_of_num @ N ) @ ( code_integer_of_num @ N ) ) ) ).

% integer_of_num(2)
thf(fact_9275_complex__mod__mult__cnj,axiom,
    ! [Z: complex] :
      ( ( real_V1022390504157884413omplex @ ( times_times_complex @ Z @ ( cnj @ Z ) ) )
      = ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% complex_mod_mult_cnj
thf(fact_9276_complex__div__gt__0,axiom,
    ! [A: complex,B2: complex] :
      ( ( ( ord_less_real @ zero_zero_real @ ( re @ ( divide1717551699836669952omplex @ A @ B2 ) ) )
        = ( ord_less_real @ zero_zero_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B2 ) ) ) ) )
      & ( ( ord_less_real @ zero_zero_real @ ( im @ ( divide1717551699836669952omplex @ A @ B2 ) ) )
        = ( ord_less_real @ zero_zero_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B2 ) ) ) ) ) ) ).

% complex_div_gt_0
thf(fact_9277_integer__of__num__triv_I2_J,axiom,
    ( ( code_integer_of_num @ ( bit0 @ one ) )
    = ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ).

% integer_of_num_triv(2)
thf(fact_9278_complex__norm__square,axiom,
    ! [Z: complex] :
      ( ( real_V4546457046886955230omplex @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( times_times_complex @ Z @ ( cnj @ Z ) ) ) ).

% complex_norm_square
thf(fact_9279_complex__add__cnj,axiom,
    ! [Z: complex] :
      ( ( plus_plus_complex @ Z @ ( cnj @ Z ) )
      = ( real_V4546457046886955230omplex @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( re @ Z ) ) ) ) ).

% complex_add_cnj
thf(fact_9280_complex__div__cnj,axiom,
    ( divide1717551699836669952omplex
    = ( ^ [A4: complex,B3: complex] : ( divide1717551699836669952omplex @ ( times_times_complex @ A4 @ ( cnj @ B3 ) ) @ ( real_V4546457046886955230omplex @ ( power_power_real @ ( real_V1022390504157884413omplex @ B3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% complex_div_cnj
thf(fact_9281_cnj__add__mult__eq__Re,axiom,
    ! [Z: complex,W: complex] :
      ( ( plus_plus_complex @ ( times_times_complex @ Z @ ( cnj @ W ) ) @ ( times_times_complex @ ( cnj @ Z ) @ W ) )
      = ( real_V4546457046886955230omplex @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( re @ ( times_times_complex @ Z @ ( cnj @ W ) ) ) ) ) ) ).

% cnj_add_mult_eq_Re
thf(fact_9282_bit__cut__integer__def,axiom,
    ( code_bit_cut_integer
    = ( ^ [K3: code_integer] :
          ( produc6677183202524767010eger_o @ ( divide6298287555418463151nteger @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
          @ ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ K3 ) ) ) ) ).

% bit_cut_integer_def
thf(fact_9283_divmod__integer__def,axiom,
    ( code_divmod_integer
    = ( ^ [K3: code_integer,L: code_integer] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ K3 @ L ) @ ( modulo364778990260209775nteger @ K3 @ L ) ) ) ) ).

% divmod_integer_def
thf(fact_9284_card__lessThan,axiom,
    ! [U: nat] :
      ( ( finite_card_nat @ ( set_ord_lessThan_nat @ U ) )
      = U ) ).

% card_lessThan
thf(fact_9285_card__Collect__less__nat,axiom,
    ! [N: nat] :
      ( ( finite_card_nat
        @ ( collect_nat
          @ ^ [I2: nat] : ( ord_less_nat @ I2 @ N ) ) )
      = N ) ).

% card_Collect_less_nat
thf(fact_9286_card__atMost,axiom,
    ! [U: nat] :
      ( ( finite_card_nat @ ( set_ord_atMost_nat @ U ) )
      = ( suc @ U ) ) ).

% card_atMost
thf(fact_9287_card__atLeastLessThan,axiom,
    ! [L2: nat,U: nat] :
      ( ( finite_card_nat @ ( set_or4665077453230672383an_nat @ L2 @ U ) )
      = ( minus_minus_nat @ U @ L2 ) ) ).

% card_atLeastLessThan
thf(fact_9288_card__Collect__le__nat,axiom,
    ! [N: nat] :
      ( ( finite_card_nat
        @ ( collect_nat
          @ ^ [I2: nat] : ( ord_less_eq_nat @ I2 @ N ) ) )
      = ( suc @ N ) ) ).

% card_Collect_le_nat
thf(fact_9289_card__atLeastAtMost,axiom,
    ! [L2: nat,U: nat] :
      ( ( finite_card_nat @ ( set_or1269000886237332187st_nat @ L2 @ U ) )
      = ( minus_minus_nat @ ( suc @ U ) @ L2 ) ) ).

% card_atLeastAtMost
thf(fact_9290_card__atLeastLessThan__int,axiom,
    ! [L2: int,U: int] :
      ( ( finite_card_int @ ( set_or4662586982721622107an_int @ L2 @ U ) )
      = ( nat2 @ ( minus_minus_int @ U @ L2 ) ) ) ).

% card_atLeastLessThan_int
thf(fact_9291_card__atLeastAtMost__int,axiom,
    ! [L2: int,U: int] :
      ( ( finite_card_int @ ( set_or1266510415728281911st_int @ L2 @ U ) )
      = ( nat2 @ ( plus_plus_int @ ( minus_minus_int @ U @ L2 ) @ one_one_int ) ) ) ).

% card_atLeastAtMost_int
thf(fact_9292_card__less__Suc2,axiom,
    ! [M7: set_nat,I: nat] :
      ( ~ ( member_nat @ zero_zero_nat @ M7 )
     => ( ( finite_card_nat
          @ ( collect_nat
            @ ^ [K3: nat] :
                ( ( member_nat @ ( suc @ K3 ) @ M7 )
                & ( ord_less_nat @ K3 @ I ) ) ) )
        = ( finite_card_nat
          @ ( collect_nat
            @ ^ [K3: nat] :
                ( ( member_nat @ K3 @ M7 )
                & ( ord_less_nat @ K3 @ ( suc @ I ) ) ) ) ) ) ) ).

% card_less_Suc2
thf(fact_9293_card__less__Suc,axiom,
    ! [M7: set_nat,I: nat] :
      ( ( member_nat @ zero_zero_nat @ M7 )
     => ( ( suc
          @ ( finite_card_nat
            @ ( collect_nat
              @ ^ [K3: nat] :
                  ( ( member_nat @ ( suc @ K3 ) @ M7 )
                  & ( ord_less_nat @ K3 @ I ) ) ) ) )
        = ( finite_card_nat
          @ ( collect_nat
            @ ^ [K3: nat] :
                ( ( member_nat @ K3 @ M7 )
                & ( ord_less_nat @ K3 @ ( suc @ I ) ) ) ) ) ) ) ).

% card_less_Suc
thf(fact_9294_card__less,axiom,
    ! [M7: set_nat,I: nat] :
      ( ( member_nat @ zero_zero_nat @ M7 )
     => ( ( finite_card_nat
          @ ( collect_nat
            @ ^ [K3: nat] :
                ( ( member_nat @ K3 @ M7 )
                & ( ord_less_nat @ K3 @ ( suc @ I ) ) ) ) )
       != zero_zero_nat ) ) ).

% card_less
thf(fact_9295_card__atLeastZeroLessThan__int,axiom,
    ! [U: int] :
      ( ( finite_card_int @ ( set_or4662586982721622107an_int @ zero_zero_int @ U ) )
      = ( nat2 @ U ) ) ).

% card_atLeastZeroLessThan_int
thf(fact_9296_subset__card__intvl__is__intvl,axiom,
    ! [A3: set_nat,K: nat] :
      ( ( ord_less_eq_set_nat @ A3 @ ( set_or4665077453230672383an_nat @ K @ ( plus_plus_nat @ K @ ( finite_card_nat @ A3 ) ) ) )
     => ( A3
        = ( set_or4665077453230672383an_nat @ K @ ( plus_plus_nat @ K @ ( finite_card_nat @ A3 ) ) ) ) ) ).

% subset_card_intvl_is_intvl
thf(fact_9297_subset__eq__atLeast0__lessThan__card,axiom,
    ! [N3: set_nat,N: nat] :
      ( ( ord_less_eq_set_nat @ N3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
     => ( ord_less_eq_nat @ ( finite_card_nat @ N3 ) @ N ) ) ).

% subset_eq_atLeast0_lessThan_card
thf(fact_9298_card__sum__le__nat__sum,axiom,
    ! [S2: set_nat] :
      ( ord_less_eq_nat
      @ ( groups3542108847815614940at_nat
        @ ^ [X: nat] : X
        @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( finite_card_nat @ S2 ) ) )
      @ ( groups3542108847815614940at_nat
        @ ^ [X: nat] : X
        @ S2 ) ) ).

% card_sum_le_nat_sum
thf(fact_9299_card__nth__roots,axiom,
    ! [C: complex,N: nat] :
      ( ( C != zero_zero_complex )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( finite_card_complex
            @ ( collect_complex
              @ ^ [Z5: complex] :
                  ( ( power_power_complex @ Z5 @ N )
                  = C ) ) )
          = N ) ) ) ).

% card_nth_roots
thf(fact_9300_card__roots__unity__eq,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( finite_card_complex
          @ ( collect_complex
            @ ^ [Z5: complex] :
                ( ( power_power_complex @ Z5 @ N )
                = one_one_complex ) ) )
        = N ) ) ).

% card_roots_unity_eq
thf(fact_9301_bit__cut__integer__code,axiom,
    ( code_bit_cut_integer
    = ( ^ [K3: code_integer] :
          ( if_Pro5737122678794959658eger_o @ ( K3 = zero_z3403309356797280102nteger ) @ ( produc6677183202524767010eger_o @ zero_z3403309356797280102nteger @ $false )
          @ ( produc9125791028180074456eger_o
            @ ^ [R5: code_integer,S4: code_integer] : ( produc6677183202524767010eger_o @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ K3 ) @ R5 @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ S4 ) ) @ ( S4 = one_one_Code_integer ) )
            @ ( code_divmod_abs @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% bit_cut_integer_code
thf(fact_9302_divmod__abs__def,axiom,
    ( code_divmod_abs
    = ( ^ [K3: code_integer,L: code_integer] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( abs_abs_Code_integer @ K3 ) @ ( abs_abs_Code_integer @ L ) ) @ ( modulo364778990260209775nteger @ ( abs_abs_Code_integer @ K3 ) @ ( abs_abs_Code_integer @ L ) ) ) ) ) ).

% divmod_abs_def
thf(fact_9303_divmod__integer__code,axiom,
    ( code_divmod_integer
    = ( ^ [K3: code_integer,L: code_integer] :
          ( if_Pro6119634080678213985nteger @ ( K3 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
          @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ L )
            @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ K3 ) @ ( code_divmod_abs @ K3 @ L )
              @ ( produc6916734918728496179nteger
                @ ^ [R5: code_integer,S4: code_integer] : ( if_Pro6119634080678213985nteger @ ( S4 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ L @ S4 ) ) )
                @ ( code_divmod_abs @ K3 @ L ) ) )
            @ ( if_Pro6119634080678213985nteger @ ( L = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ K3 )
              @ ( produc6499014454317279255nteger @ uminus1351360451143612070nteger
                @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ K3 @ zero_z3403309356797280102nteger ) @ ( code_divmod_abs @ K3 @ L )
                  @ ( produc6916734918728496179nteger
                    @ ^ [R5: code_integer,S4: code_integer] : ( if_Pro6119634080678213985nteger @ ( S4 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ L ) @ S4 ) ) )
                    @ ( code_divmod_abs @ K3 @ L ) ) ) ) ) ) ) ) ) ).

% divmod_integer_code
thf(fact_9304_vebt__maxt_Opelims,axiom,
    ! [X4: vEBT_VEBT,Y3: option_nat] :
      ( ( ( vEBT_vebt_maxt @ X4 )
        = Y3 )
     => ( ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ X4 )
       => ( ! [A5: $o,B5: $o] :
              ( ( X4
                = ( vEBT_Leaf @ A5 @ B5 ) )
             => ( ( ( B5
                   => ( Y3
                      = ( some_nat @ one_one_nat ) ) )
                  & ( ~ B5
                   => ( ( A5
                       => ( Y3
                          = ( some_nat @ zero_zero_nat ) ) )
                      & ( ~ A5
                       => ( Y3 = none_nat ) ) ) ) )
               => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Leaf @ A5 @ B5 ) ) ) )
         => ( ! [Uu2: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( X4
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv @ Uw ) )
               => ( ( Y3 = none_nat )
                 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv @ Uw ) ) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux: nat,Uy2: list_VEBT_VEBT,Uz: vEBT_VEBT] :
                  ( ( X4
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy2 @ Uz ) )
                 => ( ( Y3
                      = ( some_nat @ Ma2 ) )
                   => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy2 @ Uz ) ) ) ) ) ) ) ) ).

% vebt_maxt.pelims
thf(fact_9305_vebt__mint_Opelims,axiom,
    ! [X4: vEBT_VEBT,Y3: option_nat] :
      ( ( ( vEBT_vebt_mint @ X4 )
        = Y3 )
     => ( ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ X4 )
       => ( ! [A5: $o,B5: $o] :
              ( ( X4
                = ( vEBT_Leaf @ A5 @ B5 ) )
             => ( ( ( A5
                   => ( Y3
                      = ( some_nat @ zero_zero_nat ) ) )
                  & ( ~ A5
                   => ( ( B5
                       => ( Y3
                          = ( some_nat @ one_one_nat ) ) )
                      & ( ~ B5
                       => ( Y3 = none_nat ) ) ) ) )
               => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Leaf @ A5 @ B5 ) ) ) )
         => ( ! [Uu2: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( X4
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv @ Uw ) )
               => ( ( Y3 = none_nat )
                 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv @ Uw ) ) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux: nat,Uy2: list_VEBT_VEBT,Uz: vEBT_VEBT] :
                  ( ( X4
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy2 @ Uz ) )
                 => ( ( Y3
                      = ( some_nat @ Mi2 ) )
                   => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy2 @ Uz ) ) ) ) ) ) ) ) ).

% vebt_mint.pelims
thf(fact_9306_prod__decode__aux_Osimps,axiom,
    ( nat_prod_decode_aux
    = ( ^ [K3: nat,M6: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ M6 @ K3 ) @ ( product_Pair_nat_nat @ M6 @ ( minus_minus_nat @ K3 @ M6 ) ) @ ( nat_prod_decode_aux @ ( suc @ K3 ) @ ( minus_minus_nat @ M6 @ ( suc @ K3 ) ) ) ) ) ) ).

% prod_decode_aux.simps
thf(fact_9307_nat_Odisc__eq__case_I1_J,axiom,
    ! [Nat: nat] :
      ( ( Nat = zero_zero_nat )
      = ( case_nat_o @ $true
        @ ^ [Uu: nat] : $false
        @ Nat ) ) ).

% nat.disc_eq_case(1)
thf(fact_9308_nat_Odisc__eq__case_I2_J,axiom,
    ! [Nat: nat] :
      ( ( Nat != zero_zero_nat )
      = ( case_nat_o @ $false
        @ ^ [Uu: nat] : $true
        @ Nat ) ) ).

% nat.disc_eq_case(2)
thf(fact_9309_less__eq__nat_Osimps_I2_J,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
      = ( case_nat_o @ $false @ ( ord_less_eq_nat @ M ) @ N ) ) ).

% less_eq_nat.simps(2)
thf(fact_9310_max__Suc2,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_max_nat @ M @ ( suc @ N ) )
      = ( case_nat_nat @ ( suc @ N )
        @ ^ [M2: nat] : ( suc @ ( ord_max_nat @ M2 @ N ) )
        @ M ) ) ).

% max_Suc2
thf(fact_9311_max__Suc1,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_max_nat @ ( suc @ N ) @ M )
      = ( case_nat_nat @ ( suc @ N )
        @ ^ [M2: nat] : ( suc @ ( ord_max_nat @ N @ M2 ) )
        @ M ) ) ).

% max_Suc1
thf(fact_9312_diff__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( minus_minus_nat @ M @ ( suc @ N ) )
      = ( case_nat_nat @ zero_zero_nat
        @ ^ [K3: nat] : K3
        @ ( minus_minus_nat @ M @ N ) ) ) ).

% diff_Suc
thf(fact_9313_prod__decode__aux_Oelims,axiom,
    ! [X4: nat,Xa: nat,Y3: product_prod_nat_nat] :
      ( ( ( nat_prod_decode_aux @ X4 @ Xa )
        = Y3 )
     => ( ( ( ord_less_eq_nat @ Xa @ X4 )
         => ( Y3
            = ( product_Pair_nat_nat @ Xa @ ( minus_minus_nat @ X4 @ Xa ) ) ) )
        & ( ~ ( ord_less_eq_nat @ Xa @ X4 )
         => ( Y3
            = ( nat_prod_decode_aux @ ( suc @ X4 ) @ ( minus_minus_nat @ Xa @ ( suc @ X4 ) ) ) ) ) ) ) ).

% prod_decode_aux.elims
thf(fact_9314_pred__def,axiom,
    ( pred
    = ( case_nat_nat @ zero_zero_nat
      @ ^ [X23: nat] : X23 ) ) ).

% pred_def
thf(fact_9315_floor__real__def,axiom,
    ( archim6058952711729229775r_real
    = ( ^ [X: real] :
          ( the_int
          @ ^ [Z5: int] :
              ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z5 ) @ X )
              & ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ Z5 @ one_one_int ) ) ) ) ) ) ) ).

% floor_real_def
thf(fact_9316_floor__rat__def,axiom,
    ( archim3151403230148437115or_rat
    = ( ^ [X: rat] :
          ( the_int
          @ ^ [Z5: int] :
              ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z5 ) @ X )
              & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ Z5 @ one_one_int ) ) ) ) ) ) ) ).

% floor_rat_def
thf(fact_9317_drop__bit__numeral__minus__bit1,axiom,
    ! [L2: num,K: num] :
      ( ( bit_se8568078237143864401it_int @ ( numeral_numeral_nat @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
      = ( bit_se8568078237143864401it_int @ ( pred_numeral @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) ) ).

% drop_bit_numeral_minus_bit1
thf(fact_9318_Suc__0__mod__numeral,axiom,
    ! [K: num] :
      ( ( modulo_modulo_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ K ) )
      = ( product_snd_nat_nat @ ( unique5055182867167087721od_nat @ one @ K ) ) ) ).

% Suc_0_mod_numeral
thf(fact_9319_drop__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se8568078237143864401it_int @ N @ K ) )
      = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).

% drop_bit_nonnegative_int_iff
thf(fact_9320_drop__bit__negative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_int @ ( bit_se8568078237143864401it_int @ N @ K ) @ zero_zero_int )
      = ( ord_less_int @ K @ zero_zero_int ) ) ).

% drop_bit_negative_int_iff
thf(fact_9321_drop__bit__minus__one,axiom,
    ! [N: nat] :
      ( ( bit_se8568078237143864401it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
      = ( uminus_uminus_int @ one_one_int ) ) ).

% drop_bit_minus_one
thf(fact_9322_snd__divmod__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( product_snd_nat_nat @ ( divmod_nat @ M @ N ) )
      = ( modulo_modulo_nat @ M @ N ) ) ).

% snd_divmod_nat
thf(fact_9323_drop__bit__Suc__minus__bit0,axiom,
    ! [N: nat,K: num] :
      ( ( bit_se8568078237143864401it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
      = ( bit_se8568078237143864401it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) ) ).

% drop_bit_Suc_minus_bit0
thf(fact_9324_drop__bit__numeral__minus__bit0,axiom,
    ! [L2: num,K: num] :
      ( ( bit_se8568078237143864401it_int @ ( numeral_numeral_nat @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
      = ( bit_se8568078237143864401it_int @ ( pred_numeral @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) ) ).

% drop_bit_numeral_minus_bit0
thf(fact_9325_drop__bit__Suc__minus__bit1,axiom,
    ! [N: nat,K: num] :
      ( ( bit_se8568078237143864401it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
      = ( bit_se8568078237143864401it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) ) ).

% drop_bit_Suc_minus_bit1
thf(fact_9326_abs__rat__def,axiom,
    ( abs_abs_rat
    = ( ^ [A4: rat] : ( if_rat @ ( ord_less_rat @ A4 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A4 ) @ A4 ) ) ) ).

% abs_rat_def
thf(fact_9327_sgn__rat__def,axiom,
    ( sgn_sgn_rat
    = ( ^ [A4: rat] : ( if_rat @ ( A4 = zero_zero_rat ) @ zero_zero_rat @ ( if_rat @ ( ord_less_rat @ zero_zero_rat @ A4 ) @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ) ) ) ).

% sgn_rat_def
thf(fact_9328_obtain__pos__sum,axiom,
    ! [R2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ R2 )
     => ~ ! [S3: rat] :
            ( ( ord_less_rat @ zero_zero_rat @ S3 )
           => ! [T2: rat] :
                ( ( ord_less_rat @ zero_zero_rat @ T2 )
               => ( R2
                 != ( plus_plus_rat @ S3 @ T2 ) ) ) ) ) ).

% obtain_pos_sum
thf(fact_9329_less__eq__rat__def,axiom,
    ( ord_less_eq_rat
    = ( ^ [X: rat,Y: rat] :
          ( ( ord_less_rat @ X @ Y )
          | ( X = Y ) ) ) ) ).

% less_eq_rat_def
thf(fact_9330_drop__bit__push__bit__int,axiom,
    ! [M: nat,N: nat,K: int] :
      ( ( bit_se8568078237143864401it_int @ M @ ( bit_se545348938243370406it_int @ N @ K ) )
      = ( bit_se8568078237143864401it_int @ ( minus_minus_nat @ M @ N ) @ ( bit_se545348938243370406it_int @ ( minus_minus_nat @ N @ M ) @ K ) ) ) ).

% drop_bit_push_bit_int
thf(fact_9331_drop__bit__int__def,axiom,
    ( bit_se8568078237143864401it_int
    = ( ^ [N2: nat,K3: int] : ( divide_divide_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% drop_bit_int_def
thf(fact_9332_rat__inverse__code,axiom,
    ! [P2: rat] :
      ( ( quotient_of @ ( inverse_inverse_rat @ P2 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A4: int,B3: int] : ( if_Pro3027730157355071871nt_int @ ( A4 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ ( times_times_int @ ( sgn_sgn_int @ A4 ) @ B3 ) @ ( abs_abs_int @ A4 ) ) )
        @ ( quotient_of @ P2 ) ) ) ).

% rat_inverse_code
thf(fact_9333_normalize__negative,axiom,
    ! [Q2: int,P2: int] :
      ( ( ord_less_int @ Q2 @ zero_zero_int )
     => ( ( normalize @ ( product_Pair_int_int @ P2 @ Q2 ) )
        = ( normalize @ ( product_Pair_int_int @ ( uminus_uminus_int @ P2 ) @ ( uminus_uminus_int @ Q2 ) ) ) ) ) ).

% normalize_negative
thf(fact_9334_Suc__0__div__numeral,axiom,
    ! [K: num] :
      ( ( divide_divide_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ K ) )
      = ( product_fst_nat_nat @ ( unique5055182867167087721od_nat @ one @ K ) ) ) ).

% Suc_0_div_numeral
thf(fact_9335_snd__divmod__integer,axiom,
    ! [K: code_integer,L2: code_integer] :
      ( ( produc6174133586879617921nteger @ ( code_divmod_integer @ K @ L2 ) )
      = ( modulo364778990260209775nteger @ K @ L2 ) ) ).

% snd_divmod_integer
thf(fact_9336_snd__divmod__abs,axiom,
    ! [K: code_integer,L2: code_integer] :
      ( ( produc6174133586879617921nteger @ ( code_divmod_abs @ K @ L2 ) )
      = ( modulo364778990260209775nteger @ ( abs_abs_Code_integer @ K ) @ ( abs_abs_Code_integer @ L2 ) ) ) ).

% snd_divmod_abs
thf(fact_9337_drop__bit__of__Suc__0,axiom,
    ! [N: nat] :
      ( ( bit_se8570568707652914677it_nat @ N @ ( suc @ zero_zero_nat ) )
      = ( zero_n2687167440665602831ol_nat @ ( N = zero_zero_nat ) ) ) ).

% drop_bit_of_Suc_0
thf(fact_9338_fst__divmod__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( product_fst_nat_nat @ ( divmod_nat @ M @ N ) )
      = ( divide_divide_nat @ M @ N ) ) ).

% fst_divmod_nat
thf(fact_9339_quotient__of__denom__pos_H,axiom,
    ! [R2: rat] : ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ ( quotient_of @ R2 ) ) ) ).

% quotient_of_denom_pos'
thf(fact_9340_diff__rat__def,axiom,
    ( minus_minus_rat
    = ( ^ [Q4: rat,R5: rat] : ( plus_plus_rat @ Q4 @ ( uminus_uminus_rat @ R5 ) ) ) ) ).

% diff_rat_def
thf(fact_9341_divide__rat__def,axiom,
    ( divide_divide_rat
    = ( ^ [Q4: rat,R5: rat] : ( times_times_rat @ Q4 @ ( inverse_inverse_rat @ R5 ) ) ) ) ).

% divide_rat_def
thf(fact_9342_rat__divide__code,axiom,
    ! [P2: rat,Q2: rat] :
      ( ( quotient_of @ ( divide_divide_rat @ P2 @ Q2 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A4: int,C2: int] :
            ( produc4245557441103728435nt_int
            @ ^ [B3: int,D2: int] : ( normalize @ ( product_Pair_int_int @ ( times_times_int @ A4 @ D2 ) @ ( times_times_int @ C2 @ B3 ) ) )
            @ ( quotient_of @ Q2 ) )
        @ ( quotient_of @ P2 ) ) ) ).

% rat_divide_code
thf(fact_9343_rat__times__code,axiom,
    ! [P2: rat,Q2: rat] :
      ( ( quotient_of @ ( times_times_rat @ P2 @ Q2 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A4: int,C2: int] :
            ( produc4245557441103728435nt_int
            @ ^ [B3: int,D2: int] : ( normalize @ ( product_Pair_int_int @ ( times_times_int @ A4 @ B3 ) @ ( times_times_int @ C2 @ D2 ) ) )
            @ ( quotient_of @ Q2 ) )
        @ ( quotient_of @ P2 ) ) ) ).

% rat_times_code
thf(fact_9344_drop__bit__nat__eq,axiom,
    ! [N: nat,K: int] :
      ( ( bit_se8570568707652914677it_nat @ N @ ( nat2 @ K ) )
      = ( nat2 @ ( bit_se8568078237143864401it_int @ N @ K ) ) ) ).

% drop_bit_nat_eq
thf(fact_9345_quotient__of__div,axiom,
    ! [R2: rat,N: int,D: int] :
      ( ( ( quotient_of @ R2 )
        = ( product_Pair_int_int @ N @ D ) )
     => ( R2
        = ( divide_divide_rat @ ( ring_1_of_int_rat @ N ) @ ( ring_1_of_int_rat @ D ) ) ) ) ).

% quotient_of_div
thf(fact_9346_rat__plus__code,axiom,
    ! [P2: rat,Q2: rat] :
      ( ( quotient_of @ ( plus_plus_rat @ P2 @ Q2 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A4: int,C2: int] :
            ( produc4245557441103728435nt_int
            @ ^ [B3: int,D2: int] : ( normalize @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ A4 @ D2 ) @ ( times_times_int @ B3 @ C2 ) ) @ ( times_times_int @ C2 @ D2 ) ) )
            @ ( quotient_of @ Q2 ) )
        @ ( quotient_of @ P2 ) ) ) ).

% rat_plus_code
thf(fact_9347_rat__minus__code,axiom,
    ! [P2: rat,Q2: rat] :
      ( ( quotient_of @ ( minus_minus_rat @ P2 @ Q2 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A4: int,C2: int] :
            ( produc4245557441103728435nt_int
            @ ^ [B3: int,D2: int] : ( normalize @ ( product_Pair_int_int @ ( minus_minus_int @ ( times_times_int @ A4 @ D2 ) @ ( times_times_int @ B3 @ C2 ) ) @ ( times_times_int @ C2 @ D2 ) ) )
            @ ( quotient_of @ Q2 ) )
        @ ( quotient_of @ P2 ) ) ) ).

% rat_minus_code
thf(fact_9348_quotient__of__denom__pos,axiom,
    ! [R2: rat,P2: int,Q2: int] :
      ( ( ( quotient_of @ R2 )
        = ( product_Pair_int_int @ P2 @ Q2 ) )
     => ( ord_less_int @ zero_zero_int @ Q2 ) ) ).

% quotient_of_denom_pos
thf(fact_9349_normalize__denom__pos,axiom,
    ! [R2: product_prod_int_int,P2: int,Q2: int] :
      ( ( ( normalize @ R2 )
        = ( product_Pair_int_int @ P2 @ Q2 ) )
     => ( ord_less_int @ zero_zero_int @ Q2 ) ) ).

% normalize_denom_pos
thf(fact_9350_normalize__crossproduct,axiom,
    ! [Q2: int,S: int,P2: int,R2: int] :
      ( ( Q2 != zero_zero_int )
     => ( ( S != zero_zero_int )
       => ( ( ( normalize @ ( product_Pair_int_int @ P2 @ Q2 ) )
            = ( normalize @ ( product_Pair_int_int @ R2 @ S ) ) )
         => ( ( times_times_int @ P2 @ S )
            = ( times_times_int @ R2 @ Q2 ) ) ) ) ) ).

% normalize_crossproduct
thf(fact_9351_drop__bit__nat__def,axiom,
    ( bit_se8570568707652914677it_nat
    = ( ^ [N2: nat,M6: nat] : ( divide_divide_nat @ M6 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% drop_bit_nat_def
thf(fact_9352_one__mod__minus__numeral,axiom,
    ! [N: num] :
      ( ( modulo_modulo_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( uminus_uminus_int @ ( adjust_mod @ ( numeral_numeral_int @ N ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ one @ N ) ) ) ) ) ).

% one_mod_minus_numeral
thf(fact_9353_minus__one__mod__numeral,axiom,
    ! [N: num] :
      ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N ) )
      = ( adjust_mod @ ( numeral_numeral_int @ N ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ one @ N ) ) ) ) ).

% minus_one_mod_numeral
thf(fact_9354_numeral__mod__minus__numeral,axiom,
    ! [M: num,N: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( uminus_uminus_int @ ( adjust_mod @ ( numeral_numeral_int @ N ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ M @ N ) ) ) ) ) ).

% numeral_mod_minus_numeral
thf(fact_9355_fst__divmod__integer,axiom,
    ! [K: code_integer,L2: code_integer] :
      ( ( produc8508995932063986495nteger @ ( code_divmod_integer @ K @ L2 ) )
      = ( divide6298287555418463151nteger @ K @ L2 ) ) ).

% fst_divmod_integer
thf(fact_9356_fst__divmod__abs,axiom,
    ! [K: code_integer,L2: code_integer] :
      ( ( produc8508995932063986495nteger @ ( code_divmod_abs @ K @ L2 ) )
      = ( divide6298287555418463151nteger @ ( abs_abs_Code_integer @ K ) @ ( abs_abs_Code_integer @ L2 ) ) ) ).

% fst_divmod_abs
thf(fact_9357_minus__numeral__mod__numeral,axiom,
    ! [M: num,N: num] :
      ( ( modulo_modulo_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
      = ( adjust_mod @ ( numeral_numeral_int @ N ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ M @ N ) ) ) ) ).

% minus_numeral_mod_numeral
thf(fact_9358_Divides_Oadjust__mod__def,axiom,
    ( adjust_mod
    = ( ^ [L: int,R5: int] : ( if_int @ ( R5 = zero_zero_int ) @ zero_zero_int @ ( minus_minus_int @ L @ R5 ) ) ) ) ).

% Divides.adjust_mod_def
thf(fact_9359_bezw_Osimps,axiom,
    ( bezw
    = ( ^ [X: nat,Y: nat] : ( if_Pro3027730157355071871nt_int @ ( Y = zero_zero_nat ) @ ( product_Pair_int_int @ one_one_int @ zero_zero_int ) @ ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X @ Y ) ) ) ) ) ) ) ) ).

% bezw.simps
thf(fact_9360_bezw_Oelims,axiom,
    ! [X4: nat,Xa: nat,Y3: product_prod_int_int] :
      ( ( ( bezw @ X4 @ Xa )
        = Y3 )
     => ( ( ( Xa = zero_zero_nat )
         => ( Y3
            = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
        & ( ( Xa != zero_zero_nat )
         => ( Y3
            = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X4 @ Xa ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X4 @ Xa ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X4 @ Xa ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X4 @ Xa ) ) ) ) ) ) ) ) ) ).

% bezw.elims
thf(fact_9361_bezw__non__0,axiom,
    ! [Y3: nat,X4: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ Y3 )
     => ( ( bezw @ X4 @ Y3 )
        = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Y3 @ ( modulo_modulo_nat @ X4 @ Y3 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Y3 @ ( modulo_modulo_nat @ X4 @ Y3 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Y3 @ ( modulo_modulo_nat @ X4 @ Y3 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X4 @ Y3 ) ) ) ) ) ) ) ).

% bezw_non_0
thf(fact_9362_bezw_Opelims,axiom,
    ! [X4: nat,Xa: nat,Y3: product_prod_int_int] :
      ( ( ( bezw @ X4 @ Xa )
        = Y3 )
     => ( ( accp_P4275260045618599050at_nat @ bezw_rel @ ( product_Pair_nat_nat @ X4 @ Xa ) )
       => ~ ( ( ( ( Xa = zero_zero_nat )
               => ( Y3
                  = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
              & ( ( Xa != zero_zero_nat )
               => ( Y3
                  = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X4 @ Xa ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X4 @ Xa ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X4 @ Xa ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X4 @ Xa ) ) ) ) ) ) ) )
           => ~ ( accp_P4275260045618599050at_nat @ bezw_rel @ ( product_Pair_nat_nat @ X4 @ Xa ) ) ) ) ) ).

% bezw.pelims
thf(fact_9363_normalize__def,axiom,
    ( normalize
    = ( ^ [P3: product_prod_int_int] :
          ( if_Pro3027730157355071871nt_int @ ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ P3 ) ) @ ( product_Pair_int_int @ ( divide_divide_int @ ( product_fst_int_int @ P3 ) @ ( gcd_gcd_int @ ( product_fst_int_int @ P3 ) @ ( product_snd_int_int @ P3 ) ) ) @ ( divide_divide_int @ ( product_snd_int_int @ P3 ) @ ( gcd_gcd_int @ ( product_fst_int_int @ P3 ) @ ( product_snd_int_int @ P3 ) ) ) )
          @ ( if_Pro3027730157355071871nt_int
            @ ( ( product_snd_int_int @ P3 )
              = zero_zero_int )
            @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
            @ ( product_Pair_int_int @ ( divide_divide_int @ ( product_fst_int_int @ P3 ) @ ( uminus_uminus_int @ ( gcd_gcd_int @ ( product_fst_int_int @ P3 ) @ ( product_snd_int_int @ P3 ) ) ) ) @ ( divide_divide_int @ ( product_snd_int_int @ P3 ) @ ( uminus_uminus_int @ ( gcd_gcd_int @ ( product_fst_int_int @ P3 ) @ ( product_snd_int_int @ P3 ) ) ) ) ) ) ) ) ) ).

% normalize_def
thf(fact_9364_Frct__code__post_I5_J,axiom,
    ! [K: num] :
      ( ( frct @ ( product_Pair_int_int @ one_one_int @ ( numeral_numeral_int @ K ) ) )
      = ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ K ) ) ) ).

% Frct_code_post(5)
thf(fact_9365_gcd__pos__int,axiom,
    ! [M: int,N: int] :
      ( ( ord_less_int @ zero_zero_int @ ( gcd_gcd_int @ M @ N ) )
      = ( ( M != zero_zero_int )
        | ( N != zero_zero_int ) ) ) ).

% gcd_pos_int
thf(fact_9366_gcd__red__int,axiom,
    ( gcd_gcd_int
    = ( ^ [X: int,Y: int] : ( gcd_gcd_int @ Y @ ( modulo_modulo_int @ X @ Y ) ) ) ) ).

% gcd_red_int
thf(fact_9367_bezout__int,axiom,
    ! [X4: int,Y3: int] :
    ? [U3: int,V3: int] :
      ( ( plus_plus_int @ ( times_times_int @ U3 @ X4 ) @ ( times_times_int @ V3 @ Y3 ) )
      = ( gcd_gcd_int @ X4 @ Y3 ) ) ).

% bezout_int
thf(fact_9368_gcd__mult__distrib__int,axiom,
    ! [K: int,M: int,N: int] :
      ( ( times_times_int @ ( abs_abs_int @ K ) @ ( gcd_gcd_int @ M @ N ) )
      = ( gcd_gcd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ N ) ) ) ).

% gcd_mult_distrib_int
thf(fact_9369_gcd__le1__int,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ord_less_eq_int @ ( gcd_gcd_int @ A @ B2 ) @ A ) ) ).

% gcd_le1_int
thf(fact_9370_gcd__le2__int,axiom,
    ! [B2: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B2 )
     => ( ord_less_eq_int @ ( gcd_gcd_int @ A @ B2 ) @ B2 ) ) ).

% gcd_le2_int
thf(fact_9371_gcd__non__0__int,axiom,
    ! [Y3: int,X4: int] :
      ( ( ord_less_int @ zero_zero_int @ Y3 )
     => ( ( gcd_gcd_int @ X4 @ Y3 )
        = ( gcd_gcd_int @ Y3 @ ( modulo_modulo_int @ X4 @ Y3 ) ) ) ) ).

% gcd_non_0_int
thf(fact_9372_gcd__code__int,axiom,
    ( gcd_gcd_int
    = ( ^ [K3: int,L: int] : ( abs_abs_int @ ( if_int @ ( L = zero_zero_int ) @ K3 @ ( gcd_gcd_int @ L @ ( modulo_modulo_int @ ( abs_abs_int @ K3 ) @ ( abs_abs_int @ L ) ) ) ) ) ) ) ).

% gcd_code_int
thf(fact_9373_Frct__code__post_I6_J,axiom,
    ! [K: num,L2: num] :
      ( ( frct @ ( product_Pair_int_int @ ( numeral_numeral_int @ K ) @ ( numeral_numeral_int @ L2 ) ) )
      = ( divide_divide_rat @ ( numeral_numeral_rat @ K ) @ ( numeral_numeral_rat @ L2 ) ) ) ).

% Frct_code_post(6)
thf(fact_9374_prod__decode__aux_Opelims,axiom,
    ! [X4: nat,Xa: nat,Y3: product_prod_nat_nat] :
      ( ( ( nat_prod_decode_aux @ X4 @ Xa )
        = Y3 )
     => ( ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X4 @ Xa ) )
       => ~ ( ( ( ( ord_less_eq_nat @ Xa @ X4 )
               => ( Y3
                  = ( product_Pair_nat_nat @ Xa @ ( minus_minus_nat @ X4 @ Xa ) ) ) )
              & ( ~ ( ord_less_eq_nat @ Xa @ X4 )
               => ( Y3
                  = ( nat_prod_decode_aux @ ( suc @ X4 ) @ ( minus_minus_nat @ Xa @ ( suc @ X4 ) ) ) ) ) )
           => ~ ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X4 @ Xa ) ) ) ) ) ).

% prod_decode_aux.pelims
thf(fact_9375_finite__enumerate,axiom,
    ! [S2: set_nat] :
      ( ( finite_finite_nat @ S2 )
     => ? [R3: nat > nat] :
          ( ( strict1292158309912662752at_nat @ R3 @ ( set_ord_lessThan_nat @ ( finite_card_nat @ S2 ) ) )
          & ! [N6: nat] :
              ( ( ord_less_nat @ N6 @ ( finite_card_nat @ S2 ) )
             => ( member_nat @ ( R3 @ N6 ) @ S2 ) ) ) ) ).

% finite_enumerate
thf(fact_9376_gcd__1__nat,axiom,
    ! [M: nat] :
      ( ( gcd_gcd_nat @ M @ one_one_nat )
      = one_one_nat ) ).

% gcd_1_nat
thf(fact_9377_gcd__Suc__0,axiom,
    ! [M: nat] :
      ( ( gcd_gcd_nat @ M @ ( suc @ zero_zero_nat ) )
      = ( suc @ zero_zero_nat ) ) ).

% gcd_Suc_0
thf(fact_9378_gcd__pos__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( gcd_gcd_nat @ M @ N ) )
      = ( ( M != zero_zero_nat )
        | ( N != zero_zero_nat ) ) ) ).

% gcd_pos_nat
thf(fact_9379_gcd__red__nat,axiom,
    ( gcd_gcd_nat
    = ( ^ [X: nat,Y: nat] : ( gcd_gcd_nat @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) ) ).

% gcd_red_nat
thf(fact_9380_gcd__mult__distrib__nat,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( times_times_nat @ K @ ( gcd_gcd_nat @ M @ N ) )
      = ( gcd_gcd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).

% gcd_mult_distrib_nat
thf(fact_9381_gcd__le2__nat,axiom,
    ! [B2: nat,A: nat] :
      ( ( B2 != zero_zero_nat )
     => ( ord_less_eq_nat @ ( gcd_gcd_nat @ A @ B2 ) @ B2 ) ) ).

% gcd_le2_nat
thf(fact_9382_gcd__le1__nat,axiom,
    ! [A: nat,B2: nat] :
      ( ( A != zero_zero_nat )
     => ( ord_less_eq_nat @ ( gcd_gcd_nat @ A @ B2 ) @ A ) ) ).

% gcd_le1_nat
thf(fact_9383_gcd__diff2__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( gcd_gcd_nat @ ( minus_minus_nat @ N @ M ) @ N )
        = ( gcd_gcd_nat @ M @ N ) ) ) ).

% gcd_diff2_nat
thf(fact_9384_gcd__diff1__nat,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( gcd_gcd_nat @ ( minus_minus_nat @ M @ N ) @ N )
        = ( gcd_gcd_nat @ M @ N ) ) ) ).

% gcd_diff1_nat
thf(fact_9385_gcd__non__0__nat,axiom,
    ! [Y3: nat,X4: nat] :
      ( ( Y3 != zero_zero_nat )
     => ( ( gcd_gcd_nat @ X4 @ Y3 )
        = ( gcd_gcd_nat @ Y3 @ ( modulo_modulo_nat @ X4 @ Y3 ) ) ) ) ).

% gcd_non_0_nat
thf(fact_9386_gcd__nat_Osimps,axiom,
    ( gcd_gcd_nat
    = ( ^ [X: nat,Y: nat] : ( if_nat @ ( Y = zero_zero_nat ) @ X @ ( gcd_gcd_nat @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) ) ) ).

% gcd_nat.simps
thf(fact_9387_gcd__nat_Oelims,axiom,
    ! [X4: nat,Xa: nat,Y3: nat] :
      ( ( ( gcd_gcd_nat @ X4 @ Xa )
        = Y3 )
     => ( ( ( Xa = zero_zero_nat )
         => ( Y3 = X4 ) )
        & ( ( Xa != zero_zero_nat )
         => ( Y3
            = ( gcd_gcd_nat @ Xa @ ( modulo_modulo_nat @ X4 @ Xa ) ) ) ) ) ) ).

% gcd_nat.elims
thf(fact_9388_bezout__nat,axiom,
    ! [A: nat,B2: nat] :
      ( ( A != zero_zero_nat )
     => ? [X3: nat,Y4: nat] :
          ( ( times_times_nat @ A @ X3 )
          = ( plus_plus_nat @ ( times_times_nat @ B2 @ Y4 ) @ ( gcd_gcd_nat @ A @ B2 ) ) ) ) ).

% bezout_nat
thf(fact_9389_bezout__gcd__nat_H,axiom,
    ! [B2: nat,A: nat] :
    ? [X3: nat,Y4: nat] :
      ( ( ( ord_less_eq_nat @ ( times_times_nat @ B2 @ Y4 ) @ ( times_times_nat @ A @ X3 ) )
        & ( ( minus_minus_nat @ ( times_times_nat @ A @ X3 ) @ ( times_times_nat @ B2 @ Y4 ) )
          = ( gcd_gcd_nat @ A @ B2 ) ) )
      | ( ( ord_less_eq_nat @ ( times_times_nat @ A @ Y4 ) @ ( times_times_nat @ B2 @ X3 ) )
        & ( ( minus_minus_nat @ ( times_times_nat @ B2 @ X3 ) @ ( times_times_nat @ A @ Y4 ) )
          = ( gcd_gcd_nat @ A @ B2 ) ) ) ) ).

% bezout_gcd_nat'
thf(fact_9390_gcd__code__integer,axiom,
    ( gcd_gcd_Code_integer
    = ( ^ [K3: code_integer,L: code_integer] : ( abs_abs_Code_integer @ ( if_Code_integer @ ( L = zero_z3403309356797280102nteger ) @ K3 @ ( gcd_gcd_Code_integer @ L @ ( modulo364778990260209775nteger @ ( abs_abs_Code_integer @ K3 ) @ ( abs_abs_Code_integer @ L ) ) ) ) ) ) ) ).

% gcd_code_integer
thf(fact_9391_bezw__aux,axiom,
    ! [X4: nat,Y3: nat] :
      ( ( semiri1314217659103216013at_int @ ( gcd_gcd_nat @ X4 @ Y3 ) )
      = ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ ( bezw @ X4 @ Y3 ) ) @ ( semiri1314217659103216013at_int @ X4 ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ X4 @ Y3 ) ) @ ( semiri1314217659103216013at_int @ Y3 ) ) ) ) ).

% bezw_aux
thf(fact_9392_nat__descend__induct,axiom,
    ! [N: nat,P: nat > $o,M: nat] :
      ( ! [K2: nat] :
          ( ( ord_less_nat @ N @ K2 )
         => ( P @ K2 ) )
     => ( ! [K2: nat] :
            ( ( ord_less_eq_nat @ K2 @ N )
           => ( ! [I4: nat] :
                  ( ( ord_less_nat @ K2 @ I4 )
                 => ( P @ I4 ) )
             => ( P @ K2 ) ) )
       => ( P @ M ) ) ) ).

% nat_descend_induct
thf(fact_9393_gcd__nat_Opelims,axiom,
    ! [X4: nat,Xa: nat,Y3: nat] :
      ( ( ( gcd_gcd_nat @ X4 @ Xa )
        = Y3 )
     => ( ( accp_P4275260045618599050at_nat @ gcd_nat_rel @ ( product_Pair_nat_nat @ X4 @ Xa ) )
       => ~ ( ( ( ( Xa = zero_zero_nat )
               => ( Y3 = X4 ) )
              & ( ( Xa != zero_zero_nat )
               => ( Y3
                  = ( gcd_gcd_nat @ Xa @ ( modulo_modulo_nat @ X4 @ Xa ) ) ) ) )
           => ~ ( accp_P4275260045618599050at_nat @ gcd_nat_rel @ ( product_Pair_nat_nat @ X4 @ Xa ) ) ) ) ) ).

% gcd_nat.pelims
thf(fact_9394_divmod__integer__eq__cases,axiom,
    ( code_divmod_integer
    = ( ^ [K3: code_integer,L: code_integer] :
          ( if_Pro6119634080678213985nteger @ ( K3 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
          @ ( if_Pro6119634080678213985nteger @ ( L = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ K3 )
            @ ( comp_C1593894019821074884nteger @ ( comp_C8797469213163452608nteger @ produc6499014454317279255nteger @ times_3573771949741848930nteger ) @ sgn_sgn_Code_integer @ L
              @ ( if_Pro6119634080678213985nteger
                @ ( ( sgn_sgn_Code_integer @ K3 )
                  = ( sgn_sgn_Code_integer @ L ) )
                @ ( code_divmod_abs @ K3 @ L )
                @ ( produc6916734918728496179nteger
                  @ ^ [R5: code_integer,S4: code_integer] : ( if_Pro6119634080678213985nteger @ ( S4 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ L ) @ S4 ) ) )
                  @ ( code_divmod_abs @ K3 @ L ) ) ) ) ) ) ) ) ).

% divmod_integer_eq_cases
thf(fact_9395_card__greaterThanLessThan__int,axiom,
    ! [L2: int,U: int] :
      ( ( finite_card_int @ ( set_or5832277885323065728an_int @ L2 @ U ) )
      = ( nat2 @ ( minus_minus_int @ U @ ( plus_plus_int @ L2 @ one_one_int ) ) ) ) ).

% card_greaterThanLessThan_int
thf(fact_9396_xor__minus__numerals_I2_J,axiom,
    ! [K: int,N: num] :
      ( ( bit_se6526347334894502574or_int @ K @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( bit_ri7919022796975470100ot_int @ ( bit_se6526347334894502574or_int @ K @ ( neg_numeral_sub_int @ N @ one ) ) ) ) ).

% xor_minus_numerals(2)
thf(fact_9397_xor__minus__numerals_I1_J,axiom,
    ! [N: num,K: int] :
      ( ( bit_se6526347334894502574or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ K )
      = ( bit_ri7919022796975470100ot_int @ ( bit_se6526347334894502574or_int @ ( neg_numeral_sub_int @ N @ one ) @ K ) ) ) ).

% xor_minus_numerals(1)
thf(fact_9398_finite__greaterThanLessThan__int,axiom,
    ! [L2: int,U: int] : ( finite_finite_int @ ( set_or5832277885323065728an_int @ L2 @ U ) ) ).

% finite_greaterThanLessThan_int
thf(fact_9399_card_Ocomp__fun__commute__on,axiom,
    ( ( comp_nat_nat_nat @ suc @ suc )
    = ( comp_nat_nat_nat @ suc @ suc ) ) ).

% card.comp_fun_commute_on
thf(fact_9400_atLeastPlusOneLessThan__greaterThanLessThan__int,axiom,
    ! [L2: int,U: int] :
      ( ( set_or4662586982721622107an_int @ ( plus_plus_int @ L2 @ one_one_int ) @ U )
      = ( set_or5832277885323065728an_int @ L2 @ U ) ) ).

% atLeastPlusOneLessThan_greaterThanLessThan_int
thf(fact_9401_sub__BitM__One__eq,axiom,
    ! [N: num] :
      ( ( neg_numeral_sub_int @ ( bitM @ N ) @ one )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( neg_numeral_sub_int @ N @ one ) ) ) ).

% sub_BitM_One_eq
thf(fact_9402_finite__greaterThanLessThan,axiom,
    ! [L2: nat,U: nat] : ( finite_finite_nat @ ( set_or5834768355832116004an_nat @ L2 @ U ) ) ).

% finite_greaterThanLessThan
thf(fact_9403_card__greaterThanLessThan,axiom,
    ! [L2: nat,U: nat] :
      ( ( finite_card_nat @ ( set_or5834768355832116004an_nat @ L2 @ U ) )
      = ( minus_minus_nat @ U @ ( suc @ L2 ) ) ) ).

% card_greaterThanLessThan
thf(fact_9404_atLeastSucLessThan__greaterThanLessThan,axiom,
    ! [L2: nat,U: nat] :
      ( ( set_or4665077453230672383an_nat @ ( suc @ L2 ) @ U )
      = ( set_or5834768355832116004an_nat @ L2 @ U ) ) ).

% atLeastSucLessThan_greaterThanLessThan
thf(fact_9405_tanh__real__bounds,axiom,
    ! [X4: real] : ( member_real @ ( tanh_real @ X4 ) @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) ) ).

% tanh_real_bounds
thf(fact_9406_max__nat_Osemilattice__neutr__order__axioms,axiom,
    ( semila1623282765462674594er_nat @ ord_max_nat @ zero_zero_nat
    @ ^ [X: nat,Y: nat] : ( ord_less_eq_nat @ Y @ X )
    @ ^ [X: nat,Y: nat] : ( ord_less_nat @ Y @ X ) ) ).

% max_nat.semilattice_neutr_order_axioms
thf(fact_9407_Suc__funpow,axiom,
    ! [N: nat] :
      ( ( compow_nat_nat @ N @ suc )
      = ( plus_plus_nat @ N ) ) ).

% Suc_funpow
thf(fact_9408_int__of__integer__code,axiom,
    ( code_int_of_integer
    = ( ^ [K3: code_integer] :
          ( if_int @ ( ord_le6747313008572928689nteger @ K3 @ zero_z3403309356797280102nteger ) @ ( uminus_uminus_int @ ( code_int_of_integer @ ( uminus1351360451143612070nteger @ K3 ) ) )
          @ ( if_int @ ( K3 = zero_z3403309356797280102nteger ) @ zero_zero_int
            @ ( produc1553301316500091796er_int
              @ ^ [L: code_integer,J3: code_integer] : ( if_int @ ( J3 = zero_z3403309356797280102nteger ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( code_int_of_integer @ L ) ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( code_int_of_integer @ L ) ) @ one_one_int ) )
              @ ( code_divmod_integer @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% int_of_integer_code
thf(fact_9409_plus__integer_Orep__eq,axiom,
    ! [X4: code_integer,Xa: code_integer] :
      ( ( code_int_of_integer @ ( plus_p5714425477246183910nteger @ X4 @ Xa ) )
      = ( plus_plus_int @ ( code_int_of_integer @ X4 ) @ ( code_int_of_integer @ Xa ) ) ) ).

% plus_integer.rep_eq
thf(fact_9410_times__integer_Orep__eq,axiom,
    ! [X4: code_integer,Xa: code_integer] :
      ( ( code_int_of_integer @ ( times_3573771949741848930nteger @ X4 @ Xa ) )
      = ( times_times_int @ ( code_int_of_integer @ X4 ) @ ( code_int_of_integer @ Xa ) ) ) ).

% times_integer.rep_eq
thf(fact_9411_minus__integer_Orep__eq,axiom,
    ! [X4: code_integer,Xa: code_integer] :
      ( ( code_int_of_integer @ ( minus_8373710615458151222nteger @ X4 @ Xa ) )
      = ( minus_minus_int @ ( code_int_of_integer @ X4 ) @ ( code_int_of_integer @ Xa ) ) ) ).

% minus_integer.rep_eq
thf(fact_9412_divide__integer_Orep__eq,axiom,
    ! [X4: code_integer,Xa: code_integer] :
      ( ( code_int_of_integer @ ( divide6298287555418463151nteger @ X4 @ Xa ) )
      = ( divide_divide_int @ ( code_int_of_integer @ X4 ) @ ( code_int_of_integer @ Xa ) ) ) ).

% divide_integer.rep_eq
thf(fact_9413_modulo__integer_Orep__eq,axiom,
    ! [X4: code_integer,Xa: code_integer] :
      ( ( code_int_of_integer @ ( modulo364778990260209775nteger @ X4 @ Xa ) )
      = ( modulo_modulo_int @ ( code_int_of_integer @ X4 ) @ ( code_int_of_integer @ Xa ) ) ) ).

% modulo_integer.rep_eq
thf(fact_9414_less__integer_Orep__eq,axiom,
    ( ord_le6747313008572928689nteger
    = ( ^ [X: code_integer,Xa4: code_integer] : ( ord_less_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa4 ) ) ) ) ).

% less_integer.rep_eq
thf(fact_9415_integer__less__iff,axiom,
    ( ord_le6747313008572928689nteger
    = ( ^ [K3: code_integer,L: code_integer] : ( ord_less_int @ ( code_int_of_integer @ K3 ) @ ( code_int_of_integer @ L ) ) ) ) ).

% integer_less_iff
thf(fact_9416_times__int_Oabs__eq,axiom,
    ! [Xa: product_prod_nat_nat,X4: product_prod_nat_nat] :
      ( ( times_times_int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X4 ) )
      = ( abs_Integ
        @ ( produc27273713700761075at_nat
          @ ^ [X: nat,Y: nat] :
              ( produc2626176000494625587at_nat
              @ ^ [U4: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X @ U4 ) @ ( times_times_nat @ Y @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X @ V4 ) @ ( times_times_nat @ Y @ U4 ) ) ) )
          @ Xa
          @ X4 ) ) ) ).

% times_int.abs_eq
thf(fact_9417_Gcd__remove0__nat,axiom,
    ! [M7: set_nat] :
      ( ( finite_finite_nat @ M7 )
     => ( ( gcd_Gcd_nat @ M7 )
        = ( gcd_Gcd_nat @ ( minus_minus_set_nat @ M7 @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ) ) ).

% Gcd_remove0_nat
thf(fact_9418_Gcd__nat__eq__one,axiom,
    ! [N3: set_nat] :
      ( ( member_nat @ one_one_nat @ N3 )
     => ( ( gcd_Gcd_nat @ N3 )
        = one_one_nat ) ) ).

% Gcd_nat_eq_one
thf(fact_9419_nat_Oabs__eq,axiom,
    ! [X4: product_prod_nat_nat] :
      ( ( nat2 @ ( abs_Integ @ X4 ) )
      = ( produc6842872674320459806at_nat @ minus_minus_nat @ X4 ) ) ).

% nat.abs_eq
thf(fact_9420_one__int__def,axiom,
    ( one_one_int
    = ( abs_Integ @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) ) ) ).

% one_int_def
thf(fact_9421_less__int_Oabs__eq,axiom,
    ! [Xa: product_prod_nat_nat,X4: product_prod_nat_nat] :
      ( ( ord_less_int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X4 ) )
      = ( produc8739625826339149834_nat_o
        @ ^ [X: nat,Y: nat] :
            ( produc6081775807080527818_nat_o
            @ ^ [U4: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ U4 @ Y ) ) )
        @ Xa
        @ X4 ) ) ).

% less_int.abs_eq
thf(fact_9422_less__eq__int_Oabs__eq,axiom,
    ! [Xa: product_prod_nat_nat,X4: product_prod_nat_nat] :
      ( ( ord_less_eq_int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X4 ) )
      = ( produc8739625826339149834_nat_o
        @ ^ [X: nat,Y: nat] :
            ( produc6081775807080527818_nat_o
            @ ^ [U4: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ U4 @ Y ) ) )
        @ Xa
        @ X4 ) ) ).

% less_eq_int.abs_eq
thf(fact_9423_plus__int_Oabs__eq,axiom,
    ! [Xa: product_prod_nat_nat,X4: product_prod_nat_nat] :
      ( ( plus_plus_int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X4 ) )
      = ( abs_Integ
        @ ( produc27273713700761075at_nat
          @ ^ [X: nat,Y: nat] :
              ( produc2626176000494625587at_nat
              @ ^ [U4: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X @ U4 ) @ ( plus_plus_nat @ Y @ V4 ) ) )
          @ Xa
          @ X4 ) ) ) ).

% plus_int.abs_eq
thf(fact_9424_minus__int_Oabs__eq,axiom,
    ! [Xa: product_prod_nat_nat,X4: product_prod_nat_nat] :
      ( ( minus_minus_int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X4 ) )
      = ( abs_Integ
        @ ( produc27273713700761075at_nat
          @ ^ [X: nat,Y: nat] :
              ( produc2626176000494625587at_nat
              @ ^ [U4: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ Y @ U4 ) ) )
          @ Xa
          @ X4 ) ) ) ).

% minus_int.abs_eq
thf(fact_9425_num__of__nat_Osimps_I2_J,axiom,
    ! [N: nat] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( num_of_nat @ ( suc @ N ) )
          = ( inc @ ( num_of_nat @ N ) ) ) )
      & ( ~ ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( num_of_nat @ ( suc @ N ) )
          = one ) ) ) ).

% num_of_nat.simps(2)
thf(fact_9426_num__of__nat__numeral__eq,axiom,
    ! [Q2: num] :
      ( ( num_of_nat @ ( numeral_numeral_nat @ Q2 ) )
      = Q2 ) ).

% num_of_nat_numeral_eq
thf(fact_9427_num__of__nat_Osimps_I1_J,axiom,
    ( ( num_of_nat @ zero_zero_nat )
    = one ) ).

% num_of_nat.simps(1)
thf(fact_9428_numeral__num__of__nat,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( numeral_numeral_nat @ ( num_of_nat @ N ) )
        = N ) ) ).

% numeral_num_of_nat
thf(fact_9429_num__of__nat__One,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ N @ one_one_nat )
     => ( ( num_of_nat @ N )
        = one ) ) ).

% num_of_nat_One
thf(fact_9430_num__of__nat__double,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( num_of_nat @ ( plus_plus_nat @ N @ N ) )
        = ( bit0 @ ( num_of_nat @ N ) ) ) ) ).

% num_of_nat_double
thf(fact_9431_num__of__nat__plus__distrib,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( num_of_nat @ ( plus_plus_nat @ M @ N ) )
          = ( plus_plus_num @ ( num_of_nat @ M ) @ ( num_of_nat @ N ) ) ) ) ) ).

% num_of_nat_plus_distrib
thf(fact_9432_nth__sorted__list__of__set__greaterThanLessThan,axiom,
    ! [N: nat,J: nat,I: nat] :
      ( ( ord_less_nat @ N @ ( minus_minus_nat @ J @ ( suc @ I ) ) )
     => ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I @ J ) ) @ N )
        = ( suc @ ( plus_plus_nat @ I @ N ) ) ) ) ).

% nth_sorted_list_of_set_greaterThanLessThan
thf(fact_9433_less__eq__int_Orep__eq,axiom,
    ( ord_less_eq_int
    = ( ^ [X: int,Xa4: int] :
          ( produc8739625826339149834_nat_o
          @ ^ [Y: nat,Z5: nat] :
              ( produc6081775807080527818_nat_o
              @ ^ [U4: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ Y @ V4 ) @ ( plus_plus_nat @ U4 @ Z5 ) ) )
          @ ( rep_Integ @ X )
          @ ( rep_Integ @ Xa4 ) ) ) ) ).

% less_eq_int.rep_eq
thf(fact_9434_less__int_Orep__eq,axiom,
    ( ord_less_int
    = ( ^ [X: int,Xa4: int] :
          ( produc8739625826339149834_nat_o
          @ ^ [Y: nat,Z5: nat] :
              ( produc6081775807080527818_nat_o
              @ ^ [U4: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ Y @ V4 ) @ ( plus_plus_nat @ U4 @ Z5 ) ) )
          @ ( rep_Integ @ X )
          @ ( rep_Integ @ Xa4 ) ) ) ) ).

% less_int.rep_eq
thf(fact_9435_nat_Orep__eq,axiom,
    ( nat2
    = ( ^ [X: int] : ( produc6842872674320459806at_nat @ minus_minus_nat @ ( rep_Integ @ X ) ) ) ) ).

% nat.rep_eq
thf(fact_9436_nth__sorted__list__of__set__greaterThanAtMost,axiom,
    ! [N: nat,J: nat,I: nat] :
      ( ( ord_less_nat @ N @ ( minus_minus_nat @ J @ I ) )
     => ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I @ J ) ) @ N )
        = ( suc @ ( plus_plus_nat @ I @ N ) ) ) ) ).

% nth_sorted_list_of_set_greaterThanAtMost
thf(fact_9437_prod__encode__def,axiom,
    ( nat_prod_encode
    = ( produc6842872674320459806at_nat
      @ ^ [M6: nat,N2: nat] : ( plus_plus_nat @ ( nat_triangle @ ( plus_plus_nat @ M6 @ N2 ) ) @ M6 ) ) ) ).

% prod_encode_def
thf(fact_9438_pow_Osimps_I3_J,axiom,
    ! [X4: num,Y3: num] :
      ( ( pow @ X4 @ ( bit1 @ Y3 ) )
      = ( times_times_num @ ( sqr @ ( pow @ X4 @ Y3 ) ) @ X4 ) ) ).

% pow.simps(3)
thf(fact_9439_finite__greaterThanAtMost,axiom,
    ! [L2: nat,U: nat] : ( finite_finite_nat @ ( set_or6659071591806873216st_nat @ L2 @ U ) ) ).

% finite_greaterThanAtMost
thf(fact_9440_card__greaterThanAtMost,axiom,
    ! [L2: nat,U: nat] :
      ( ( finite_card_nat @ ( set_or6659071591806873216st_nat @ L2 @ U ) )
      = ( minus_minus_nat @ U @ L2 ) ) ).

% card_greaterThanAtMost
thf(fact_9441_sqr_Osimps_I2_J,axiom,
    ! [N: num] :
      ( ( sqr @ ( bit0 @ N ) )
      = ( bit0 @ ( bit0 @ ( sqr @ N ) ) ) ) ).

% sqr.simps(2)
thf(fact_9442_sqr_Osimps_I1_J,axiom,
    ( ( sqr @ one )
    = one ) ).

% sqr.simps(1)
thf(fact_9443_atLeastSucAtMost__greaterThanAtMost,axiom,
    ! [L2: nat,U: nat] :
      ( ( set_or1269000886237332187st_nat @ ( suc @ L2 ) @ U )
      = ( set_or6659071591806873216st_nat @ L2 @ U ) ) ).

% atLeastSucAtMost_greaterThanAtMost
thf(fact_9444_sqr__conv__mult,axiom,
    ( sqr
    = ( ^ [X: num] : ( times_times_num @ X @ X ) ) ) ).

% sqr_conv_mult
thf(fact_9445_le__prod__encode__1,axiom,
    ! [A: nat,B2: nat] : ( ord_less_eq_nat @ A @ ( nat_prod_encode @ ( product_Pair_nat_nat @ A @ B2 ) ) ) ).

% le_prod_encode_1
thf(fact_9446_le__prod__encode__2,axiom,
    ! [B2: nat,A: nat] : ( ord_less_eq_nat @ B2 @ ( nat_prod_encode @ ( product_Pair_nat_nat @ A @ B2 ) ) ) ).

% le_prod_encode_2
thf(fact_9447_pow_Osimps_I2_J,axiom,
    ! [X4: num,Y3: num] :
      ( ( pow @ X4 @ ( bit0 @ Y3 ) )
      = ( sqr @ ( pow @ X4 @ Y3 ) ) ) ).

% pow.simps(2)
thf(fact_9448_sqr_Osimps_I3_J,axiom,
    ! [N: num] :
      ( ( sqr @ ( bit1 @ N ) )
      = ( bit1 @ ( bit0 @ ( plus_plus_num @ ( sqr @ N ) @ N ) ) ) ) ).

% sqr.simps(3)
thf(fact_9449_prod__encode__prod__decode__aux,axiom,
    ! [K: nat,M: nat] :
      ( ( nat_prod_encode @ ( nat_prod_decode_aux @ K @ M ) )
      = ( plus_plus_nat @ ( nat_triangle @ K ) @ M ) ) ).

% prod_encode_prod_decode_aux
thf(fact_9450_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
    ! [X4: vEBT_VEBT,Xa: nat,Y3: $o] :
      ( ( ( vEBT_VEBT_valid @ X4 @ Xa )
        = Y3 )
     => ( ( ? [Uu2: $o,Uv: $o] :
              ( X4
              = ( vEBT_Leaf @ Uu2 @ Uv ) )
         => ( Y3
            = ( Xa != one_one_nat ) ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X4
                = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
             => ( Y3
                = ( ~ ( ( Deg2 = Xa )
                      & ! [X: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                         => ( vEBT_VEBT_valid @ X @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                      & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                      & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
                        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      & ( case_o184042715313410164at_nat
                        @ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X6 )
                          & ! [X: vEBT_VEBT] :
                              ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                             => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) )
                        @ ( produc6081775807080527818_nat_o
                          @ ^ [Mi3: nat,Ma3: nat] :
                              ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
                              & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                              & ! [I2: nat] :
                                  ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I2 ) @ X6 ) )
                                    = ( vEBT_V8194947554948674370ptions @ Summary2 @ I2 ) ) )
                              & ( ( Mi3 = Ma3 )
                               => ! [X: vEBT_VEBT] :
                                    ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                                   => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) )
                              & ( ( Mi3 != Ma3 )
                               => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
                                  & ! [X: nat] :
                                      ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                     => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X )
                                       => ( ( ord_less_nat @ Mi3 @ X )
                                          & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
                        @ Mima ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(1)
thf(fact_9451_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom,
    ! [X4: vEBT_VEBT,Xa: nat] :
      ( ( vEBT_VEBT_valid @ X4 @ Xa )
     => ( ( ? [Uu2: $o,Uv: $o] :
              ( X4
              = ( vEBT_Leaf @ Uu2 @ Uv ) )
         => ( Xa != one_one_nat ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X4
                = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
             => ~ ( ( Deg2 = Xa )
                  & ! [X5: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                     => ( vEBT_VEBT_valid @ X5 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                  & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                  & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
                    = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  & ( case_o184042715313410164at_nat
                    @ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X6 )
                      & ! [X: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                         => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) )
                    @ ( produc6081775807080527818_nat_o
                      @ ^ [Mi3: nat,Ma3: nat] :
                          ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
                          & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                          & ! [I2: nat] :
                              ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                             => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I2 ) @ X6 ) )
                                = ( vEBT_V8194947554948674370ptions @ Summary2 @ I2 ) ) )
                          & ( ( Mi3 = Ma3 )
                           => ! [X: vEBT_VEBT] :
                                ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                               => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) )
                          & ( ( Mi3 != Ma3 )
                           => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
                              & ! [X: nat] :
                                  ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X )
                                   => ( ( ord_less_nat @ Mi3 @ X )
                                      & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
                    @ Mima ) ) ) ) ) ).

% VEBT_internal.valid'.elims(2)
thf(fact_9452_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom,
    ! [X4: vEBT_VEBT,Xa: nat] :
      ( ~ ( vEBT_VEBT_valid @ X4 @ Xa )
     => ( ( ? [Uu2: $o,Uv: $o] :
              ( X4
              = ( vEBT_Leaf @ Uu2 @ Uv ) )
         => ( Xa = one_one_nat ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X4
                = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
             => ( ( Deg2 = Xa )
                & ! [X3: vEBT_VEBT] :
                    ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                   => ( vEBT_VEBT_valid @ X3 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
                  = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                & ( case_o184042715313410164at_nat
                  @ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X6 )
                    & ! [X: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                       => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) )
                  @ ( produc6081775807080527818_nat_o
                    @ ^ [Mi3: nat,Ma3: nat] :
                        ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
                        & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                        & ! [I2: nat] :
                            ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                           => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I2 ) @ X6 ) )
                              = ( vEBT_V8194947554948674370ptions @ Summary2 @ I2 ) ) )
                        & ( ( Mi3 = Ma3 )
                         => ! [X: vEBT_VEBT] :
                              ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                             => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) )
                        & ( ( Mi3 != Ma3 )
                         => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
                            & ! [X: nat] :
                                ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                               => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X )
                                 => ( ( ord_less_nat @ Mi3 @ X )
                                    & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
                  @ Mima ) ) ) ) ) ).

% VEBT_internal.valid'.elims(3)
thf(fact_9453_finite__greaterThanAtMost__int,axiom,
    ! [L2: int,U: int] : ( finite_finite_int @ ( set_or6656581121297822940st_int @ L2 @ U ) ) ).

% finite_greaterThanAtMost_int
thf(fact_9454_card__greaterThanAtMost__int,axiom,
    ! [L2: int,U: int] :
      ( ( finite_card_int @ ( set_or6656581121297822940st_int @ L2 @ U ) )
      = ( nat2 @ ( minus_minus_int @ U @ L2 ) ) ) ).

% card_greaterThanAtMost_int
thf(fact_9455_atLeastPlusOneAtMost__greaterThanAtMost__int,axiom,
    ! [L2: int,U: int] :
      ( ( set_or1266510415728281911st_int @ ( plus_plus_int @ L2 @ one_one_int ) @ U )
      = ( set_or6656581121297822940st_int @ L2 @ U ) ) ).

% atLeastPlusOneAtMost_greaterThanAtMost_int
thf(fact_9456_VEBT__internal_Ovalid_H_Osimps_I2_J,axiom,
    ! [Mima2: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,Deg3: nat] :
      ( ( vEBT_VEBT_valid @ ( vEBT_Node @ Mima2 @ Deg @ TreeList2 @ Summary ) @ Deg3 )
      = ( ( Deg = Deg3 )
        & ! [X: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
           => ( vEBT_VEBT_valid @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
        & ( vEBT_VEBT_valid @ Summary @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
        & ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
        & ( case_o184042715313410164at_nat
          @ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X6 )
            & ! [X: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
               => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) )
          @ ( produc6081775807080527818_nat_o
            @ ^ [Mi3: nat,Ma3: nat] :
                ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
                & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
                & ! [I2: nat] :
                    ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                   => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I2 ) @ X6 ) )
                      = ( vEBT_V8194947554948674370ptions @ Summary @ I2 ) ) )
                & ( ( Mi3 = Ma3 )
                 => ! [X: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                     => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) )
                & ( ( Mi3 != Ma3 )
                 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ Ma3 )
                    & ! [X: nat] :
                        ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
                       => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ X )
                         => ( ( ord_less_nat @ Mi3 @ X )
                            & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
          @ Mima2 ) ) ) ).

% VEBT_internal.valid'.simps(2)
thf(fact_9457_VEBT__internal_Ovalid_H_Opelims_I1_J,axiom,
    ! [X4: vEBT_VEBT,Xa: nat,Y3: $o] :
      ( ( ( vEBT_VEBT_valid @ X4 @ Xa )
        = Y3 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X4 @ Xa ) )
       => ( ! [Uu2: $o,Uv: $o] :
              ( ( X4
                = ( vEBT_Leaf @ Uu2 @ Uv ) )
             => ( ( Y3
                  = ( Xa = one_one_nat ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv ) @ Xa ) ) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X4
                  = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
               => ( ( Y3
                    = ( ( Deg2 = Xa )
                      & ! [X: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                         => ( vEBT_VEBT_valid @ X @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                      & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                      & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
                        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      & ( case_o184042715313410164at_nat
                        @ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X6 )
                          & ! [X: vEBT_VEBT] :
                              ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                             => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) )
                        @ ( produc6081775807080527818_nat_o
                          @ ^ [Mi3: nat,Ma3: nat] :
                              ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
                              & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                              & ! [I2: nat] :
                                  ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I2 ) @ X6 ) )
                                    = ( vEBT_V8194947554948674370ptions @ Summary2 @ I2 ) ) )
                              & ( ( Mi3 = Ma3 )
                               => ! [X: vEBT_VEBT] :
                                    ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                                   => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) )
                              & ( ( Mi3 != Ma3 )
                               => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
                                  & ! [X: nat] :
                                      ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                     => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X )
                                       => ( ( ord_less_nat @ Mi3 @ X )
                                          & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
                        @ Mima ) ) )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(1)
thf(fact_9458_VEBT__internal_Ovalid_H_Opelims_I2_J,axiom,
    ! [X4: vEBT_VEBT,Xa: nat] :
      ( ( vEBT_VEBT_valid @ X4 @ Xa )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X4 @ Xa ) )
       => ( ! [Uu2: $o,Uv: $o] :
              ( ( X4
                = ( vEBT_Leaf @ Uu2 @ Uv ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv ) @ Xa ) )
               => ( Xa != one_one_nat ) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X4
                  = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
               => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) @ Xa ) )
                 => ~ ( ( Deg2 = Xa )
                      & ! [X5: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                         => ( vEBT_VEBT_valid @ X5 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                      & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                      & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
                        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      & ( case_o184042715313410164at_nat
                        @ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X6 )
                          & ! [X: vEBT_VEBT] :
                              ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                             => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) )
                        @ ( produc6081775807080527818_nat_o
                          @ ^ [Mi3: nat,Ma3: nat] :
                              ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
                              & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                              & ! [I2: nat] :
                                  ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I2 ) @ X6 ) )
                                    = ( vEBT_V8194947554948674370ptions @ Summary2 @ I2 ) ) )
                              & ( ( Mi3 = Ma3 )
                               => ! [X: vEBT_VEBT] :
                                    ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                                   => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) )
                              & ( ( Mi3 != Ma3 )
                               => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
                                  & ! [X: nat] :
                                      ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                     => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X )
                                       => ( ( ord_less_nat @ Mi3 @ X )
                                          & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
                        @ Mima ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(2)
thf(fact_9459_VEBT__internal_Ovalid_H_Opelims_I3_J,axiom,
    ! [X4: vEBT_VEBT,Xa: nat] :
      ( ~ ( vEBT_VEBT_valid @ X4 @ Xa )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X4 @ Xa ) )
       => ( ! [Uu2: $o,Uv: $o] :
              ( ( X4
                = ( vEBT_Leaf @ Uu2 @ Uv ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv ) @ Xa ) )
               => ( Xa = one_one_nat ) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X4
                  = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
               => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) @ Xa ) )
                 => ( ( Deg2 = Xa )
                    & ! [X3: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                       => ( vEBT_VEBT_valid @ X3 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                    & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                    & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
                      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                    & ( case_o184042715313410164at_nat
                      @ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X6 )
                        & ! [X: vEBT_VEBT] :
                            ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                           => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) )
                      @ ( produc6081775807080527818_nat_o
                        @ ^ [Mi3: nat,Ma3: nat] :
                            ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
                            & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                            & ! [I2: nat] :
                                ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                               => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I2 ) @ X6 ) )
                                  = ( vEBT_V8194947554948674370ptions @ Summary2 @ I2 ) ) )
                            & ( ( Mi3 = Ma3 )
                             => ! [X: vEBT_VEBT] :
                                  ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                                 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) )
                            & ( ( Mi3 != Ma3 )
                             => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
                                & ! [X: nat] :
                                    ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                   => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X )
                                     => ( ( ord_less_nat @ Mi3 @ X )
                                        & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
                      @ Mima ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(3)
thf(fact_9460_GreatestI__nat,axiom,
    ! [P: nat > $o,K: nat,B2: nat] :
      ( ( P @ K )
     => ( ! [Y4: nat] :
            ( ( P @ Y4 )
           => ( ord_less_eq_nat @ Y4 @ B2 ) )
       => ( P @ ( order_Greatest_nat @ P ) ) ) ) ).

% GreatestI_nat
thf(fact_9461_Greatest__le__nat,axiom,
    ! [P: nat > $o,K: nat,B2: nat] :
      ( ( P @ K )
     => ( ! [Y4: nat] :
            ( ( P @ Y4 )
           => ( ord_less_eq_nat @ Y4 @ B2 ) )
       => ( ord_less_eq_nat @ K @ ( order_Greatest_nat @ P ) ) ) ) ).

% Greatest_le_nat
thf(fact_9462_GreatestI__ex__nat,axiom,
    ! [P: nat > $o,B2: nat] :
      ( ? [X_12: nat] : ( P @ X_12 )
     => ( ! [Y4: nat] :
            ( ( P @ Y4 )
           => ( ord_less_eq_nat @ Y4 @ B2 ) )
       => ( P @ ( order_Greatest_nat @ P ) ) ) ) ).

% GreatestI_ex_nat
thf(fact_9463_rat__floor__lemma,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( divide_divide_int @ A @ B2 ) ) @ ( fract @ A @ B2 ) )
      & ( ord_less_rat @ ( fract @ A @ B2 ) @ ( ring_1_of_int_rat @ ( plus_plus_int @ ( divide_divide_int @ A @ B2 ) @ one_one_int ) ) ) ) ).

% rat_floor_lemma
thf(fact_9464_image__minus__const__atLeastLessThan__nat,axiom,
    ! [C: nat,Y3: nat,X4: nat] :
      ( ( ( ord_less_nat @ C @ Y3 )
       => ( ( image_nat_nat
            @ ^ [I2: nat] : ( minus_minus_nat @ I2 @ C )
            @ ( set_or4665077453230672383an_nat @ X4 @ Y3 ) )
          = ( set_or4665077453230672383an_nat @ ( minus_minus_nat @ X4 @ C ) @ ( minus_minus_nat @ Y3 @ C ) ) ) )
      & ( ~ ( ord_less_nat @ C @ Y3 )
       => ( ( ( ord_less_nat @ X4 @ Y3 )
           => ( ( image_nat_nat
                @ ^ [I2: nat] : ( minus_minus_nat @ I2 @ C )
                @ ( set_or4665077453230672383an_nat @ X4 @ Y3 ) )
              = ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) )
          & ( ~ ( ord_less_nat @ X4 @ Y3 )
           => ( ( image_nat_nat
                @ ^ [I2: nat] : ( minus_minus_nat @ I2 @ C )
                @ ( set_or4665077453230672383an_nat @ X4 @ Y3 ) )
              = bot_bot_set_nat ) ) ) ) ) ).

% image_minus_const_atLeastLessThan_nat
thf(fact_9465_take__bit__numeral__minus__numeral__int,axiom,
    ! [M: num,N: num] :
      ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( case_option_int_num @ zero_zero_int
        @ ^ [Q4: num] : ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ M ) @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_int @ Q4 ) ) )
        @ ( bit_take_bit_num @ ( numeral_numeral_nat @ M ) @ N ) ) ) ).

% take_bit_numeral_minus_numeral_int
thf(fact_9466_take__bit__num__simps_I1_J,axiom,
    ! [M: num] :
      ( ( bit_take_bit_num @ zero_zero_nat @ M )
      = none_num ) ).

% take_bit_num_simps(1)
thf(fact_9467_bij__betw__Suc,axiom,
    ! [M7: set_nat,N3: set_nat] :
      ( ( bij_betw_nat_nat @ suc @ M7 @ N3 )
      = ( ( image_nat_nat @ suc @ M7 )
        = N3 ) ) ).

% bij_betw_Suc
thf(fact_9468_take__bit__num__simps_I2_J,axiom,
    ! [N: nat] :
      ( ( bit_take_bit_num @ ( suc @ N ) @ one )
      = ( some_num @ one ) ) ).

% take_bit_num_simps(2)
thf(fact_9469_take__bit__num__simps_I5_J,axiom,
    ! [R2: num] :
      ( ( bit_take_bit_num @ ( numeral_numeral_nat @ R2 ) @ one )
      = ( some_num @ one ) ) ).

% take_bit_num_simps(5)
thf(fact_9470_image__Suc__atLeastAtMost,axiom,
    ! [I: nat,J: nat] :
      ( ( image_nat_nat @ suc @ ( set_or1269000886237332187st_nat @ I @ J ) )
      = ( set_or1269000886237332187st_nat @ ( suc @ I ) @ ( suc @ J ) ) ) ).

% image_Suc_atLeastAtMost
thf(fact_9471_image__Suc__atLeastLessThan,axiom,
    ! [I: nat,J: nat] :
      ( ( image_nat_nat @ suc @ ( set_or4665077453230672383an_nat @ I @ J ) )
      = ( set_or4665077453230672383an_nat @ ( suc @ I ) @ ( suc @ J ) ) ) ).

% image_Suc_atLeastLessThan
thf(fact_9472_mult__rat,axiom,
    ! [A: int,B2: int,C: int,D: int] :
      ( ( times_times_rat @ ( fract @ A @ B2 ) @ ( fract @ C @ D ) )
      = ( fract @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ D ) ) ) ).

% mult_rat
thf(fact_9473_divide__rat,axiom,
    ! [A: int,B2: int,C: int,D: int] :
      ( ( divide_divide_rat @ ( fract @ A @ B2 ) @ ( fract @ C @ D ) )
      = ( fract @ ( times_times_int @ A @ D ) @ ( times_times_int @ B2 @ C ) ) ) ).

% divide_rat
thf(fact_9474_floor__Fract,axiom,
    ! [A: int,B2: int] :
      ( ( archim3151403230148437115or_rat @ ( fract @ A @ B2 ) )
      = ( divide_divide_int @ A @ B2 ) ) ).

% floor_Fract
thf(fact_9475_less__rat,axiom,
    ! [B2: int,D: int,A: int,C: int] :
      ( ( B2 != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( ( ord_less_rat @ ( fract @ A @ B2 ) @ ( fract @ C @ D ) )
          = ( ord_less_int @ ( times_times_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ B2 @ D ) ) @ ( times_times_int @ ( times_times_int @ C @ B2 ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ) ).

% less_rat
thf(fact_9476_add__rat,axiom,
    ! [B2: int,D: int,A: int,C: int] :
      ( ( B2 != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( ( plus_plus_rat @ ( fract @ A @ B2 ) @ ( fract @ C @ D ) )
          = ( fract @ ( plus_plus_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ C @ B2 ) ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ).

% add_rat
thf(fact_9477_le__rat,axiom,
    ! [B2: int,D: int,A: int,C: int] :
      ( ( B2 != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( ( ord_less_eq_rat @ ( fract @ A @ B2 ) @ ( fract @ C @ D ) )
          = ( ord_less_eq_int @ ( times_times_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ B2 @ D ) ) @ ( times_times_int @ ( times_times_int @ C @ B2 ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ) ).

% le_rat
thf(fact_9478_diff__rat,axiom,
    ! [B2: int,D: int,A: int,C: int] :
      ( ( B2 != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( ( minus_minus_rat @ ( fract @ A @ B2 ) @ ( fract @ C @ D ) )
          = ( fract @ ( minus_minus_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ C @ B2 ) ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ).

% diff_rat
thf(fact_9479_sgn__rat,axiom,
    ! [A: int,B2: int] :
      ( ( sgn_sgn_rat @ ( fract @ A @ B2 ) )
      = ( ring_1_of_int_rat @ ( times_times_int @ ( sgn_sgn_int @ A ) @ ( sgn_sgn_int @ B2 ) ) ) ) ).

% sgn_rat
thf(fact_9480_Code__Abstract__Nat_Otake__bit__num__code_I1_J,axiom,
    ! [N: nat] :
      ( ( bit_take_bit_num @ N @ one )
      = ( case_nat_option_num @ none_num
        @ ^ [N2: nat] : ( some_num @ one )
        @ N ) ) ).

% Code_Abstract_Nat.take_bit_num_code(1)
thf(fact_9481_zero__notin__Suc__image,axiom,
    ! [A3: set_nat] :
      ~ ( member_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ A3 ) ) ).

% zero_notin_Suc_image
thf(fact_9482_Rat__induct__pos,axiom,
    ! [P: rat > $o,Q2: rat] :
      ( ! [A5: int,B5: int] :
          ( ( ord_less_int @ zero_zero_int @ B5 )
         => ( P @ ( fract @ A5 @ B5 ) ) )
     => ( P @ Q2 ) ) ).

% Rat_induct_pos
thf(fact_9483_eq__rat_I1_J,axiom,
    ! [B2: int,D: int,A: int,C: int] :
      ( ( B2 != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( ( ( fract @ A @ B2 )
            = ( fract @ C @ D ) )
          = ( ( times_times_int @ A @ D )
            = ( times_times_int @ C @ B2 ) ) ) ) ) ).

% eq_rat(1)
thf(fact_9484_mult__rat__cancel,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( C != zero_zero_int )
     => ( ( fract @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
        = ( fract @ A @ B2 ) ) ) ).

% mult_rat_cancel
thf(fact_9485_Fract__coprime,axiom,
    ! [A: int,B2: int] :
      ( ( fract @ ( divide_divide_int @ A @ ( gcd_gcd_int @ A @ B2 ) ) @ ( divide_divide_int @ B2 @ ( gcd_gcd_int @ A @ B2 ) ) )
      = ( fract @ A @ B2 ) ) ).

% Fract_coprime
thf(fact_9486_Fract__of__int__quotient,axiom,
    ( fract
    = ( ^ [K3: int,L: int] : ( divide_divide_rat @ ( ring_1_of_int_rat @ K3 ) @ ( ring_1_of_int_rat @ L ) ) ) ) ).

% Fract_of_int_quotient
thf(fact_9487_image__Suc__lessThan,axiom,
    ! [N: nat] :
      ( ( image_nat_nat @ suc @ ( set_ord_lessThan_nat @ N ) )
      = ( set_or1269000886237332187st_nat @ one_one_nat @ N ) ) ).

% image_Suc_lessThan
thf(fact_9488_image__Suc__atMost,axiom,
    ! [N: nat] :
      ( ( image_nat_nat @ suc @ ( set_ord_atMost_nat @ N ) )
      = ( set_or1269000886237332187st_nat @ one_one_nat @ ( suc @ N ) ) ) ).

% image_Suc_atMost
thf(fact_9489_atLeast0__atMost__Suc__eq__insert__0,axiom,
    ! [N: nat] :
      ( ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) )
      = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ) ).

% atLeast0_atMost_Suc_eq_insert_0
thf(fact_9490_atLeast0__lessThan__Suc__eq__insert__0,axiom,
    ! [N: nat] :
      ( ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) )
      = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ).

% atLeast0_lessThan_Suc_eq_insert_0
thf(fact_9491_lessThan__Suc__eq__insert__0,axiom,
    ! [N: nat] :
      ( ( set_ord_lessThan_nat @ ( suc @ N ) )
      = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% lessThan_Suc_eq_insert_0
thf(fact_9492_atMost__Suc__eq__insert__0,axiom,
    ! [N: nat] :
      ( ( set_ord_atMost_nat @ ( suc @ N ) )
      = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_ord_atMost_nat @ N ) ) ) ) ).

% atMost_Suc_eq_insert_0
thf(fact_9493_Fract__less__zero__iff,axiom,
    ! [B2: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B2 )
     => ( ( ord_less_rat @ ( fract @ A @ B2 ) @ zero_zero_rat )
        = ( ord_less_int @ A @ zero_zero_int ) ) ) ).

% Fract_less_zero_iff
thf(fact_9494_zero__less__Fract__iff,axiom,
    ! [B2: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B2 )
     => ( ( ord_less_rat @ zero_zero_rat @ ( fract @ A @ B2 ) )
        = ( ord_less_int @ zero_zero_int @ A ) ) ) ).

% zero_less_Fract_iff
thf(fact_9495_Fract__less__one__iff,axiom,
    ! [B2: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B2 )
     => ( ( ord_less_rat @ ( fract @ A @ B2 ) @ one_one_rat )
        = ( ord_less_int @ A @ B2 ) ) ) ).

% Fract_less_one_iff
thf(fact_9496_one__less__Fract__iff,axiom,
    ! [B2: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B2 )
     => ( ( ord_less_rat @ one_one_rat @ ( fract @ A @ B2 ) )
        = ( ord_less_int @ B2 @ A ) ) ) ).

% one_less_Fract_iff
thf(fact_9497_Fract__add__one,axiom,
    ! [N: int,M: int] :
      ( ( N != zero_zero_int )
     => ( ( fract @ ( plus_plus_int @ M @ N ) @ N )
        = ( plus_plus_rat @ ( fract @ M @ N ) @ one_one_rat ) ) ) ).

% Fract_add_one
thf(fact_9498_Fract__le__zero__iff,axiom,
    ! [B2: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B2 )
     => ( ( ord_less_eq_rat @ ( fract @ A @ B2 ) @ zero_zero_rat )
        = ( ord_less_eq_int @ A @ zero_zero_int ) ) ) ).

% Fract_le_zero_iff
thf(fact_9499_zero__le__Fract__iff,axiom,
    ! [B2: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B2 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ ( fract @ A @ B2 ) )
        = ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).

% zero_le_Fract_iff
thf(fact_9500_one__le__Fract__iff,axiom,
    ! [B2: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B2 )
     => ( ( ord_less_eq_rat @ one_one_rat @ ( fract @ A @ B2 ) )
        = ( ord_less_eq_int @ B2 @ A ) ) ) ).

% one_le_Fract_iff
thf(fact_9501_Fract__le__one__iff,axiom,
    ! [B2: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B2 )
     => ( ( ord_less_eq_rat @ ( fract @ A @ B2 ) @ one_one_rat )
        = ( ord_less_eq_int @ A @ B2 ) ) ) ).

% Fract_le_one_iff
thf(fact_9502_take__bit__num__def,axiom,
    ( bit_take_bit_num
    = ( ^ [N2: nat,M6: num] :
          ( if_option_num
          @ ( ( bit_se2925701944663578781it_nat @ N2 @ ( numeral_numeral_nat @ M6 ) )
            = zero_zero_nat )
          @ none_num
          @ ( some_num @ ( num_of_nat @ ( bit_se2925701944663578781it_nat @ N2 @ ( numeral_numeral_nat @ M6 ) ) ) ) ) ) ) ).

% take_bit_num_def
thf(fact_9503_and__minus__numerals_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
      = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bitM @ N ) ) ) ) ).

% and_minus_numerals(3)
thf(fact_9504_and__minus__numerals_I7_J,axiom,
    ! [N: num,M: num] :
      ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ ( 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)
thf(fact_9505_and__minus__numerals_I4_J,axiom,
    ! [M: num,N: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
      = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bit0 @ N ) ) ) ) ).

% and_minus_numerals(4)
thf(fact_9506_take__bit__num__simps_I4_J,axiom,
    ! [N: nat,M: num] :
      ( ( bit_take_bit_num @ ( suc @ N ) @ ( bit1 @ M ) )
      = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ N @ M ) ) ) ) ).

% take_bit_num_simps(4)
thf(fact_9507_take__bit__num__simps_I3_J,axiom,
    ! [N: nat,M: num] :
      ( ( bit_take_bit_num @ ( suc @ N ) @ ( bit0 @ M ) )
      = ( case_o6005452278849405969um_num @ none_num
        @ ^ [Q4: num] : ( some_num @ ( bit0 @ Q4 ) )
        @ ( bit_take_bit_num @ N @ M ) ) ) ).

% take_bit_num_simps(3)
thf(fact_9508_take__bit__num__simps_I7_J,axiom,
    ! [R2: num,M: num] :
      ( ( bit_take_bit_num @ ( numeral_numeral_nat @ R2 ) @ ( bit1 @ M ) )
      = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ ( pred_numeral @ R2 ) @ M ) ) ) ) ).

% take_bit_num_simps(7)
thf(fact_9509_take__bit__num__simps_I6_J,axiom,
    ! [R2: num,M: num] :
      ( ( bit_take_bit_num @ ( numeral_numeral_nat @ R2 ) @ ( bit0 @ M ) )
      = ( case_o6005452278849405969um_num @ none_num
        @ ^ [Q4: num] : ( some_num @ ( bit0 @ Q4 ) )
        @ ( bit_take_bit_num @ ( pred_numeral @ R2 ) @ M ) ) ) ).

% take_bit_num_simps(6)
thf(fact_9510_and__minus__numerals_I8_J,axiom,
    ! [N: num,M: num] :
      ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ ( numeral_numeral_int @ M ) )
      = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bit0 @ N ) ) ) ) ).

% and_minus_numerals(8)
thf(fact_9511_and__not__num_Osimps_I8_J,axiom,
    ! [M: num,N: num] :
      ( ( bit_and_not_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
      = ( case_o6005452278849405969um_num @ ( some_num @ one )
        @ ^ [N9: num] : ( some_num @ ( bit1 @ N9 ) )
        @ ( bit_and_not_num @ M @ N ) ) ) ).

% and_not_num.simps(8)
thf(fact_9512_and__not__num_Osimps_I1_J,axiom,
    ( ( bit_and_not_num @ one @ one )
    = none_num ) ).

% and_not_num.simps(1)
thf(fact_9513_and__not__num_Osimps_I2_J,axiom,
    ! [N: num] :
      ( ( bit_and_not_num @ one @ ( bit0 @ N ) )
      = ( some_num @ one ) ) ).

% and_not_num.simps(2)
thf(fact_9514_and__not__num_Osimps_I4_J,axiom,
    ! [M: num] :
      ( ( bit_and_not_num @ ( bit0 @ M ) @ one )
      = ( some_num @ ( bit0 @ M ) ) ) ).

% and_not_num.simps(4)
thf(fact_9515_and__not__num_Osimps_I3_J,axiom,
    ! [N: num] :
      ( ( bit_and_not_num @ one @ ( bit1 @ N ) )
      = none_num ) ).

% and_not_num.simps(3)
thf(fact_9516_Code__Abstract__Nat_Otake__bit__num__code_I2_J,axiom,
    ! [N: nat,M: num] :
      ( ( bit_take_bit_num @ N @ ( bit0 @ M ) )
      = ( case_nat_option_num @ none_num
        @ ^ [N2: nat] :
            ( case_o6005452278849405969um_num @ none_num
            @ ^ [Q4: num] : ( some_num @ ( bit0 @ Q4 ) )
            @ ( bit_take_bit_num @ N2 @ M ) )
        @ N ) ) ).

% Code_Abstract_Nat.take_bit_num_code(2)
thf(fact_9517_image__int__atLeastAtMost,axiom,
    ! [A: nat,B2: nat] :
      ( ( image_nat_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ A @ B2 ) )
      = ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ).

% image_int_atLeastAtMost
thf(fact_9518_image__int__atLeastLessThan,axiom,
    ! [A: nat,B2: nat] :
      ( ( image_nat_int @ semiri1314217659103216013at_int @ ( set_or4665077453230672383an_nat @ A @ B2 ) )
      = ( set_or4662586982721622107an_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ).

% image_int_atLeastLessThan
thf(fact_9519_and__not__num_Osimps_I7_J,axiom,
    ! [M: num] :
      ( ( bit_and_not_num @ ( bit1 @ M ) @ one )
      = ( some_num @ ( bit0 @ M ) ) ) ).

% and_not_num.simps(7)
thf(fact_9520_and__not__num__eq__Some__iff,axiom,
    ! [M: num,N: num,Q2: num] :
      ( ( ( bit_and_not_num @ M @ N )
        = ( some_num @ Q2 ) )
      = ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
        = ( numeral_numeral_int @ Q2 ) ) ) ).

% and_not_num_eq_Some_iff
thf(fact_9521_Code__Abstract__Nat_Otake__bit__num__code_I3_J,axiom,
    ! [N: nat,M: num] :
      ( ( bit_take_bit_num @ N @ ( bit1 @ M ) )
      = ( case_nat_option_num @ none_num
        @ ^ [N2: nat] : ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ N2 @ M ) ) )
        @ N ) ) ).

% Code_Abstract_Nat.take_bit_num_code(3)
thf(fact_9522_image__add__int__atLeastLessThan,axiom,
    ! [L2: int,U: int] :
      ( ( image_int_int
        @ ^ [X: int] : ( plus_plus_int @ X @ L2 )
        @ ( set_or4662586982721622107an_int @ zero_zero_int @ ( minus_minus_int @ U @ L2 ) ) )
      = ( set_or4662586982721622107an_int @ L2 @ U ) ) ).

% image_add_int_atLeastLessThan
thf(fact_9523_and__not__num__eq__None__iff,axiom,
    ! [M: num,N: num] :
      ( ( ( bit_and_not_num @ M @ N )
        = none_num )
      = ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
        = zero_zero_int ) ) ).

% and_not_num_eq_None_iff
thf(fact_9524_image__atLeastZeroLessThan__int,axiom,
    ! [U: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ U )
     => ( ( set_or4662586982721622107an_int @ zero_zero_int @ U )
        = ( image_nat_int @ semiri1314217659103216013at_int @ ( set_ord_lessThan_nat @ ( nat2 @ U ) ) ) ) ) ).

% image_atLeastZeroLessThan_int
thf(fact_9525_int__numeral__and__not__num,axiom,
    ! [M: num,N: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_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
thf(fact_9526_int__numeral__not__and__num,axiom,
    ! [M: num,N: num] :
      ( ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ M ) ) @ ( 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
thf(fact_9527_positive__rat,axiom,
    ! [A: int,B2: int] :
      ( ( positive @ ( fract @ A @ B2 ) )
      = ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B2 ) ) ) ).

% positive_rat
thf(fact_9528_Rat_Opositive__add,axiom,
    ! [X4: rat,Y3: rat] :
      ( ( positive @ X4 )
     => ( ( positive @ Y3 )
       => ( positive @ ( plus_plus_rat @ X4 @ Y3 ) ) ) ) ).

% Rat.positive_add
thf(fact_9529_Rat_Opositive__mult,axiom,
    ! [X4: rat,Y3: rat] :
      ( ( positive @ X4 )
     => ( ( positive @ Y3 )
       => ( positive @ ( times_times_rat @ X4 @ Y3 ) ) ) ) ).

% Rat.positive_mult
thf(fact_9530_less__rat__def,axiom,
    ( ord_less_rat
    = ( ^ [X: rat,Y: rat] : ( positive @ ( minus_minus_rat @ Y @ X ) ) ) ) ).

% less_rat_def
thf(fact_9531_Rat_Opositive_Orep__eq,axiom,
    ( positive
    = ( ^ [X: rat] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ ( rep_Rat @ X ) ) @ ( product_snd_int_int @ ( rep_Rat @ X ) ) ) ) ) ) ).

% Rat.positive.rep_eq
thf(fact_9532_and__not__num_Oelims,axiom,
    ! [X4: num,Xa: num,Y3: option_num] :
      ( ( ( bit_and_not_num @ X4 @ Xa )
        = Y3 )
     => ( ( ( X4 = one )
         => ( ( Xa = one )
           => ( Y3 != none_num ) ) )
       => ( ( ( X4 = one )
           => ( ? [N4: num] :
                  ( Xa
                  = ( bit0 @ N4 ) )
             => ( Y3
               != ( some_num @ one ) ) ) )
         => ( ( ( X4 = one )
             => ( ? [N4: num] :
                    ( Xa
                    = ( bit1 @ N4 ) )
               => ( Y3 != none_num ) ) )
           => ( ! [M4: num] :
                  ( ( X4
                    = ( bit0 @ M4 ) )
                 => ( ( Xa = one )
                   => ( Y3
                     != ( some_num @ ( bit0 @ M4 ) ) ) ) )
             => ( ! [M4: num] :
                    ( ( X4
                      = ( bit0 @ M4 ) )
                   => ! [N4: num] :
                        ( ( Xa
                          = ( bit0 @ N4 ) )
                       => ( Y3
                         != ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M4 @ N4 ) ) ) ) )
               => ( ! [M4: num] :
                      ( ( X4
                        = ( bit0 @ M4 ) )
                     => ! [N4: num] :
                          ( ( Xa
                            = ( bit1 @ N4 ) )
                         => ( Y3
                           != ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M4 @ N4 ) ) ) ) )
                 => ( ! [M4: num] :
                        ( ( X4
                          = ( bit1 @ M4 ) )
                       => ( ( Xa = one )
                         => ( Y3
                           != ( some_num @ ( bit0 @ M4 ) ) ) ) )
                   => ( ! [M4: num] :
                          ( ( X4
                            = ( bit1 @ M4 ) )
                         => ! [N4: num] :
                              ( ( Xa
                                = ( bit0 @ N4 ) )
                             => ( Y3
                               != ( case_o6005452278849405969um_num @ ( some_num @ one )
                                  @ ^ [N9: num] : ( some_num @ ( bit1 @ N9 ) )
                                  @ ( bit_and_not_num @ M4 @ N4 ) ) ) ) )
                     => ~ ! [M4: num] :
                            ( ( X4
                              = ( bit1 @ M4 ) )
                           => ! [N4: num] :
                                ( ( Xa
                                  = ( bit1 @ N4 ) )
                               => ( Y3
                                 != ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M4 @ N4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_not_num.elims
thf(fact_9533_Bit__Operations_Otake__bit__num__code,axiom,
    ( bit_take_bit_num
    = ( ^ [N2: nat,M6: num] :
          ( produc478579273971653890on_num
          @ ^ [A4: nat,X: num] :
              ( case_nat_option_num @ none_num
              @ ^ [O: nat] :
                  ( case_num_option_num @ ( some_num @ one )
                  @ ^ [P3: num] :
                      ( case_o6005452278849405969um_num @ none_num
                      @ ^ [Q4: num] : ( some_num @ ( bit0 @ Q4 ) )
                      @ ( bit_take_bit_num @ O @ P3 ) )
                  @ ^ [P3: num] : ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ O @ P3 ) ) )
                  @ X )
              @ A4 )
          @ ( product_Pair_nat_num @ N2 @ M6 ) ) ) ) ).

% Bit_Operations.take_bit_num_code
thf(fact_9534_and__not__num_Osimps_I5_J,axiom,
    ! [M: num,N: num] :
      ( ( bit_and_not_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
      = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M @ N ) ) ) ).

% and_not_num.simps(5)
thf(fact_9535_and__not__num_Osimps_I9_J,axiom,
    ! [M: num,N: num] :
      ( ( bit_and_not_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
      = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M @ N ) ) ) ).

% and_not_num.simps(9)
thf(fact_9536_and__not__num_Osimps_I6_J,axiom,
    ! [M: num,N: num] :
      ( ( bit_and_not_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
      = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M @ N ) ) ) ).

% and_not_num.simps(6)
thf(fact_9537_and__num_Oelims,axiom,
    ! [X4: num,Xa: num,Y3: option_num] :
      ( ( ( bit_un7362597486090784418nd_num @ X4 @ Xa )
        = Y3 )
     => ( ( ( X4 = one )
         => ( ( Xa = one )
           => ( Y3
             != ( some_num @ one ) ) ) )
       => ( ( ( X4 = one )
           => ( ? [N4: num] :
                  ( Xa
                  = ( bit0 @ N4 ) )
             => ( Y3 != none_num ) ) )
         => ( ( ( X4 = one )
             => ( ? [N4: num] :
                    ( Xa
                    = ( bit1 @ N4 ) )
               => ( Y3
                 != ( some_num @ one ) ) ) )
           => ( ( ? [M4: num] :
                    ( X4
                    = ( bit0 @ M4 ) )
               => ( ( Xa = one )
                 => ( Y3 != none_num ) ) )
             => ( ! [M4: num] :
                    ( ( X4
                      = ( bit0 @ M4 ) )
                   => ! [N4: num] :
                        ( ( Xa
                          = ( bit0 @ N4 ) )
                       => ( Y3
                         != ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M4 @ N4 ) ) ) ) )
               => ( ! [M4: num] :
                      ( ( X4
                        = ( bit0 @ M4 ) )
                     => ! [N4: num] :
                          ( ( Xa
                            = ( bit1 @ N4 ) )
                         => ( Y3
                           != ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M4 @ N4 ) ) ) ) )
                 => ( ( ? [M4: num] :
                          ( X4
                          = ( bit1 @ M4 ) )
                     => ( ( Xa = one )
                       => ( Y3
                         != ( some_num @ one ) ) ) )
                   => ( ! [M4: num] :
                          ( ( X4
                            = ( bit1 @ M4 ) )
                         => ! [N4: num] :
                              ( ( Xa
                                = ( bit0 @ N4 ) )
                             => ( Y3
                               != ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M4 @ N4 ) ) ) ) )
                     => ~ ! [M4: num] :
                            ( ( X4
                              = ( bit1 @ M4 ) )
                           => ! [N4: num] :
                                ( ( Xa
                                  = ( bit1 @ N4 ) )
                               => ( Y3
                                 != ( case_o6005452278849405969um_num @ ( some_num @ one )
                                    @ ^ [N9: num] : ( some_num @ ( bit1 @ N9 ) )
                                    @ ( bit_un7362597486090784418nd_num @ M4 @ N4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_num.elims
thf(fact_9538_xor__num_Oelims,axiom,
    ! [X4: num,Xa: num,Y3: option_num] :
      ( ( ( bit_un2480387367778600638or_num @ X4 @ Xa )
        = Y3 )
     => ( ( ( X4 = one )
         => ( ( Xa = one )
           => ( Y3 != none_num ) ) )
       => ( ( ( X4 = one )
           => ! [N4: num] :
                ( ( Xa
                  = ( bit0 @ N4 ) )
               => ( Y3
                 != ( some_num @ ( bit1 @ N4 ) ) ) ) )
         => ( ( ( X4 = one )
             => ! [N4: num] :
                  ( ( Xa
                    = ( bit1 @ N4 ) )
                 => ( Y3
                   != ( some_num @ ( bit0 @ N4 ) ) ) ) )
           => ( ! [M4: num] :
                  ( ( X4
                    = ( bit0 @ M4 ) )
                 => ( ( Xa = one )
                   => ( Y3
                     != ( some_num @ ( bit1 @ M4 ) ) ) ) )
             => ( ! [M4: num] :
                    ( ( X4
                      = ( bit0 @ M4 ) )
                   => ! [N4: num] :
                        ( ( Xa
                          = ( bit0 @ N4 ) )
                       => ( Y3
                         != ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M4 @ N4 ) ) ) ) )
               => ( ! [M4: num] :
                      ( ( X4
                        = ( bit0 @ M4 ) )
                     => ! [N4: num] :
                          ( ( Xa
                            = ( bit1 @ N4 ) )
                         => ( Y3
                           != ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M4 @ N4 ) ) ) ) ) )
                 => ( ! [M4: num] :
                        ( ( X4
                          = ( bit1 @ M4 ) )
                       => ( ( Xa = one )
                         => ( Y3
                           != ( some_num @ ( bit0 @ M4 ) ) ) ) )
                   => ( ! [M4: num] :
                          ( ( X4
                            = ( bit1 @ M4 ) )
                         => ! [N4: num] :
                              ( ( Xa
                                = ( bit0 @ N4 ) )
                             => ( Y3
                               != ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M4 @ N4 ) ) ) ) ) )
                     => ~ ! [M4: num] :
                            ( ( X4
                              = ( bit1 @ M4 ) )
                           => ! [N4: num] :
                                ( ( Xa
                                  = ( bit1 @ N4 ) )
                               => ( Y3
                                 != ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M4 @ N4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% xor_num.elims
thf(fact_9539_xor__num_Osimps_I8_J,axiom,
    ! [M: num,N: num] :
      ( ( bit_un2480387367778600638or_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
      = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M @ N ) ) ) ) ).

% xor_num.simps(8)
thf(fact_9540_and__num_Osimps_I1_J,axiom,
    ( ( bit_un7362597486090784418nd_num @ one @ one )
    = ( some_num @ one ) ) ).

% and_num.simps(1)
thf(fact_9541_xor__num_Osimps_I1_J,axiom,
    ( ( bit_un2480387367778600638or_num @ one @ one )
    = none_num ) ).

% xor_num.simps(1)
thf(fact_9542_and__num_Osimps_I5_J,axiom,
    ! [M: num,N: num] :
      ( ( bit_un7362597486090784418nd_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
      = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M @ N ) ) ) ).

% and_num.simps(5)
thf(fact_9543_xor__num_Osimps_I5_J,axiom,
    ! [M: num,N: num] :
      ( ( bit_un2480387367778600638or_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
      = ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M @ N ) ) ) ).

% xor_num.simps(5)
thf(fact_9544_and__num_Osimps_I7_J,axiom,
    ! [M: num] :
      ( ( bit_un7362597486090784418nd_num @ ( bit1 @ M ) @ one )
      = ( some_num @ one ) ) ).

% and_num.simps(7)
thf(fact_9545_and__num_Osimps_I3_J,axiom,
    ! [N: num] :
      ( ( bit_un7362597486090784418nd_num @ one @ ( bit1 @ N ) )
      = ( some_num @ one ) ) ).

% and_num.simps(3)
thf(fact_9546_and__num_Osimps_I2_J,axiom,
    ! [N: num] :
      ( ( bit_un7362597486090784418nd_num @ one @ ( bit0 @ N ) )
      = none_num ) ).

% and_num.simps(2)
thf(fact_9547_and__num_Osimps_I4_J,axiom,
    ! [M: num] :
      ( ( bit_un7362597486090784418nd_num @ ( bit0 @ M ) @ one )
      = none_num ) ).

% and_num.simps(4)
thf(fact_9548_xor__num_Osimps_I9_J,axiom,
    ! [M: num,N: num] :
      ( ( bit_un2480387367778600638or_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
      = ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M @ N ) ) ) ).

% xor_num.simps(9)
thf(fact_9549_and__num_Osimps_I6_J,axiom,
    ! [M: num,N: num] :
      ( ( bit_un7362597486090784418nd_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
      = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M @ N ) ) ) ).

% and_num.simps(6)
thf(fact_9550_and__num_Osimps_I8_J,axiom,
    ! [M: num,N: num] :
      ( ( bit_un7362597486090784418nd_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
      = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M @ N ) ) ) ).

% and_num.simps(8)
thf(fact_9551_xor__num_Osimps_I7_J,axiom,
    ! [M: num] :
      ( ( bit_un2480387367778600638or_num @ ( bit1 @ M ) @ one )
      = ( some_num @ ( bit0 @ M ) ) ) ).

% xor_num.simps(7)
thf(fact_9552_xor__num_Osimps_I4_J,axiom,
    ! [M: num] :
      ( ( bit_un2480387367778600638or_num @ ( bit0 @ M ) @ one )
      = ( some_num @ ( bit1 @ M ) ) ) ).

% xor_num.simps(4)
thf(fact_9553_xor__num_Osimps_I3_J,axiom,
    ! [N: num] :
      ( ( bit_un2480387367778600638or_num @ one @ ( bit1 @ N ) )
      = ( some_num @ ( bit0 @ N ) ) ) ).

% xor_num.simps(3)
thf(fact_9554_xor__num_Osimps_I2_J,axiom,
    ! [N: num] :
      ( ( bit_un2480387367778600638or_num @ one @ ( bit0 @ N ) )
      = ( some_num @ ( bit1 @ N ) ) ) ).

% xor_num.simps(2)
thf(fact_9555_and__num_Osimps_I9_J,axiom,
    ! [M: num,N: num] :
      ( ( bit_un7362597486090784418nd_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
      = ( case_o6005452278849405969um_num @ ( some_num @ one )
        @ ^ [N9: num] : ( some_num @ ( bit1 @ N9 ) )
        @ ( bit_un7362597486090784418nd_num @ M @ N ) ) ) ).

% and_num.simps(9)
thf(fact_9556_xor__num_Osimps_I6_J,axiom,
    ! [M: num,N: num] :
      ( ( bit_un2480387367778600638or_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
      = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M @ N ) ) ) ) ).

% xor_num.simps(6)
thf(fact_9557_and__num__dict,axiom,
    bit_un7362597486090784418nd_num = bit_un1837492267222099188nd_num ).

% and_num_dict
thf(fact_9558_xor__num__dict,axiom,
    bit_un2480387367778600638or_num = bit_un6178654185764691216or_num ).

% xor_num_dict
thf(fact_9559_num__of__integer__code,axiom,
    ( code_num_of_integer
    = ( ^ [K3: code_integer] :
          ( if_num @ ( ord_le3102999989581377725nteger @ K3 @ one_one_Code_integer ) @ one
          @ ( produc7336495610019696514er_num
            @ ^ [L: code_integer,J3: code_integer] : ( if_num @ ( J3 = zero_z3403309356797280102nteger ) @ ( plus_plus_num @ ( code_num_of_integer @ L ) @ ( code_num_of_integer @ L ) ) @ ( plus_plus_num @ ( plus_plus_num @ ( code_num_of_integer @ L ) @ ( code_num_of_integer @ L ) ) @ one ) )
            @ ( code_divmod_integer @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% num_of_integer_code
thf(fact_9560_min__Suc__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_min_nat @ ( suc @ M ) @ ( suc @ N ) )
      = ( suc @ ( ord_min_nat @ M @ N ) ) ) ).

% min_Suc_Suc
thf(fact_9561_min__0R,axiom,
    ! [N: nat] :
      ( ( ord_min_nat @ N @ zero_zero_nat )
      = zero_zero_nat ) ).

% min_0R
thf(fact_9562_min__0L,axiom,
    ! [N: nat] :
      ( ( ord_min_nat @ zero_zero_nat @ N )
      = zero_zero_nat ) ).

% min_0L
thf(fact_9563_min__numeral__Suc,axiom,
    ! [K: num,N: nat] :
      ( ( ord_min_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
      = ( suc @ ( ord_min_nat @ ( pred_numeral @ K ) @ N ) ) ) ).

% min_numeral_Suc
thf(fact_9564_min__Suc__numeral,axiom,
    ! [N: nat,K: num] :
      ( ( ord_min_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
      = ( suc @ ( ord_min_nat @ N @ ( pred_numeral @ K ) ) ) ) ).

% min_Suc_numeral
thf(fact_9565_nat__mult__min__right,axiom,
    ! [M: nat,N: nat,Q2: nat] :
      ( ( times_times_nat @ M @ ( ord_min_nat @ N @ Q2 ) )
      = ( ord_min_nat @ ( times_times_nat @ M @ N ) @ ( times_times_nat @ M @ Q2 ) ) ) ).

% nat_mult_min_right
thf(fact_9566_nat__mult__min__left,axiom,
    ! [M: nat,N: nat,Q2: nat] :
      ( ( times_times_nat @ ( ord_min_nat @ M @ N ) @ Q2 )
      = ( ord_min_nat @ ( times_times_nat @ M @ Q2 ) @ ( times_times_nat @ N @ Q2 ) ) ) ).

% nat_mult_min_left
thf(fact_9567_min__diff,axiom,
    ! [M: nat,I: nat,N: nat] :
      ( ( ord_min_nat @ ( minus_minus_nat @ M @ I ) @ ( minus_minus_nat @ N @ I ) )
      = ( minus_minus_nat @ ( ord_min_nat @ M @ N ) @ I ) ) ).

% min_diff
thf(fact_9568_concat__bit__assoc__sym,axiom,
    ! [M: nat,N: nat,K: int,L2: int,R2: int] :
      ( ( bit_concat_bit @ M @ ( bit_concat_bit @ N @ K @ L2 ) @ R2 )
      = ( bit_concat_bit @ ( ord_min_nat @ M @ N ) @ K @ ( bit_concat_bit @ ( minus_minus_nat @ M @ N ) @ L2 @ R2 ) ) ) ).

% concat_bit_assoc_sym
thf(fact_9569_take__bit__concat__bit__eq,axiom,
    ! [M: nat,N: nat,K: int,L2: int] :
      ( ( bit_se2923211474154528505it_int @ M @ ( bit_concat_bit @ N @ K @ L2 ) )
      = ( bit_concat_bit @ ( ord_min_nat @ M @ N ) @ K @ ( bit_se2923211474154528505it_int @ ( minus_minus_nat @ M @ N ) @ L2 ) ) ) ).

% take_bit_concat_bit_eq
thf(fact_9570_min__Suc1,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_min_nat @ ( suc @ N ) @ M )
      = ( case_nat_nat @ zero_zero_nat
        @ ^ [M2: nat] : ( suc @ ( ord_min_nat @ N @ M2 ) )
        @ M ) ) ).

% min_Suc1
thf(fact_9571_min__Suc2,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_min_nat @ M @ ( suc @ N ) )
      = ( case_nat_nat @ zero_zero_nat
        @ ^ [M2: nat] : ( suc @ ( ord_min_nat @ M2 @ N ) )
        @ M ) ) ).

% min_Suc2
thf(fact_9572_card__le__Suc__Max,axiom,
    ! [S2: set_nat] :
      ( ( finite_finite_nat @ S2 )
     => ( ord_less_eq_nat @ ( finite_card_nat @ S2 ) @ ( suc @ ( lattic8265883725875713057ax_nat @ S2 ) ) ) ) ).

% card_le_Suc_Max
thf(fact_9573_divide__nat__def,axiom,
    ( divide_divide_nat
    = ( ^ [M6: nat,N2: nat] :
          ( if_nat @ ( N2 = zero_zero_nat ) @ zero_zero_nat
          @ ( lattic8265883725875713057ax_nat
            @ ( collect_nat
              @ ^ [K3: nat] : ( ord_less_eq_nat @ ( times_times_nat @ K3 @ N2 ) @ M6 ) ) ) ) ) ) ).

% divide_nat_def
thf(fact_9574_gcd__is__Max__divisors__nat,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( gcd_gcd_nat @ M @ N )
        = ( lattic8265883725875713057ax_nat
          @ ( collect_nat
            @ ^ [D2: nat] :
                ( ( dvd_dvd_nat @ D2 @ M )
                & ( dvd_dvd_nat @ D2 @ N ) ) ) ) ) ) ).

% gcd_is_Max_divisors_nat
thf(fact_9575_min__enat__simps_I2_J,axiom,
    ! [Q2: extended_enat] :
      ( ( ord_mi8085742599997312461d_enat @ Q2 @ zero_z5237406670263579293d_enat )
      = zero_z5237406670263579293d_enat ) ).

% min_enat_simps(2)
thf(fact_9576_min__enat__simps_I3_J,axiom,
    ! [Q2: extended_enat] :
      ( ( ord_mi8085742599997312461d_enat @ zero_z5237406670263579293d_enat @ Q2 )
      = zero_z5237406670263579293d_enat ) ).

% min_enat_simps(3)
thf(fact_9577_sorted__list__of__set__greaterThanAtMost,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_eq_nat @ ( suc @ I ) @ J )
     => ( ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I @ J ) )
        = ( cons_nat @ ( suc @ I ) @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ ( suc @ I ) @ J ) ) ) ) ) ).

% sorted_list_of_set_greaterThanAtMost
thf(fact_9578_sorted__list__of__set__greaterThanLessThan,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_nat @ ( suc @ I ) @ J )
     => ( ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I @ J ) )
        = ( cons_nat @ ( suc @ I ) @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ ( suc @ I ) @ J ) ) ) ) ) ).

% sorted_list_of_set_greaterThanLessThan
thf(fact_9579_upto__aux__rec,axiom,
    ( upto_aux
    = ( ^ [I2: int,J3: int,Js: list_int] : ( if_list_int @ ( ord_less_int @ J3 @ I2 ) @ Js @ ( upto_aux @ I2 @ ( minus_minus_int @ J3 @ one_one_int ) @ ( cons_int @ J3 @ Js ) ) ) ) ) ).

% upto_aux_rec
thf(fact_9580_list__encode_Osimps_I2_J,axiom,
    ! [X4: nat,Xs2: list_nat] :
      ( ( nat_list_encode @ ( cons_nat @ X4 @ Xs2 ) )
      = ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X4 @ ( nat_list_encode @ Xs2 ) ) ) ) ) ).

% list_encode.simps(2)
thf(fact_9581_list__encode_Oelims,axiom,
    ! [X4: list_nat,Y3: nat] :
      ( ( ( nat_list_encode @ X4 )
        = Y3 )
     => ( ( ( X4 = nil_nat )
         => ( Y3 != zero_zero_nat ) )
       => ~ ! [X3: nat,Xs3: list_nat] :
              ( ( X4
                = ( cons_nat @ X3 @ Xs3 ) )
             => ( Y3
               != ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X3 @ ( nat_list_encode @ Xs3 ) ) ) ) ) ) ) ) ).

% list_encode.elims
thf(fact_9582_quotient__of__def,axiom,
    ( quotient_of
    = ( ^ [X: rat] :
          ( the_Pr4378521158711661632nt_int
          @ ^ [Pair: product_prod_int_int] :
              ( ( X
                = ( fract @ ( product_fst_int_int @ Pair ) @ ( product_snd_int_int @ Pair ) ) )
              & ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ Pair ) )
              & ( algebr932160517623751201me_int @ ( product_fst_int_int @ Pair ) @ ( product_snd_int_int @ Pair ) ) ) ) ) ) ).

% quotient_of_def
thf(fact_9583_coprime__abs__right__iff,axiom,
    ! [K: int,L2: int] :
      ( ( algebr932160517623751201me_int @ K @ ( abs_abs_int @ L2 ) )
      = ( algebr932160517623751201me_int @ K @ L2 ) ) ).

% coprime_abs_right_iff
thf(fact_9584_coprime__abs__left__iff,axiom,
    ! [K: int,L2: int] :
      ( ( algebr932160517623751201me_int @ ( abs_abs_int @ K ) @ L2 )
      = ( algebr932160517623751201me_int @ K @ L2 ) ) ).

% coprime_abs_left_iff
thf(fact_9585_normalize__stable,axiom,
    ! [Q2: int,P2: int] :
      ( ( ord_less_int @ zero_zero_int @ Q2 )
     => ( ( algebr932160517623751201me_int @ P2 @ Q2 )
       => ( ( normalize @ ( product_Pair_int_int @ P2 @ Q2 ) )
          = ( product_Pair_int_int @ P2 @ Q2 ) ) ) ) ).

% normalize_stable
thf(fact_9586_coprime__crossproduct__int,axiom,
    ! [A: int,D: int,B2: int,C: int] :
      ( ( algebr932160517623751201me_int @ A @ D )
     => ( ( algebr932160517623751201me_int @ B2 @ C )
       => ( ( ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ C ) )
            = ( times_times_int @ ( abs_abs_int @ B2 ) @ ( abs_abs_int @ D ) ) )
          = ( ( ( abs_abs_int @ A )
              = ( abs_abs_int @ B2 ) )
            & ( ( abs_abs_int @ C )
              = ( abs_abs_int @ D ) ) ) ) ) ) ).

% coprime_crossproduct_int
thf(fact_9587_Rat__cases,axiom,
    ! [Q2: rat] :
      ~ ! [A5: int,B5: int] :
          ( ( Q2
            = ( fract @ A5 @ B5 ) )
         => ( ( ord_less_int @ zero_zero_int @ B5 )
           => ~ ( algebr932160517623751201me_int @ A5 @ B5 ) ) ) ).

% Rat_cases
thf(fact_9588_Rat__induct,axiom,
    ! [P: rat > $o,Q2: rat] :
      ( ! [A5: int,B5: int] :
          ( ( ord_less_int @ zero_zero_int @ B5 )
         => ( ( algebr932160517623751201me_int @ A5 @ B5 )
           => ( P @ ( fract @ A5 @ B5 ) ) ) )
     => ( P @ Q2 ) ) ).

% Rat_induct
thf(fact_9589_coprime__common__divisor__int,axiom,
    ! [A: int,B2: int,X4: int] :
      ( ( algebr932160517623751201me_int @ A @ B2 )
     => ( ( dvd_dvd_int @ X4 @ A )
       => ( ( dvd_dvd_int @ X4 @ B2 )
         => ( ( abs_abs_int @ X4 )
            = one_one_int ) ) ) ) ).

% coprime_common_divisor_int
thf(fact_9590_Rat__cases__nonzero,axiom,
    ! [Q2: rat] :
      ( ! [A5: int,B5: int] :
          ( ( Q2
            = ( fract @ A5 @ B5 ) )
         => ( ( ord_less_int @ zero_zero_int @ B5 )
           => ( ( A5 != zero_zero_int )
             => ~ ( algebr932160517623751201me_int @ A5 @ B5 ) ) ) )
     => ( Q2 = zero_zero_rat ) ) ).

% Rat_cases_nonzero
thf(fact_9591_card__length__sum__list__rec,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ M )
     => ( ( finite_card_list_nat
          @ ( collect_list_nat
            @ ^ [L: list_nat] :
                ( ( ( size_size_list_nat @ L )
                  = M )
                & ( ( groups4561878855575611511st_nat @ L )
                  = N3 ) ) ) )
        = ( plus_plus_nat
          @ ( finite_card_list_nat
            @ ( collect_list_nat
              @ ^ [L: list_nat] :
                  ( ( ( size_size_list_nat @ L )
                    = ( minus_minus_nat @ M @ one_one_nat ) )
                  & ( ( groups4561878855575611511st_nat @ L )
                    = N3 ) ) ) )
          @ ( finite_card_list_nat
            @ ( collect_list_nat
              @ ^ [L: list_nat] :
                  ( ( ( size_size_list_nat @ L )
                    = M )
                  & ( ( plus_plus_nat @ ( groups4561878855575611511st_nat @ L ) @ one_one_nat )
                    = N3 ) ) ) ) ) ) ) ).

% card_length_sum_list_rec
thf(fact_9592_card__length__sum__list,axiom,
    ! [M: nat,N3: nat] :
      ( ( finite_card_list_nat
        @ ( collect_list_nat
          @ ^ [L: list_nat] :
              ( ( ( size_size_list_nat @ L )
                = M )
              & ( ( groups4561878855575611511st_nat @ L )
                = N3 ) ) ) )
      = ( binomial @ ( minus_minus_nat @ ( plus_plus_nat @ N3 @ M ) @ one_one_nat ) @ N3 ) ) ).

% card_length_sum_list
thf(fact_9593_quotient__of__unique,axiom,
    ! [R2: rat] :
    ? [X3: product_prod_int_int] :
      ( ( R2
        = ( fract @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ X3 ) ) )
      & ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ X3 ) )
      & ( algebr932160517623751201me_int @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ X3 ) )
      & ! [Y5: product_prod_int_int] :
          ( ( ( R2
              = ( fract @ ( product_fst_int_int @ Y5 ) @ ( product_snd_int_int @ Y5 ) ) )
            & ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ Y5 ) )
            & ( algebr932160517623751201me_int @ ( product_fst_int_int @ Y5 ) @ ( product_snd_int_int @ Y5 ) ) )
         => ( Y5 = X3 ) ) ) ).

% quotient_of_unique
thf(fact_9594_sorted__list__of__set__atMost__Suc,axiom,
    ! [K: nat] :
      ( ( linord2614967742042102400et_nat @ ( set_ord_atMost_nat @ ( suc @ K ) ) )
      = ( append_nat @ ( linord2614967742042102400et_nat @ ( set_ord_atMost_nat @ K ) ) @ ( cons_nat @ ( suc @ K ) @ nil_nat ) ) ) ).

% sorted_list_of_set_atMost_Suc
thf(fact_9595_sorted__list__of__set__lessThan__Suc,axiom,
    ! [K: nat] :
      ( ( linord2614967742042102400et_nat @ ( set_ord_lessThan_nat @ ( suc @ K ) ) )
      = ( append_nat @ ( linord2614967742042102400et_nat @ ( set_ord_lessThan_nat @ K ) ) @ ( cons_nat @ K @ nil_nat ) ) ) ).

% sorted_list_of_set_lessThan_Suc
thf(fact_9596_coprime__int__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( algebr932160517623751201me_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
      = ( algebr934650988132801477me_nat @ M @ N ) ) ).

% coprime_int_iff
thf(fact_9597_coprime__nat__abs__right__iff,axiom,
    ! [N: nat,K: int] :
      ( ( algebr934650988132801477me_nat @ N @ ( nat2 @ ( abs_abs_int @ K ) ) )
      = ( algebr932160517623751201me_int @ ( semiri1314217659103216013at_int @ N ) @ K ) ) ).

% coprime_nat_abs_right_iff
thf(fact_9598_coprime__nat__abs__left__iff,axiom,
    ! [K: int,N: nat] :
      ( ( algebr934650988132801477me_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ N )
      = ( algebr932160517623751201me_int @ K @ ( semiri1314217659103216013at_int @ N ) ) ) ).

% coprime_nat_abs_left_iff
thf(fact_9599_coprime__common__divisor__nat,axiom,
    ! [A: nat,B2: nat,X4: nat] :
      ( ( algebr934650988132801477me_nat @ A @ B2 )
     => ( ( dvd_dvd_nat @ X4 @ A )
       => ( ( dvd_dvd_nat @ X4 @ B2 )
         => ( X4 = one_one_nat ) ) ) ) ).

% coprime_common_divisor_nat
thf(fact_9600_coprime__Suc__0__right,axiom,
    ! [N: nat] : ( algebr934650988132801477me_nat @ N @ ( suc @ zero_zero_nat ) ) ).

% coprime_Suc_0_right
thf(fact_9601_coprime__Suc__0__left,axiom,
    ! [N: nat] : ( algebr934650988132801477me_nat @ ( suc @ zero_zero_nat ) @ N ) ).

% coprime_Suc_0_left
thf(fact_9602_coprime__Suc__left__nat,axiom,
    ! [N: nat] : ( algebr934650988132801477me_nat @ ( suc @ N ) @ N ) ).

% coprime_Suc_left_nat
thf(fact_9603_coprime__Suc__right__nat,axiom,
    ! [N: nat] : ( algebr934650988132801477me_nat @ N @ ( suc @ N ) ) ).

% coprime_Suc_right_nat
thf(fact_9604_coprime__crossproduct__nat,axiom,
    ! [A: nat,D: nat,B2: nat,C: nat] :
      ( ( algebr934650988132801477me_nat @ A @ D )
     => ( ( algebr934650988132801477me_nat @ B2 @ C )
       => ( ( ( times_times_nat @ A @ C )
            = ( times_times_nat @ B2 @ D ) )
          = ( ( A = B2 )
            & ( C = D ) ) ) ) ) ).

% coprime_crossproduct_nat
thf(fact_9605_coprime__diff__one__right__nat,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( algebr934650988132801477me_nat @ N @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ).

% coprime_diff_one_right_nat
thf(fact_9606_coprime__diff__one__left__nat,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( algebr934650988132801477me_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ N ) ) ).

% coprime_diff_one_left_nat
thf(fact_9607_upto_Opsimps,axiom,
    ! [I: int,J: int] :
      ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I @ J ) )
     => ( ( ( ord_less_eq_int @ I @ J )
         => ( ( upto @ I @ J )
            = ( cons_int @ I @ ( upto @ ( plus_plus_int @ I @ one_one_int ) @ J ) ) ) )
        & ( ~ ( ord_less_eq_int @ I @ J )
         => ( ( upto @ I @ J )
            = nil_int ) ) ) ) ).

% upto.psimps
thf(fact_9608_upto__Nil,axiom,
    ! [I: int,J: int] :
      ( ( ( upto @ I @ J )
        = nil_int )
      = ( ord_less_int @ J @ I ) ) ).

% upto_Nil
thf(fact_9609_upto__Nil2,axiom,
    ! [I: int,J: int] :
      ( ( nil_int
        = ( upto @ I @ J ) )
      = ( ord_less_int @ J @ I ) ) ).

% upto_Nil2
thf(fact_9610_upto__empty,axiom,
    ! [J: int,I: int] :
      ( ( ord_less_int @ J @ I )
     => ( ( upto @ I @ J )
        = nil_int ) ) ).

% upto_empty
thf(fact_9611_nth__upto,axiom,
    ! [I: int,K: nat,J: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ I @ ( semiri1314217659103216013at_int @ K ) ) @ J )
     => ( ( nth_int @ ( upto @ I @ J ) @ K )
        = ( plus_plus_int @ I @ ( semiri1314217659103216013at_int @ K ) ) ) ) ).

% nth_upto
thf(fact_9612_length__upto,axiom,
    ! [I: int,J: int] :
      ( ( size_size_list_int @ ( upto @ I @ J ) )
      = ( nat2 @ ( plus_plus_int @ ( minus_minus_int @ J @ I ) @ one_one_int ) ) ) ).

% length_upto
thf(fact_9613_upto__rec__numeral_I1_J,axiom,
    ! [M: num,N: num] :
      ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
       => ( ( upto @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
          = ( cons_int @ ( numeral_numeral_int @ M ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( numeral_numeral_int @ N ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
       => ( ( upto @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
          = nil_int ) ) ) ).

% upto_rec_numeral(1)
thf(fact_9614_upto__rec__numeral_I2_J,axiom,
    ! [M: num,N: num] :
      ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
       => ( ( upto @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
          = ( cons_int @ ( numeral_numeral_int @ M ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
       => ( ( upto @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
          = nil_int ) ) ) ).

% upto_rec_numeral(2)
thf(fact_9615_upto__rec__numeral_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
       => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
          = ( cons_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( upto @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) @ ( numeral_numeral_int @ N ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
       => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
          = nil_int ) ) ) ).

% upto_rec_numeral(3)
thf(fact_9616_upto__rec__numeral_I4_J,axiom,
    ! [M: num,N: num] :
      ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
       => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
          = ( cons_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( upto @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
       => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
          = nil_int ) ) ) ).

% upto_rec_numeral(4)
thf(fact_9617_upto__split2,axiom,
    ! [I: int,J: int,K: int] :
      ( ( ord_less_eq_int @ I @ J )
     => ( ( ord_less_eq_int @ J @ K )
       => ( ( upto @ I @ K )
          = ( append_int @ ( upto @ I @ J ) @ ( upto @ ( plus_plus_int @ J @ one_one_int ) @ K ) ) ) ) ) ).

% upto_split2
thf(fact_9618_upto__split1,axiom,
    ! [I: int,J: int,K: int] :
      ( ( ord_less_eq_int @ I @ J )
     => ( ( ord_less_eq_int @ J @ K )
       => ( ( upto @ I @ K )
          = ( append_int @ ( upto @ I @ ( minus_minus_int @ J @ one_one_int ) ) @ ( upto @ J @ K ) ) ) ) ) ).

% upto_split1
thf(fact_9619_atLeastLessThan__upto,axiom,
    ( set_or4662586982721622107an_int
    = ( ^ [I2: int,J3: int] : ( set_int2 @ ( upto @ I2 @ ( minus_minus_int @ J3 @ one_one_int ) ) ) ) ) ).

% atLeastLessThan_upto
thf(fact_9620_greaterThanAtMost__upto,axiom,
    ( set_or6656581121297822940st_int
    = ( ^ [I2: int,J3: int] : ( set_int2 @ ( upto @ ( plus_plus_int @ I2 @ one_one_int ) @ J3 ) ) ) ) ).

% greaterThanAtMost_upto
thf(fact_9621_upto__rec1,axiom,
    ! [I: int,J: int] :
      ( ( ord_less_eq_int @ I @ J )
     => ( ( upto @ I @ J )
        = ( cons_int @ I @ ( upto @ ( plus_plus_int @ I @ one_one_int ) @ J ) ) ) ) ).

% upto_rec1
thf(fact_9622_upto_Oelims,axiom,
    ! [X4: int,Xa: int,Y3: list_int] :
      ( ( ( upto @ X4 @ Xa )
        = Y3 )
     => ( ( ( ord_less_eq_int @ X4 @ Xa )
         => ( Y3
            = ( cons_int @ X4 @ ( upto @ ( plus_plus_int @ X4 @ one_one_int ) @ Xa ) ) ) )
        & ( ~ ( ord_less_eq_int @ X4 @ Xa )
         => ( Y3 = nil_int ) ) ) ) ).

% upto.elims
thf(fact_9623_upto_Osimps,axiom,
    ( upto
    = ( ^ [I2: int,J3: int] : ( if_list_int @ ( ord_less_eq_int @ I2 @ J3 ) @ ( cons_int @ I2 @ ( upto @ ( plus_plus_int @ I2 @ one_one_int ) @ J3 ) ) @ nil_int ) ) ) ).

% upto.simps
thf(fact_9624_upto__rec2,axiom,
    ! [I: int,J: int] :
      ( ( ord_less_eq_int @ I @ J )
     => ( ( upto @ I @ J )
        = ( append_int @ ( upto @ I @ ( minus_minus_int @ J @ one_one_int ) ) @ ( cons_int @ J @ nil_int ) ) ) ) ).

% upto_rec2
thf(fact_9625_greaterThanLessThan__upto,axiom,
    ( set_or5832277885323065728an_int
    = ( ^ [I2: int,J3: int] : ( set_int2 @ ( upto @ ( plus_plus_int @ I2 @ one_one_int ) @ ( minus_minus_int @ J3 @ one_one_int ) ) ) ) ) ).

% greaterThanLessThan_upto
thf(fact_9626_upto__split3,axiom,
    ! [I: int,J: int,K: int] :
      ( ( ord_less_eq_int @ I @ J )
     => ( ( ord_less_eq_int @ J @ K )
       => ( ( upto @ I @ K )
          = ( append_int @ ( upto @ I @ ( minus_minus_int @ J @ one_one_int ) ) @ ( cons_int @ J @ ( upto @ ( plus_plus_int @ J @ one_one_int ) @ K ) ) ) ) ) ) ).

% upto_split3
thf(fact_9627_upto_Opelims,axiom,
    ! [X4: int,Xa: int,Y3: list_int] :
      ( ( ( upto @ X4 @ Xa )
        = Y3 )
     => ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X4 @ Xa ) )
       => ~ ( ( ( ( ord_less_eq_int @ X4 @ Xa )
               => ( Y3
                  = ( cons_int @ X4 @ ( upto @ ( plus_plus_int @ X4 @ one_one_int ) @ Xa ) ) ) )
              & ( ~ ( ord_less_eq_int @ X4 @ Xa )
               => ( Y3 = nil_int ) ) )
           => ~ ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X4 @ Xa ) ) ) ) ) ).

% upto.pelims
thf(fact_9628_list__encode_Opelims,axiom,
    ! [X4: list_nat,Y3: nat] :
      ( ( ( nat_list_encode @ X4 )
        = Y3 )
     => ( ( accp_list_nat @ nat_list_encode_rel @ X4 )
       => ( ( ( X4 = nil_nat )
           => ( ( Y3 = zero_zero_nat )
             => ~ ( accp_list_nat @ nat_list_encode_rel @ nil_nat ) ) )
         => ~ ! [X3: nat,Xs3: list_nat] :
                ( ( X4
                  = ( cons_nat @ X3 @ Xs3 ) )
               => ( ( Y3
                    = ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X3 @ ( nat_list_encode @ Xs3 ) ) ) ) )
                 => ~ ( accp_list_nat @ nat_list_encode_rel @ ( cons_nat @ X3 @ Xs3 ) ) ) ) ) ) ) ).

% list_encode.pelims
thf(fact_9629_Rats__abs__iff,axiom,
    ! [X4: real] :
      ( ( member_real @ ( abs_abs_real @ X4 ) @ field_5140801741446780682s_real )
      = ( member_real @ X4 @ field_5140801741446780682s_real ) ) ).

% Rats_abs_iff
thf(fact_9630_Rats__no__top__le,axiom,
    ! [X4: real] :
    ? [X3: real] :
      ( ( member_real @ X3 @ field_5140801741446780682s_real )
      & ( ord_less_eq_real @ X4 @ X3 ) ) ).

% Rats_no_top_le
thf(fact_9631_Rats__no__bot__less,axiom,
    ! [X4: real] :
    ? [X3: real] :
      ( ( member_real @ X3 @ field_5140801741446780682s_real )
      & ( ord_less_real @ X3 @ X4 ) ) ).

% Rats_no_bot_less
thf(fact_9632_Rats__dense__in__real,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_real @ X4 @ Y3 )
     => ? [X3: real] :
          ( ( member_real @ X3 @ field_5140801741446780682s_real )
          & ( ord_less_real @ X4 @ X3 )
          & ( ord_less_real @ X3 @ Y3 ) ) ) ).

% Rats_dense_in_real
thf(fact_9633_Rats__abs__nat__div__natE,axiom,
    ! [X4: real] :
      ( ( member_real @ X4 @ field_5140801741446780682s_real )
     => ~ ! [M4: nat,N4: nat] :
            ( ( N4 != zero_zero_nat )
           => ( ( ( abs_abs_real @ X4 )
                = ( divide_divide_real @ ( semiri5074537144036343181t_real @ M4 ) @ ( semiri5074537144036343181t_real @ N4 ) ) )
             => ~ ( algebr934650988132801477me_nat @ M4 @ N4 ) ) ) ) ).

% Rats_abs_nat_div_natE
thf(fact_9634_upt__rec__numeral,axiom,
    ! [M: num,N: num] :
      ( ( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
       => ( ( upt @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
          = ( cons_nat @ ( numeral_numeral_nat @ M ) @ ( upt @ ( suc @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) ) ) ) )
      & ( ~ ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
       => ( ( upt @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
          = nil_nat ) ) ) ).

% upt_rec_numeral
thf(fact_9635_hd__upt,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( hd_nat @ ( upt @ I @ J ) )
        = I ) ) ).

% hd_upt
thf(fact_9636_drop__upt,axiom,
    ! [M: nat,I: nat,J: nat] :
      ( ( drop_nat @ M @ ( upt @ I @ J ) )
      = ( upt @ ( plus_plus_nat @ I @ M ) @ J ) ) ).

% drop_upt
thf(fact_9637_take__upt,axiom,
    ! [I: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ M ) @ N )
     => ( ( take_nat @ M @ ( upt @ I @ N ) )
        = ( upt @ I @ ( plus_plus_nat @ I @ M ) ) ) ) ).

% take_upt
thf(fact_9638_upt__conv__Nil,axiom,
    ! [J: nat,I: nat] :
      ( ( ord_less_eq_nat @ J @ I )
     => ( ( upt @ I @ J )
        = nil_nat ) ) ).

% upt_conv_Nil
thf(fact_9639_length__upt,axiom,
    ! [I: nat,J: nat] :
      ( ( size_size_list_nat @ ( upt @ I @ J ) )
      = ( minus_minus_nat @ J @ I ) ) ).

% length_upt
thf(fact_9640_upt__eq__Nil__conv,axiom,
    ! [I: nat,J: nat] :
      ( ( ( upt @ I @ J )
        = nil_nat )
      = ( ( J = zero_zero_nat )
        | ( ord_less_eq_nat @ J @ I ) ) ) ).

% upt_eq_Nil_conv
thf(fact_9641_nth__upt,axiom,
    ! [I: nat,K: nat,J: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J )
     => ( ( nth_nat @ ( upt @ I @ J ) @ K )
        = ( plus_plus_nat @ I @ K ) ) ) ).

% nth_upt
thf(fact_9642_sum__list__upt,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( groups4561878855575611511st_nat @ ( upt @ M @ N ) )
        = ( groups3542108847815614940at_nat
          @ ^ [X: nat] : X
          @ ( set_or4665077453230672383an_nat @ M @ N ) ) ) ) ).

% sum_list_upt
thf(fact_9643_atLeastAtMost__upt,axiom,
    ( set_or1269000886237332187st_nat
    = ( ^ [N2: nat,M6: nat] : ( set_nat2 @ ( upt @ N2 @ ( suc @ M6 ) ) ) ) ) ).

% atLeastAtMost_upt
thf(fact_9644_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 @ ( suc @ M ) @ Q2 ) ) ) ).

% upt_conv_Cons_Cons
thf(fact_9645_greaterThanAtMost__upt,axiom,
    ( set_or6659071591806873216st_nat
    = ( ^ [N2: nat,M6: nat] : ( set_nat2 @ ( upt @ ( suc @ N2 ) @ ( suc @ M6 ) ) ) ) ) ).

% greaterThanAtMost_upt
thf(fact_9646_greaterThanLessThan__upt,axiom,
    ( set_or5834768355832116004an_nat
    = ( ^ [N2: nat,M6: nat] : ( set_nat2 @ ( upt @ ( suc @ N2 ) @ M6 ) ) ) ) ).

% greaterThanLessThan_upt
thf(fact_9647_atMost__upto,axiom,
    ( set_ord_atMost_nat
    = ( ^ [N2: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ ( suc @ N2 ) ) ) ) ) ).

% atMost_upto
thf(fact_9648_upt__conv__Cons,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( upt @ I @ J )
        = ( cons_nat @ I @ ( upt @ ( suc @ I ) @ J ) ) ) ) ).

% upt_conv_Cons
thf(fact_9649_upt__add__eq__append,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( upt @ I @ ( plus_plus_nat @ J @ K ) )
        = ( append_nat @ ( upt @ I @ J ) @ ( upt @ J @ ( plus_plus_nat @ J @ K ) ) ) ) ) ).

% upt_add_eq_append
thf(fact_9650_upt__eq__Cons__conv,axiom,
    ! [I: nat,J: nat,X4: nat,Xs2: list_nat] :
      ( ( ( upt @ I @ J )
        = ( cons_nat @ X4 @ Xs2 ) )
      = ( ( ord_less_nat @ I @ J )
        & ( I = X4 )
        & ( ( upt @ ( plus_plus_nat @ I @ one_one_nat ) @ J )
          = Xs2 ) ) ) ).

% upt_eq_Cons_conv
thf(fact_9651_upt__rec,axiom,
    ( upt
    = ( ^ [I2: nat,J3: nat] : ( if_list_nat @ ( ord_less_nat @ I2 @ J3 ) @ ( cons_nat @ I2 @ ( upt @ ( suc @ I2 ) @ J3 ) ) @ nil_nat ) ) ) ).

% upt_rec
thf(fact_9652_upt__Suc__append,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( upt @ I @ ( suc @ J ) )
        = ( append_nat @ ( upt @ I @ J ) @ ( cons_nat @ J @ nil_nat ) ) ) ) ).

% upt_Suc_append
thf(fact_9653_upt__Suc,axiom,
    ! [I: nat,J: nat] :
      ( ( ( ord_less_eq_nat @ I @ J )
       => ( ( upt @ I @ ( suc @ J ) )
          = ( append_nat @ ( upt @ I @ J ) @ ( cons_nat @ J @ nil_nat ) ) ) )
      & ( ~ ( ord_less_eq_nat @ I @ J )
       => ( ( upt @ I @ ( suc @ J ) )
          = nil_nat ) ) ) ).

% upt_Suc
thf(fact_9654_mono__Suc,axiom,
    order_mono_nat_nat @ suc ).

% mono_Suc
thf(fact_9655_mono__times__nat,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( order_mono_nat_nat @ ( times_times_nat @ N ) ) ) ).

% mono_times_nat
thf(fact_9656_mono__ge2__power__minus__self,axiom,
    ! [K: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
     => ( order_mono_nat_nat
        @ ^ [M6: nat] : ( minus_minus_nat @ ( power_power_nat @ K @ M6 ) @ M6 ) ) ) ).

% mono_ge2_power_minus_self
thf(fact_9657_map__Suc__upt,axiom,
    ! [M: nat,N: nat] :
      ( ( map_nat_nat @ suc @ ( upt @ M @ N ) )
      = ( upt @ ( suc @ M ) @ ( suc @ N ) ) ) ).

% map_Suc_upt
thf(fact_9658_sorted__upt,axiom,
    ! [M: nat,N: nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( upt @ M @ N ) ) ).

% sorted_upt
thf(fact_9659_sorted__wrt__upt,axiom,
    ! [M: nat,N: nat] : ( sorted_wrt_nat @ ord_less_nat @ ( upt @ M @ N ) ) ).

% sorted_wrt_upt
thf(fact_9660_map__add__upt,axiom,
    ! [N: nat,M: nat] :
      ( ( map_nat_nat
        @ ^ [I2: nat] : ( plus_plus_nat @ I2 @ N )
        @ ( upt @ zero_zero_nat @ M ) )
      = ( upt @ N @ ( plus_plus_nat @ M @ N ) ) ) ).

% map_add_upt
thf(fact_9661_map__decr__upt,axiom,
    ! [M: nat,N: nat] :
      ( ( map_nat_nat
        @ ^ [N2: nat] : ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) )
        @ ( upt @ ( suc @ M ) @ ( suc @ N ) ) )
      = ( upt @ M @ N ) ) ).

% map_decr_upt
thf(fact_9662_sorted__wrt__less__idx,axiom,
    ! [Ns: list_nat,I: nat] :
      ( ( sorted_wrt_nat @ ord_less_nat @ Ns )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ns ) )
       => ( ord_less_eq_nat @ I @ ( nth_nat @ Ns @ I ) ) ) ) ).

% sorted_wrt_less_idx
thf(fact_9663_sorted__wrt__upto,axiom,
    ! [I: int,J: int] : ( sorted_wrt_int @ ord_less_int @ ( upto @ I @ J ) ) ).

% sorted_wrt_upto
thf(fact_9664_Rats__eq__int__div__int,axiom,
    ( field_5140801741446780682s_real
    = ( collect_real
      @ ^ [Uu: real] :
        ? [I2: int,J3: int] :
          ( ( Uu
            = ( divide_divide_real @ ( ring_1_of_int_real @ I2 ) @ ( ring_1_of_int_real @ J3 ) ) )
          & ( J3 != zero_zero_int ) ) ) ) ).

% Rats_eq_int_div_int
thf(fact_9665_Rats__eq__int__div__nat,axiom,
    ( field_5140801741446780682s_real
    = ( collect_real
      @ ^ [Uu: real] :
        ? [I2: int,N2: nat] :
          ( ( Uu
            = ( divide_divide_real @ ( ring_1_of_int_real @ I2 ) @ ( semiri5074537144036343181t_real @ N2 ) ) )
          & ( N2 != zero_zero_nat ) ) ) ) ).

% Rats_eq_int_div_nat
thf(fact_9666_range__mod,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( image_nat_nat
          @ ^ [M6: nat] : ( modulo_modulo_nat @ M6 @ N )
          @ top_top_set_nat )
        = ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ).

% range_mod
thf(fact_9667_UNIV__nat__eq,axiom,
    ( top_top_set_nat
    = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ top_top_set_nat ) ) ) ).

% UNIV_nat_eq
thf(fact_9668_card__UNIV__unit,axiom,
    ( ( finite410649719033368117t_unit @ top_to1996260823553986621t_unit )
    = one_one_nat ) ).

% card_UNIV_unit
thf(fact_9669_range__mult,axiom,
    ! [A: real] :
      ( ( ( A = zero_zero_real )
       => ( ( image_real_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 @ ( times_times_real @ A ) @ top_top_set_real )
          = top_top_set_real ) ) ) ).

% range_mult
thf(fact_9670_card__UNIV__bool,axiom,
    ( ( finite_card_o @ top_top_set_o )
    = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% card_UNIV_bool
thf(fact_9671_root__def,axiom,
    ( root
    = ( ^ [N2: nat,X: real] :
          ( if_real @ ( N2 = zero_zero_nat ) @ zero_zero_real
          @ ( the_in5290026491893676941l_real @ top_top_set_real
            @ ^ [Y: real] : ( times_times_real @ ( sgn_sgn_real @ Y ) @ ( power_power_real @ ( abs_abs_real @ Y ) @ N2 ) )
            @ X ) ) ) ) ).

% root_def
thf(fact_9672_card__UNIV__char,axiom,
    ( ( finite_card_char @ top_top_set_char )
    = ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% card_UNIV_char
thf(fact_9673_UNIV__char__of__nat,axiom,
    ( top_top_set_char
    = ( image_nat_char @ unique3096191561947761185of_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ).

% UNIV_char_of_nat
thf(fact_9674_nat__of__char__less__256,axiom,
    ! [C: char] : ( ord_less_nat @ ( comm_s629917340098488124ar_nat @ C ) @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% nat_of_char_less_256
thf(fact_9675_range__nat__of__char,axiom,
    ( ( image_char_nat @ comm_s629917340098488124ar_nat @ top_top_set_char )
    = ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).

% range_nat_of_char
thf(fact_9676_integer__of__char__code,axiom,
    ! [B0: $o,B1: $o,B22: $o,B32: $o,B42: $o,B52: $o,B62: $o,B72: $o] :
      ( ( integer_of_char @ ( char2 @ B0 @ B1 @ B22 @ B32 @ B42 @ B52 @ B62 @ B72 ) )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ B72 ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B62 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B52 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B42 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B32 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B22 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B1 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B0 ) ) ) ).

% integer_of_char_code
thf(fact_9677_String_Ochar__of__ascii__of,axiom,
    ! [C: char] :
      ( ( comm_s629917340098488124ar_nat @ ( ascii_of @ C ) )
      = ( bit_se2925701944663578781it_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ one ) ) ) @ ( comm_s629917340098488124ar_nat @ C ) ) ) ).

% String.char_of_ascii_of
thf(fact_9678_DERIV__even__real__root,axiom,
    ! [N: nat,X4: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( ord_less_real @ X4 @ zero_zero_real )
         => ( has_fi5821293074295781190e_real @ ( root @ N ) @ ( inverse_inverse_real @ ( times_times_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ ( power_power_real @ ( root @ N @ X4 ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) ) ) ).

% DERIV_even_real_root
thf(fact_9679_DERIV__real__root__generic,axiom,
    ! [N: nat,X4: real,D4: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( X4 != zero_zero_real )
       => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
           => ( ( ord_less_real @ zero_zero_real @ X4 )
             => ( D4
                = ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X4 ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) )
         => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
             => ( ( ord_less_real @ X4 @ zero_zero_real )
               => ( D4
                  = ( uminus_uminus_real @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X4 ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ) )
           => ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
               => ( D4
                  = ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X4 ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) )
             => ( has_fi5821293074295781190e_real @ ( root @ N ) @ D4 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) ) ) ) ) ).

% DERIV_real_root_generic
thf(fact_9680_DERIV__neg__dec__right,axiom,
    ! [F: real > real,L2: real,X4: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
     => ( ( ord_less_real @ L2 @ zero_zero_real )
       => ? [D3: real] :
            ( ( ord_less_real @ zero_zero_real @ D3 )
            & ! [H4: real] :
                ( ( ord_less_real @ zero_zero_real @ H4 )
               => ( ( ord_less_real @ H4 @ D3 )
                 => ( ord_less_real @ ( F @ ( plus_plus_real @ X4 @ H4 ) ) @ ( F @ X4 ) ) ) ) ) ) ) ).

% DERIV_neg_dec_right
thf(fact_9681_DERIV__pos__inc__right,axiom,
    ! [F: real > real,L2: real,X4: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ L2 )
       => ? [D3: real] :
            ( ( ord_less_real @ zero_zero_real @ D3 )
            & ! [H4: real] :
                ( ( ord_less_real @ zero_zero_real @ H4 )
               => ( ( ord_less_real @ H4 @ D3 )
                 => ( ord_less_real @ ( F @ X4 ) @ ( F @ ( plus_plus_real @ X4 @ H4 ) ) ) ) ) ) ) ) ).

% DERIV_pos_inc_right
thf(fact_9682_DERIV__pos__imp__increasing,axiom,
    ! [A: real,B2: real,F: real > real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ! [X3: real] :
            ( ( ord_less_eq_real @ A @ X3 )
           => ( ( ord_less_eq_real @ X3 @ B2 )
             => ? [Y5: real] :
                  ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
                  & ( ord_less_real @ zero_zero_real @ Y5 ) ) ) )
       => ( ord_less_real @ ( F @ A ) @ ( F @ B2 ) ) ) ) ).

% DERIV_pos_imp_increasing
thf(fact_9683_DERIV__neg__imp__decreasing,axiom,
    ! [A: real,B2: real,F: real > real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ! [X3: real] :
            ( ( ord_less_eq_real @ A @ X3 )
           => ( ( ord_less_eq_real @ X3 @ B2 )
             => ? [Y5: real] :
                  ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
                  & ( ord_less_real @ Y5 @ zero_zero_real ) ) ) )
       => ( ord_less_real @ ( F @ B2 ) @ ( F @ A ) ) ) ) ).

% DERIV_neg_imp_decreasing
thf(fact_9684_DERIV__nonpos__imp__nonincreasing,axiom,
    ! [A: real,B2: real,F: real > real] :
      ( ( ord_less_eq_real @ A @ B2 )
     => ( ! [X3: real] :
            ( ( ord_less_eq_real @ A @ X3 )
           => ( ( ord_less_eq_real @ X3 @ B2 )
             => ? [Y5: real] :
                  ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
                  & ( ord_less_eq_real @ Y5 @ zero_zero_real ) ) ) )
       => ( ord_less_eq_real @ ( F @ B2 ) @ ( F @ A ) ) ) ) ).

% DERIV_nonpos_imp_nonincreasing
thf(fact_9685_DERIV__nonneg__imp__nondecreasing,axiom,
    ! [A: real,B2: real,F: real > real] :
      ( ( ord_less_eq_real @ A @ B2 )
     => ( ! [X3: real] :
            ( ( ord_less_eq_real @ A @ X3 )
           => ( ( ord_less_eq_real @ X3 @ B2 )
             => ? [Y5: real] :
                  ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
                  & ( ord_less_eq_real @ zero_zero_real @ Y5 ) ) ) )
       => ( ord_less_eq_real @ ( F @ A ) @ ( F @ B2 ) ) ) ) ).

% DERIV_nonneg_imp_nondecreasing
thf(fact_9686_deriv__nonneg__imp__mono,axiom,
    ! [A: real,B2: real,G: real > real,G2: real > real] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ ( set_or1222579329274155063t_real @ A @ B2 ) )
         => ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ ( set_or1222579329274155063t_real @ A @ B2 ) )
           => ( ord_less_eq_real @ zero_zero_real @ ( G2 @ X3 ) ) )
       => ( ( ord_less_eq_real @ A @ B2 )
         => ( ord_less_eq_real @ ( G @ A ) @ ( G @ B2 ) ) ) ) ) ).

% deriv_nonneg_imp_mono
thf(fact_9687_DERIV__isconst__all,axiom,
    ! [F: real > real,X4: real,Y3: real] :
      ( ! [X3: real] : ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
     => ( ( F @ X4 )
        = ( F @ Y3 ) ) ) ).

% DERIV_isconst_all
thf(fact_9688_DERIV__mirror,axiom,
    ! [F: real > real,Y3: real,X4: real] :
      ( ( has_fi5821293074295781190e_real @ F @ Y3 @ ( topolo2177554685111907308n_real @ ( uminus_uminus_real @ X4 ) @ top_top_set_real ) )
      = ( has_fi5821293074295781190e_real
        @ ^ [X: real] : ( F @ ( uminus_uminus_real @ X ) )
        @ ( uminus_uminus_real @ Y3 )
        @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) ).

% DERIV_mirror
thf(fact_9689_DERIV__const__ratio__const2,axiom,
    ! [A: real,B2: real,F: real > real,K: real] :
      ( ( A != B2 )
     => ( ! [X3: real] : ( has_fi5821293074295781190e_real @ F @ K @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
       => ( ( divide_divide_real @ ( minus_minus_real @ ( F @ B2 ) @ ( F @ A ) ) @ ( minus_minus_real @ B2 @ A ) )
          = K ) ) ) ).

% DERIV_const_ratio_const2
thf(fact_9690_DERIV__const__ratio__const,axiom,
    ! [A: real,B2: real,F: real > real,K: real] :
      ( ( A != B2 )
     => ( ! [X3: real] : ( has_fi5821293074295781190e_real @ F @ K @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
       => ( ( minus_minus_real @ ( F @ B2 ) @ ( F @ A ) )
          = ( times_times_real @ ( minus_minus_real @ B2 @ A ) @ K ) ) ) ) ).

% DERIV_const_ratio_const
thf(fact_9691_DERIV__pos__inc__left,axiom,
    ! [F: real > real,L2: real,X4: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ L2 )
       => ? [D3: real] :
            ( ( ord_less_real @ zero_zero_real @ D3 )
            & ! [H4: real] :
                ( ( ord_less_real @ zero_zero_real @ H4 )
               => ( ( ord_less_real @ H4 @ D3 )
                 => ( ord_less_real @ ( F @ ( minus_minus_real @ X4 @ H4 ) ) @ ( F @ X4 ) ) ) ) ) ) ) ).

% DERIV_pos_inc_left
thf(fact_9692_DERIV__neg__dec__left,axiom,
    ! [F: real > real,L2: real,X4: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
     => ( ( ord_less_real @ L2 @ zero_zero_real )
       => ? [D3: real] :
            ( ( ord_less_real @ zero_zero_real @ D3 )
            & ! [H4: real] :
                ( ( ord_less_real @ zero_zero_real @ H4 )
               => ( ( ord_less_real @ H4 @ D3 )
                 => ( ord_less_real @ ( F @ X4 ) @ ( F @ ( minus_minus_real @ X4 @ H4 ) ) ) ) ) ) ) ) ).

% DERIV_neg_dec_left
thf(fact_9693_DERIV__isconst3,axiom,
    ! [A: real,B2: real,X4: real,Y3: real,F: real > real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ( member_real @ X4 @ ( set_or1633881224788618240n_real @ A @ B2 ) )
       => ( ( member_real @ Y3 @ ( set_or1633881224788618240n_real @ A @ B2 ) )
         => ( ! [X3: real] :
                ( ( member_real @ X3 @ ( set_or1633881224788618240n_real @ A @ B2 ) )
               => ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) )
           => ( ( F @ X4 )
              = ( F @ Y3 ) ) ) ) ) ) ).

% DERIV_isconst3
thf(fact_9694_has__real__derivative__pos__inc__left,axiom,
    ! [F: real > real,L2: real,X4: real,S2: set_real] :
      ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X4 @ S2 ) )
     => ( ( ord_less_real @ zero_zero_real @ L2 )
       => ? [D3: real] :
            ( ( ord_less_real @ zero_zero_real @ D3 )
            & ! [H4: real] :
                ( ( ord_less_real @ zero_zero_real @ H4 )
               => ( ( member_real @ ( minus_minus_real @ X4 @ H4 ) @ S2 )
                 => ( ( ord_less_real @ H4 @ D3 )
                   => ( ord_less_real @ ( F @ ( minus_minus_real @ X4 @ H4 ) ) @ ( F @ X4 ) ) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_left
thf(fact_9695_has__real__derivative__neg__dec__left,axiom,
    ! [F: real > real,L2: real,X4: real,S2: set_real] :
      ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X4 @ S2 ) )
     => ( ( ord_less_real @ L2 @ zero_zero_real )
       => ? [D3: real] :
            ( ( ord_less_real @ zero_zero_real @ D3 )
            & ! [H4: real] :
                ( ( ord_less_real @ zero_zero_real @ H4 )
               => ( ( member_real @ ( minus_minus_real @ X4 @ H4 ) @ S2 )
                 => ( ( ord_less_real @ H4 @ D3 )
                   => ( ord_less_real @ ( F @ X4 ) @ ( F @ ( minus_minus_real @ X4 @ H4 ) ) ) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_left
thf(fact_9696_has__real__derivative__pos__inc__right,axiom,
    ! [F: real > real,L2: real,X4: real,S2: set_real] :
      ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X4 @ S2 ) )
     => ( ( ord_less_real @ zero_zero_real @ L2 )
       => ? [D3: real] :
            ( ( ord_less_real @ zero_zero_real @ D3 )
            & ! [H4: real] :
                ( ( ord_less_real @ zero_zero_real @ H4 )
               => ( ( member_real @ ( plus_plus_real @ X4 @ H4 ) @ S2 )
                 => ( ( ord_less_real @ H4 @ D3 )
                   => ( ord_less_real @ ( F @ X4 ) @ ( F @ ( plus_plus_real @ X4 @ H4 ) ) ) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_right
thf(fact_9697_has__real__derivative__neg__dec__right,axiom,
    ! [F: real > real,L2: real,X4: real,S2: set_real] :
      ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X4 @ S2 ) )
     => ( ( ord_less_real @ L2 @ zero_zero_real )
       => ? [D3: real] :
            ( ( ord_less_real @ zero_zero_real @ D3 )
            & ! [H4: real] :
                ( ( ord_less_real @ zero_zero_real @ H4 )
               => ( ( member_real @ ( plus_plus_real @ X4 @ H4 ) @ S2 )
                 => ( ( ord_less_real @ H4 @ D3 )
                   => ( ord_less_real @ ( F @ ( plus_plus_real @ X4 @ H4 ) ) @ ( F @ X4 ) ) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_right
thf(fact_9698_MVT2,axiom,
    ! [A: real,B2: real,F: real > real,F4: real > real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ! [X3: real] :
            ( ( ord_less_eq_real @ A @ X3 )
           => ( ( ord_less_eq_real @ X3 @ B2 )
             => ( has_fi5821293074295781190e_real @ F @ ( F4 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) )
       => ? [Z2: real] :
            ( ( ord_less_real @ A @ Z2 )
            & ( ord_less_real @ Z2 @ B2 )
            & ( ( minus_minus_real @ ( F @ B2 ) @ ( F @ A ) )
              = ( times_times_real @ ( minus_minus_real @ B2 @ A ) @ ( F4 @ Z2 ) ) ) ) ) ) ).

% MVT2
thf(fact_9699_DERIV__local__const,axiom,
    ! [F: real > real,L2: real,X4: real,D: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ D )
       => ( ! [Y4: real] :
              ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X4 @ Y4 ) ) @ D )
             => ( ( F @ X4 )
                = ( F @ Y4 ) ) )
         => ( L2 = zero_zero_real ) ) ) ) ).

% DERIV_local_const
thf(fact_9700_DERIV__ln,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( has_fi5821293074295781190e_real @ ln_ln_real @ ( inverse_inverse_real @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) ).

% DERIV_ln
thf(fact_9701_DERIV__const__average,axiom,
    ! [A: real,B2: real,V: real > real,K: real] :
      ( ( A != B2 )
     => ( ! [X3: real] : ( has_fi5821293074295781190e_real @ V @ K @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
       => ( ( V @ ( divide_divide_real @ ( plus_plus_real @ A @ B2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
          = ( divide_divide_real @ ( plus_plus_real @ ( V @ A ) @ ( V @ B2 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% DERIV_const_average
thf(fact_9702_DERIV__local__max,axiom,
    ! [F: real > real,L2: real,X4: real,D: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ D )
       => ( ! [Y4: real] :
              ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X4 @ Y4 ) ) @ D )
             => ( ord_less_eq_real @ ( F @ Y4 ) @ ( F @ X4 ) ) )
         => ( L2 = zero_zero_real ) ) ) ) ).

% DERIV_local_max
thf(fact_9703_DERIV__local__min,axiom,
    ! [F: real > real,L2: real,X4: real,D: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ D )
       => ( ! [Y4: real] :
              ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X4 @ Y4 ) ) @ D )
             => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
         => ( L2 = zero_zero_real ) ) ) ) ).

% DERIV_local_min
thf(fact_9704_DERIV__ln__divide,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( has_fi5821293074295781190e_real @ ln_ln_real @ ( divide_divide_real @ one_one_real @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) ).

% DERIV_ln_divide
thf(fact_9705_DERIV__pow,axiom,
    ! [N: nat,X4: real,S: set_real] :
      ( has_fi5821293074295781190e_real
      @ ^ [X: real] : ( power_power_real @ X @ N )
      @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ X4 @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) )
      @ ( topolo2177554685111907308n_real @ X4 @ S ) ) ).

% DERIV_pow
thf(fact_9706_DERIV__fun__pow,axiom,
    ! [G: real > real,M: real,X4: real,N: nat] :
      ( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
     => ( has_fi5821293074295781190e_real
        @ ^ [X: real] : ( power_power_real @ ( G @ X ) @ N )
        @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( G @ X4 ) @ ( minus_minus_nat @ N @ one_one_nat ) ) ) @ M )
        @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) ).

% DERIV_fun_pow
thf(fact_9707_has__real__derivative__powr,axiom,
    ! [Z: real,R2: real] :
      ( ( ord_less_real @ zero_zero_real @ Z )
     => ( has_fi5821293074295781190e_real
        @ ^ [Z5: real] : ( powr_real @ Z5 @ R2 )
        @ ( times_times_real @ R2 @ ( powr_real @ Z @ ( minus_minus_real @ R2 @ one_one_real ) ) )
        @ ( topolo2177554685111907308n_real @ Z @ top_top_set_real ) ) ) ).

% has_real_derivative_powr
thf(fact_9708_DERIV__log,axiom,
    ! [X4: real,B2: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( has_fi5821293074295781190e_real @ ( log @ B2 ) @ ( divide_divide_real @ one_one_real @ ( times_times_real @ ( ln_ln_real @ B2 ) @ X4 ) ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) ).

% DERIV_log
thf(fact_9709_DERIV__fun__powr,axiom,
    ! [G: real > real,M: real,X4: real,R2: real] :
      ( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ ( G @ X4 ) )
       => ( has_fi5821293074295781190e_real
          @ ^ [X: real] : ( powr_real @ ( G @ X ) @ R2 )
          @ ( times_times_real @ ( times_times_real @ R2 @ ( powr_real @ ( G @ X4 ) @ ( minus_minus_real @ R2 @ ( semiri5074537144036343181t_real @ one_one_nat ) ) ) ) @ M )
          @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) ) ).

% DERIV_fun_powr
thf(fact_9710_DERIV__powr,axiom,
    ! [G: real > real,M: real,X4: real,F: real > real,R2: real] :
      ( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ ( G @ X4 ) )
       => ( ( has_fi5821293074295781190e_real @ F @ R2 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
         => ( has_fi5821293074295781190e_real
            @ ^ [X: real] : ( powr_real @ ( G @ X ) @ ( F @ X ) )
            @ ( times_times_real @ ( powr_real @ ( G @ X4 ) @ ( F @ X4 ) ) @ ( plus_plus_real @ ( times_times_real @ R2 @ ( ln_ln_real @ ( G @ X4 ) ) ) @ ( divide_divide_real @ ( times_times_real @ M @ ( F @ X4 ) ) @ ( G @ X4 ) ) ) )
            @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) ) ) ).

% DERIV_powr
thf(fact_9711_DERIV__real__sqrt,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( has_fi5821293074295781190e_real @ sqrt @ ( divide_divide_real @ ( inverse_inverse_real @ ( sqrt @ X4 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) ).

% DERIV_real_sqrt
thf(fact_9712_DERIV__series_H,axiom,
    ! [F: real > nat > real,F4: real > nat > real,X0: real,A: real,B2: real,L5: nat > real] :
      ( ! [N4: nat] :
          ( has_fi5821293074295781190e_real
          @ ^ [X: real] : ( F @ X @ N4 )
          @ ( F4 @ X0 @ N4 )
          @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ ( set_or1633881224788618240n_real @ A @ B2 ) )
           => ( summable_real @ ( F @ X3 ) ) )
       => ( ( member_real @ X0 @ ( set_or1633881224788618240n_real @ A @ B2 ) )
         => ( ( summable_real @ ( F4 @ X0 ) )
           => ( ( summable_real @ L5 )
             => ( ! [N4: nat,X3: real,Y4: real] :
                    ( ( member_real @ X3 @ ( set_or1633881224788618240n_real @ A @ B2 ) )
                   => ( ( member_real @ Y4 @ ( set_or1633881224788618240n_real @ A @ B2 ) )
                     => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( F @ X3 @ N4 ) @ ( F @ Y4 @ N4 ) ) ) @ ( times_times_real @ ( L5 @ N4 ) @ ( abs_abs_real @ ( minus_minus_real @ X3 @ Y4 ) ) ) ) ) )
               => ( has_fi5821293074295781190e_real
                  @ ^ [X: real] : ( suminf_real @ ( F @ X ) )
                  @ ( suminf_real @ ( F4 @ X0 ) )
                  @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) ) ) ) ) ) ) ) ).

% DERIV_series'
thf(fact_9713_DERIV__arctan,axiom,
    ! [X4: real] : ( has_fi5821293074295781190e_real @ arctan @ ( inverse_inverse_real @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ).

% DERIV_arctan
thf(fact_9714_arsinh__real__has__field__derivative,axiom,
    ! [X4: real,A3: set_real] : ( has_fi5821293074295781190e_real @ arsinh_real @ ( divide_divide_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) @ ( topolo2177554685111907308n_real @ X4 @ A3 ) ) ).

% arsinh_real_has_field_derivative
thf(fact_9715_DERIV__real__sqrt__generic,axiom,
    ! [X4: real,D4: real] :
      ( ( X4 != zero_zero_real )
     => ( ( ( ord_less_real @ zero_zero_real @ X4 )
         => ( D4
            = ( divide_divide_real @ ( inverse_inverse_real @ ( sqrt @ X4 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
       => ( ( ( ord_less_real @ X4 @ zero_zero_real )
           => ( D4
              = ( divide_divide_real @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( sqrt @ X4 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
         => ( has_fi5821293074295781190e_real @ sqrt @ D4 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) ) ) ).

% DERIV_real_sqrt_generic
thf(fact_9716_arcosh__real__has__field__derivative,axiom,
    ! [X4: real,A3: set_real] :
      ( ( ord_less_real @ one_one_real @ X4 )
     => ( has_fi5821293074295781190e_real @ arcosh_real @ ( divide_divide_real @ one_one_real @ ( sqrt @ ( minus_minus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) @ ( topolo2177554685111907308n_real @ X4 @ A3 ) ) ) ).

% arcosh_real_has_field_derivative
thf(fact_9717_artanh__real__has__field__derivative,axiom,
    ! [X4: real,A3: set_real] :
      ( ( ord_less_real @ ( abs_abs_real @ X4 ) @ one_one_real )
     => ( has_fi5821293074295781190e_real @ artanh_real @ ( divide_divide_real @ one_one_real @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X4 @ A3 ) ) ) ).

% artanh_real_has_field_derivative
thf(fact_9718_DERIV__power__series_H,axiom,
    ! [R: real,F: nat > real,X0: real] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ R ) @ R ) )
         => ( summable_real
            @ ^ [N2: nat] : ( times_times_real @ ( times_times_real @ ( F @ N2 ) @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) @ ( power_power_real @ X3 @ N2 ) ) ) )
     => ( ( member_real @ X0 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ R ) @ R ) )
       => ( ( ord_less_real @ zero_zero_real @ R )
         => ( has_fi5821293074295781190e_real
            @ ^ [X: real] :
                ( suminf_real
                @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ X @ ( suc @ N2 ) ) ) )
            @ ( suminf_real
              @ ^ [N2: nat] : ( times_times_real @ ( times_times_real @ ( F @ N2 ) @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) @ ( power_power_real @ X0 @ N2 ) ) )
            @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) ) ) ) ) ).

% DERIV_power_series'
thf(fact_9719_DERIV__real__root,axiom,
    ! [N: nat,X4: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ X4 )
       => ( has_fi5821293074295781190e_real @ ( root @ N ) @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X4 ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) ) ).

% DERIV_real_root
thf(fact_9720_DERIV__arccos,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X4 )
     => ( ( ord_less_real @ X4 @ one_one_real )
       => ( has_fi5821293074295781190e_real @ arccos @ ( inverse_inverse_real @ ( uminus_uminus_real @ ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) ) ).

% DERIV_arccos
thf(fact_9721_DERIV__arcsin,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X4 )
     => ( ( ord_less_real @ X4 @ one_one_real )
       => ( has_fi5821293074295781190e_real @ arcsin @ ( inverse_inverse_real @ ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) ) ).

% DERIV_arcsin
thf(fact_9722_Maclaurin__all__le__objl,axiom,
    ! [Diff: nat > real > real,F: real > real,X4: real,N: nat] :
      ( ( ( ( Diff @ zero_zero_nat )
          = F )
        & ! [M4: nat,X3: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) )
     => ? [T2: real] :
          ( ( ord_less_eq_real @ ( abs_abs_real @ T2 ) @ ( abs_abs_real @ X4 ) )
          & ( ( F @ X4 )
            = ( plus_plus_real
              @ ( groups6591440286371151544t_real
                @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ X4 @ M6 ) )
                @ ( set_ord_lessThan_nat @ N ) )
              @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T2 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X4 @ N ) ) ) ) ) ) ).

% Maclaurin_all_le_objl
thf(fact_9723_Maclaurin__all__le,axiom,
    ! [Diff: nat > real > real,F: real > real,X4: real,N: nat] :
      ( ( ( Diff @ zero_zero_nat )
        = F )
     => ( ! [M4: nat,X3: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
       => ? [T2: real] :
            ( ( ord_less_eq_real @ ( abs_abs_real @ T2 ) @ ( abs_abs_real @ X4 ) )
            & ( ( F @ X4 )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ X4 @ M6 ) )
                  @ ( set_ord_lessThan_nat @ N ) )
                @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T2 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X4 @ N ) ) ) ) ) ) ) ).

% Maclaurin_all_le
thf(fact_9724_DERIV__odd__real__root,axiom,
    ! [N: nat,X4: real] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( X4 != zero_zero_real )
       => ( has_fi5821293074295781190e_real @ ( root @ N ) @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X4 ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) ) ).

% DERIV_odd_real_root
thf(fact_9725_Maclaurin,axiom,
    ! [H2: real,N: nat,Diff: nat > real > real,F: real > real] :
      ( ( ord_less_real @ zero_zero_real @ H2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( ( Diff @ zero_zero_nat )
            = F )
         => ( ! [M4: nat,T2: real] :
                ( ( ( ord_less_nat @ M4 @ N )
                  & ( ord_less_eq_real @ zero_zero_real @ T2 )
                  & ( ord_less_eq_real @ T2 @ H2 ) )
               => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T2 ) @ ( topolo2177554685111907308n_real @ T2 @ top_top_set_real ) ) )
           => ? [T2: real] :
                ( ( ord_less_real @ zero_zero_real @ T2 )
                & ( ord_less_real @ T2 @ H2 )
                & ( ( F @ H2 )
                  = ( plus_plus_real
                    @ ( groups6591440286371151544t_real
                      @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ H2 @ M6 ) )
                      @ ( set_ord_lessThan_nat @ N ) )
                    @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T2 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H2 @ N ) ) ) ) ) ) ) ) ) ).

% Maclaurin
thf(fact_9726_Maclaurin2,axiom,
    ! [H2: real,Diff: nat > real > real,F: real > real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ H2 )
     => ( ( ( Diff @ zero_zero_nat )
          = F )
       => ( ! [M4: nat,T2: real] :
              ( ( ( ord_less_nat @ M4 @ N )
                & ( ord_less_eq_real @ zero_zero_real @ T2 )
                & ( ord_less_eq_real @ T2 @ H2 ) )
             => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T2 ) @ ( topolo2177554685111907308n_real @ T2 @ top_top_set_real ) ) )
         => ? [T2: real] :
              ( ( ord_less_real @ zero_zero_real @ T2 )
              & ( ord_less_eq_real @ T2 @ H2 )
              & ( ( F @ H2 )
                = ( plus_plus_real
                  @ ( groups6591440286371151544t_real
                    @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ H2 @ M6 ) )
                    @ ( set_ord_lessThan_nat @ N ) )
                  @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T2 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H2 @ N ) ) ) ) ) ) ) ) ).

% Maclaurin2
thf(fact_9727_Maclaurin__minus,axiom,
    ! [H2: real,N: nat,Diff: nat > real > real,F: real > real] :
      ( ( ord_less_real @ H2 @ zero_zero_real )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( ( Diff @ zero_zero_nat )
            = F )
         => ( ! [M4: nat,T2: real] :
                ( ( ( ord_less_nat @ M4 @ N )
                  & ( ord_less_eq_real @ H2 @ T2 )
                  & ( ord_less_eq_real @ T2 @ zero_zero_real ) )
               => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T2 ) @ ( topolo2177554685111907308n_real @ T2 @ top_top_set_real ) ) )
           => ? [T2: real] :
                ( ( ord_less_real @ H2 @ T2 )
                & ( ord_less_real @ T2 @ zero_zero_real )
                & ( ( F @ H2 )
                  = ( plus_plus_real
                    @ ( groups6591440286371151544t_real
                      @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ H2 @ M6 ) )
                      @ ( set_ord_lessThan_nat @ N ) )
                    @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T2 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H2 @ N ) ) ) ) ) ) ) ) ) ).

% Maclaurin_minus
thf(fact_9728_Maclaurin__all__lt,axiom,
    ! [Diff: nat > real > real,F: real > real,N: nat,X4: real] :
      ( ( ( Diff @ zero_zero_nat )
        = F )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( X4 != zero_zero_real )
         => ( ! [M4: nat,X3: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
           => ? [T2: real] :
                ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ T2 ) )
                & ( ord_less_real @ ( abs_abs_real @ T2 ) @ ( abs_abs_real @ X4 ) )
                & ( ( F @ X4 )
                  = ( plus_plus_real
                    @ ( groups6591440286371151544t_real
                      @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ X4 @ M6 ) )
                      @ ( set_ord_lessThan_nat @ N ) )
                    @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T2 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X4 @ N ) ) ) ) ) ) ) ) ) ).

% Maclaurin_all_lt
thf(fact_9729_Maclaurin__bi__le,axiom,
    ! [Diff: nat > real > real,F: real > real,N: nat,X4: real] :
      ( ( ( Diff @ zero_zero_nat )
        = F )
     => ( ! [M4: nat,T2: real] :
            ( ( ( ord_less_nat @ M4 @ N )
              & ( ord_less_eq_real @ ( abs_abs_real @ T2 ) @ ( abs_abs_real @ X4 ) ) )
           => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T2 ) @ ( topolo2177554685111907308n_real @ T2 @ top_top_set_real ) ) )
       => ? [T2: real] :
            ( ( ord_less_eq_real @ ( abs_abs_real @ T2 ) @ ( abs_abs_real @ X4 ) )
            & ( ( F @ X4 )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ X4 @ M6 ) )
                  @ ( set_ord_lessThan_nat @ N ) )
                @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T2 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X4 @ N ) ) ) ) ) ) ) ).

% Maclaurin_bi_le
thf(fact_9730_Taylor,axiom,
    ! [N: nat,Diff: nat > real > real,F: real > real,A: real,B2: real,C: real,X4: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ( Diff @ zero_zero_nat )
          = F )
       => ( ! [M4: nat,T2: real] :
              ( ( ( ord_less_nat @ M4 @ N )
                & ( ord_less_eq_real @ A @ T2 )
                & ( ord_less_eq_real @ T2 @ B2 ) )
             => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T2 ) @ ( topolo2177554685111907308n_real @ T2 @ top_top_set_real ) ) )
         => ( ( ord_less_eq_real @ A @ C )
           => ( ( ord_less_eq_real @ C @ B2 )
             => ( ( ord_less_eq_real @ A @ X4 )
               => ( ( ord_less_eq_real @ X4 @ B2 )
                 => ( ( X4 != C )
                   => ? [T2: real] :
                        ( ( ( ord_less_real @ X4 @ C )
                         => ( ( ord_less_real @ X4 @ T2 )
                            & ( ord_less_real @ T2 @ C ) ) )
                        & ( ~ ( ord_less_real @ X4 @ C )
                         => ( ( ord_less_real @ C @ T2 )
                            & ( ord_less_real @ T2 @ X4 ) ) )
                        & ( ( F @ X4 )
                          = ( plus_plus_real
                            @ ( groups6591440286371151544t_real
                              @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ C ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ ( minus_minus_real @ X4 @ C ) @ M6 ) )
                              @ ( set_ord_lessThan_nat @ N ) )
                            @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T2 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ X4 @ C ) @ N ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% Taylor
thf(fact_9731_Taylor__up,axiom,
    ! [N: nat,Diff: nat > real > real,F: real > real,A: real,B2: real,C: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ( Diff @ zero_zero_nat )
          = F )
       => ( ! [M4: nat,T2: real] :
              ( ( ( ord_less_nat @ M4 @ N )
                & ( ord_less_eq_real @ A @ T2 )
                & ( ord_less_eq_real @ T2 @ B2 ) )
             => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T2 ) @ ( topolo2177554685111907308n_real @ T2 @ top_top_set_real ) ) )
         => ( ( ord_less_eq_real @ A @ C )
           => ( ( ord_less_real @ C @ B2 )
             => ? [T2: real] :
                  ( ( ord_less_real @ C @ T2 )
                  & ( ord_less_real @ T2 @ B2 )
                  & ( ( F @ B2 )
                    = ( plus_plus_real
                      @ ( groups6591440286371151544t_real
                        @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ C ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ ( minus_minus_real @ B2 @ C ) @ M6 ) )
                        @ ( set_ord_lessThan_nat @ N ) )
                      @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T2 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ B2 @ C ) @ N ) ) ) ) ) ) ) ) ) ) ).

% Taylor_up
thf(fact_9732_Taylor__down,axiom,
    ! [N: nat,Diff: nat > real > real,F: real > real,A: real,B2: real,C: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ( Diff @ zero_zero_nat )
          = F )
       => ( ! [M4: nat,T2: real] :
              ( ( ( ord_less_nat @ M4 @ N )
                & ( ord_less_eq_real @ A @ T2 )
                & ( ord_less_eq_real @ T2 @ B2 ) )
             => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T2 ) @ ( topolo2177554685111907308n_real @ T2 @ top_top_set_real ) ) )
         => ( ( ord_less_real @ A @ C )
           => ( ( ord_less_eq_real @ C @ B2 )
             => ? [T2: real] :
                  ( ( ord_less_real @ A @ T2 )
                  & ( ord_less_real @ T2 @ C )
                  & ( ( F @ A )
                    = ( plus_plus_real
                      @ ( groups6591440286371151544t_real
                        @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ C ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ ( minus_minus_real @ A @ C ) @ M6 ) )
                        @ ( set_ord_lessThan_nat @ N ) )
                      @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T2 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ A @ C ) @ N ) ) ) ) ) ) ) ) ) ) ).

% Taylor_down
thf(fact_9733_Maclaurin__lemma2,axiom,
    ! [N: nat,H2: real,Diff: nat > real > real,K: nat,B4: real] :
      ( ! [M4: nat,T2: real] :
          ( ( ( ord_less_nat @ M4 @ N )
            & ( ord_less_eq_real @ zero_zero_real @ T2 )
            & ( ord_less_eq_real @ T2 @ H2 ) )
         => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T2 ) @ ( topolo2177554685111907308n_real @ T2 @ top_top_set_real ) ) )
     => ( ( N
          = ( suc @ K ) )
       => ! [M5: nat,T4: real] :
            ( ( ( ord_less_nat @ M5 @ N )
              & ( ord_less_eq_real @ zero_zero_real @ T4 )
              & ( ord_less_eq_real @ T4 @ H2 ) )
           => ( has_fi5821293074295781190e_real
              @ ^ [U4: real] :
                  ( minus_minus_real @ ( Diff @ M5 @ U4 )
                  @ ( plus_plus_real
                    @ ( groups6591440286371151544t_real
                      @ ^ [P3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ ( plus_plus_nat @ M5 @ P3 ) @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ P3 ) ) @ ( power_power_real @ U4 @ P3 ) )
                      @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ M5 ) ) )
                    @ ( times_times_real @ B4 @ ( divide_divide_real @ ( power_power_real @ U4 @ ( minus_minus_nat @ N @ M5 ) ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ M5 ) ) ) ) ) )
              @ ( minus_minus_real @ ( Diff @ ( suc @ M5 ) @ T4 )
                @ ( plus_plus_real
                  @ ( groups6591440286371151544t_real
                    @ ^ [P3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ ( plus_plus_nat @ ( suc @ M5 ) @ P3 ) @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ P3 ) ) @ ( power_power_real @ T4 @ P3 ) )
                    @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ ( suc @ M5 ) ) ) )
                  @ ( times_times_real @ B4 @ ( divide_divide_real @ ( power_power_real @ T4 @ ( minus_minus_nat @ N @ ( suc @ M5 ) ) ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ ( suc @ M5 ) ) ) ) ) ) )
              @ ( topolo2177554685111907308n_real @ T4 @ top_top_set_real ) ) ) ) ) ).

% Maclaurin_lemma2
thf(fact_9734_DERIV__arctan__series,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X4 ) @ one_one_real )
     => ( has_fi5821293074295781190e_real
        @ ^ [X9: real] :
            ( suminf_real
            @ ^ [K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X9 @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) )
        @ ( suminf_real
          @ ^ [K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( power_power_real @ X4 @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
        @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) ).

% DERIV_arctan_series
thf(fact_9735_isCont__Lb__Ub,axiom,
    ! [A: real,B2: real,F: real > real] :
      ( ( ord_less_eq_real @ A @ B2 )
     => ( ! [X3: real] :
            ( ( ( ord_less_eq_real @ A @ X3 )
              & ( ord_less_eq_real @ X3 @ B2 ) )
           => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) @ F ) )
       => ? [L6: real,M9: real] :
            ( ! [X5: real] :
                ( ( ( ord_less_eq_real @ A @ X5 )
                  & ( ord_less_eq_real @ X5 @ B2 ) )
               => ( ( ord_less_eq_real @ L6 @ ( F @ X5 ) )
                  & ( ord_less_eq_real @ ( F @ X5 ) @ M9 ) ) )
            & ! [Y5: real] :
                ( ( ( ord_less_eq_real @ L6 @ Y5 )
                  & ( ord_less_eq_real @ Y5 @ M9 ) )
               => ? [X3: real] :
                    ( ( ord_less_eq_real @ A @ X3 )
                    & ( ord_less_eq_real @ X3 @ B2 )
                    & ( ( F @ X3 )
                      = Y5 ) ) ) ) ) ) ).

% isCont_Lb_Ub
thf(fact_9736_LIM__fun__less__zero,axiom,
    ! [F: real > real,L2: real,C: real] :
      ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L2 ) @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
     => ( ( ord_less_real @ L2 @ zero_zero_real )
       => ? [R3: real] :
            ( ( ord_less_real @ zero_zero_real @ R3 )
            & ! [X5: real] :
                ( ( ( X5 != C )
                  & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X5 ) ) @ R3 ) )
               => ( ord_less_real @ ( F @ X5 ) @ zero_zero_real ) ) ) ) ) ).

% LIM_fun_less_zero
thf(fact_9737_LIM__fun__not__zero,axiom,
    ! [F: real > real,L2: real,C: real] :
      ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L2 ) @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
     => ( ( L2 != zero_zero_real )
       => ? [R3: real] :
            ( ( ord_less_real @ zero_zero_real @ R3 )
            & ! [X5: real] :
                ( ( ( X5 != C )
                  & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X5 ) ) @ R3 ) )
               => ( ( F @ X5 )
                 != zero_zero_real ) ) ) ) ) ).

% LIM_fun_not_zero
thf(fact_9738_LIM__fun__gt__zero,axiom,
    ! [F: real > real,L2: real,C: real] :
      ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L2 ) @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ L2 )
       => ? [R3: real] :
            ( ( ord_less_real @ zero_zero_real @ R3 )
            & ! [X5: real] :
                ( ( ( X5 != C )
                  & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X5 ) ) @ R3 ) )
               => ( ord_less_real @ zero_zero_real @ ( F @ X5 ) ) ) ) ) ) ).

% LIM_fun_gt_zero
thf(fact_9739_isCont__real__sqrt,axiom,
    ! [X4: real] : ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) @ sqrt ) ).

% isCont_real_sqrt
thf(fact_9740_isCont__real__root,axiom,
    ! [X4: real,N: nat] : ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) @ ( root @ N ) ) ).

% isCont_real_root
thf(fact_9741_continuous__frac,axiom,
    ! [X4: real] :
      ( ~ ( member_real @ X4 @ ring_1_Ints_real )
     => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) @ archim2898591450579166408c_real ) ) ).

% continuous_frac
thf(fact_9742_isCont__inverse__function2,axiom,
    ! [A: real,X4: real,B2: real,G: real > real,F: real > real] :
      ( ( ord_less_real @ A @ X4 )
     => ( ( ord_less_real @ X4 @ B2 )
       => ( ! [Z2: real] :
              ( ( ord_less_eq_real @ A @ Z2 )
             => ( ( ord_less_eq_real @ Z2 @ B2 )
               => ( ( G @ ( F @ Z2 ) )
                  = Z2 ) ) )
         => ( ! [Z2: real] :
                ( ( ord_less_eq_real @ A @ Z2 )
               => ( ( ord_less_eq_real @ Z2 @ B2 )
                 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) @ F ) ) )
           => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ ( F @ X4 ) @ top_top_set_real ) @ G ) ) ) ) ) ).

% isCont_inverse_function2
thf(fact_9743_isCont__arcosh,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ one_one_real @ X4 )
     => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) @ arcosh_real ) ) ).

% isCont_arcosh
thf(fact_9744_LIM__cos__div__sin,axiom,
    ( filterlim_real_real
    @ ^ [X: real] : ( divide_divide_real @ ( cos_real @ X ) @ ( sin_real @ X ) )
    @ ( topolo2815343760600316023s_real @ zero_zero_real )
    @ ( topolo2177554685111907308n_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ top_top_set_real ) ) ).

% LIM_cos_div_sin
thf(fact_9745_continuous__floor,axiom,
    ! [X4: real] :
      ( ~ ( member_real @ X4 @ ring_1_Ints_real )
     => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) @ ( comp_int_real_real @ ring_1_of_int_real @ archim6058952711729229775r_real ) ) ) ).

% continuous_floor
thf(fact_9746_DERIV__inverse__function,axiom,
    ! [F: real > real,D4: real,G: real > real,X4: real,A: real,B2: real] :
      ( ( has_fi5821293074295781190e_real @ F @ D4 @ ( topolo2177554685111907308n_real @ ( G @ X4 ) @ top_top_set_real ) )
     => ( ( D4 != zero_zero_real )
       => ( ( ord_less_real @ A @ X4 )
         => ( ( ord_less_real @ X4 @ B2 )
           => ( ! [Y4: real] :
                  ( ( ord_less_real @ A @ Y4 )
                 => ( ( ord_less_real @ Y4 @ B2 )
                   => ( ( F @ ( G @ Y4 ) )
                      = Y4 ) ) )
             => ( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) @ G )
               => ( has_fi5821293074295781190e_real @ G @ ( inverse_inverse_real @ D4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) ) ) ) ) ) ).

% DERIV_inverse_function
thf(fact_9747_isCont__arccos,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X4 )
     => ( ( ord_less_real @ X4 @ one_one_real )
       => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) @ arccos ) ) ) ).

% isCont_arccos
thf(fact_9748_isCont__arcsin,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X4 )
     => ( ( ord_less_real @ X4 @ one_one_real )
       => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) @ arcsin ) ) ) ).

% isCont_arcsin
thf(fact_9749_LIM__less__bound,axiom,
    ! [B2: real,X4: real,F: real > real] :
      ( ( ord_less_real @ B2 @ X4 )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ ( set_or1633881224788618240n_real @ B2 @ X4 ) )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
       => ( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) @ F )
         => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) ) ) ) ).

% LIM_less_bound
thf(fact_9750_isCont__artanh,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X4 )
     => ( ( ord_less_real @ X4 @ one_one_real )
       => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) @ artanh_real ) ) ) ).

% isCont_artanh
thf(fact_9751_isCont__inverse__function,axiom,
    ! [D: real,X4: real,G: real > real,F: real > real] :
      ( ( ord_less_real @ zero_zero_real @ D )
     => ( ! [Z2: real] :
            ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z2 @ X4 ) ) @ D )
           => ( ( G @ ( F @ Z2 ) )
              = Z2 ) )
       => ( ! [Z2: real] :
              ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z2 @ X4 ) ) @ D )
             => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) @ F ) )
         => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ ( F @ X4 ) @ top_top_set_real ) @ G ) ) ) ) ).

% isCont_inverse_function
thf(fact_9752_GMVT_H,axiom,
    ! [A: real,B2: real,F: real > real,G: real > real,G2: real > real,F4: real > real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ! [Z2: real] :
            ( ( ord_less_eq_real @ A @ Z2 )
           => ( ( ord_less_eq_real @ Z2 @ B2 )
             => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) @ F ) ) )
       => ( ! [Z2: real] :
              ( ( ord_less_eq_real @ A @ Z2 )
             => ( ( ord_less_eq_real @ Z2 @ B2 )
               => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) @ G ) ) )
         => ( ! [Z2: real] :
                ( ( ord_less_real @ A @ Z2 )
               => ( ( ord_less_real @ Z2 @ B2 )
                 => ( has_fi5821293074295781190e_real @ G @ ( G2 @ Z2 ) @ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) ) ) )
           => ( ! [Z2: real] :
                  ( ( ord_less_real @ A @ Z2 )
                 => ( ( ord_less_real @ Z2 @ B2 )
                   => ( has_fi5821293074295781190e_real @ F @ ( F4 @ Z2 ) @ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) ) ) )
             => ? [C3: real] :
                  ( ( ord_less_real @ A @ C3 )
                  & ( ord_less_real @ C3 @ B2 )
                  & ( ( times_times_real @ ( minus_minus_real @ ( F @ B2 ) @ ( F @ A ) ) @ ( G2 @ C3 ) )
                    = ( times_times_real @ ( minus_minus_real @ ( G @ B2 ) @ ( G @ A ) ) @ ( F4 @ C3 ) ) ) ) ) ) ) ) ) ).

% GMVT'
thf(fact_9753_summable__Leibniz_I2_J,axiom,
    ! [A: nat > real] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ( topolo6980174941875973593q_real @ A )
       => ( ( ord_less_real @ zero_zero_real @ ( A @ zero_zero_nat ) )
         => ! [N6: nat] :
              ( member_real
              @ ( suminf_real
                @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) ) )
              @ ( set_or1222579329274155063t_real
                @ ( groups6591440286371151544t_real
                  @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
                  @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N6 ) ) )
                @ ( groups6591440286371151544t_real
                  @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
                  @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N6 ) @ one_one_nat ) ) ) ) ) ) ) ) ).

% summable_Leibniz(2)
thf(fact_9754_summable__Leibniz_I3_J,axiom,
    ! [A: nat > real] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ( topolo6980174941875973593q_real @ A )
       => ( ( ord_less_real @ ( A @ zero_zero_nat ) @ zero_zero_real )
         => ! [N6: nat] :
              ( member_real
              @ ( suminf_real
                @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) ) )
              @ ( set_or1222579329274155063t_real
                @ ( groups6591440286371151544t_real
                  @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
                  @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N6 ) @ one_one_nat ) ) )
                @ ( groups6591440286371151544t_real
                  @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
                  @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N6 ) ) ) ) ) ) ) ) ).

% summable_Leibniz(3)
thf(fact_9755_incseq__convergent,axiom,
    ! [X8: nat > real,B4: real] :
      ( ( order_mono_nat_real @ X8 )
     => ( ! [I3: nat] : ( ord_less_eq_real @ ( X8 @ I3 ) @ B4 )
       => ~ ! [L6: real] :
              ( ( filterlim_nat_real @ X8 @ ( topolo2815343760600316023s_real @ L6 ) @ at_top_nat )
             => ~ ! [I4: nat] : ( ord_less_eq_real @ ( X8 @ I4 ) @ L6 ) ) ) ) ).

% incseq_convergent
thf(fact_9756_mult__nat__left__at__top,axiom,
    ! [C: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ C )
     => ( filterlim_nat_nat @ ( times_times_nat @ C ) @ at_top_nat @ at_top_nat ) ) ).

% mult_nat_left_at_top
thf(fact_9757_mult__nat__right__at__top,axiom,
    ! [C: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ C )
     => ( filterlim_nat_nat
        @ ^ [X: nat] : ( times_times_nat @ X @ C )
        @ at_top_nat
        @ at_top_nat ) ) ).

% mult_nat_right_at_top
thf(fact_9758_monoseq__convergent,axiom,
    ! [X8: nat > real,B4: real] :
      ( ( topolo6980174941875973593q_real @ X8 )
     => ( ! [I3: nat] : ( ord_less_eq_real @ ( abs_abs_real @ ( X8 @ I3 ) ) @ B4 )
       => ~ ! [L6: real] :
              ~ ( filterlim_nat_real @ X8 @ ( topolo2815343760600316023s_real @ L6 ) @ at_top_nat ) ) ) ).

% monoseq_convergent
thf(fact_9759_LIMSEQ__root,axiom,
    ( filterlim_nat_real
    @ ^ [N2: nat] : ( root @ N2 @ ( semiri5074537144036343181t_real @ N2 ) )
    @ ( topolo2815343760600316023s_real @ one_one_real )
    @ at_top_nat ) ).

% LIMSEQ_root
thf(fact_9760_nested__sequence__unique,axiom,
    ! [F: nat > real,G: nat > real] :
      ( ! [N4: nat] : ( ord_less_eq_real @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
     => ( ! [N4: nat] : ( ord_less_eq_real @ ( G @ ( suc @ N4 ) ) @ ( G @ N4 ) )
       => ( ! [N4: nat] : ( ord_less_eq_real @ ( F @ N4 ) @ ( G @ N4 ) )
         => ( ( filterlim_nat_real
              @ ^ [N2: nat] : ( minus_minus_real @ ( F @ N2 ) @ ( G @ N2 ) )
              @ ( topolo2815343760600316023s_real @ zero_zero_real )
              @ at_top_nat )
           => ? [L3: real] :
                ( ! [N6: nat] : ( ord_less_eq_real @ ( F @ N6 ) @ L3 )
                & ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L3 ) @ at_top_nat )
                & ! [N6: nat] : ( ord_less_eq_real @ L3 @ ( G @ N6 ) )
                & ( filterlim_nat_real @ G @ ( topolo2815343760600316023s_real @ L3 ) @ at_top_nat ) ) ) ) ) ) ).

% nested_sequence_unique
thf(fact_9761_LIMSEQ__inverse__zero,axiom,
    ! [X8: nat > real] :
      ( ! [R3: real] :
        ? [N7: nat] :
        ! [N4: nat] :
          ( ( ord_less_eq_nat @ N7 @ N4 )
         => ( ord_less_real @ R3 @ ( X8 @ N4 ) ) )
     => ( filterlim_nat_real
        @ ^ [N2: nat] : ( inverse_inverse_real @ ( X8 @ N2 ) )
        @ ( topolo2815343760600316023s_real @ zero_zero_real )
        @ at_top_nat ) ) ).

% LIMSEQ_inverse_zero
thf(fact_9762_lim__inverse__n_H,axiom,
    ( filterlim_nat_real
    @ ^ [N2: nat] : ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N2 ) )
    @ ( topolo2815343760600316023s_real @ zero_zero_real )
    @ at_top_nat ) ).

% lim_inverse_n'
thf(fact_9763_LIMSEQ__root__const,axiom,
    ! [C: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( filterlim_nat_real
        @ ^ [N2: nat] : ( root @ N2 @ C )
        @ ( topolo2815343760600316023s_real @ one_one_real )
        @ at_top_nat ) ) ).

% LIMSEQ_root_const
thf(fact_9764_LIMSEQ__inverse__real__of__nat,axiom,
    ( filterlim_nat_real
    @ ^ [N2: nat] : ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) )
    @ ( topolo2815343760600316023s_real @ zero_zero_real )
    @ at_top_nat ) ).

% LIMSEQ_inverse_real_of_nat
thf(fact_9765_LIMSEQ__inverse__real__of__nat__add,axiom,
    ! [R2: real] :
      ( filterlim_nat_real
      @ ^ [N2: nat] : ( plus_plus_real @ R2 @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) )
      @ ( topolo2815343760600316023s_real @ R2 )
      @ at_top_nat ) ).

% LIMSEQ_inverse_real_of_nat_add
thf(fact_9766_increasing__LIMSEQ,axiom,
    ! [F: nat > real,L2: real] :
      ( ! [N4: nat] : ( ord_less_eq_real @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
     => ( ! [N4: nat] : ( ord_less_eq_real @ ( F @ N4 ) @ L2 )
       => ( ! [E2: real] :
              ( ( ord_less_real @ zero_zero_real @ E2 )
             => ? [N6: nat] : ( ord_less_eq_real @ L2 @ ( plus_plus_real @ ( F @ N6 ) @ E2 ) ) )
         => ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L2 ) @ at_top_nat ) ) ) ) ).

% increasing_LIMSEQ
thf(fact_9767_LIMSEQ__realpow__zero,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_real @ X4 @ one_one_real )
       => ( filterlim_nat_real @ ( power_power_real @ X4 ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ) ).

% LIMSEQ_realpow_zero
thf(fact_9768_LIMSEQ__divide__realpow__zero,axiom,
    ! [X4: real,A: real] :
      ( ( ord_less_real @ one_one_real @ X4 )
     => ( filterlim_nat_real
        @ ^ [N2: nat] : ( divide_divide_real @ A @ ( power_power_real @ X4 @ N2 ) )
        @ ( topolo2815343760600316023s_real @ zero_zero_real )
        @ at_top_nat ) ) ).

% LIMSEQ_divide_realpow_zero
thf(fact_9769_LIMSEQ__abs__realpow__zero,axiom,
    ! [C: real] :
      ( ( ord_less_real @ ( abs_abs_real @ C ) @ one_one_real )
     => ( filterlim_nat_real @ ( power_power_real @ ( abs_abs_real @ C ) ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ).

% LIMSEQ_abs_realpow_zero
thf(fact_9770_LIMSEQ__abs__realpow__zero2,axiom,
    ! [C: real] :
      ( ( ord_less_real @ ( abs_abs_real @ C ) @ one_one_real )
     => ( filterlim_nat_real @ ( power_power_real @ C ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ).

% LIMSEQ_abs_realpow_zero2
thf(fact_9771_LIMSEQ__inverse__realpow__zero,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ one_one_real @ X4 )
     => ( filterlim_nat_real
        @ ^ [N2: nat] : ( inverse_inverse_real @ ( power_power_real @ X4 @ N2 ) )
        @ ( topolo2815343760600316023s_real @ zero_zero_real )
        @ at_top_nat ) ) ).

% LIMSEQ_inverse_realpow_zero
thf(fact_9772_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
    ! [R2: real] :
      ( filterlim_nat_real
      @ ^ [N2: nat] : ( plus_plus_real @ R2 @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) ) )
      @ ( topolo2815343760600316023s_real @ R2 )
      @ at_top_nat ) ).

% LIMSEQ_inverse_real_of_nat_add_minus
thf(fact_9773_tendsto__exp__limit__sequentially,axiom,
    ! [X4: real] :
      ( filterlim_nat_real
      @ ^ [N2: nat] : ( power_power_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X4 @ ( semiri5074537144036343181t_real @ N2 ) ) ) @ N2 )
      @ ( topolo2815343760600316023s_real @ ( exp_real @ X4 ) )
      @ at_top_nat ) ).

% tendsto_exp_limit_sequentially
thf(fact_9774_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
    ! [R2: real] :
      ( filterlim_nat_real
      @ ^ [N2: nat] : ( times_times_real @ R2 @ ( plus_plus_real @ one_one_real @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) ) ) )
      @ ( topolo2815343760600316023s_real @ R2 )
      @ at_top_nat ) ).

% LIMSEQ_inverse_real_of_nat_add_minus_mult
thf(fact_9775_summable__Leibniz_I1_J,axiom,
    ! [A: nat > real] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ( topolo6980174941875973593q_real @ A )
       => ( summable_real
          @ ^ [N2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( A @ N2 ) ) ) ) ) ).

% summable_Leibniz(1)
thf(fact_9776_summable,axiom,
    ! [A: nat > real] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ! [N4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N4 ) )
       => ( ! [N4: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N4 ) ) @ ( A @ N4 ) )
         => ( summable_real
            @ ^ [N2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( A @ N2 ) ) ) ) ) ) ).

% summable
thf(fact_9777_cos__diff__limit__1,axiom,
    ! [Theta: nat > real,Theta2: real] :
      ( ( filterlim_nat_real
        @ ^ [J3: nat] : ( cos_real @ ( minus_minus_real @ ( Theta @ J3 ) @ Theta2 ) )
        @ ( topolo2815343760600316023s_real @ one_one_real )
        @ at_top_nat )
     => ~ ! [K2: nat > int] :
            ~ ( filterlim_nat_real
              @ ^ [J3: nat] : ( minus_minus_real @ ( Theta @ J3 ) @ ( times_times_real @ ( ring_1_of_int_real @ ( K2 @ J3 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
              @ ( topolo2815343760600316023s_real @ Theta2 )
              @ at_top_nat ) ) ).

% cos_diff_limit_1
thf(fact_9778_cos__limit__1,axiom,
    ! [Theta: nat > real] :
      ( ( filterlim_nat_real
        @ ^ [J3: nat] : ( cos_real @ ( Theta @ J3 ) )
        @ ( topolo2815343760600316023s_real @ one_one_real )
        @ at_top_nat )
     => ? [K2: nat > int] :
          ( filterlim_nat_real
          @ ^ [J3: nat] : ( minus_minus_real @ ( Theta @ J3 ) @ ( times_times_real @ ( ring_1_of_int_real @ ( K2 @ J3 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
          @ ( topolo2815343760600316023s_real @ zero_zero_real )
          @ at_top_nat ) ) ).

% cos_limit_1
thf(fact_9779_summable__Leibniz_I4_J,axiom,
    ! [A: nat > real] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ( topolo6980174941875973593q_real @ A )
       => ( filterlim_nat_real
          @ ^ [N2: nat] :
              ( groups6591440286371151544t_real
              @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
              @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
          @ ( topolo2815343760600316023s_real
            @ ( suminf_real
              @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) ) ) )
          @ at_top_nat ) ) ) ).

% summable_Leibniz(4)
thf(fact_9780_zeroseq__arctan__series,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X4 ) @ one_one_real )
     => ( filterlim_nat_real
        @ ^ [N2: nat] : ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X4 @ ( plus_plus_nat @ ( times_times_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) )
        @ ( topolo2815343760600316023s_real @ zero_zero_real )
        @ at_top_nat ) ) ).

% zeroseq_arctan_series
thf(fact_9781_summable__Leibniz_H_I2_J,axiom,
    ! [A: nat > real,N: nat] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ! [N4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N4 ) )
       => ( ! [N4: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N4 ) ) @ ( A @ N4 ) )
         => ( ord_less_eq_real
            @ ( groups6591440286371151544t_real
              @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
              @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
            @ ( suminf_real
              @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) ) ) ) ) ) ) ).

% summable_Leibniz'(2)
thf(fact_9782_summable__Leibniz_H_I3_J,axiom,
    ! [A: nat > real] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ! [N4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N4 ) )
       => ( ! [N4: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N4 ) ) @ ( A @ N4 ) )
         => ( filterlim_nat_real
            @ ^ [N2: nat] :
                ( groups6591440286371151544t_real
                @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
                @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
            @ ( topolo2815343760600316023s_real
              @ ( suminf_real
                @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) ) ) )
            @ at_top_nat ) ) ) ) ).

% summable_Leibniz'(3)
thf(fact_9783_sums__alternating__upper__lower,axiom,
    ! [A: nat > real] :
      ( ! [N4: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N4 ) ) @ ( A @ N4 ) )
     => ( ! [N4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N4 ) )
       => ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
         => ? [L3: real] :
              ( ! [N6: nat] :
                  ( ord_less_eq_real
                  @ ( groups6591440286371151544t_real
                    @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
                    @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N6 ) ) )
                  @ L3 )
              & ( filterlim_nat_real
                @ ^ [N2: nat] :
                    ( groups6591440286371151544t_real
                    @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
                    @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
                @ ( topolo2815343760600316023s_real @ L3 )
                @ at_top_nat )
              & ! [N6: nat] :
                  ( ord_less_eq_real @ L3
                  @ ( groups6591440286371151544t_real
                    @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
                    @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N6 ) @ one_one_nat ) ) ) )
              & ( filterlim_nat_real
                @ ^ [N2: nat] :
                    ( groups6591440286371151544t_real
                    @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
                    @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) )
                @ ( topolo2815343760600316023s_real @ L3 )
                @ at_top_nat ) ) ) ) ) ).

% sums_alternating_upper_lower
thf(fact_9784_summable__Leibniz_I5_J,axiom,
    ! [A: nat > real] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ( topolo6980174941875973593q_real @ A )
       => ( filterlim_nat_real
          @ ^ [N2: nat] :
              ( groups6591440286371151544t_real
              @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
              @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) )
          @ ( topolo2815343760600316023s_real
            @ ( suminf_real
              @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) ) ) )
          @ at_top_nat ) ) ) ).

% summable_Leibniz(5)
thf(fact_9785_summable__Leibniz_H_I5_J,axiom,
    ! [A: nat > real] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ! [N4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N4 ) )
       => ( ! [N4: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N4 ) ) @ ( A @ N4 ) )
         => ( filterlim_nat_real
            @ ^ [N2: nat] :
                ( groups6591440286371151544t_real
                @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
                @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) )
            @ ( topolo2815343760600316023s_real
              @ ( suminf_real
                @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) ) ) )
            @ at_top_nat ) ) ) ) ).

% summable_Leibniz'(5)
thf(fact_9786_summable__Leibniz_H_I4_J,axiom,
    ! [A: nat > real,N: nat] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ! [N4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N4 ) )
       => ( ! [N4: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N4 ) ) @ ( A @ N4 ) )
         => ( ord_less_eq_real
            @ ( suminf_real
              @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) ) )
            @ ( groups6591440286371151544t_real
              @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
              @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ) ) ) ).

% summable_Leibniz'(4)
thf(fact_9787_eventually__sequentially__Suc,axiom,
    ! [P: nat > $o] :
      ( ( eventually_nat
        @ ^ [I2: nat] : ( P @ ( suc @ I2 ) )
        @ at_top_nat )
      = ( eventually_nat @ P @ at_top_nat ) ) ).

% eventually_sequentially_Suc
thf(fact_9788_eventually__sequentially__seg,axiom,
    ! [P: nat > $o,K: nat] :
      ( ( eventually_nat
        @ ^ [N2: nat] : ( P @ ( plus_plus_nat @ N2 @ K ) )
        @ at_top_nat )
      = ( eventually_nat @ P @ at_top_nat ) ) ).

% eventually_sequentially_seg
thf(fact_9789_sequentially__offset,axiom,
    ! [P: nat > $o,K: nat] :
      ( ( eventually_nat @ P @ at_top_nat )
     => ( eventually_nat
        @ ^ [I2: nat] : ( P @ ( plus_plus_nat @ I2 @ K ) )
        @ at_top_nat ) ) ).

% sequentially_offset
thf(fact_9790_eventually__sequentially,axiom,
    ! [P: nat > $o] :
      ( ( eventually_nat @ P @ at_top_nat )
      = ( ? [N8: nat] :
          ! [N2: nat] :
            ( ( ord_less_eq_nat @ N8 @ N2 )
           => ( P @ N2 ) ) ) ) ).

% eventually_sequentially
thf(fact_9791_eventually__sequentiallyI,axiom,
    ! [C: nat,P: nat > $o] :
      ( ! [X3: nat] :
          ( ( ord_less_eq_nat @ C @ X3 )
         => ( P @ X3 ) )
     => ( eventually_nat @ P @ at_top_nat ) ) ).

% eventually_sequentiallyI
thf(fact_9792_le__sequentially,axiom,
    ! [F5: filter_nat] :
      ( ( ord_le2510731241096832064er_nat @ F5 @ at_top_nat )
      = ( ! [N8: nat] : ( eventually_nat @ ( ord_less_eq_nat @ N8 ) @ F5 ) ) ) ).

% le_sequentially
thf(fact_9793_filterlim__Suc,axiom,
    filterlim_nat_nat @ suc @ at_top_nat @ at_top_nat ).

% filterlim_Suc
thf(fact_9794_real__bounded__linear,axiom,
    ( real_V5970128139526366754l_real
    = ( ^ [F2: real > real] :
        ? [C2: real] :
          ( F2
          = ( ^ [X: real] : ( times_times_real @ X @ C2 ) ) ) ) ) ).

% real_bounded_linear
thf(fact_9795_sqrt__at__top,axiom,
    filterlim_real_real @ sqrt @ at_top_real @ at_top_real ).

% sqrt_at_top
thf(fact_9796_lhopital__left__at__top__at__top,axiom,
    ! [F: real > real,A: real,G: real > real,F4: real > real,G2: 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
            @ ^ [X: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
            @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
         => ( ( eventually_real
              @ ^ [X: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
              @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
           => ( ( filterlim_real_real
                @ ^ [X: real] : ( divide_divide_real @ ( F4 @ X ) @ ( G2 @ X ) )
                @ at_top_real
                @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
             => ( filterlim_real_real
                @ ^ [X: real] : ( divide_divide_real @ ( F @ X ) @ ( G @ X ) )
                @ at_top_real
                @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) ) ) ) ) ) ) ).

% lhopital_left_at_top_at_top
thf(fact_9797_lhopital__at__top__at__top,axiom,
    ! [F: real > real,A: real,G: real > real,F4: real > real,G2: 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
            @ ^ [X: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
            @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
         => ( ( eventually_real
              @ ^ [X: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
              @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
           => ( ( filterlim_real_real
                @ ^ [X: real] : ( divide_divide_real @ ( F4 @ X ) @ ( G2 @ X ) )
                @ at_top_real
                @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
             => ( filterlim_real_real
                @ ^ [X: real] : ( divide_divide_real @ ( F @ X ) @ ( G @ X ) )
                @ at_top_real
                @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) ) ) ) ) ) ) ).

% lhopital_at_top_at_top
thf(fact_9798_eventually__at__left__real,axiom,
    ! [B2: real,A: real] :
      ( ( ord_less_real @ B2 @ A )
     => ( eventually_real
        @ ^ [X: real] : ( member_real @ X @ ( set_or1633881224788618240n_real @ B2 @ A ) )
        @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) ) ) ).

% eventually_at_left_real
thf(fact_9799_lhopital__left__at__top,axiom,
    ! [G: real > real,X4: real,G2: real > real,F: real > real,F4: real > real,Y3: real] :
      ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ X4 @ ( set_or5984915006950818249n_real @ X4 ) ) )
     => ( ( eventually_real
          @ ^ [X: real] :
              ( ( G2 @ X )
             != zero_zero_real )
          @ ( topolo2177554685111907308n_real @ X4 @ ( set_or5984915006950818249n_real @ X4 ) ) )
       => ( ( eventually_real
            @ ^ [X: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
            @ ( topolo2177554685111907308n_real @ X4 @ ( set_or5984915006950818249n_real @ X4 ) ) )
         => ( ( eventually_real
              @ ^ [X: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
              @ ( topolo2177554685111907308n_real @ X4 @ ( set_or5984915006950818249n_real @ X4 ) ) )
           => ( ( filterlim_real_real
                @ ^ [X: real] : ( divide_divide_real @ ( F4 @ X ) @ ( G2 @ X ) )
                @ ( topolo2815343760600316023s_real @ Y3 )
                @ ( topolo2177554685111907308n_real @ X4 @ ( set_or5984915006950818249n_real @ X4 ) ) )
             => ( filterlim_real_real
                @ ^ [X: real] : ( divide_divide_real @ ( F @ X ) @ ( G @ X ) )
                @ ( topolo2815343760600316023s_real @ Y3 )
                @ ( topolo2177554685111907308n_real @ X4 @ ( set_or5984915006950818249n_real @ X4 ) ) ) ) ) ) ) ) ).

% lhopital_left_at_top
thf(fact_9800_lhospital__at__top__at__top,axiom,
    ! [G: real > real,G2: real > real,F: real > real,F4: real > real,X4: real] :
      ( ( filterlim_real_real @ G @ at_top_real @ at_top_real )
     => ( ( eventually_real
          @ ^ [X: real] :
              ( ( G2 @ X )
             != zero_zero_real )
          @ at_top_real )
       => ( ( eventually_real
            @ ^ [X: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
            @ at_top_real )
         => ( ( eventually_real
              @ ^ [X: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
              @ at_top_real )
           => ( ( filterlim_real_real
                @ ^ [X: real] : ( divide_divide_real @ ( F4 @ X ) @ ( G2 @ X ) )
                @ ( topolo2815343760600316023s_real @ X4 )
                @ at_top_real )
             => ( filterlim_real_real
                @ ^ [X: real] : ( divide_divide_real @ ( F @ X ) @ ( G @ X ) )
                @ ( topolo2815343760600316023s_real @ X4 )
                @ at_top_real ) ) ) ) ) ) ).

% lhospital_at_top_at_top
thf(fact_9801_lhopital__at__top,axiom,
    ! [G: real > real,X4: real,G2: real > real,F: real > real,F4: real > real,Y3: real] :
      ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
     => ( ( eventually_real
          @ ^ [X: real] :
              ( ( G2 @ X )
             != zero_zero_real )
          @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
       => ( ( eventually_real
            @ ^ [X: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
            @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
         => ( ( eventually_real
              @ ^ [X: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
              @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
           => ( ( filterlim_real_real
                @ ^ [X: real] : ( divide_divide_real @ ( F4 @ X ) @ ( G2 @ X ) )
                @ ( topolo2815343760600316023s_real @ Y3 )
                @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
             => ( filterlim_real_real
                @ ^ [X: real] : ( divide_divide_real @ ( F @ X ) @ ( G @ X ) )
                @ ( topolo2815343760600316023s_real @ Y3 )
                @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) ) ) ) ) ).

% lhopital_at_top
thf(fact_9802_tanh__real__at__top,axiom,
    filterlim_real_real @ tanh_real @ ( topolo2815343760600316023s_real @ one_one_real ) @ at_top_real ).

% tanh_real_at_top
thf(fact_9803_artanh__real__at__left__1,axiom,
    filterlim_real_real @ artanh_real @ at_top_real @ ( topolo2177554685111907308n_real @ one_one_real @ ( set_or5984915006950818249n_real @ one_one_real ) ) ).

% artanh_real_at_left_1
thf(fact_9804_ln__x__over__x__tendsto__0,axiom,
    ( filterlim_real_real
    @ ^ [X: real] : ( divide_divide_real @ ( ln_ln_real @ X ) @ X )
    @ ( topolo2815343760600316023s_real @ zero_zero_real )
    @ at_top_real ) ).

% ln_x_over_x_tendsto_0
thf(fact_9805_tendsto__power__div__exp__0,axiom,
    ! [K: nat] :
      ( filterlim_real_real
      @ ^ [X: real] : ( divide_divide_real @ ( power_power_real @ X @ K ) @ ( exp_real @ X ) )
      @ ( topolo2815343760600316023s_real @ zero_zero_real )
      @ at_top_real ) ).

% tendsto_power_div_exp_0
thf(fact_9806_lhopital__left,axiom,
    ! [F: real > real,X4: real,G: real > real,G2: real > real,F4: real > real,F5: filter_real] :
      ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X4 @ ( set_or5984915006950818249n_real @ X4 ) ) )
     => ( ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X4 @ ( set_or5984915006950818249n_real @ X4 ) ) )
       => ( ( eventually_real
            @ ^ [X: real] :
                ( ( G @ X )
               != zero_zero_real )
            @ ( topolo2177554685111907308n_real @ X4 @ ( set_or5984915006950818249n_real @ X4 ) ) )
         => ( ( eventually_real
              @ ^ [X: real] :
                  ( ( G2 @ X )
                 != zero_zero_real )
              @ ( topolo2177554685111907308n_real @ X4 @ ( set_or5984915006950818249n_real @ X4 ) ) )
           => ( ( eventually_real
                @ ^ [X: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
                @ ( topolo2177554685111907308n_real @ X4 @ ( set_or5984915006950818249n_real @ X4 ) ) )
             => ( ( eventually_real
                  @ ^ [X: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
                  @ ( topolo2177554685111907308n_real @ X4 @ ( set_or5984915006950818249n_real @ X4 ) ) )
               => ( ( filterlim_real_real
                    @ ^ [X: real] : ( divide_divide_real @ ( F4 @ X ) @ ( G2 @ X ) )
                    @ F5
                    @ ( topolo2177554685111907308n_real @ X4 @ ( set_or5984915006950818249n_real @ X4 ) ) )
                 => ( filterlim_real_real
                    @ ^ [X: real] : ( divide_divide_real @ ( F @ X ) @ ( G @ X ) )
                    @ F5
                    @ ( topolo2177554685111907308n_real @ X4 @ ( set_or5984915006950818249n_real @ X4 ) ) ) ) ) ) ) ) ) ) ).

% lhopital_left
thf(fact_9807_lhopital,axiom,
    ! [F: real > real,X4: real,G: real > real,G2: real > real,F4: real > real,F5: filter_real] :
      ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
     => ( ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
       => ( ( eventually_real
            @ ^ [X: real] :
                ( ( G @ X )
               != zero_zero_real )
            @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
         => ( ( eventually_real
              @ ^ [X: real] :
                  ( ( G2 @ X )
                 != zero_zero_real )
              @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
           => ( ( eventually_real
                @ ^ [X: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
                @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
             => ( ( eventually_real
                  @ ^ [X: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
                  @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
               => ( ( filterlim_real_real
                    @ ^ [X: real] : ( divide_divide_real @ ( F4 @ X ) @ ( G2 @ X ) )
                    @ F5
                    @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
                 => ( filterlim_real_real
                    @ ^ [X: real] : ( divide_divide_real @ ( F @ X ) @ ( G @ X ) )
                    @ F5
                    @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) ) ) ) ) ) ) ).

% lhopital
thf(fact_9808_tendsto__exp__limit__at__top,axiom,
    ! [X4: real] :
      ( filterlim_real_real
      @ ^ [Y: real] : ( powr_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X4 @ Y ) ) @ Y )
      @ ( topolo2815343760600316023s_real @ ( exp_real @ X4 ) )
      @ at_top_real ) ).

% tendsto_exp_limit_at_top
thf(fact_9809_filterlim__tan__at__left,axiom,
    filterlim_real_real @ tan_real @ at_top_real @ ( topolo2177554685111907308n_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( set_or5984915006950818249n_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% filterlim_tan_at_left
thf(fact_9810_DERIV__neg__imp__decreasing__at__top,axiom,
    ! [B2: real,F: real > real,Flim: real] :
      ( ! [X3: real] :
          ( ( ord_less_eq_real @ B2 @ X3 )
         => ? [Y5: real] :
              ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
              & ( ord_less_real @ Y5 @ zero_zero_real ) ) )
     => ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ Flim ) @ at_top_real )
       => ( ord_less_real @ Flim @ ( F @ B2 ) ) ) ) ).

% DERIV_neg_imp_decreasing_at_top
thf(fact_9811_tendsto__arctan__at__top,axiom,
    filterlim_real_real @ arctan @ ( topolo2815343760600316023s_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ at_top_real ).

% tendsto_arctan_at_top
thf(fact_9812_filterlim__pow__at__bot__even,axiom,
    ! [N: nat,F: real > real,F5: filter_real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( filterlim_real_real @ F @ at_bot_real @ F5 )
       => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
         => ( filterlim_real_real
            @ ^ [X: real] : ( power_power_real @ ( F @ X ) @ N )
            @ at_top_real
            @ F5 ) ) ) ) ).

% filterlim_pow_at_bot_even
thf(fact_9813_dist__real__def,axiom,
    ( real_V975177566351809787t_real
    = ( ^ [X: real,Y: real] : ( abs_abs_real @ ( minus_minus_real @ X @ Y ) ) ) ) ).

% dist_real_def
thf(fact_9814_dist__complex__def,axiom,
    ( real_V3694042436643373181omplex
    = ( ^ [X: complex,Y: complex] : ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Y ) ) ) ) ).

% dist_complex_def
thf(fact_9815_tanh__real__at__bot,axiom,
    filterlim_real_real @ tanh_real @ ( topolo2815343760600316023s_real @ ( uminus_uminus_real @ one_one_real ) ) @ at_bot_real ).

% tanh_real_at_bot
thf(fact_9816_lhopital__at__top__at__bot,axiom,
    ! [F: real > real,A: real,G: real > real,F4: real > real,G2: 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
            @ ^ [X: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
            @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
         => ( ( eventually_real
              @ ^ [X: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
              @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
           => ( ( filterlim_real_real
                @ ^ [X: real] : ( divide_divide_real @ ( F4 @ X ) @ ( G2 @ X ) )
                @ at_bot_real
                @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
             => ( filterlim_real_real
                @ ^ [X: real] : ( divide_divide_real @ ( F @ X ) @ ( G @ X ) )
                @ at_bot_real
                @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) ) ) ) ) ) ) ).

% lhopital_at_top_at_bot
thf(fact_9817_DERIV__pos__imp__increasing__at__bot,axiom,
    ! [B2: real,F: real > real,Flim: real] :
      ( ! [X3: real] :
          ( ( ord_less_eq_real @ X3 @ B2 )
         => ? [Y5: real] :
              ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
              & ( ord_less_real @ zero_zero_real @ Y5 ) ) )
     => ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ Flim ) @ at_bot_real )
       => ( ord_less_real @ Flim @ ( F @ B2 ) ) ) ) ).

% DERIV_pos_imp_increasing_at_bot
thf(fact_9818_filterlim__pow__at__bot__odd,axiom,
    ! [N: nat,F: real > real,F5: filter_real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( filterlim_real_real @ F @ at_bot_real @ F5 )
       => ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
         => ( filterlim_real_real
            @ ^ [X: real] : ( power_power_real @ ( F @ X ) @ N )
            @ at_bot_real
            @ F5 ) ) ) ) ).

% filterlim_pow_at_bot_odd
thf(fact_9819_lhopital__left__at__top__at__bot,axiom,
    ! [F: real > real,A: real,G: real > real,F4: real > real,G2: 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
            @ ^ [X: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
            @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
         => ( ( eventually_real
              @ ^ [X: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
              @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
           => ( ( filterlim_real_real
                @ ^ [X: real] : ( divide_divide_real @ ( F4 @ X ) @ ( G2 @ X ) )
                @ at_bot_real
                @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
             => ( filterlim_real_real
                @ ^ [X: real] : ( divide_divide_real @ ( F @ X ) @ ( G @ X ) )
                @ at_bot_real
                @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) ) ) ) ) ) ) ).

% lhopital_left_at_top_at_bot
thf(fact_9820_tendsto__arctan__at__bot,axiom,
    filterlim_real_real @ arctan @ ( topolo2815343760600316023s_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ at_bot_real ).

% tendsto_arctan_at_bot
thf(fact_9821_incseq__bounded,axiom,
    ! [X8: nat > real,B4: real] :
      ( ( order_mono_nat_real @ X8 )
     => ( ! [I3: nat] : ( ord_less_eq_real @ ( X8 @ I3 ) @ B4 )
       => ( bfun_nat_real @ X8 @ at_top_nat ) ) ) ).

% incseq_bounded
thf(fact_9822_Bseq__realpow,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_eq_real @ X4 @ one_one_real )
       => ( bfun_nat_real @ ( power_power_real @ X4 ) @ at_top_nat ) ) ) ).

% Bseq_realpow
thf(fact_9823_tendsto__exp__limit__at__right,axiom,
    ! [X4: real] :
      ( filterlim_real_real
      @ ^ [Y: real] : ( powr_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ X4 @ Y ) ) @ ( divide_divide_real @ one_one_real @ Y ) )
      @ ( topolo2815343760600316023s_real @ ( exp_real @ X4 ) )
      @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ).

% tendsto_exp_limit_at_right
thf(fact_9824_filterlim__tan__at__right,axiom,
    filterlim_real_real @ tan_real @ at_bot_real @ ( topolo2177554685111907308n_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( set_or5849166863359141190n_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% filterlim_tan_at_right
thf(fact_9825_eventually__at__right__to__0,axiom,
    ! [P: real > $o,A: real] :
      ( ( eventually_real @ P @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
      = ( eventually_real
        @ ^ [X: real] : ( P @ ( plus_plus_real @ X @ A ) )
        @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).

% eventually_at_right_to_0
thf(fact_9826_eventually__at__right__real,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ A @ B2 )
     => ( eventually_real
        @ ^ [X: real] : ( member_real @ X @ ( set_or1633881224788618240n_real @ A @ B2 ) )
        @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) ) ) ).

% eventually_at_right_real
thf(fact_9827_tendsto__arcosh__at__left__1,axiom,
    filterlim_real_real @ arcosh_real @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ one_one_real @ ( set_or5849166863359141190n_real @ one_one_real ) ) ).

% tendsto_arcosh_at_left_1
thf(fact_9828_artanh__real__at__right__1,axiom,
    filterlim_real_real @ artanh_real @ at_bot_real @ ( topolo2177554685111907308n_real @ ( uminus_uminus_real @ one_one_real ) @ ( set_or5849166863359141190n_real @ ( uminus_uminus_real @ one_one_real ) ) ) ).

% artanh_real_at_right_1
thf(fact_9829_lhopital__right__at__top__at__top,axiom,
    ! [F: real > real,A: real,G: real > real,F4: real > real,G2: 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
            @ ^ [X: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
            @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
         => ( ( eventually_real
              @ ^ [X: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
              @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
           => ( ( filterlim_real_real
                @ ^ [X: real] : ( divide_divide_real @ ( F4 @ X ) @ ( G2 @ X ) )
                @ at_top_real
                @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
             => ( filterlim_real_real
                @ ^ [X: real] : ( divide_divide_real @ ( F @ X ) @ ( G @ X ) )
                @ at_top_real
                @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) ) ) ) ) ) ) ).

% lhopital_right_at_top_at_top
thf(fact_9830_lhopital__right__0,axiom,
    ! [F0: real > real,G0: real > real,G2: real > real,F4: real > real,F5: 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
            @ ^ [X: real] :
                ( ( G0 @ X )
               != zero_zero_real )
            @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
         => ( ( eventually_real
              @ ^ [X: real] :
                  ( ( G2 @ X )
                 != zero_zero_real )
              @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
           => ( ( eventually_real
                @ ^ [X: real] : ( has_fi5821293074295781190e_real @ F0 @ ( F4 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
                @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
             => ( ( eventually_real
                  @ ^ [X: real] : ( has_fi5821293074295781190e_real @ G0 @ ( G2 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
                  @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
               => ( ( filterlim_real_real
                    @ ^ [X: real] : ( divide_divide_real @ ( F4 @ X ) @ ( G2 @ X ) )
                    @ F5
                    @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
                 => ( filterlim_real_real
                    @ ^ [X: real] : ( divide_divide_real @ ( F0 @ X ) @ ( G0 @ X ) )
                    @ F5
                    @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ) ) ) ) ) ) ).

% lhopital_right_0
thf(fact_9831_lhopital__right,axiom,
    ! [F: real > real,X4: real,G: real > real,G2: real > real,F4: real > real,F5: filter_real] :
      ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X4 @ ( set_or5849166863359141190n_real @ X4 ) ) )
     => ( ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X4 @ ( set_or5849166863359141190n_real @ X4 ) ) )
       => ( ( eventually_real
            @ ^ [X: real] :
                ( ( G @ X )
               != zero_zero_real )
            @ ( topolo2177554685111907308n_real @ X4 @ ( set_or5849166863359141190n_real @ X4 ) ) )
         => ( ( eventually_real
              @ ^ [X: real] :
                  ( ( G2 @ X )
                 != zero_zero_real )
              @ ( topolo2177554685111907308n_real @ X4 @ ( set_or5849166863359141190n_real @ X4 ) ) )
           => ( ( eventually_real
                @ ^ [X: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
                @ ( topolo2177554685111907308n_real @ X4 @ ( set_or5849166863359141190n_real @ X4 ) ) )
             => ( ( eventually_real
                  @ ^ [X: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
                  @ ( topolo2177554685111907308n_real @ X4 @ ( set_or5849166863359141190n_real @ X4 ) ) )
               => ( ( filterlim_real_real
                    @ ^ [X: real] : ( divide_divide_real @ ( F4 @ X ) @ ( G2 @ X ) )
                    @ F5
                    @ ( topolo2177554685111907308n_real @ X4 @ ( set_or5849166863359141190n_real @ X4 ) ) )
                 => ( filterlim_real_real
                    @ ^ [X: real] : ( divide_divide_real @ ( F @ X ) @ ( G @ X ) )
                    @ F5
                    @ ( topolo2177554685111907308n_real @ X4 @ ( set_or5849166863359141190n_real @ X4 ) ) ) ) ) ) ) ) ) ) ).

% lhopital_right
thf(fact_9832_lhopital__right__at__top__at__bot,axiom,
    ! [F: real > real,A: real,G: real > real,F4: real > real,G2: 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
            @ ^ [X: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
            @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
         => ( ( eventually_real
              @ ^ [X: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
              @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
           => ( ( filterlim_real_real
                @ ^ [X: real] : ( divide_divide_real @ ( F4 @ X ) @ ( G2 @ X ) )
                @ at_bot_real
                @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
             => ( filterlim_real_real
                @ ^ [X: real] : ( divide_divide_real @ ( F @ X ) @ ( G @ X ) )
                @ at_bot_real
                @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) ) ) ) ) ) ) ).

% lhopital_right_at_top_at_bot
thf(fact_9833_lhopital__right__at__top,axiom,
    ! [G: real > real,X4: real,G2: real > real,F: real > real,F4: real > real,Y3: real] :
      ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ X4 @ ( set_or5849166863359141190n_real @ X4 ) ) )
     => ( ( eventually_real
          @ ^ [X: real] :
              ( ( G2 @ X )
             != zero_zero_real )
          @ ( topolo2177554685111907308n_real @ X4 @ ( set_or5849166863359141190n_real @ X4 ) ) )
       => ( ( eventually_real
            @ ^ [X: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
            @ ( topolo2177554685111907308n_real @ X4 @ ( set_or5849166863359141190n_real @ X4 ) ) )
         => ( ( eventually_real
              @ ^ [X: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
              @ ( topolo2177554685111907308n_real @ X4 @ ( set_or5849166863359141190n_real @ X4 ) ) )
           => ( ( filterlim_real_real
                @ ^ [X: real] : ( divide_divide_real @ ( F4 @ X ) @ ( G2 @ X ) )
                @ ( topolo2815343760600316023s_real @ Y3 )
                @ ( topolo2177554685111907308n_real @ X4 @ ( set_or5849166863359141190n_real @ X4 ) ) )
             => ( filterlim_real_real
                @ ^ [X: real] : ( divide_divide_real @ ( F @ X ) @ ( G @ X ) )
                @ ( topolo2815343760600316023s_real @ Y3 )
                @ ( topolo2177554685111907308n_real @ X4 @ ( set_or5849166863359141190n_real @ X4 ) ) ) ) ) ) ) ) ).

% lhopital_right_at_top
thf(fact_9834_lhopital__right__0__at__top,axiom,
    ! [G: real > real,G2: real > real,F: real > real,F4: real > real,X4: real] :
      ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
     => ( ( eventually_real
          @ ^ [X: real] :
              ( ( G2 @ X )
             != zero_zero_real )
          @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
       => ( ( eventually_real
            @ ^ [X: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
            @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
         => ( ( eventually_real
              @ ^ [X: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
              @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
           => ( ( filterlim_real_real
                @ ^ [X: real] : ( divide_divide_real @ ( F4 @ X ) @ ( G2 @ X ) )
                @ ( topolo2815343760600316023s_real @ X4 )
                @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
             => ( filterlim_real_real
                @ ^ [X: real] : ( divide_divide_real @ ( F @ X ) @ ( G @ X ) )
                @ ( topolo2815343760600316023s_real @ X4 )
                @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ) ) ) ) ).

% lhopital_right_0_at_top
thf(fact_9835_atLeast__0,axiom,
    ( ( set_ord_atLeast_nat @ zero_zero_nat )
    = top_top_set_nat ) ).

% atLeast_0
thf(fact_9836_atLeast__Suc__greaterThan,axiom,
    ! [K: nat] :
      ( ( set_ord_atLeast_nat @ ( suc @ K ) )
      = ( set_or1210151606488870762an_nat @ K ) ) ).

% atLeast_Suc_greaterThan
thf(fact_9837_decseq__bounded,axiom,
    ! [X8: nat > real,B4: real] :
      ( ( order_9091379641038594480t_real @ X8 )
     => ( ! [I3: nat] : ( ord_less_eq_real @ B4 @ ( X8 @ I3 ) )
       => ( bfun_nat_real @ X8 @ at_top_nat ) ) ) ).

% decseq_bounded
thf(fact_9838_greaterThan__0,axiom,
    ( ( set_or1210151606488870762an_nat @ zero_zero_nat )
    = ( image_nat_nat @ suc @ top_top_set_nat ) ) ).

% greaterThan_0
thf(fact_9839_greaterThan__Suc,axiom,
    ! [K: nat] :
      ( ( set_or1210151606488870762an_nat @ ( suc @ K ) )
      = ( minus_minus_set_nat @ ( set_or1210151606488870762an_nat @ K ) @ ( insert_nat @ ( suc @ K ) @ bot_bot_set_nat ) ) ) ).

% greaterThan_Suc
thf(fact_9840_decseq__convergent,axiom,
    ! [X8: nat > real,B4: real] :
      ( ( order_9091379641038594480t_real @ X8 )
     => ( ! [I3: nat] : ( ord_less_eq_real @ B4 @ ( X8 @ I3 ) )
       => ~ ! [L6: real] :
              ( ( filterlim_nat_real @ X8 @ ( topolo2815343760600316023s_real @ L6 ) @ at_top_nat )
             => ~ ! [I4: nat] : ( ord_less_eq_real @ L6 @ ( X8 @ I4 ) ) ) ) ) ).

% decseq_convergent
thf(fact_9841_atLeast__Suc,axiom,
    ! [K: nat] :
      ( ( set_ord_atLeast_nat @ ( suc @ K ) )
      = ( minus_minus_set_nat @ ( set_ord_atLeast_nat @ K ) @ ( insert_nat @ K @ bot_bot_set_nat ) ) ) ).

% atLeast_Suc
thf(fact_9842_GMVT,axiom,
    ! [A: real,B2: real,F: real > real,G: real > real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ! [X3: real] :
            ( ( ( ord_less_eq_real @ A @ X3 )
              & ( ord_less_eq_real @ X3 @ B2 ) )
           => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) @ F ) )
       => ( ! [X3: real] :
              ( ( ( ord_less_real @ A @ X3 )
                & ( ord_less_real @ X3 @ B2 ) )
             => ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) )
         => ( ! [X3: real] :
                ( ( ( ord_less_eq_real @ A @ X3 )
                  & ( ord_less_eq_real @ X3 @ B2 ) )
               => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) @ G ) )
           => ( ! [X3: real] :
                  ( ( ( ord_less_real @ A @ X3 )
                    & ( ord_less_real @ X3 @ B2 ) )
                 => ( differ6690327859849518006l_real @ G @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) )
             => ? [G_c: real,F_c: real,C3: real] :
                  ( ( has_fi5821293074295781190e_real @ G @ G_c @ ( topolo2177554685111907308n_real @ C3 @ top_top_set_real ) )
                  & ( has_fi5821293074295781190e_real @ F @ F_c @ ( topolo2177554685111907308n_real @ C3 @ top_top_set_real ) )
                  & ( ord_less_real @ A @ C3 )
                  & ( ord_less_real @ C3 @ B2 )
                  & ( ( times_times_real @ ( minus_minus_real @ ( F @ B2 ) @ ( F @ A ) ) @ G_c )
                    = ( times_times_real @ ( minus_minus_real @ ( G @ B2 ) @ ( G @ A ) ) @ F_c ) ) ) ) ) ) ) ) ).

% GMVT
thf(fact_9843_real__differentiableE,axiom,
    ! [F: real > real,X4: real,S: set_real] :
      ( ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X4 @ S ) )
     => ~ ! [Df: real] :
            ~ ( has_fi5821293074295781190e_real @ F @ Df @ ( topolo2177554685111907308n_real @ X4 @ S ) ) ) ).

% real_differentiableE
thf(fact_9844_real__differentiable__def,axiom,
    ! [F: real > real,X4: real,S: set_real] :
      ( ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X4 @ S ) )
      = ( ? [D6: real] : ( has_fi5821293074295781190e_real @ F @ D6 @ ( topolo2177554685111907308n_real @ X4 @ S ) ) ) ) ).

% real_differentiable_def
thf(fact_9845_MVT,axiom,
    ! [A: real,B2: real,F: real > real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B2 ) @ F )
       => ( ! [X3: real] :
              ( ( ord_less_real @ A @ X3 )
             => ( ( ord_less_real @ X3 @ B2 )
               => ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) )
         => ? [L3: real,Z2: real] :
              ( ( ord_less_real @ A @ Z2 )
              & ( ord_less_real @ Z2 @ B2 )
              & ( has_fi5821293074295781190e_real @ F @ L3 @ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) )
              & ( ( minus_minus_real @ ( F @ B2 ) @ ( F @ A ) )
                = ( times_times_real @ ( minus_minus_real @ B2 @ A ) @ L3 ) ) ) ) ) ) ).

% MVT
thf(fact_9846_suminf__eq__SUP__real,axiom,
    ! [X8: nat > real] :
      ( ( summable_real @ X8 )
     => ( ! [I3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( X8 @ I3 ) )
       => ( ( suminf_real @ X8 )
          = ( comple1385675409528146559p_real
            @ ( image_nat_real
              @ ^ [I2: nat] : ( groups6591440286371151544t_real @ X8 @ ( set_ord_lessThan_nat @ I2 ) )
              @ top_top_set_nat ) ) ) ) ) ).

% suminf_eq_SUP_real
thf(fact_9847_continuous__on__arcosh,axiom,
    ! [A3: set_real] :
      ( ( ord_less_eq_set_real @ A3 @ ( set_ord_atLeast_real @ one_one_real ) )
     => ( topolo5044208981011980120l_real @ A3 @ arcosh_real ) ) ).

% continuous_on_arcosh
thf(fact_9848_continuous__on__arcosh_H,axiom,
    ! [A3: set_real,F: real > real] :
      ( ( topolo5044208981011980120l_real @ A3 @ F )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ A3 )
           => ( ord_less_eq_real @ one_one_real @ ( F @ X3 ) ) )
       => ( topolo5044208981011980120l_real @ A3
          @ ^ [X: real] : ( arcosh_real @ ( F @ X ) ) ) ) ) ).

% continuous_on_arcosh'
thf(fact_9849_continuous__image__closed__interval,axiom,
    ! [A: real,B2: real,F: real > real] :
      ( ( ord_less_eq_real @ A @ B2 )
     => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B2 ) @ F )
       => ? [C3: real,D3: real] :
            ( ( ( image_real_real @ F @ ( set_or1222579329274155063t_real @ A @ B2 ) )
              = ( set_or1222579329274155063t_real @ C3 @ D3 ) )
            & ( ord_less_eq_real @ C3 @ D3 ) ) ) ) ).

% continuous_image_closed_interval
thf(fact_9850_continuous__on__arccos_H,axiom,
    topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) @ arccos ).

% continuous_on_arccos'
thf(fact_9851_continuous__on__arcsin_H,axiom,
    topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) @ arcsin ).

% continuous_on_arcsin'
thf(fact_9852_continuous__on__artanh,axiom,
    ! [A3: set_real] :
      ( ( ord_less_eq_set_real @ A3 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) )
     => ( topolo5044208981011980120l_real @ A3 @ artanh_real ) ) ).

% continuous_on_artanh
thf(fact_9853_continuous__on__artanh_H,axiom,
    ! [A3: set_real,F: real > real] :
      ( ( topolo5044208981011980120l_real @ A3 @ F )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ A3 )
           => ( member_real @ ( F @ X3 ) @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) ) )
       => ( topolo5044208981011980120l_real @ A3
          @ ^ [X: real] : ( artanh_real @ ( F @ X ) ) ) ) ) ).

% continuous_on_artanh'
thf(fact_9854_Rolle__deriv,axiom,
    ! [A: real,B2: real,F: real > real,F4: real > real > real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ( ( F @ A )
          = ( F @ B2 ) )
       => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B2 ) @ F )
         => ( ! [X3: real] :
                ( ( ord_less_real @ A @ X3 )
               => ( ( ord_less_real @ X3 @ B2 )
                 => ( has_de1759254742604945161l_real @ F @ ( F4 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) )
           => ? [Z2: real] :
                ( ( ord_less_real @ A @ Z2 )
                & ( ord_less_real @ Z2 @ B2 )
                & ( ( F4 @ Z2 )
                  = ( ^ [V4: real] : zero_zero_real ) ) ) ) ) ) ) ).

% Rolle_deriv
thf(fact_9855_mvt,axiom,
    ! [A: real,B2: real,F: real > real,F4: real > real > real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B2 ) @ F )
       => ( ! [X3: real] :
              ( ( ord_less_real @ A @ X3 )
             => ( ( ord_less_real @ X3 @ B2 )
               => ( has_de1759254742604945161l_real @ F @ ( F4 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) )
         => ~ ! [Xi: real] :
                ( ( ord_less_real @ A @ Xi )
               => ( ( ord_less_real @ Xi @ B2 )
                 => ( ( minus_minus_real @ ( F @ B2 ) @ ( F @ A ) )
                   != ( F4 @ Xi @ ( minus_minus_real @ B2 @ A ) ) ) ) ) ) ) ) ).

% mvt
thf(fact_9856_DERIV__pos__imp__increasing__open,axiom,
    ! [A: real,B2: real,F: real > real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ! [X3: real] :
            ( ( ord_less_real @ A @ X3 )
           => ( ( ord_less_real @ X3 @ B2 )
             => ? [Y5: real] :
                  ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
                  & ( ord_less_real @ zero_zero_real @ Y5 ) ) ) )
       => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B2 ) @ F )
         => ( ord_less_real @ ( F @ A ) @ ( F @ B2 ) ) ) ) ) ).

% DERIV_pos_imp_increasing_open
thf(fact_9857_DERIV__neg__imp__decreasing__open,axiom,
    ! [A: real,B2: real,F: real > real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ! [X3: real] :
            ( ( ord_less_real @ A @ X3 )
           => ( ( ord_less_real @ X3 @ B2 )
             => ? [Y5: real] :
                  ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
                  & ( ord_less_real @ Y5 @ zero_zero_real ) ) ) )
       => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B2 ) @ F )
         => ( ord_less_real @ ( F @ B2 ) @ ( F @ A ) ) ) ) ) ).

% DERIV_neg_imp_decreasing_open
thf(fact_9858_DERIV__isconst__end,axiom,
    ! [A: real,B2: real,F: real > real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B2 ) @ F )
       => ( ! [X3: real] :
              ( ( ord_less_real @ A @ X3 )
             => ( ( ord_less_real @ X3 @ B2 )
               => ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) )
         => ( ( F @ B2 )
            = ( F @ A ) ) ) ) ) ).

% DERIV_isconst_end
thf(fact_9859_DERIV__isconst2,axiom,
    ! [A: real,B2: real,F: real > real,X4: real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B2 ) @ F )
       => ( ! [X3: real] :
              ( ( ord_less_real @ A @ X3 )
             => ( ( ord_less_real @ X3 @ B2 )
               => ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) )
         => ( ( ord_less_eq_real @ A @ X4 )
           => ( ( ord_less_eq_real @ X4 @ B2 )
             => ( ( F @ X4 )
                = ( F @ A ) ) ) ) ) ) ) ).

% DERIV_isconst2
thf(fact_9860_Rolle,axiom,
    ! [A: real,B2: real,F: real > real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ( ( F @ A )
          = ( F @ B2 ) )
       => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B2 ) @ F )
         => ( ! [X3: real] :
                ( ( ord_less_real @ A @ X3 )
               => ( ( ord_less_real @ X3 @ B2 )
                 => ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) )
           => ? [Z2: real] :
                ( ( ord_less_real @ A @ Z2 )
                & ( ord_less_real @ Z2 @ B2 )
                & ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) ) ) ) ) ) ) ).

% Rolle
thf(fact_9861_UN__lessThan__UNIV,axiom,
    ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ set_ord_lessThan_nat @ top_top_set_nat ) )
    = top_top_set_nat ) ).

% UN_lessThan_UNIV
thf(fact_9862_UN__atMost__UNIV,axiom,
    ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ set_ord_atMost_nat @ top_top_set_nat ) )
    = top_top_set_nat ) ).

% UN_atMost_UNIV
thf(fact_9863_UN__atLeast__UNIV,axiom,
    ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ set_ord_atLeast_nat @ top_top_set_nat ) )
    = top_top_set_nat ) ).

% UN_atLeast_UNIV
thf(fact_9864_INT__greaterThan__UNIV,axiom,
    ( ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ set_or1210151606488870762an_nat @ top_top_set_nat ) )
    = bot_bot_set_nat ) ).

% INT_greaterThan_UNIV
thf(fact_9865_Inf__real__def,axiom,
    ( comple4887499456419720421f_real
    = ( ^ [X6: set_real] : ( uminus_uminus_real @ ( comple1385675409528146559p_real @ ( image_real_real @ uminus_uminus_real @ X6 ) ) ) ) ) ).

% Inf_real_def
thf(fact_9866_inf__enat__def,axiom,
    inf_in1870772243966228564d_enat = ord_mi8085742599997312461d_enat ).

% inf_enat_def
thf(fact_9867_inf__nat__def,axiom,
    inf_inf_nat = ord_min_nat ).

% inf_nat_def
thf(fact_9868_rat__less__code,axiom,
    ( ord_less_rat
    = ( ^ [P3: rat,Q4: rat] :
          ( produc4947309494688390418_int_o
          @ ^ [A4: int,C2: int] :
              ( produc4947309494688390418_int_o
              @ ^ [B3: int,D2: int] : ( ord_less_int @ ( times_times_int @ A4 @ D2 ) @ ( times_times_int @ C2 @ B3 ) )
              @ ( quotient_of @ Q4 ) )
          @ ( quotient_of @ P3 ) ) ) ) ).

% rat_less_code
thf(fact_9869_rat__less__eq__code,axiom,
    ( ord_less_eq_rat
    = ( ^ [P3: rat,Q4: rat] :
          ( produc4947309494688390418_int_o
          @ ^ [A4: int,C2: int] :
              ( produc4947309494688390418_int_o
              @ ^ [B3: int,D2: int] : ( ord_less_eq_int @ ( times_times_int @ A4 @ D2 ) @ ( times_times_int @ C2 @ B3 ) )
              @ ( quotient_of @ Q4 ) )
          @ ( quotient_of @ P3 ) ) ) ) ).

% rat_less_eq_code
thf(fact_9870_int__ge__less__than2__def,axiom,
    ( int_ge_less_than2
    = ( ^ [D2: int] :
          ( collec213857154873943460nt_int
          @ ( produc4947309494688390418_int_o
            @ ^ [Z7: int,Z5: int] :
                ( ( ord_less_eq_int @ D2 @ Z5 )
                & ( ord_less_int @ Z7 @ Z5 ) ) ) ) ) ) ).

% int_ge_less_than2_def
thf(fact_9871_int__ge__less__than__def,axiom,
    ( int_ge_less_than
    = ( ^ [D2: int] :
          ( collec213857154873943460nt_int
          @ ( produc4947309494688390418_int_o
            @ ^ [Z7: int,Z5: int] :
                ( ( ord_less_eq_int @ D2 @ Z7 )
                & ( ord_less_int @ Z7 @ Z5 ) ) ) ) ) ) ).

% int_ge_less_than_def
thf(fact_9872_uniformity__real__def,axiom,
    ( topolo1511823702728130853y_real
    = ( comple2936214249959783750l_real
      @ ( image_2178119161166701260l_real
        @ ^ [E3: real] :
            ( princi6114159922880469582l_real
            @ ( collec3799799289383736868l_real
              @ ( produc5414030515140494994real_o
                @ ^ [X: real,Y: real] : ( ord_less_real @ ( real_V975177566351809787t_real @ X @ Y ) @ E3 ) ) ) )
        @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).

% uniformity_real_def
thf(fact_9873_uniformity__complex__def,axiom,
    ( topolo896644834953643431omplex
    = ( comple8358262395181532106omplex
      @ ( image_5971271580939081552omplex
        @ ^ [E3: real] :
            ( princi3496590319149328850omplex
            @ ( collec8663557070575231912omplex
              @ ( produc6771430404735790350plex_o
                @ ^ [X: complex,Y: complex] : ( ord_less_real @ ( real_V3694042436643373181omplex @ X @ Y ) @ E3 ) ) ) )
        @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).

% uniformity_complex_def
thf(fact_9874_nonneg__incseq__Bseq__subseq__iff,axiom,
    ! [F: nat > real,G: nat > nat] :
      ( ! [X3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) )
     => ( ( order_mono_nat_real @ F )
       => ( ( order_5726023648592871131at_nat @ G )
         => ( ( bfun_nat_real
              @ ^ [X: nat] : ( F @ ( G @ X ) )
              @ at_top_nat )
            = ( bfun_nat_real @ F @ at_top_nat ) ) ) ) ) ).

% nonneg_incseq_Bseq_subseq_iff
thf(fact_9875_inj__sgn__power,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( inj_on_real_real
        @ ^ [Y: real] : ( times_times_real @ ( sgn_sgn_real @ Y ) @ ( power_power_real @ ( abs_abs_real @ Y ) @ N ) )
        @ top_top_set_real ) ) ).

% inj_sgn_power
thf(fact_9876_strict__mono__imp__increasing,axiom,
    ! [F: nat > nat,N: nat] :
      ( ( order_5726023648592871131at_nat @ F )
     => ( ord_less_eq_nat @ N @ ( F @ N ) ) ) ).

% strict_mono_imp_increasing
thf(fact_9877_log__inj,axiom,
    ! [B2: real] :
      ( ( ord_less_real @ one_one_real @ B2 )
     => ( inj_on_real_real @ ( log @ B2 ) @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ).

% log_inj
thf(fact_9878_inj__Suc,axiom,
    ! [N3: set_nat] : ( inj_on_nat_nat @ suc @ N3 ) ).

% inj_Suc
thf(fact_9879_inj__on__diff__nat,axiom,
    ! [N3: set_nat,K: nat] :
      ( ! [N4: nat] :
          ( ( member_nat @ N4 @ N3 )
         => ( ord_less_eq_nat @ K @ N4 ) )
     => ( inj_on_nat_nat
        @ ^ [N2: nat] : ( minus_minus_nat @ N2 @ K )
        @ N3 ) ) ).

% inj_on_diff_nat
thf(fact_9880_summable__reindex,axiom,
    ! [F: nat > real,G: nat > nat] :
      ( ( summable_real @ F )
     => ( ( inj_on_nat_nat @ G @ top_top_set_nat )
       => ( ! [X3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) )
         => ( summable_real @ ( comp_nat_real_nat @ F @ G ) ) ) ) ) ).

% summable_reindex
thf(fact_9881_suminf__reindex__mono,axiom,
    ! [F: nat > real,G: nat > nat] :
      ( ( summable_real @ F )
     => ( ( inj_on_nat_nat @ G @ top_top_set_nat )
       => ( ! [X3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) )
         => ( ord_less_eq_real @ ( suminf_real @ ( comp_nat_real_nat @ F @ G ) ) @ ( suminf_real @ F ) ) ) ) ) ).

% suminf_reindex_mono
thf(fact_9882_powr__real__of__int_H,axiom,
    ! [X4: real,N: int] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ( X4 != zero_zero_real )
          | ( ord_less_int @ zero_zero_int @ N ) )
       => ( ( powr_real @ X4 @ ( ring_1_of_int_real @ N ) )
          = ( power_int_real @ X4 @ N ) ) ) ) ).

% powr_real_of_int'
thf(fact_9883_inj__on__char__of__nat,axiom,
    inj_on_nat_char @ unique3096191561947761185of_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% inj_on_char_of_nat
thf(fact_9884_suminf__reindex,axiom,
    ! [F: nat > real,G: nat > nat] :
      ( ( summable_real @ F )
     => ( ( inj_on_nat_nat @ G @ top_top_set_nat )
       => ( ! [X3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) )
         => ( ! [X3: nat] :
                ( ~ ( member_nat @ X3 @ ( image_nat_nat @ G @ top_top_set_nat ) )
               => ( ( F @ X3 )
                  = zero_zero_real ) )
           => ( ( suminf_real @ ( comp_nat_real_nat @ F @ G ) )
              = ( suminf_real @ F ) ) ) ) ) ) ).

% suminf_reindex
thf(fact_9885_pos__deriv__imp__strict__mono,axiom,
    ! [F: real > real,F4: real > real] :
      ( ! [X3: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
     => ( ! [X3: real] : ( ord_less_real @ zero_zero_real @ ( F4 @ X3 ) )
       => ( order_7092887310737990675l_real @ F ) ) ) ).

% pos_deriv_imp_strict_mono
thf(fact_9886_pred__nat__def,axiom,
    ( pred_nat
    = ( collec3392354462482085612at_nat
      @ ( produc6081775807080527818_nat_o
        @ ^ [M6: nat,N2: nat] :
            ( N2
            = ( suc @ M6 ) ) ) ) ) ).

% pred_nat_def
thf(fact_9887_atLeastLessThan__add__Un,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( set_or4665077453230672383an_nat @ I @ ( plus_plus_nat @ J @ K ) )
        = ( sup_sup_set_nat @ ( set_or4665077453230672383an_nat @ I @ J ) @ ( set_or4665077453230672383an_nat @ J @ ( plus_plus_nat @ J @ K ) ) ) ) ) ).

% atLeastLessThan_add_Un
thf(fact_9888_sup__enat__def,axiom,
    sup_su3973961784419623482d_enat = ord_ma741700101516333627d_enat ).

% sup_enat_def
thf(fact_9889_sup__nat__def,axiom,
    sup_sup_nat = ord_max_nat ).

% sup_nat_def
thf(fact_9890_VEBT_Osize_I3_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] :
      ( ( size_size_VEBT_VEBT @ ( vEBT_Node @ X11 @ X12 @ X13 @ X14 ) )
      = ( plus_plus_nat @ ( plus_plus_nat @ ( size_list_VEBT_VEBT @ size_size_VEBT_VEBT @ X13 ) @ ( size_size_VEBT_VEBT @ X14 ) ) @ ( suc @ zero_zero_nat ) ) ) ).

% VEBT.size(3)
thf(fact_9891_VEBT_Osize__gen_I1_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] :
      ( ( vEBT_size_VEBT @ ( vEBT_Node @ X11 @ X12 @ X13 @ X14 ) )
      = ( plus_plus_nat @ ( plus_plus_nat @ ( size_list_VEBT_VEBT @ vEBT_size_VEBT @ X13 ) @ ( vEBT_size_VEBT @ X14 ) ) @ ( suc @ zero_zero_nat ) ) ) ).

% VEBT.size_gen(1)
thf(fact_9892_tl__upt,axiom,
    ! [M: nat,N: nat] :
      ( ( tl_nat @ ( upt @ M @ N ) )
      = ( upt @ ( suc @ M ) @ N ) ) ).

% tl_upt
thf(fact_9893_vanishes__mult__bounded,axiom,
    ! [X8: nat > rat,Y7: nat > rat] :
      ( ? [A7: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ A7 )
          & ! [N4: nat] : ( ord_less_rat @ ( abs_abs_rat @ ( X8 @ N4 ) ) @ A7 ) )
     => ( ( vanishes @ Y7 )
       => ( vanishes
          @ ^ [N2: nat] : ( times_times_rat @ ( X8 @ N2 ) @ ( Y7 @ N2 ) ) ) ) ) ).

% vanishes_mult_bounded
thf(fact_9894_vanishes__const,axiom,
    ! [C: rat] :
      ( ( vanishes
        @ ^ [N2: nat] : C )
      = ( C = zero_zero_rat ) ) ).

% vanishes_const
thf(fact_9895_vanishes__minus,axiom,
    ! [X8: nat > rat] :
      ( ( vanishes @ X8 )
     => ( vanishes
        @ ^ [N2: nat] : ( uminus_uminus_rat @ ( X8 @ N2 ) ) ) ) ).

% vanishes_minus
thf(fact_9896_vanishes__add,axiom,
    ! [X8: nat > rat,Y7: nat > rat] :
      ( ( vanishes @ X8 )
     => ( ( vanishes @ Y7 )
       => ( vanishes
          @ ^ [N2: nat] : ( plus_plus_rat @ ( X8 @ N2 ) @ ( Y7 @ N2 ) ) ) ) ) ).

% vanishes_add
thf(fact_9897_vanishes__diff,axiom,
    ! [X8: nat > rat,Y7: nat > rat] :
      ( ( vanishes @ X8 )
     => ( ( vanishes @ Y7 )
       => ( vanishes
          @ ^ [N2: nat] : ( minus_minus_rat @ ( X8 @ N2 ) @ ( Y7 @ N2 ) ) ) ) ) ).

% vanishes_diff
thf(fact_9898_vanishesD,axiom,
    ! [X8: nat > rat,R2: rat] :
      ( ( vanishes @ X8 )
     => ( ( ord_less_rat @ zero_zero_rat @ R2 )
       => ? [K2: nat] :
          ! [N6: nat] :
            ( ( ord_less_eq_nat @ K2 @ N6 )
           => ( ord_less_rat @ ( abs_abs_rat @ ( X8 @ N6 ) ) @ R2 ) ) ) ) ).

% vanishesD
thf(fact_9899_vanishesI,axiom,
    ! [X8: nat > rat] :
      ( ! [R3: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ R3 )
         => ? [K4: nat] :
            ! [N4: nat] :
              ( ( ord_less_eq_nat @ K4 @ N4 )
             => ( ord_less_rat @ ( abs_abs_rat @ ( X8 @ N4 ) ) @ R3 ) ) )
     => ( vanishes @ X8 ) ) ).

% vanishesI
thf(fact_9900_vanishes__def,axiom,
    ( vanishes
    = ( ^ [X6: nat > rat] :
        ! [R5: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ R5 )
         => ? [K3: nat] :
            ! [N2: nat] :
              ( ( ord_less_eq_nat @ K3 @ N2 )
             => ( ord_less_rat @ ( abs_abs_rat @ ( X6 @ N2 ) ) @ R5 ) ) ) ) ) ).

% vanishes_def
thf(fact_9901_and__not__num_Opelims,axiom,
    ! [X4: num,Xa: num,Y3: option_num] :
      ( ( ( bit_and_not_num @ X4 @ Xa )
        = Y3 )
     => ( ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ X4 @ Xa ) )
       => ( ( ( X4 = one )
           => ( ( Xa = one )
             => ( ( Y3 = none_num )
               => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
         => ( ( ( X4 = one )
             => ! [N4: num] :
                  ( ( Xa
                    = ( bit0 @ N4 ) )
                 => ( ( Y3
                      = ( some_num @ one ) )
                   => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ one @ ( bit0 @ N4 ) ) ) ) ) )
           => ( ( ( X4 = one )
               => ! [N4: num] :
                    ( ( Xa
                      = ( bit1 @ N4 ) )
                   => ( ( Y3 = none_num )
                     => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ one @ ( bit1 @ N4 ) ) ) ) ) )
             => ( ! [M4: num] :
                    ( ( X4
                      = ( bit0 @ M4 ) )
                   => ( ( Xa = one )
                     => ( ( Y3
                          = ( some_num @ ( bit0 @ M4 ) ) )
                       => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ one ) ) ) ) )
               => ( ! [M4: num] :
                      ( ( X4
                        = ( bit0 @ M4 ) )
                     => ! [N4: num] :
                          ( ( Xa
                            = ( bit0 @ N4 ) )
                         => ( ( Y3
                              = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M4 @ N4 ) ) )
                           => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit0 @ N4 ) ) ) ) ) )
                 => ( ! [M4: num] :
                        ( ( X4
                          = ( bit0 @ M4 ) )
                       => ! [N4: num] :
                            ( ( Xa
                              = ( bit1 @ N4 ) )
                           => ( ( Y3
                                = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M4 @ N4 ) ) )
                             => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit1 @ N4 ) ) ) ) ) )
                   => ( ! [M4: num] :
                          ( ( X4
                            = ( bit1 @ M4 ) )
                         => ( ( Xa = one )
                           => ( ( Y3
                                = ( some_num @ ( bit0 @ M4 ) ) )
                             => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ one ) ) ) ) )
                     => ( ! [M4: num] :
                            ( ( X4
                              = ( bit1 @ M4 ) )
                           => ! [N4: num] :
                                ( ( Xa
                                  = ( bit0 @ N4 ) )
                               => ( ( Y3
                                    = ( case_o6005452278849405969um_num @ ( some_num @ one )
                                      @ ^ [N9: num] : ( some_num @ ( bit1 @ N9 ) )
                                      @ ( bit_and_not_num @ M4 @ N4 ) ) )
                                 => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit0 @ N4 ) ) ) ) ) )
                       => ~ ! [M4: num] :
                              ( ( X4
                                = ( bit1 @ M4 ) )
                             => ! [N4: num] :
                                  ( ( Xa
                                    = ( bit1 @ N4 ) )
                                 => ( ( Y3
                                      = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M4 @ N4 ) ) )
                                   => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit1 @ N4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_not_num.pelims
thf(fact_9902_and__num_Opelims,axiom,
    ! [X4: num,Xa: num,Y3: option_num] :
      ( ( ( bit_un7362597486090784418nd_num @ X4 @ Xa )
        = Y3 )
     => ( ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ X4 @ Xa ) )
       => ( ( ( X4 = one )
           => ( ( Xa = one )
             => ( ( Y3
                  = ( some_num @ one ) )
               => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
         => ( ( ( X4 = one )
             => ! [N4: num] :
                  ( ( Xa
                    = ( bit0 @ N4 ) )
                 => ( ( Y3 = none_num )
                   => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ one @ ( bit0 @ N4 ) ) ) ) ) )
           => ( ( ( X4 = one )
               => ! [N4: num] :
                    ( ( Xa
                      = ( bit1 @ N4 ) )
                   => ( ( Y3
                        = ( some_num @ one ) )
                     => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ one @ ( bit1 @ N4 ) ) ) ) ) )
             => ( ! [M4: num] :
                    ( ( X4
                      = ( bit0 @ M4 ) )
                   => ( ( Xa = one )
                     => ( ( Y3 = none_num )
                       => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ one ) ) ) ) )
               => ( ! [M4: num] :
                      ( ( X4
                        = ( bit0 @ M4 ) )
                     => ! [N4: num] :
                          ( ( Xa
                            = ( bit0 @ N4 ) )
                         => ( ( Y3
                              = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M4 @ N4 ) ) )
                           => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit0 @ N4 ) ) ) ) ) )
                 => ( ! [M4: num] :
                        ( ( X4
                          = ( bit0 @ M4 ) )
                       => ! [N4: num] :
                            ( ( Xa
                              = ( bit1 @ N4 ) )
                           => ( ( Y3
                                = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M4 @ N4 ) ) )
                             => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit1 @ N4 ) ) ) ) ) )
                   => ( ! [M4: num] :
                          ( ( X4
                            = ( bit1 @ M4 ) )
                         => ( ( Xa = one )
                           => ( ( Y3
                                = ( some_num @ one ) )
                             => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ one ) ) ) ) )
                     => ( ! [M4: num] :
                            ( ( X4
                              = ( bit1 @ M4 ) )
                           => ! [N4: num] :
                                ( ( Xa
                                  = ( bit0 @ N4 ) )
                               => ( ( Y3
                                    = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M4 @ N4 ) ) )
                                 => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit0 @ N4 ) ) ) ) ) )
                       => ~ ! [M4: num] :
                              ( ( X4
                                = ( bit1 @ M4 ) )
                             => ! [N4: num] :
                                  ( ( Xa
                                    = ( bit1 @ N4 ) )
                                 => ( ( Y3
                                      = ( case_o6005452278849405969um_num @ ( some_num @ one )
                                        @ ^ [N9: num] : ( some_num @ ( bit1 @ N9 ) )
                                        @ ( bit_un7362597486090784418nd_num @ M4 @ N4 ) ) )
                                   => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit1 @ N4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_num.pelims
thf(fact_9903_xor__num_Opelims,axiom,
    ! [X4: num,Xa: num,Y3: option_num] :
      ( ( ( bit_un2480387367778600638or_num @ X4 @ Xa )
        = Y3 )
     => ( ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ X4 @ Xa ) )
       => ( ( ( X4 = one )
           => ( ( Xa = one )
             => ( ( Y3 = none_num )
               => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
         => ( ( ( X4 = one )
             => ! [N4: num] :
                  ( ( Xa
                    = ( bit0 @ N4 ) )
                 => ( ( Y3
                      = ( some_num @ ( bit1 @ N4 ) ) )
                   => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ one @ ( bit0 @ N4 ) ) ) ) ) )
           => ( ( ( X4 = one )
               => ! [N4: num] :
                    ( ( Xa
                      = ( bit1 @ N4 ) )
                   => ( ( Y3
                        = ( some_num @ ( bit0 @ N4 ) ) )
                     => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ one @ ( bit1 @ N4 ) ) ) ) ) )
             => ( ! [M4: num] :
                    ( ( X4
                      = ( bit0 @ M4 ) )
                   => ( ( Xa = one )
                     => ( ( Y3
                          = ( some_num @ ( bit1 @ M4 ) ) )
                       => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ one ) ) ) ) )
               => ( ! [M4: num] :
                      ( ( X4
                        = ( bit0 @ M4 ) )
                     => ! [N4: num] :
                          ( ( Xa
                            = ( bit0 @ N4 ) )
                         => ( ( Y3
                              = ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M4 @ N4 ) ) )
                           => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit0 @ N4 ) ) ) ) ) )
                 => ( ! [M4: num] :
                        ( ( X4
                          = ( bit0 @ M4 ) )
                       => ! [N4: num] :
                            ( ( Xa
                              = ( bit1 @ N4 ) )
                           => ( ( Y3
                                = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M4 @ N4 ) ) ) )
                             => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit1 @ N4 ) ) ) ) ) )
                   => ( ! [M4: num] :
                          ( ( X4
                            = ( bit1 @ M4 ) )
                         => ( ( Xa = one )
                           => ( ( Y3
                                = ( some_num @ ( bit0 @ M4 ) ) )
                             => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ one ) ) ) ) )
                     => ( ! [M4: num] :
                            ( ( X4
                              = ( bit1 @ M4 ) )
                           => ! [N4: num] :
                                ( ( Xa
                                  = ( bit0 @ N4 ) )
                               => ( ( Y3
                                    = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M4 @ N4 ) ) ) )
                                 => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit0 @ N4 ) ) ) ) ) )
                       => ~ ! [M4: num] :
                              ( ( X4
                                = ( bit1 @ M4 ) )
                             => ! [N4: num] :
                                  ( ( Xa
                                    = ( bit1 @ N4 ) )
                                 => ( ( Y3
                                      = ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M4 @ N4 ) ) )
                                   => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit1 @ N4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% xor_num.pelims
thf(fact_9904_or__not__num__neg_Opelims,axiom,
    ! [X4: num,Xa: num,Y3: num] :
      ( ( ( bit_or_not_num_neg @ X4 @ Xa )
        = Y3 )
     => ( ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ X4 @ Xa ) )
       => ( ( ( X4 = one )
           => ( ( Xa = one )
             => ( ( Y3 = one )
               => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
         => ( ( ( X4 = one )
             => ! [M4: num] :
                  ( ( Xa
                    = ( bit0 @ M4 ) )
                 => ( ( Y3
                      = ( bit1 @ M4 ) )
                   => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ one @ ( bit0 @ M4 ) ) ) ) ) )
           => ( ( ( X4 = one )
               => ! [M4: num] :
                    ( ( Xa
                      = ( bit1 @ M4 ) )
                   => ( ( Y3
                        = ( bit1 @ M4 ) )
                     => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ one @ ( bit1 @ M4 ) ) ) ) ) )
             => ( ! [N4: num] :
                    ( ( X4
                      = ( bit0 @ N4 ) )
                   => ( ( Xa = one )
                     => ( ( Y3
                          = ( bit0 @ one ) )
                       => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit0 @ N4 ) @ one ) ) ) ) )
               => ( ! [N4: num] :
                      ( ( X4
                        = ( bit0 @ N4 ) )
                     => ! [M4: num] :
                          ( ( Xa
                            = ( bit0 @ M4 ) )
                         => ( ( Y3
                              = ( bitM @ ( bit_or_not_num_neg @ N4 @ M4 ) ) )
                           => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit0 @ N4 ) @ ( bit0 @ M4 ) ) ) ) ) )
                 => ( ! [N4: num] :
                        ( ( X4
                          = ( bit0 @ N4 ) )
                       => ! [M4: num] :
                            ( ( Xa
                              = ( bit1 @ M4 ) )
                           => ( ( Y3
                                = ( bit0 @ ( bit_or_not_num_neg @ N4 @ M4 ) ) )
                             => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit0 @ N4 ) @ ( bit1 @ M4 ) ) ) ) ) )
                   => ( ! [N4: num] :
                          ( ( X4
                            = ( bit1 @ N4 ) )
                         => ( ( Xa = one )
                           => ( ( Y3 = one )
                             => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit1 @ N4 ) @ one ) ) ) ) )
                     => ( ! [N4: num] :
                            ( ( X4
                              = ( bit1 @ N4 ) )
                           => ! [M4: num] :
                                ( ( Xa
                                  = ( bit0 @ M4 ) )
                               => ( ( Y3
                                    = ( bitM @ ( bit_or_not_num_neg @ N4 @ M4 ) ) )
                                 => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit1 @ N4 ) @ ( bit0 @ M4 ) ) ) ) ) )
                       => ~ ! [N4: num] :
                              ( ( X4
                                = ( bit1 @ N4 ) )
                             => ! [M4: num] :
                                  ( ( Xa
                                    = ( bit1 @ M4 ) )
                                 => ( ( Y3
                                      = ( bitM @ ( bit_or_not_num_neg @ N4 @ M4 ) ) )
                                   => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit1 @ N4 ) @ ( bit1 @ M4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% or_not_num_neg.pelims
thf(fact_9905_xor__num__rel__dict,axiom,
    bit_un2901131394128224187um_rel = bit_un3595099601533988841um_rel ).

% xor_num_rel_dict
thf(fact_9906_and__num__rel__dict,axiom,
    bit_un4731106466462545111um_rel = bit_un5425074673868309765um_rel ).

% and_num_rel_dict
thf(fact_9907_rcis__inverse,axiom,
    ! [R2: real,A: real] :
      ( ( invers8013647133539491842omplex @ ( rcis @ R2 @ A ) )
      = ( rcis @ ( divide_divide_real @ one_one_real @ R2 ) @ ( uminus_uminus_real @ A ) ) ) ).

% rcis_inverse
thf(fact_9908_Re__rcis,axiom,
    ! [R2: real,A: real] :
      ( ( re @ ( rcis @ R2 @ A ) )
      = ( times_times_real @ R2 @ ( cos_real @ A ) ) ) ).

% Re_rcis
thf(fact_9909_Im__rcis,axiom,
    ! [R2: real,A: real] :
      ( ( im @ ( rcis @ R2 @ A ) )
      = ( times_times_real @ R2 @ ( sin_real @ A ) ) ) ).

% Im_rcis
thf(fact_9910_cis__rcis__eq,axiom,
    ( cis
    = ( rcis @ one_one_real ) ) ).

% cis_rcis_eq
thf(fact_9911_rcis__mult,axiom,
    ! [R1: real,A: real,R22: real,B2: real] :
      ( ( times_times_complex @ ( rcis @ R1 @ A ) @ ( rcis @ R22 @ B2 ) )
      = ( rcis @ ( times_times_real @ R1 @ R22 ) @ ( plus_plus_real @ A @ B2 ) ) ) ).

% rcis_mult
thf(fact_9912_rcis__divide,axiom,
    ! [R1: real,A: real,R22: real,B2: real] :
      ( ( divide1717551699836669952omplex @ ( rcis @ R1 @ A ) @ ( rcis @ R22 @ B2 ) )
      = ( rcis @ ( divide_divide_real @ R1 @ R22 ) @ ( minus_minus_real @ A @ B2 ) ) ) ).

% rcis_divide
thf(fact_9913_rcis__def,axiom,
    ( rcis
    = ( ^ [R5: real,A4: real] : ( times_times_complex @ ( real_V4546457046886955230omplex @ R5 ) @ ( cis @ A4 ) ) ) ) ).

% rcis_def
thf(fact_9914_DeMoivre2,axiom,
    ! [R2: real,A: real,N: nat] :
      ( ( power_power_complex @ ( rcis @ R2 @ A ) @ N )
      = ( rcis @ ( power_power_real @ R2 @ N ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ A ) ) ) ).

% DeMoivre2
thf(fact_9915_nat__of__integer__code,axiom,
    ( code_nat_of_integer
    = ( ^ [K3: code_integer] :
          ( if_nat @ ( ord_le3102999989581377725nteger @ K3 @ zero_z3403309356797280102nteger ) @ zero_zero_nat
          @ ( produc1555791787009142072er_nat
            @ ^ [L: code_integer,J3: code_integer] : ( if_nat @ ( J3 = zero_z3403309356797280102nteger ) @ ( plus_plus_nat @ ( code_nat_of_integer @ L ) @ ( code_nat_of_integer @ L ) ) @ ( plus_plus_nat @ ( plus_plus_nat @ ( code_nat_of_integer @ L ) @ ( code_nat_of_integer @ L ) ) @ one_one_nat ) )
            @ ( code_divmod_integer @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% nat_of_integer_code
thf(fact_9916_nat__of__integer__code__post_I3_J,axiom,
    ! [K: num] :
      ( ( code_nat_of_integer @ ( numera6620942414471956472nteger @ K ) )
      = ( numeral_numeral_nat @ K ) ) ).

% nat_of_integer_code_post(3)
thf(fact_9917_nat__of__integer__code__post_I2_J,axiom,
    ( ( code_nat_of_integer @ one_one_Code_integer )
    = one_one_nat ) ).

% nat_of_integer_code_post(2)
thf(fact_9918_cauchy__def,axiom,
    ( cauchy
    = ( ^ [X6: nat > rat] :
        ! [R5: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ R5 )
         => ? [K3: nat] :
            ! [M6: nat] :
              ( ( ord_less_eq_nat @ K3 @ M6 )
             => ! [N2: nat] :
                  ( ( ord_less_eq_nat @ K3 @ N2 )
                 => ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( X6 @ M6 ) @ ( X6 @ N2 ) ) ) @ R5 ) ) ) ) ) ) ).

% cauchy_def
thf(fact_9919_cauchyI,axiom,
    ! [X8: nat > rat] :
      ( ! [R3: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ R3 )
         => ? [K4: nat] :
            ! [M4: nat] :
              ( ( ord_less_eq_nat @ K4 @ M4 )
             => ! [N4: nat] :
                  ( ( ord_less_eq_nat @ K4 @ N4 )
                 => ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( X8 @ M4 ) @ ( X8 @ N4 ) ) ) @ R3 ) ) ) )
     => ( cauchy @ X8 ) ) ).

% cauchyI
thf(fact_9920_cauchy__inverse,axiom,
    ! [X8: nat > rat] :
      ( ( cauchy @ X8 )
     => ( ~ ( vanishes @ X8 )
       => ( cauchy
          @ ^ [N2: nat] : ( inverse_inverse_rat @ ( X8 @ N2 ) ) ) ) ) ).

% cauchy_inverse
thf(fact_9921_cauchy__diff,axiom,
    ! [X8: nat > rat,Y7: nat > rat] :
      ( ( cauchy @ X8 )
     => ( ( cauchy @ Y7 )
       => ( cauchy
          @ ^ [N2: nat] : ( minus_minus_rat @ ( X8 @ N2 ) @ ( Y7 @ N2 ) ) ) ) ) ).

% cauchy_diff
thf(fact_9922_cauchy__mult,axiom,
    ! [X8: nat > rat,Y7: nat > rat] :
      ( ( cauchy @ X8 )
     => ( ( cauchy @ Y7 )
       => ( cauchy
          @ ^ [N2: nat] : ( times_times_rat @ ( X8 @ N2 ) @ ( Y7 @ N2 ) ) ) ) ) ).

% cauchy_mult
thf(fact_9923_cauchy__const,axiom,
    ! [X4: rat] :
      ( cauchy
      @ ^ [N2: nat] : X4 ) ).

% cauchy_const
thf(fact_9924_cauchy__minus,axiom,
    ! [X8: nat > rat] :
      ( ( cauchy @ X8 )
     => ( cauchy
        @ ^ [N2: nat] : ( uminus_uminus_rat @ ( X8 @ N2 ) ) ) ) ).

% cauchy_minus
thf(fact_9925_cauchy__add,axiom,
    ! [X8: nat > rat,Y7: nat > rat] :
      ( ( cauchy @ X8 )
     => ( ( cauchy @ Y7 )
       => ( cauchy
          @ ^ [N2: nat] : ( plus_plus_rat @ ( X8 @ N2 ) @ ( Y7 @ N2 ) ) ) ) ) ).

% cauchy_add
thf(fact_9926_cauchy__imp__bounded,axiom,
    ! [X8: nat > rat] :
      ( ( cauchy @ X8 )
     => ? [B5: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ B5 )
          & ! [N6: nat] : ( ord_less_rat @ ( abs_abs_rat @ ( X8 @ N6 ) ) @ B5 ) ) ) ).

% cauchy_imp_bounded
thf(fact_9927_vanishes__diff__inverse,axiom,
    ! [X8: nat > rat,Y7: nat > rat] :
      ( ( cauchy @ X8 )
     => ( ~ ( vanishes @ X8 )
       => ( ( cauchy @ Y7 )
         => ( ~ ( vanishes @ Y7 )
           => ( ( vanishes
                @ ^ [N2: nat] : ( minus_minus_rat @ ( X8 @ N2 ) @ ( Y7 @ N2 ) ) )
             => ( vanishes
                @ ^ [N2: nat] : ( minus_minus_rat @ ( inverse_inverse_rat @ ( X8 @ N2 ) ) @ ( inverse_inverse_rat @ ( Y7 @ N2 ) ) ) ) ) ) ) ) ) ).

% vanishes_diff_inverse
thf(fact_9928_cauchy__not__vanishes__cases,axiom,
    ! [X8: nat > rat] :
      ( ( cauchy @ X8 )
     => ( ~ ( vanishes @ X8 )
       => ? [B5: rat] :
            ( ( ord_less_rat @ zero_zero_rat @ B5 )
            & ? [K2: nat] :
                ( ! [N6: nat] :
                    ( ( ord_less_eq_nat @ K2 @ N6 )
                   => ( ord_less_rat @ B5 @ ( uminus_uminus_rat @ ( X8 @ N6 ) ) ) )
                | ! [N6: nat] :
                    ( ( ord_less_eq_nat @ K2 @ N6 )
                   => ( ord_less_rat @ B5 @ ( X8 @ N6 ) ) ) ) ) ) ) ).

% cauchy_not_vanishes_cases
thf(fact_9929_cauchy__not__vanishes,axiom,
    ! [X8: nat > rat] :
      ( ( cauchy @ X8 )
     => ( ~ ( vanishes @ X8 )
       => ? [B5: rat] :
            ( ( ord_less_rat @ zero_zero_rat @ B5 )
            & ? [K2: nat] :
              ! [N6: nat] :
                ( ( ord_less_eq_nat @ K2 @ N6 )
               => ( ord_less_rat @ B5 @ ( abs_abs_rat @ ( X8 @ N6 ) ) ) ) ) ) ) ).

% cauchy_not_vanishes
thf(fact_9930_cauchyD,axiom,
    ! [X8: nat > rat,R2: rat] :
      ( ( cauchy @ X8 )
     => ( ( ord_less_rat @ zero_zero_rat @ R2 )
       => ? [K2: nat] :
          ! [M5: nat] :
            ( ( ord_less_eq_nat @ K2 @ M5 )
           => ! [N6: nat] :
                ( ( ord_less_eq_nat @ K2 @ N6 )
               => ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( X8 @ M5 ) @ ( X8 @ N6 ) ) ) @ R2 ) ) ) ) ) ).

% cauchyD
thf(fact_9931_le__Real,axiom,
    ! [X8: nat > rat,Y7: nat > rat] :
      ( ( cauchy @ X8 )
     => ( ( cauchy @ Y7 )
       => ( ( ord_less_eq_real @ ( real2 @ X8 ) @ ( real2 @ Y7 ) )
          = ( ! [R5: rat] :
                ( ( ord_less_rat @ zero_zero_rat @ R5 )
               => ? [K3: nat] :
                  ! [N2: nat] :
                    ( ( ord_less_eq_nat @ K3 @ N2 )
                   => ( ord_less_eq_rat @ ( X8 @ N2 ) @ ( plus_plus_rat @ ( Y7 @ N2 ) @ R5 ) ) ) ) ) ) ) ) ).

% le_Real
thf(fact_9932_inverse__Real,axiom,
    ! [X8: nat > rat] :
      ( ( cauchy @ X8 )
     => ( ( ( vanishes @ X8 )
         => ( ( inverse_inverse_real @ ( real2 @ X8 ) )
            = zero_zero_real ) )
        & ( ~ ( vanishes @ X8 )
         => ( ( inverse_inverse_real @ ( real2 @ X8 ) )
            = ( real2
              @ ^ [N2: nat] : ( inverse_inverse_rat @ ( X8 @ N2 ) ) ) ) ) ) ) ).

% inverse_Real
thf(fact_9933_Real__induct,axiom,
    ! [P: real > $o,X4: real] :
      ( ! [X10: nat > rat] :
          ( ( cauchy @ X10 )
         => ( P @ ( real2 @ X10 ) ) )
     => ( P @ X4 ) ) ).

% Real_induct
thf(fact_9934_one__real__def,axiom,
    ( one_one_real
    = ( real2
      @ ^ [N2: nat] : one_one_rat ) ) ).

% one_real_def
thf(fact_9935_of__nat__Real,axiom,
    ( semiri5074537144036343181t_real
    = ( ^ [X: nat] :
          ( real2
          @ ^ [N2: nat] : ( semiri681578069525770553at_rat @ X ) ) ) ) ).

% of_nat_Real
thf(fact_9936_zero__real__def,axiom,
    ( zero_zero_real
    = ( real2
      @ ^ [N2: nat] : zero_zero_rat ) ) ).

% zero_real_def
thf(fact_9937_of__int__Real,axiom,
    ( ring_1_of_int_real
    = ( ^ [X: int] :
          ( real2
          @ ^ [N2: nat] : ( ring_1_of_int_rat @ X ) ) ) ) ).

% of_int_Real
thf(fact_9938_minus__Real,axiom,
    ! [X8: nat > rat] :
      ( ( cauchy @ X8 )
     => ( ( uminus_uminus_real @ ( real2 @ X8 ) )
        = ( real2
          @ ^ [N2: nat] : ( uminus_uminus_rat @ ( X8 @ N2 ) ) ) ) ) ).

% minus_Real
thf(fact_9939_add__Real,axiom,
    ! [X8: nat > rat,Y7: nat > rat] :
      ( ( cauchy @ X8 )
     => ( ( cauchy @ Y7 )
       => ( ( plus_plus_real @ ( real2 @ X8 ) @ ( real2 @ Y7 ) )
          = ( real2
            @ ^ [N2: nat] : ( plus_plus_rat @ ( X8 @ N2 ) @ ( Y7 @ N2 ) ) ) ) ) ) ).

% add_Real
thf(fact_9940_mult__Real,axiom,
    ! [X8: nat > rat,Y7: nat > rat] :
      ( ( cauchy @ X8 )
     => ( ( cauchy @ Y7 )
       => ( ( times_times_real @ ( real2 @ X8 ) @ ( real2 @ Y7 ) )
          = ( real2
            @ ^ [N2: nat] : ( times_times_rat @ ( X8 @ N2 ) @ ( Y7 @ N2 ) ) ) ) ) ) ).

% mult_Real
thf(fact_9941_diff__Real,axiom,
    ! [X8: nat > rat,Y7: nat > rat] :
      ( ( cauchy @ X8 )
     => ( ( cauchy @ Y7 )
       => ( ( minus_minus_real @ ( real2 @ X8 ) @ ( real2 @ Y7 ) )
          = ( real2
            @ ^ [N2: nat] : ( minus_minus_rat @ ( X8 @ N2 ) @ ( Y7 @ N2 ) ) ) ) ) ) ).

% diff_Real
thf(fact_9942_eq__Real,axiom,
    ! [X8: nat > rat,Y7: nat > rat] :
      ( ( cauchy @ X8 )
     => ( ( cauchy @ Y7 )
       => ( ( ( real2 @ X8 )
            = ( real2 @ Y7 ) )
          = ( vanishes
            @ ^ [N2: nat] : ( minus_minus_rat @ ( X8 @ N2 ) @ ( Y7 @ N2 ) ) ) ) ) ) ).

% eq_Real
thf(fact_9943_not__positive__Real,axiom,
    ! [X8: nat > rat] :
      ( ( cauchy @ X8 )
     => ( ( ~ ( positive2 @ ( real2 @ X8 ) ) )
        = ( ! [R5: rat] :
              ( ( ord_less_rat @ zero_zero_rat @ R5 )
             => ? [K3: nat] :
                ! [N2: nat] :
                  ( ( ord_less_eq_nat @ K3 @ N2 )
                 => ( ord_less_eq_rat @ ( X8 @ N2 ) @ R5 ) ) ) ) ) ) ).

% not_positive_Real
thf(fact_9944_positive__Real,axiom,
    ! [X8: nat > rat] :
      ( ( cauchy @ X8 )
     => ( ( positive2 @ ( real2 @ X8 ) )
        = ( ? [R5: rat] :
              ( ( ord_less_rat @ zero_zero_rat @ R5 )
              & ? [K3: nat] :
                ! [N2: nat] :
                  ( ( ord_less_eq_nat @ K3 @ N2 )
                 => ( ord_less_rat @ R5 @ ( X8 @ N2 ) ) ) ) ) ) ) ).

% positive_Real
thf(fact_9945_Real_Opositive__mult,axiom,
    ! [X4: real,Y3: real] :
      ( ( positive2 @ X4 )
     => ( ( positive2 @ Y3 )
       => ( positive2 @ ( times_times_real @ X4 @ Y3 ) ) ) ) ).

% Real.positive_mult
thf(fact_9946_Real_Opositive__add,axiom,
    ! [X4: real,Y3: real] :
      ( ( positive2 @ X4 )
     => ( ( positive2 @ Y3 )
       => ( positive2 @ ( plus_plus_real @ X4 @ Y3 ) ) ) ) ).

% Real.positive_add
thf(fact_9947_Real_Opositive__zero,axiom,
    ~ ( positive2 @ zero_zero_real ) ).

% Real.positive_zero
thf(fact_9948_Real_Opositive__minus,axiom,
    ! [X4: real] :
      ( ~ ( positive2 @ X4 )
     => ( ( X4 != zero_zero_real )
       => ( positive2 @ ( uminus_uminus_real @ X4 ) ) ) ) ).

% Real.positive_minus
thf(fact_9949_less__real__def,axiom,
    ( ord_less_real
    = ( ^ [X: real,Y: real] : ( positive2 @ ( minus_minus_real @ Y @ X ) ) ) ) ).

% less_real_def
thf(fact_9950_Real_Opositive_Orep__eq,axiom,
    ( positive2
    = ( ^ [X: real] :
        ? [R5: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ R5 )
          & ? [K3: nat] :
            ! [N2: nat] :
              ( ( ord_less_eq_nat @ K3 @ N2 )
             => ( ord_less_rat @ R5 @ ( rep_real @ X @ N2 ) ) ) ) ) ) ).

% Real.positive.rep_eq
thf(fact_9951_inverse__real_Oabs__eq,axiom,
    ! [X4: nat > rat] :
      ( ( realrel @ X4 @ X4 )
     => ( ( inverse_inverse_real @ ( real2 @ X4 ) )
        = ( real2
          @ ( if_nat_rat @ ( vanishes @ X4 )
            @ ^ [N2: nat] : zero_zero_rat
            @ ^ [N2: nat] : ( inverse_inverse_rat @ ( X4 @ N2 ) ) ) ) ) ) ).

% inverse_real.abs_eq
thf(fact_9952_realrel__refl,axiom,
    ! [X8: nat > rat] :
      ( ( cauchy @ X8 )
     => ( realrel @ X8 @ X8 ) ) ).

% realrel_refl
thf(fact_9953_one__real_Orsp,axiom,
    ( realrel
    @ ^ [N2: nat] : one_one_rat
    @ ^ [N2: nat] : one_one_rat ) ).

% one_real.rsp
thf(fact_9954_zero__real_Orsp,axiom,
    ( realrel
    @ ^ [N2: nat] : zero_zero_rat
    @ ^ [N2: nat] : zero_zero_rat ) ).

% zero_real.rsp
thf(fact_9955_real_Oabs__induct,axiom,
    ! [P: real > $o,X4: real] :
      ( ! [Y4: nat > rat] :
          ( ( realrel @ Y4 @ Y4 )
         => ( P @ ( real2 @ Y4 ) ) )
     => ( P @ X4 ) ) ).

% real.abs_induct
thf(fact_9956_uminus__real_Oabs__eq,axiom,
    ! [X4: nat > rat] :
      ( ( realrel @ X4 @ X4 )
     => ( ( uminus_uminus_real @ ( real2 @ X4 ) )
        = ( real2
          @ ^ [N2: nat] : ( uminus_uminus_rat @ ( X4 @ N2 ) ) ) ) ) ).

% uminus_real.abs_eq
thf(fact_9957_plus__real_Oabs__eq,axiom,
    ! [Xa: nat > rat,X4: nat > rat] :
      ( ( realrel @ Xa @ Xa )
     => ( ( realrel @ X4 @ X4 )
       => ( ( plus_plus_real @ ( real2 @ Xa ) @ ( real2 @ X4 ) )
          = ( real2
            @ ^ [N2: nat] : ( plus_plus_rat @ ( Xa @ N2 ) @ ( X4 @ N2 ) ) ) ) ) ) ).

% plus_real.abs_eq
thf(fact_9958_times__real_Oabs__eq,axiom,
    ! [Xa: nat > rat,X4: nat > rat] :
      ( ( realrel @ Xa @ Xa )
     => ( ( realrel @ X4 @ X4 )
       => ( ( times_times_real @ ( real2 @ Xa ) @ ( real2 @ X4 ) )
          = ( real2
            @ ^ [N2: nat] : ( times_times_rat @ ( Xa @ N2 ) @ ( X4 @ N2 ) ) ) ) ) ) ).

% times_real.abs_eq
thf(fact_9959_realrelI,axiom,
    ! [X8: nat > rat,Y7: nat > rat] :
      ( ( cauchy @ X8 )
     => ( ( cauchy @ Y7 )
       => ( ( vanishes
            @ ^ [N2: nat] : ( minus_minus_rat @ ( X8 @ N2 ) @ ( Y7 @ N2 ) ) )
         => ( realrel @ X8 @ Y7 ) ) ) ) ).

% realrelI
thf(fact_9960_realrel__def,axiom,
    ( realrel
    = ( ^ [X6: nat > rat,Y8: nat > rat] :
          ( ( cauchy @ X6 )
          & ( cauchy @ Y8 )
          & ( vanishes
            @ ^ [N2: nat] : ( minus_minus_rat @ ( X6 @ N2 ) @ ( Y8 @ N2 ) ) ) ) ) ) ).

% realrel_def
thf(fact_9961_Real_Opositive_Oabs__eq,axiom,
    ! [X4: nat > rat] :
      ( ( realrel @ X4 @ X4 )
     => ( ( positive2 @ ( real2 @ X4 ) )
        = ( ? [R5: rat] :
              ( ( ord_less_rat @ zero_zero_rat @ R5 )
              & ? [K3: nat] :
                ! [N2: nat] :
                  ( ( ord_less_eq_nat @ K3 @ N2 )
                 => ( ord_less_rat @ R5 @ ( X4 @ N2 ) ) ) ) ) ) ) ).

% Real.positive.abs_eq
thf(fact_9962_inverse__real__def,axiom,
    ( inverse_inverse_real
    = ( map_fu7146612038024189824t_real @ rep_real @ real2
      @ ^ [X6: nat > rat] :
          ( if_nat_rat @ ( vanishes @ X6 )
          @ ^ [N2: nat] : zero_zero_rat
          @ ^ [N2: nat] : ( inverse_inverse_rat @ ( X6 @ N2 ) ) ) ) ) ).

% inverse_real_def
thf(fact_9963_cr__real__def,axiom,
    ( cr_real
    = ( ^ [X: nat > rat,Y: real] :
          ( ( realrel @ X @ X )
          & ( ( real2 @ X )
            = Y ) ) ) ) ).

% cr_real_def
thf(fact_9964_uminus__real__def,axiom,
    ( uminus_uminus_real
    = ( map_fu7146612038024189824t_real @ rep_real @ real2
      @ ^ [X6: nat > rat,N2: nat] : ( uminus_uminus_rat @ ( X6 @ N2 ) ) ) ) ).

% uminus_real_def
thf(fact_9965_times__real__def,axiom,
    ( times_times_real
    = ( map_fu1532550112467129777l_real @ rep_real @ ( map_fu7146612038024189824t_real @ rep_real @ real2 )
      @ ^ [X6: nat > rat,Y8: nat > rat,N2: nat] : ( times_times_rat @ ( X6 @ N2 ) @ ( Y8 @ N2 ) ) ) ) ).

% times_real_def
thf(fact_9966_plus__real__def,axiom,
    ( plus_plus_real
    = ( map_fu1532550112467129777l_real @ rep_real @ ( map_fu7146612038024189824t_real @ rep_real @ real2 )
      @ ^ [X6: nat > rat,Y8: nat > rat,N2: nat] : ( plus_plus_rat @ ( X6 @ N2 ) @ ( Y8 @ N2 ) ) ) ) ).

% plus_real_def
thf(fact_9967_Real_Opositive_Orsp,axiom,
    ( bNF_re728719798268516973at_o_o @ realrel
    @ ^ [Y6: $o,Z4: $o] : Y6 = Z4
    @ ^ [X6: nat > rat] :
      ? [R5: rat] :
        ( ( ord_less_rat @ zero_zero_rat @ R5 )
        & ? [K3: nat] :
          ! [N2: nat] :
            ( ( ord_less_eq_nat @ K3 @ N2 )
           => ( ord_less_rat @ R5 @ ( X6 @ N2 ) ) ) )
    @ ^ [X6: nat > rat] :
      ? [R5: rat] :
        ( ( ord_less_rat @ zero_zero_rat @ R5 )
        & ? [K3: nat] :
          ! [N2: nat] :
            ( ( ord_less_eq_nat @ K3 @ N2 )
           => ( ord_less_rat @ R5 @ ( X6 @ N2 ) ) ) ) ) ).

% Real.positive.rsp
thf(fact_9968_plus__real_Orsp,axiom,
    ( bNF_re1962705104956426057at_rat @ realrel @ ( bNF_re895249473297799549at_rat @ realrel @ realrel )
    @ ^ [X6: nat > rat,Y8: nat > rat,N2: nat] : ( plus_plus_rat @ ( X6 @ N2 ) @ ( Y8 @ N2 ) )
    @ ^ [X6: nat > rat,Y8: nat > rat,N2: nat] : ( plus_plus_rat @ ( X6 @ N2 ) @ ( Y8 @ N2 ) ) ) ).

% plus_real.rsp
thf(fact_9969_uminus__real_Orsp,axiom,
    ( bNF_re895249473297799549at_rat @ realrel @ realrel
    @ ^ [X6: nat > rat,N2: nat] : ( uminus_uminus_rat @ ( X6 @ N2 ) )
    @ ^ [X6: nat > rat,N2: nat] : ( uminus_uminus_rat @ ( X6 @ N2 ) ) ) ).

% uminus_real.rsp
thf(fact_9970_times__real_Orsp,axiom,
    ( bNF_re1962705104956426057at_rat @ realrel @ ( bNF_re895249473297799549at_rat @ realrel @ realrel )
    @ ^ [X6: nat > rat,Y8: nat > rat,N2: nat] : ( times_times_rat @ ( X6 @ N2 ) @ ( Y8 @ N2 ) )
    @ ^ [X6: nat > rat,Y8: nat > rat,N2: nat] : ( times_times_rat @ ( X6 @ N2 ) @ ( Y8 @ N2 ) ) ) ).

% times_real.rsp
thf(fact_9971_sub_Orsp,axiom,
    ( bNF_re8402795839162346335um_int
    @ ^ [Y6: num,Z4: num] : Y6 = Z4
    @ ( bNF_re1822329894187522285nt_int
      @ ^ [Y6: num,Z4: num] : Y6 = Z4
      @ ^ [Y6: int,Z4: int] : Y6 = Z4 )
    @ ^ [M6: num,N2: num] : ( minus_minus_int @ ( numeral_numeral_int @ M6 ) @ ( numeral_numeral_int @ N2 ) )
    @ ^ [M6: num,N2: num] : ( minus_minus_int @ ( numeral_numeral_int @ M6 ) @ ( numeral_numeral_int @ N2 ) ) ) ).

% sub.rsp
thf(fact_9972_times__natural_Orsp,axiom,
    ( bNF_re1345281282404953727at_nat
    @ ^ [Y6: nat,Z4: nat] : Y6 = Z4
    @ ( bNF_re5653821019739307937at_nat
      @ ^ [Y6: nat,Z4: nat] : Y6 = Z4
      @ ^ [Y6: nat,Z4: nat] : Y6 = Z4 )
    @ times_times_nat
    @ times_times_nat ) ).

% times_natural.rsp
thf(fact_9973_times__integer_Orsp,axiom,
    ( bNF_re711492959462206631nt_int
    @ ^ [Y6: int,Z4: int] : Y6 = Z4
    @ ( bNF_re4712519889275205905nt_int
      @ ^ [Y6: int,Z4: int] : Y6 = Z4
      @ ^ [Y6: int,Z4: int] : Y6 = Z4 )
    @ times_times_int
    @ times_times_int ) ).

% times_integer.rsp
thf(fact_9974_minus__integer_Orsp,axiom,
    ( bNF_re711492959462206631nt_int
    @ ^ [Y6: int,Z4: int] : Y6 = Z4
    @ ( bNF_re4712519889275205905nt_int
      @ ^ [Y6: int,Z4: int] : Y6 = Z4
      @ ^ [Y6: int,Z4: int] : Y6 = Z4 )
    @ minus_minus_int
    @ minus_minus_int ) ).

% minus_integer.rsp
thf(fact_9975_minus__natural_Orsp,axiom,
    ( bNF_re1345281282404953727at_nat
    @ ^ [Y6: nat,Z4: nat] : Y6 = Z4
    @ ( bNF_re5653821019739307937at_nat
      @ ^ [Y6: nat,Z4: nat] : Y6 = Z4
      @ ^ [Y6: nat,Z4: nat] : Y6 = Z4 )
    @ minus_minus_nat
    @ minus_minus_nat ) ).

% minus_natural.rsp
thf(fact_9976_divide__natural_Orsp,axiom,
    ( bNF_re1345281282404953727at_nat
    @ ^ [Y6: nat,Z4: nat] : Y6 = Z4
    @ ( bNF_re5653821019739307937at_nat
      @ ^ [Y6: nat,Z4: nat] : Y6 = Z4
      @ ^ [Y6: nat,Z4: nat] : Y6 = Z4 )
    @ divide_divide_nat
    @ divide_divide_nat ) ).

% divide_natural.rsp
thf(fact_9977_divide__integer_Orsp,axiom,
    ( bNF_re711492959462206631nt_int
    @ ^ [Y6: int,Z4: int] : Y6 = Z4
    @ ( bNF_re4712519889275205905nt_int
      @ ^ [Y6: int,Z4: int] : Y6 = Z4
      @ ^ [Y6: int,Z4: int] : Y6 = Z4 )
    @ divide_divide_int
    @ divide_divide_int ) ).

% divide_integer.rsp
thf(fact_9978_modulo__natural_Orsp,axiom,
    ( bNF_re1345281282404953727at_nat
    @ ^ [Y6: nat,Z4: nat] : Y6 = Z4
    @ ( bNF_re5653821019739307937at_nat
      @ ^ [Y6: nat,Z4: nat] : Y6 = Z4
      @ ^ [Y6: nat,Z4: nat] : Y6 = Z4 )
    @ modulo_modulo_nat
    @ modulo_modulo_nat ) ).

% modulo_natural.rsp
thf(fact_9979_modulo__integer_Orsp,axiom,
    ( bNF_re711492959462206631nt_int
    @ ^ [Y6: int,Z4: int] : Y6 = Z4
    @ ( bNF_re4712519889275205905nt_int
      @ ^ [Y6: int,Z4: int] : Y6 = Z4
      @ ^ [Y6: int,Z4: int] : Y6 = Z4 )
    @ modulo_modulo_int
    @ modulo_modulo_int ) ).

% modulo_integer.rsp
thf(fact_9980_Suc_Orsp,axiom,
    ( bNF_re5653821019739307937at_nat
    @ ^ [Y6: nat,Z4: nat] : Y6 = Z4
    @ ^ [Y6: nat,Z4: nat] : Y6 = Z4
    @ suc
    @ suc ) ).

% Suc.rsp
thf(fact_9981_dup_Orsp,axiom,
    ( bNF_re4712519889275205905nt_int
    @ ^ [Y6: int,Z4: int] : Y6 = Z4
    @ ^ [Y6: int,Z4: int] : Y6 = Z4
    @ ^ [K3: int] : ( plus_plus_int @ K3 @ K3 )
    @ ^ [K3: int] : ( plus_plus_int @ K3 @ K3 ) ) ).

% dup.rsp
thf(fact_9982_plus__natural_Orsp,axiom,
    ( bNF_re1345281282404953727at_nat
    @ ^ [Y6: nat,Z4: nat] : Y6 = Z4
    @ ( bNF_re5653821019739307937at_nat
      @ ^ [Y6: nat,Z4: nat] : Y6 = Z4
      @ ^ [Y6: nat,Z4: nat] : Y6 = Z4 )
    @ plus_plus_nat
    @ plus_plus_nat ) ).

% plus_natural.rsp
thf(fact_9983_plus__integer_Orsp,axiom,
    ( bNF_re711492959462206631nt_int
    @ ^ [Y6: int,Z4: int] : Y6 = Z4
    @ ( bNF_re4712519889275205905nt_int
      @ ^ [Y6: int,Z4: int] : Y6 = Z4
      @ ^ [Y6: int,Z4: int] : Y6 = Z4 )
    @ plus_plus_int
    @ plus_plus_int ) ).

% plus_integer.rsp
thf(fact_9984_less__integer_Orsp,axiom,
    ( bNF_re3403563459893282935_int_o
    @ ^ [Y6: int,Z4: int] : Y6 = Z4
    @ ( bNF_re5089333283451836215nt_o_o
      @ ^ [Y6: int,Z4: int] : Y6 = Z4
      @ ^ [Y6: $o,Z4: $o] : Y6 = Z4 )
    @ ord_less_int
    @ ord_less_int ) ).

% less_integer.rsp
thf(fact_9985_less__natural_Orsp,axiom,
    ( bNF_re578469030762574527_nat_o
    @ ^ [Y6: nat,Z4: nat] : Y6 = Z4
    @ ( bNF_re4705727531993890431at_o_o
      @ ^ [Y6: nat,Z4: nat] : Y6 = Z4
      @ ^ [Y6: $o,Z4: $o] : Y6 = Z4 )
    @ ord_less_nat
    @ ord_less_nat ) ).

% less_natural.rsp
thf(fact_9986_less__eq__natural_Orsp,axiom,
    ( bNF_re578469030762574527_nat_o
    @ ^ [Y6: nat,Z4: nat] : Y6 = Z4
    @ ( bNF_re4705727531993890431at_o_o
      @ ^ [Y6: nat,Z4: nat] : Y6 = Z4
      @ ^ [Y6: $o,Z4: $o] : Y6 = Z4 )
    @ ord_less_eq_nat
    @ ord_less_eq_nat ) ).

% less_eq_natural.rsp
thf(fact_9987_le__enumerate,axiom,
    ! [S2: set_nat,N: nat] :
      ( ~ ( finite_finite_nat @ S2 )
     => ( ord_less_eq_nat @ N @ ( infini8530281810654367211te_nat @ S2 @ N ) ) ) ).

% le_enumerate
thf(fact_9988_inverse__real_Orsp,axiom,
    ( bNF_re895249473297799549at_rat @ realrel @ realrel
    @ ^ [X6: nat > rat] :
        ( if_nat_rat @ ( vanishes @ X6 )
        @ ^ [N2: nat] : zero_zero_rat
        @ ^ [N2: nat] : ( inverse_inverse_rat @ ( X6 @ N2 ) ) )
    @ ^ [X6: nat > rat] :
        ( if_nat_rat @ ( vanishes @ X6 )
        @ ^ [N2: nat] : zero_zero_rat
        @ ^ [N2: nat] : ( inverse_inverse_rat @ ( X6 @ N2 ) ) ) ) ).

% inverse_real.rsp
thf(fact_9989_finite__le__enumerate,axiom,
    ! [S2: set_nat,N: nat] :
      ( ( finite_finite_nat @ S2 )
     => ( ( ord_less_nat @ N @ ( finite_card_nat @ S2 ) )
       => ( ord_less_eq_nat @ N @ ( infini8530281810654367211te_nat @ S2 @ N ) ) ) ) ).

% finite_le_enumerate
thf(fact_9990_Real_Opositive_Otransfer,axiom,
    ( bNF_re4297313714947099218al_o_o @ pcr_real
    @ ^ [Y6: $o,Z4: $o] : Y6 = Z4
    @ ^ [X6: nat > rat] :
      ? [R5: rat] :
        ( ( ord_less_rat @ zero_zero_rat @ R5 )
        & ? [K3: nat] :
          ! [N2: nat] :
            ( ( ord_less_eq_nat @ K3 @ N2 )
           => ( ord_less_rat @ R5 @ ( X6 @ N2 ) ) ) )
    @ positive2 ) ).

% Real.positive.transfer
thf(fact_9991_plus__rat_Otransfer,axiom,
    ( bNF_re7627151682743391978at_rat @ pcr_rat @ ( bNF_re8279943556446156061nt_rat @ pcr_rat @ pcr_rat )
    @ ^ [X: product_prod_int_int,Y: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ Y ) ) @ ( times_times_int @ ( product_fst_int_int @ Y ) @ ( product_snd_int_int @ X ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X ) @ ( product_snd_int_int @ Y ) ) )
    @ plus_plus_rat ) ).

% plus_rat.transfer
thf(fact_9992_real_Orel__eq__transfer,axiom,
    ( bNF_re4521903465945308077real_o @ pcr_real
    @ ( bNF_re4297313714947099218al_o_o @ pcr_real
      @ ^ [Y6: $o,Z4: $o] : Y6 = Z4 )
    @ realrel
    @ ^ [Y6: real,Z4: real] : Y6 = Z4 ) ).

% real.rel_eq_transfer
thf(fact_9993_real_Opcr__cr__eq,axiom,
    pcr_real = cr_real ).

% real.pcr_cr_eq
thf(fact_9994_zero__real_Otransfer,axiom,
    ( pcr_real
    @ ^ [N2: nat] : zero_zero_rat
    @ zero_zero_real ) ).

% zero_real.transfer
thf(fact_9995_one__real_Otransfer,axiom,
    ( pcr_real
    @ ^ [N2: nat] : one_one_rat
    @ one_one_real ) ).

% one_real.transfer
thf(fact_9996_cr__real__eq,axiom,
    ( pcr_real
    = ( ^ [X: nat > rat,Y: real] :
          ( ( cauchy @ X )
          & ( ( real2 @ X )
            = Y ) ) ) ) ).

% cr_real_eq
thf(fact_9997_uminus__real_Otransfer,axiom,
    ( bNF_re3023117138289059399t_real @ pcr_real @ pcr_real
    @ ^ [X6: nat > rat,N2: nat] : ( uminus_uminus_rat @ ( X6 @ N2 ) )
    @ uminus_uminus_real ) ).

% uminus_real.transfer
thf(fact_9998_plus__real_Otransfer,axiom,
    ( bNF_re4695409256820837752l_real @ pcr_real @ ( bNF_re3023117138289059399t_real @ pcr_real @ pcr_real )
    @ ^ [X6: nat > rat,Y8: nat > rat,N2: nat] : ( plus_plus_rat @ ( X6 @ N2 ) @ ( Y8 @ N2 ) )
    @ plus_plus_real ) ).

% plus_real.transfer
thf(fact_9999_times__real_Otransfer,axiom,
    ( bNF_re4695409256820837752l_real @ pcr_real @ ( bNF_re3023117138289059399t_real @ pcr_real @ pcr_real )
    @ ^ [X6: nat > rat,Y8: nat > rat,N2: nat] : ( times_times_rat @ ( X6 @ N2 ) @ ( Y8 @ N2 ) )
    @ times_times_real ) ).

% times_real.transfer
thf(fact_10000_inverse__real_Otransfer,axiom,
    ( bNF_re3023117138289059399t_real @ pcr_real @ pcr_real
    @ ^ [X6: nat > rat] :
        ( if_nat_rat @ ( vanishes @ X6 )
        @ ^ [N2: nat] : zero_zero_rat
        @ ^ [N2: nat] : ( inverse_inverse_rat @ ( X6 @ N2 ) ) )
    @ inverse_inverse_real ) ).

% inverse_real.transfer
thf(fact_10001_times__rat_Otransfer,axiom,
    ( bNF_re7627151682743391978at_rat @ pcr_rat @ ( bNF_re8279943556446156061nt_rat @ pcr_rat @ pcr_rat )
    @ ^ [X: product_prod_int_int,Y: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_fst_int_int @ Y ) ) @ ( times_times_int @ ( product_snd_int_int @ X ) @ ( product_snd_int_int @ Y ) ) )
    @ times_times_rat ) ).

% times_rat.transfer
thf(fact_10002_Rat_Opositive_Otransfer,axiom,
    ( bNF_re1494630372529172596at_o_o @ pcr_rat
    @ ^ [Y6: $o,Z4: $o] : Y6 = Z4
    @ ^ [X: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ X ) ) )
    @ positive ) ).

% Rat.positive.transfer
thf(fact_10003_times__int_Otransfer,axiom,
    ( bNF_re7408651293131936558nt_int @ pcr_int @ ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int )
    @ ( produc27273713700761075at_nat
      @ ^ [X: nat,Y: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U4: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X @ U4 ) @ ( times_times_nat @ Y @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X @ V4 ) @ ( times_times_nat @ Y @ U4 ) ) ) ) )
    @ times_times_int ) ).

% times_int.transfer
thf(fact_10004_Least__eq__0,axiom,
    ! [P: nat > $o] :
      ( ( P @ zero_zero_nat )
     => ( ( ord_Least_nat @ P )
        = zero_zero_nat ) ) ).

% Least_eq_0
thf(fact_10005_Least__Suc,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ N )
     => ( ~ ( P @ zero_zero_nat )
       => ( ( ord_Least_nat @ P )
          = ( suc
            @ ( ord_Least_nat
              @ ^ [M6: nat] : ( P @ ( suc @ M6 ) ) ) ) ) ) ) ).

% Least_Suc
thf(fact_10006_Least__Suc2,axiom,
    ! [P: nat > $o,N: nat,Q: nat > $o,M: nat] :
      ( ( P @ N )
     => ( ( Q @ M )
       => ( ~ ( P @ zero_zero_nat )
         => ( ! [K2: nat] :
                ( ( P @ ( suc @ K2 ) )
                = ( Q @ K2 ) )
           => ( ( ord_Least_nat @ P )
              = ( suc @ ( ord_Least_nat @ Q ) ) ) ) ) ) ) ).

% Least_Suc2
thf(fact_10007_Sup__real__def,axiom,
    ( comple1385675409528146559p_real
    = ( ^ [X6: set_real] :
          ( ord_Least_real
          @ ^ [Z5: real] :
            ! [X: real] :
              ( ( member_real @ X @ X6 )
             => ( ord_less_eq_real @ X @ Z5 ) ) ) ) ) ).

% Sup_real_def
thf(fact_10008_nat_Otransfer,axiom,
    ( bNF_re4555766996558763186at_nat @ pcr_int
    @ ^ [Y6: nat,Z4: nat] : Y6 = Z4
    @ ( produc6842872674320459806at_nat @ minus_minus_nat )
    @ nat2 ) ).

% nat.transfer
thf(fact_10009_one__int_Otransfer,axiom,
    pcr_int @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) @ one_one_int ).

% one_int.transfer
thf(fact_10010_less__int_Otransfer,axiom,
    ( bNF_re717283939379294677_int_o @ pcr_int
    @ ( bNF_re6644619430987730960nt_o_o @ pcr_int
      @ ^ [Y6: $o,Z4: $o] : Y6 = Z4 )
    @ ( produc8739625826339149834_nat_o
      @ ^ [X: nat,Y: nat] :
          ( produc6081775807080527818_nat_o
          @ ^ [U4: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ U4 @ Y ) ) ) )
    @ ord_less_int ) ).

% less_int.transfer
thf(fact_10011_less__eq__int_Otransfer,axiom,
    ( bNF_re717283939379294677_int_o @ pcr_int
    @ ( bNF_re6644619430987730960nt_o_o @ pcr_int
      @ ^ [Y6: $o,Z4: $o] : Y6 = Z4 )
    @ ( produc8739625826339149834_nat_o
      @ ^ [X: nat,Y: nat] :
          ( produc6081775807080527818_nat_o
          @ ^ [U4: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ U4 @ Y ) ) ) )
    @ ord_less_eq_int ) ).

% less_eq_int.transfer
thf(fact_10012_plus__int_Otransfer,axiom,
    ( bNF_re7408651293131936558nt_int @ pcr_int @ ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int )
    @ ( produc27273713700761075at_nat
      @ ^ [X: nat,Y: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U4: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X @ U4 ) @ ( plus_plus_nat @ Y @ V4 ) ) ) )
    @ plus_plus_int ) ).

% plus_int.transfer
thf(fact_10013_minus__int_Otransfer,axiom,
    ( bNF_re7408651293131936558nt_int @ pcr_int @ ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int )
    @ ( produc27273713700761075at_nat
      @ ^ [X: nat,Y: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U4: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ Y @ U4 ) ) ) )
    @ minus_minus_int ) ).

% minus_int.transfer
thf(fact_10014_of__rat__Real,axiom,
    ( field_7254667332652039916t_real
    = ( ^ [X: rat] :
          ( real2
          @ ^ [N2: nat] : X ) ) ) ).

% of_rat_Real
thf(fact_10015_of__rat__dense,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_real @ X4 @ Y3 )
     => ? [Q3: rat] :
          ( ( ord_less_real @ X4 @ ( field_7254667332652039916t_real @ Q3 ) )
          & ( ord_less_real @ ( field_7254667332652039916t_real @ Q3 ) @ Y3 ) ) ) ).

% of_rat_dense
thf(fact_10016_less__RealD,axiom,
    ! [Y7: nat > rat,X4: real] :
      ( ( cauchy @ Y7 )
     => ( ( ord_less_real @ X4 @ ( real2 @ Y7 ) )
       => ? [N4: nat] : ( ord_less_real @ X4 @ ( field_7254667332652039916t_real @ ( Y7 @ N4 ) ) ) ) ) ).

% less_RealD
thf(fact_10017_le__RealI,axiom,
    ! [Y7: nat > rat,X4: real] :
      ( ( cauchy @ Y7 )
     => ( ! [N4: nat] : ( ord_less_eq_real @ X4 @ ( field_7254667332652039916t_real @ ( Y7 @ N4 ) ) )
       => ( ord_less_eq_real @ X4 @ ( real2 @ Y7 ) ) ) ) ).

% le_RealI
thf(fact_10018_Real__leI,axiom,
    ! [X8: nat > rat,Y3: real] :
      ( ( cauchy @ X8 )
     => ( ! [N4: nat] : ( ord_less_eq_real @ ( field_7254667332652039916t_real @ ( X8 @ N4 ) ) @ Y3 )
       => ( ord_less_eq_real @ ( real2 @ X8 ) @ Y3 ) ) ) ).

% Real_leI
thf(fact_10019_times__int_Orsp,axiom,
    ( bNF_re3099431351363272937at_nat @ intrel @ ( bNF_re2241393799969408733at_nat @ intrel @ intrel )
    @ ( produc27273713700761075at_nat
      @ ^ [X: nat,Y: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U4: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X @ U4 ) @ ( times_times_nat @ Y @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X @ V4 ) @ ( times_times_nat @ Y @ U4 ) ) ) ) )
    @ ( produc27273713700761075at_nat
      @ ^ [X: nat,Y: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U4: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X @ U4 ) @ ( times_times_nat @ Y @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X @ V4 ) @ ( times_times_nat @ Y @ U4 ) ) ) ) ) ) ).

% times_int.rsp
thf(fact_10020_intrel__iff,axiom,
    ! [X4: nat,Y3: nat,U: nat,V: nat] :
      ( ( intrel @ ( product_Pair_nat_nat @ X4 @ Y3 ) @ ( product_Pair_nat_nat @ U @ V ) )
      = ( ( plus_plus_nat @ X4 @ V )
        = ( plus_plus_nat @ U @ Y3 ) ) ) ).

% intrel_iff
thf(fact_10021_nat_Orsp,axiom,
    ( bNF_re8246922863344978751at_nat @ intrel
    @ ^ [Y6: nat,Z4: nat] : Y6 = Z4
    @ ( produc6842872674320459806at_nat @ minus_minus_nat )
    @ ( produc6842872674320459806at_nat @ minus_minus_nat ) ) ).

% nat.rsp
thf(fact_10022_one__int_Orsp,axiom,
    intrel @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) ).

% one_int.rsp
thf(fact_10023_intrel__def,axiom,
    ( intrel
    = ( produc8739625826339149834_nat_o
      @ ^ [X: nat,Y: nat] :
          ( produc6081775807080527818_nat_o
          @ ^ [U4: nat,V4: nat] :
              ( ( plus_plus_nat @ X @ V4 )
              = ( plus_plus_nat @ U4 @ Y ) ) ) ) ) ).

% intrel_def
thf(fact_10024_less__int_Orsp,axiom,
    ( bNF_re4202695980764964119_nat_o @ intrel
    @ ( bNF_re3666534408544137501at_o_o @ intrel
      @ ^ [Y6: $o,Z4: $o] : Y6 = Z4 )
    @ ( produc8739625826339149834_nat_o
      @ ^ [X: nat,Y: nat] :
          ( produc6081775807080527818_nat_o
          @ ^ [U4: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ U4 @ Y ) ) ) )
    @ ( produc8739625826339149834_nat_o
      @ ^ [X: nat,Y: nat] :
          ( produc6081775807080527818_nat_o
          @ ^ [U4: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ U4 @ Y ) ) ) ) ) ).

% less_int.rsp
thf(fact_10025_less__eq__int_Orsp,axiom,
    ( bNF_re4202695980764964119_nat_o @ intrel
    @ ( bNF_re3666534408544137501at_o_o @ intrel
      @ ^ [Y6: $o,Z4: $o] : Y6 = Z4 )
    @ ( produc8739625826339149834_nat_o
      @ ^ [X: nat,Y: nat] :
          ( produc6081775807080527818_nat_o
          @ ^ [U4: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ U4 @ Y ) ) ) )
    @ ( produc8739625826339149834_nat_o
      @ ^ [X: nat,Y: nat] :
          ( produc6081775807080527818_nat_o
          @ ^ [U4: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ U4 @ Y ) ) ) ) ) ).

% less_eq_int.rsp
thf(fact_10026_plus__int_Orsp,axiom,
    ( bNF_re3099431351363272937at_nat @ intrel @ ( bNF_re2241393799969408733at_nat @ intrel @ intrel )
    @ ( produc27273713700761075at_nat
      @ ^ [X: nat,Y: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U4: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X @ U4 ) @ ( plus_plus_nat @ Y @ V4 ) ) ) )
    @ ( produc27273713700761075at_nat
      @ ^ [X: nat,Y: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U4: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X @ U4 ) @ ( plus_plus_nat @ Y @ V4 ) ) ) ) ) ).

% plus_int.rsp
thf(fact_10027_minus__int_Orsp,axiom,
    ( bNF_re3099431351363272937at_nat @ intrel @ ( bNF_re2241393799969408733at_nat @ intrel @ intrel )
    @ ( produc27273713700761075at_nat
      @ ^ [X: nat,Y: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U4: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ Y @ U4 ) ) ) )
    @ ( produc27273713700761075at_nat
      @ ^ [X: nat,Y: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U4: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ Y @ U4 ) ) ) ) ) ).

% minus_int.rsp
thf(fact_10028_has__vector__derivative__id,axiom,
    ! [Net: filter_real] :
      ( has_ve631408500373753343e_real
      @ ^ [X: real] : X
      @ one_one_real
      @ Net ) ).

% has_vector_derivative_id
thf(fact_10029_has__field__derivative__iff__has__vector__derivative,axiom,
    has_fi5821293074295781190e_real = has_ve631408500373753343e_real ).

% has_field_derivative_iff_has_vector_derivative
thf(fact_10030_plus__rat_Orsp,axiom,
    ( bNF_re5228765855967844073nt_int @ ratrel @ ( bNF_re7145576690424134365nt_int @ ratrel @ ratrel )
    @ ^ [X: product_prod_int_int,Y: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ Y ) ) @ ( times_times_int @ ( product_fst_int_int @ Y ) @ ( product_snd_int_int @ X ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X ) @ ( product_snd_int_int @ Y ) ) )
    @ ^ [X: product_prod_int_int,Y: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ Y ) ) @ ( times_times_int @ ( product_fst_int_int @ Y ) @ ( product_snd_int_int @ X ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X ) @ ( product_snd_int_int @ Y ) ) ) ) ).

% plus_rat.rsp
thf(fact_10031_ratrel__iff,axiom,
    ( ratrel
    = ( ^ [X: product_prod_int_int,Y: product_prod_int_int] :
          ( ( ( product_snd_int_int @ X )
           != zero_zero_int )
          & ( ( product_snd_int_int @ Y )
           != zero_zero_int )
          & ( ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ Y ) )
            = ( times_times_int @ ( product_fst_int_int @ Y ) @ ( product_snd_int_int @ X ) ) ) ) ) ) ).

% ratrel_iff
thf(fact_10032_ratrel__def,axiom,
    ( ratrel
    = ( ^ [X: product_prod_int_int,Y: product_prod_int_int] :
          ( ( ( product_snd_int_int @ X )
           != zero_zero_int )
          & ( ( product_snd_int_int @ Y )
           != zero_zero_int )
          & ( ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ Y ) )
            = ( times_times_int @ ( product_fst_int_int @ Y ) @ ( product_snd_int_int @ X ) ) ) ) ) ) ).

% ratrel_def
thf(fact_10033_times__rat_Orsp,axiom,
    ( bNF_re5228765855967844073nt_int @ ratrel @ ( bNF_re7145576690424134365nt_int @ ratrel @ ratrel )
    @ ^ [X: product_prod_int_int,Y: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_fst_int_int @ Y ) ) @ ( times_times_int @ ( product_snd_int_int @ X ) @ ( product_snd_int_int @ Y ) ) )
    @ ^ [X: product_prod_int_int,Y: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_fst_int_int @ Y ) ) @ ( times_times_int @ ( product_snd_int_int @ X ) @ ( product_snd_int_int @ Y ) ) ) ) ).

% times_rat.rsp
thf(fact_10034_Rat_Opositive_Orsp,axiom,
    ( bNF_re8699439704749558557nt_o_o @ ratrel
    @ ^ [Y6: $o,Z4: $o] : Y6 = Z4
    @ ^ [X: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ X ) ) )
    @ ^ [X: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ X ) ) ) ) ).

% Rat.positive.rsp
thf(fact_10035_plus__rat_Oabs__eq,axiom,
    ! [Xa: product_prod_int_int,X4: product_prod_int_int] :
      ( ( ratrel @ Xa @ Xa )
     => ( ( ratrel @ X4 @ X4 )
       => ( ( plus_plus_rat @ ( abs_Rat @ Xa ) @ ( abs_Rat @ X4 ) )
          = ( abs_Rat @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ Xa ) @ ( product_snd_int_int @ X4 ) ) @ ( times_times_int @ ( product_fst_int_int @ X4 ) @ ( product_snd_int_int @ Xa ) ) ) @ ( times_times_int @ ( product_snd_int_int @ Xa ) @ ( product_snd_int_int @ X4 ) ) ) ) ) ) ) ).

% plus_rat.abs_eq
thf(fact_10036_Rat_Opositive_Oabs__eq,axiom,
    ! [X4: product_prod_int_int] :
      ( ( ratrel @ X4 @ X4 )
     => ( ( positive @ ( abs_Rat @ X4 ) )
        = ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X4 ) @ ( product_snd_int_int @ X4 ) ) ) ) ) ).

% Rat.positive.abs_eq
thf(fact_10037_times__rat_Oabs__eq,axiom,
    ! [Xa: product_prod_int_int,X4: product_prod_int_int] :
      ( ( ratrel @ Xa @ Xa )
     => ( ( ratrel @ X4 @ X4 )
       => ( ( times_times_rat @ ( abs_Rat @ Xa ) @ ( abs_Rat @ X4 ) )
          = ( abs_Rat @ ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ Xa ) @ ( product_fst_int_int @ X4 ) ) @ ( times_times_int @ ( product_snd_int_int @ Xa ) @ ( product_snd_int_int @ X4 ) ) ) ) ) ) ) ).

% times_rat.abs_eq
thf(fact_10038_last__upt,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( last_nat @ ( upt @ I @ J ) )
        = ( minus_minus_nat @ J @ one_one_nat ) ) ) ).

% last_upt
thf(fact_10039_Bseq__monoseq__convergent_H__dec,axiom,
    ! [F: nat > real,M7: nat] :
      ( ( bfun_nat_real
        @ ^ [N2: nat] : ( F @ ( plus_plus_nat @ N2 @ M7 ) )
        @ at_top_nat )
     => ( ! [M4: nat,N4: nat] :
            ( ( ord_less_eq_nat @ M7 @ M4 )
           => ( ( ord_less_eq_nat @ M4 @ N4 )
             => ( ord_less_eq_real @ ( F @ N4 ) @ ( F @ M4 ) ) ) )
       => ( topolo7531315842566124627t_real @ F ) ) ) ).

% Bseq_monoseq_convergent'_dec
thf(fact_10040_Bseq__monoseq__convergent_H__inc,axiom,
    ! [F: nat > real,M7: nat] :
      ( ( bfun_nat_real
        @ ^ [N2: nat] : ( F @ ( plus_plus_nat @ N2 @ M7 ) )
        @ at_top_nat )
     => ( ! [M4: nat,N4: nat] :
            ( ( ord_less_eq_nat @ M7 @ M4 )
           => ( ( ord_less_eq_nat @ M4 @ N4 )
             => ( ord_less_eq_real @ ( F @ M4 ) @ ( F @ N4 ) ) ) )
       => ( topolo7531315842566124627t_real @ F ) ) ) ).

% Bseq_monoseq_convergent'_inc
thf(fact_10041_Bseq__mono__convergent,axiom,
    ! [X8: nat > real] :
      ( ( bfun_nat_real @ X8 @ at_top_nat )
     => ( ! [M4: nat,N4: nat] :
            ( ( ord_less_eq_nat @ M4 @ N4 )
           => ( ord_less_eq_real @ ( X8 @ M4 ) @ ( X8 @ N4 ) ) )
       => ( topolo7531315842566124627t_real @ X8 ) ) ) ).

% Bseq_mono_convergent
thf(fact_10042_convergent__realpow,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_eq_real @ X4 @ one_one_real )
       => ( topolo7531315842566124627t_real @ ( power_power_real @ X4 ) ) ) ) ).

% convergent_realpow
thf(fact_10043_filtermap__at__right__shift,axiom,
    ! [D: real,A: real] :
      ( ( filtermap_real_real
        @ ^ [X: real] : ( minus_minus_real @ X @ D )
        @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
      = ( topolo2177554685111907308n_real @ ( minus_minus_real @ A @ D ) @ ( set_or5849166863359141190n_real @ ( minus_minus_real @ A @ D ) ) ) ) ).

% filtermap_at_right_shift
thf(fact_10044_at__right__to__0,axiom,
    ! [A: real] :
      ( ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) )
      = ( filtermap_real_real
        @ ^ [X: real] : ( plus_plus_real @ X @ A )
        @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).

% at_right_to_0
thf(fact_10045_pair__lessI2,axiom,
    ! [A: nat,B2: nat,S: nat,T: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( ord_less_nat @ S @ T )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B2 @ T ) ) @ fun_pair_less ) ) ) ).

% pair_lessI2
thf(fact_10046_pair__less__iff1,axiom,
    ! [X4: nat,Y3: nat,Z: nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ X4 @ Y3 ) @ ( product_Pair_nat_nat @ X4 @ Z ) ) @ fun_pair_less )
      = ( ord_less_nat @ Y3 @ Z ) ) ).

% pair_less_iff1
thf(fact_10047_pair__lessI1,axiom,
    ! [A: nat,B2: nat,S: nat,T: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B2 @ T ) ) @ fun_pair_less ) ) ).

% pair_lessI1
thf(fact_10048_pair__leqI2,axiom,
    ! [A: nat,B2: nat,S: nat,T: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( ord_less_eq_nat @ S @ T )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B2 @ T ) ) @ fun_pair_leq ) ) ) ).

% pair_leqI2
thf(fact_10049_pair__leqI1,axiom,
    ! [A: nat,B2: nat,S: nat,T: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B2 @ T ) ) @ fun_pair_leq ) ) ).

% pair_leqI1
thf(fact_10050_bot__nat__0_Oordering__top__axioms,axiom,
    ( ordering_top_nat
    @ ^ [X: nat,Y: nat] : ( ord_less_eq_nat @ Y @ X )
    @ ^ [X: nat,Y: nat] : ( ord_less_nat @ Y @ X )
    @ zero_zero_nat ) ).

% bot_nat_0.ordering_top_axioms
thf(fact_10051_set__encode__vimage__Suc,axiom,
    ! [A3: set_nat] :
      ( ( nat_set_encode @ ( vimage_nat_nat @ suc @ A3 ) )
      = ( divide_divide_nat @ ( nat_set_encode @ A3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% set_encode_vimage_Suc
thf(fact_10052_euclidean__size__int__def,axiom,
    ( euclid4774559944035922753ze_int
    = ( comp_int_nat_int @ nat2 @ abs_abs_int ) ) ).

% euclidean_size_int_def
thf(fact_10053_finite__vimage__Suc__iff,axiom,
    ! [F5: set_nat] :
      ( ( finite_finite_nat @ ( vimage_nat_nat @ suc @ F5 ) )
      = ( finite_finite_nat @ F5 ) ) ).

% finite_vimage_Suc_iff
thf(fact_10054_vimage__Suc__insert__Suc,axiom,
    ! [N: nat,A3: set_nat] :
      ( ( vimage_nat_nat @ suc @ ( insert_nat @ ( suc @ N ) @ A3 ) )
      = ( insert_nat @ N @ ( vimage_nat_nat @ suc @ A3 ) ) ) ).

% vimage_Suc_insert_Suc
thf(fact_10055_vimage__Suc__insert__0,axiom,
    ! [A3: set_nat] :
      ( ( vimage_nat_nat @ suc @ ( insert_nat @ zero_zero_nat @ A3 ) )
      = ( vimage_nat_nat @ suc @ A3 ) ) ).

% vimage_Suc_insert_0
thf(fact_10056_set__decode__div__2,axiom,
    ! [X4: nat] :
      ( ( nat_set_decode @ ( divide_divide_nat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( vimage_nat_nat @ suc @ ( nat_set_decode @ X4 ) ) ) ).

% set_decode_div_2
thf(fact_10057_abs__division__segment,axiom,
    ! [K: int] :
      ( ( abs_abs_int @ ( euclid3395696857347342551nt_int @ K ) )
      = one_one_int ) ).

% abs_division_segment
thf(fact_10058_division__segment__nat__def,axiom,
    ( euclid3398187327856392827nt_nat
    = ( ^ [N2: nat] : one_one_nat ) ) ).

% division_segment_nat_def
thf(fact_10059_division__segment__eq__sgn,axiom,
    ! [K: int] :
      ( ( K != zero_zero_int )
     => ( ( euclid3395696857347342551nt_int @ K )
        = ( sgn_sgn_int @ K ) ) ) ).

% division_segment_eq_sgn
thf(fact_10060_division__segment__int__def,axiom,
    ( euclid3395696857347342551nt_int
    = ( ^ [K3: int] : ( if_int @ ( ord_less_eq_int @ zero_zero_int @ K3 ) @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ) ) ).

% division_segment_int_def
thf(fact_10061_less__eq__enat__def,axiom,
    ( ord_le2932123472753598470d_enat
    = ( ^ [M6: extended_enat] :
          ( extended_case_enat_o
          @ ^ [N1: nat] :
              ( extended_case_enat_o
              @ ^ [M1: nat] : ( ord_less_eq_nat @ M1 @ N1 )
              @ $false
              @ M6 )
          @ $true ) ) ) ).

% less_eq_enat_def
thf(fact_10062_transp__realrel,axiom,
    transp_nat_rat @ realrel ).

% transp_realrel
thf(fact_10063_less__enat__def,axiom,
    ( ord_le72135733267957522d_enat
    = ( ^ [M6: extended_enat,N2: extended_enat] :
          ( extended_case_enat_o
          @ ^ [M1: nat] : ( extended_case_enat_o @ ( ord_less_nat @ M1 ) @ $true @ N2 )
          @ $false
          @ M6 ) ) ) ).

% less_enat_def
thf(fact_10064_Divides_Oadjust__div__def,axiom,
    ( adjust_div
    = ( produc8211389475949308722nt_int
      @ ^ [Q4: int,R5: int] : ( plus_plus_int @ Q4 @ ( zero_n2684676970156552555ol_int @ ( R5 != zero_zero_int ) ) ) ) ) ).

% Divides.adjust_div_def
thf(fact_10065_rat__floor__code,axiom,
    ( archim3151403230148437115or_rat
    = ( ^ [P3: rat] : ( produc8211389475949308722nt_int @ divide_divide_int @ ( quotient_of @ P3 ) ) ) ) ).

% rat_floor_code
thf(fact_10066_prod__decode__triangle__add,axiom,
    ! [K: nat,M: nat] :
      ( ( nat_prod_decode @ ( plus_plus_nat @ ( nat_triangle @ K ) @ M ) )
      = ( nat_prod_decode_aux @ K @ M ) ) ).

% prod_decode_triangle_add
thf(fact_10067_list__decode_Opinduct,axiom,
    ! [A0: nat,P: nat > $o] :
      ( ( accp_nat @ nat_list_decode_rel @ A0 )
     => ( ( ( accp_nat @ nat_list_decode_rel @ zero_zero_nat )
         => ( P @ zero_zero_nat ) )
       => ( ! [N4: nat] :
              ( ( accp_nat @ nat_list_decode_rel @ ( suc @ N4 ) )
             => ( ! [X5: nat,Y5: nat] :
                    ( ( ( product_Pair_nat_nat @ X5 @ Y5 )
                      = ( nat_prod_decode @ N4 ) )
                   => ( P @ Y5 ) )
               => ( P @ ( suc @ N4 ) ) ) )
         => ( P @ A0 ) ) ) ) ).

% list_decode.pinduct
thf(fact_10068_list__decode_Oelims,axiom,
    ! [X4: nat,Y3: list_nat] :
      ( ( ( nat_list_decode @ X4 )
        = Y3 )
     => ( ( ( X4 = zero_zero_nat )
         => ( Y3 != nil_nat ) )
       => ~ ! [N4: nat] :
              ( ( X4
                = ( suc @ N4 ) )
             => ( Y3
               != ( produc2761476792215241774st_nat
                  @ ^ [X: nat,Y: nat] : ( cons_nat @ X @ ( nat_list_decode @ Y ) )
                  @ ( nat_prod_decode @ N4 ) ) ) ) ) ) ).

% list_decode.elims
thf(fact_10069_list__decode_Opsimps_I2_J,axiom,
    ! [N: nat] :
      ( ( accp_nat @ nat_list_decode_rel @ ( suc @ N ) )
     => ( ( nat_list_decode @ ( suc @ N ) )
        = ( produc2761476792215241774st_nat
          @ ^ [X: nat,Y: nat] : ( cons_nat @ X @ ( nat_list_decode @ Y ) )
          @ ( nat_prod_decode @ N ) ) ) ) ).

% list_decode.psimps(2)
thf(fact_10070_list__decode_Osimps_I2_J,axiom,
    ! [N: nat] :
      ( ( nat_list_decode @ ( suc @ N ) )
      = ( produc2761476792215241774st_nat
        @ ^ [X: nat,Y: nat] : ( cons_nat @ X @ ( nat_list_decode @ Y ) )
        @ ( nat_prod_decode @ N ) ) ) ).

% list_decode.simps(2)
thf(fact_10071_list__decode_Opelims,axiom,
    ! [X4: nat,Y3: list_nat] :
      ( ( ( nat_list_decode @ X4 )
        = Y3 )
     => ( ( accp_nat @ nat_list_decode_rel @ X4 )
       => ( ( ( X4 = zero_zero_nat )
           => ( ( Y3 = nil_nat )
             => ~ ( accp_nat @ nat_list_decode_rel @ zero_zero_nat ) ) )
         => ~ ! [N4: nat] :
                ( ( X4
                  = ( suc @ N4 ) )
               => ( ( Y3
                    = ( produc2761476792215241774st_nat
                      @ ^ [X: nat,Y: nat] : ( cons_nat @ X @ ( nat_list_decode @ Y ) )
                      @ ( nat_prod_decode @ N4 ) ) )
                 => ~ ( accp_nat @ nat_list_decode_rel @ ( suc @ N4 ) ) ) ) ) ) ) ).

% list_decode.pelims
thf(fact_10072_compute__powr__real,axiom,
    ( powr_real2
    = ( ^ [B3: real,I2: real] :
          ( if_real @ ( ord_less_eq_real @ B3 @ zero_zero_real )
          @ ( abort_real @ ( literal2 @ $false @ $false @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $false @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $true @ $false @ $true @ ( literal2 @ $false @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $false @ $true @ $false @ ( literal2 @ $true @ $true @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $false @ $true @ $false @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $false @ $true @ $false @ ( literal2 @ $false @ $true @ $false @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ zero_zero_literal ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
            @ ^ [Uu: product_unit] : ( powr_real2 @ B3 @ I2 ) )
          @ ( if_real
            @ ( ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ I2 ) )
              = I2 )
            @ ( if_real @ ( ord_less_eq_real @ zero_zero_real @ I2 ) @ ( power_power_real @ B3 @ ( nat2 @ ( archim6058952711729229775r_real @ I2 ) ) ) @ ( divide_divide_real @ one_one_real @ ( power_power_real @ B3 @ ( nat2 @ ( archim6058952711729229775r_real @ ( uminus_uminus_real @ I2 ) ) ) ) ) )
            @ ( abort_real @ ( literal2 @ $false @ $false @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $false @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $true @ $false @ $true @ ( literal2 @ $false @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $false @ $true @ $false @ ( literal2 @ $true @ $true @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $false @ $true @ $false @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $true @ $false @ $true @ $false @ ( literal2 @ $true @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $false @ $true @ $false @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $true @ $true @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $false @ $true @ $true @ $true @ zero_zero_literal ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
              @ ^ [Uu: product_unit] : ( powr_real2 @ B3 @ I2 ) ) ) ) ) ) ).

% compute_powr_real
thf(fact_10073_pairs__le__eq__Sigma,axiom,
    ! [M: nat] :
      ( ( collec3392354462482085612at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [I2: nat,J3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ J3 ) @ M ) ) )
      = ( produc457027306803732586at_nat @ ( set_ord_atMost_nat @ M )
        @ ^ [R5: nat] : ( set_ord_atMost_nat @ ( minus_minus_nat @ M @ R5 ) ) ) ) ).

% pairs_le_eq_Sigma
thf(fact_10074_product__atMost__eq__Un,axiom,
    ! [A3: set_nat,M: nat] :
      ( ( produc457027306803732586at_nat @ A3
        @ ^ [Uu: nat] : ( set_ord_atMost_nat @ M ) )
      = ( sup_su6327502436637775413at_nat
        @ ( produc457027306803732586at_nat @ A3
          @ ^ [I2: nat] : ( set_ord_atMost_nat @ ( minus_minus_nat @ M @ I2 ) ) )
        @ ( produc457027306803732586at_nat @ A3
          @ ^ [I2: nat] : ( set_or6659071591806873216st_nat @ ( minus_minus_nat @ M @ I2 ) @ M ) ) ) ) ).

% product_atMost_eq_Un
thf(fact_10075_of__nat__eq__id,axiom,
    semiri1316708129612266289at_nat = id_nat ).

% of_nat_eq_id
thf(fact_10076_Real_Opositive__def,axiom,
    ( positive2
    = ( map_fu1856342031159181835at_o_o @ rep_real @ id_o
      @ ^ [X6: nat > rat] :
        ? [R5: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ R5 )
          & ? [K3: nat] :
            ! [N2: nat] :
              ( ( ord_less_eq_nat @ K3 @ N2 )
             => ( ord_less_rat @ R5 @ ( X6 @ N2 ) ) ) ) ) ) ).

% Real.positive_def
thf(fact_10077_euclidean__size__nat__def,axiom,
    euclid4777050414544973029ze_nat = id_nat ).

% euclidean_size_nat_def
thf(fact_10078_times__rat__def,axiom,
    ( times_times_rat
    = ( map_fu4333342158222067775at_rat @ rep_Rat @ ( map_fu5673905371560938248nt_rat @ rep_Rat @ abs_Rat )
      @ ^ [X: product_prod_int_int,Y: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_fst_int_int @ Y ) ) @ ( times_times_int @ ( product_snd_int_int @ X ) @ ( product_snd_int_int @ Y ) ) ) ) ) ).

% times_rat_def
thf(fact_10079_plus__rat__def,axiom,
    ( plus_plus_rat
    = ( map_fu4333342158222067775at_rat @ rep_Rat @ ( map_fu5673905371560938248nt_rat @ rep_Rat @ abs_Rat )
      @ ^ [X: product_prod_int_int,Y: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ Y ) ) @ ( times_times_int @ ( product_fst_int_int @ Y ) @ ( product_snd_int_int @ X ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X ) @ ( product_snd_int_int @ Y ) ) ) ) ) ).

% plus_rat_def
thf(fact_10080_Rat_Opositive__def,axiom,
    ( positive
    = ( map_fu898904425404107465nt_o_o @ rep_Rat @ id_o
      @ ^ [X: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ X ) ) ) ) ) ).

% Rat.positive_def
thf(fact_10081_nat__def,axiom,
    ( nat2
    = ( map_fu2345160673673942751at_nat @ rep_Integ @ id_nat @ ( produc6842872674320459806at_nat @ minus_minus_nat ) ) ) ).

% nat_def
thf(fact_10082_less__int__def,axiom,
    ( ord_less_int
    = ( map_fu434086159418415080_int_o @ rep_Integ @ ( map_fu4826362097070443709at_o_o @ rep_Integ @ id_o )
      @ ( produc8739625826339149834_nat_o
        @ ^ [X: nat,Y: nat] :
            ( produc6081775807080527818_nat_o
            @ ^ [U4: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ U4 @ Y ) ) ) ) ) ) ).

% less_int_def
thf(fact_10083_less__eq__int__def,axiom,
    ( ord_less_eq_int
    = ( map_fu434086159418415080_int_o @ rep_Integ @ ( map_fu4826362097070443709at_o_o @ rep_Integ @ id_o )
      @ ( produc8739625826339149834_nat_o
        @ ^ [X: nat,Y: nat] :
            ( produc6081775807080527818_nat_o
            @ ^ [U4: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ U4 @ Y ) ) ) ) ) ) ).

% less_eq_int_def
thf(fact_10084_plus__int__def,axiom,
    ( plus_plus_int
    = ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
      @ ( produc27273713700761075at_nat
        @ ^ [X: nat,Y: nat] :
            ( produc2626176000494625587at_nat
            @ ^ [U4: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X @ U4 ) @ ( plus_plus_nat @ Y @ V4 ) ) ) ) ) ) ).

% plus_int_def
thf(fact_10085_minus__int__def,axiom,
    ( minus_minus_int
    = ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
      @ ( produc27273713700761075at_nat
        @ ^ [X: nat,Y: nat] :
            ( produc2626176000494625587at_nat
            @ ^ [U4: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ Y @ U4 ) ) ) ) ) ) ).

% minus_int_def
thf(fact_10086_times__int__def,axiom,
    ( times_times_int
    = ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
      @ ( produc27273713700761075at_nat
        @ ^ [X: nat,Y: nat] :
            ( produc2626176000494625587at_nat
            @ ^ [U4: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X @ U4 ) @ ( times_times_nat @ Y @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X @ V4 ) @ ( times_times_nat @ Y @ U4 ) ) ) ) ) ) ) ).

% times_int_def
thf(fact_10087_eventually__prod__sequentially,axiom,
    ! [P: product_prod_nat_nat > $o] :
      ( ( eventu1038000079068216329at_nat @ P @ ( prod_filter_nat_nat @ at_top_nat @ at_top_nat ) )
      = ( ? [N8: nat] :
          ! [M6: nat] :
            ( ( ord_less_eq_nat @ N8 @ M6 )
           => ! [N2: nat] :
                ( ( ord_less_eq_nat @ N8 @ N2 )
               => ( P @ ( product_Pair_nat_nat @ N2 @ M6 ) ) ) ) ) ) ).

% eventually_prod_sequentially
thf(fact_10088_less__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ M @ N ) @ ( transi6264000038957366511cl_nat @ pred_nat ) )
      = ( ord_less_nat @ M @ N ) ) ).

% less_eq
thf(fact_10089_numeral__le__enat__iff,axiom,
    ! [M: num,N: nat] :
      ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( extended_enat2 @ N ) )
      = ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ N ) ) ).

% numeral_le_enat_iff
thf(fact_10090_enat_Oinject,axiom,
    ! [Nat: nat,Nat2: nat] :
      ( ( ( extended_enat2 @ Nat )
        = ( extended_enat2 @ Nat2 ) )
      = ( Nat = Nat2 ) ) ).

% enat.inject
thf(fact_10091_enat__ord__simps_I2_J,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_le72135733267957522d_enat @ ( extended_enat2 @ M ) @ ( extended_enat2 @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% enat_ord_simps(2)
thf(fact_10092_enat__ord__simps_I1_J,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_le2932123472753598470d_enat @ ( extended_enat2 @ M ) @ ( extended_enat2 @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% enat_ord_simps(1)
thf(fact_10093_plus__enat__simps_I1_J,axiom,
    ! [M: nat,N: nat] :
      ( ( plus_p3455044024723400733d_enat @ ( extended_enat2 @ M ) @ ( extended_enat2 @ N ) )
      = ( extended_enat2 @ ( plus_plus_nat @ M @ N ) ) ) ).

% plus_enat_simps(1)
thf(fact_10094_idiff__enat__0__right,axiom,
    ! [N: extended_enat] :
      ( ( minus_3235023915231533773d_enat @ N @ ( extended_enat2 @ zero_zero_nat ) )
      = N ) ).

% idiff_enat_0_right
thf(fact_10095_idiff__enat__0,axiom,
    ! [N: extended_enat] :
      ( ( minus_3235023915231533773d_enat @ ( extended_enat2 @ zero_zero_nat ) @ N )
      = ( extended_enat2 @ zero_zero_nat ) ) ).

% idiff_enat_0
thf(fact_10096_idiff__enat__enat,axiom,
    ! [A: nat,B2: nat] :
      ( ( minus_3235023915231533773d_enat @ ( extended_enat2 @ A ) @ ( extended_enat2 @ B2 ) )
      = ( extended_enat2 @ ( minus_minus_nat @ A @ B2 ) ) ) ).

% idiff_enat_enat
thf(fact_10097_times__enat__simps_I1_J,axiom,
    ! [M: nat,N: nat] :
      ( ( times_7803423173614009249d_enat @ ( extended_enat2 @ M ) @ ( extended_enat2 @ N ) )
      = ( extended_enat2 @ ( times_times_nat @ M @ N ) ) ) ).

% times_enat_simps(1)
thf(fact_10098_max__enat__simps_I1_J,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_ma741700101516333627d_enat @ ( extended_enat2 @ M ) @ ( extended_enat2 @ N ) )
      = ( extended_enat2 @ ( ord_max_nat @ M @ N ) ) ) ).

% max_enat_simps(1)
thf(fact_10099_min__enat__simps_I1_J,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_mi8085742599997312461d_enat @ ( extended_enat2 @ M ) @ ( extended_enat2 @ N ) )
      = ( extended_enat2 @ ( ord_min_nat @ M @ N ) ) ) ).

% min_enat_simps(1)
thf(fact_10100_numeral__less__enat__iff,axiom,
    ! [M: num,N: nat] :
      ( ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( extended_enat2 @ N ) )
      = ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ N ) ) ).

% numeral_less_enat_iff
thf(fact_10101_zero__enat__def,axiom,
    ( zero_z5237406670263579293d_enat
    = ( extended_enat2 @ zero_zero_nat ) ) ).

% zero_enat_def
thf(fact_10102_enat__0__iff_I1_J,axiom,
    ! [X4: nat] :
      ( ( ( extended_enat2 @ X4 )
        = zero_z5237406670263579293d_enat )
      = ( X4 = zero_zero_nat ) ) ).

% enat_0_iff(1)
thf(fact_10103_enat__0__iff_I2_J,axiom,
    ! [X4: nat] :
      ( ( zero_z5237406670263579293d_enat
        = ( extended_enat2 @ X4 ) )
      = ( X4 = zero_zero_nat ) ) ).

% enat_0_iff(2)
thf(fact_10104_of__nat__eq__enat,axiom,
    semiri4216267220026989637d_enat = extended_enat2 ).

% of_nat_eq_enat
thf(fact_10105_enat__iless,axiom,
    ! [N: extended_enat,M: nat] :
      ( ( ord_le72135733267957522d_enat @ N @ ( extended_enat2 @ M ) )
     => ? [K2: nat] :
          ( N
          = ( extended_enat2 @ K2 ) ) ) ).

% enat_iless
thf(fact_10106_less__enatE,axiom,
    ! [N: extended_enat,M: nat] :
      ( ( ord_le72135733267957522d_enat @ N @ ( extended_enat2 @ M ) )
     => ~ ! [K2: nat] :
            ( ( N
              = ( extended_enat2 @ K2 ) )
           => ~ ( ord_less_nat @ K2 @ M ) ) ) ).

% less_enatE
thf(fact_10107_one__enat__def,axiom,
    ( one_on7984719198319812577d_enat
    = ( extended_enat2 @ one_one_nat ) ) ).

% one_enat_def
thf(fact_10108_enat__1__iff_I1_J,axiom,
    ! [X4: nat] :
      ( ( ( extended_enat2 @ X4 )
        = one_on7984719198319812577d_enat )
      = ( X4 = one_one_nat ) ) ).

% enat_1_iff(1)
thf(fact_10109_enat__1__iff_I2_J,axiom,
    ! [X4: nat] :
      ( ( one_on7984719198319812577d_enat
        = ( extended_enat2 @ X4 ) )
      = ( X4 = one_one_nat ) ) ).

% enat_1_iff(2)
thf(fact_10110_numeral__eq__enat,axiom,
    ( numera1916890842035813515d_enat
    = ( ^ [K3: num] : ( extended_enat2 @ ( numeral_numeral_nat @ K3 ) ) ) ) ).

% numeral_eq_enat
thf(fact_10111_Suc__ile__eq,axiom,
    ! [M: nat,N: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ ( extended_enat2 @ ( suc @ M ) ) @ N )
      = ( ord_le72135733267957522d_enat @ ( extended_enat2 @ M ) @ N ) ) ).

% Suc_ile_eq
thf(fact_10112_finite__enat__bounded,axiom,
    ! [A3: set_Extended_enat,N: nat] :
      ( ! [Y4: extended_enat] :
          ( ( member_Extended_enat @ Y4 @ A3 )
         => ( ord_le2932123472753598470d_enat @ Y4 @ ( extended_enat2 @ N ) ) )
     => ( finite4001608067531595151d_enat @ A3 ) ) ).

% finite_enat_bounded
thf(fact_10113_enat__ile,axiom,
    ! [N: extended_enat,M: nat] :
      ( ( ord_le2932123472753598470d_enat @ N @ ( extended_enat2 @ M ) )
     => ? [K2: nat] :
          ( N
          = ( extended_enat2 @ K2 ) ) ) ).

% enat_ile
thf(fact_10114_pred__nat__trancl__eq__le,axiom,
    ! [M: nat,N: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ M @ N ) @ ( transi2905341329935302413cl_nat @ pred_nat ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% pred_nat_trancl_eq_le
thf(fact_10115_iadd__le__enat__iff,axiom,
    ! [X4: extended_enat,Y3: extended_enat,N: nat] :
      ( ( ord_le2932123472753598470d_enat @ ( plus_p3455044024723400733d_enat @ X4 @ Y3 ) @ ( extended_enat2 @ N ) )
      = ( ? [Y9: nat,X9: nat] :
            ( ( X4
              = ( extended_enat2 @ X9 ) )
            & ( Y3
              = ( extended_enat2 @ Y9 ) )
            & ( ord_less_eq_nat @ ( plus_plus_nat @ X9 @ Y9 ) @ N ) ) ) ) ).

% iadd_le_enat_iff
thf(fact_10116_elimnum,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ N )
     => ( ( vEBT_VEBT_elim_dead @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ ( extended_enat2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
        = ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) ) ) ).

% elimnum
thf(fact_10117_VEBT__internal_Oelim__dead_Osimps_I3_J,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,L2: nat] :
      ( ( vEBT_VEBT_elim_dead @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ ( extended_enat2 @ L2 ) )
      = ( vEBT_Node @ Info @ Deg
        @ ( take_VEBT_VEBT @ ( divide_divide_nat @ L2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
          @ ( map_VE8901447254227204932T_VEBT
            @ ^ [T3: vEBT_VEBT] : ( vEBT_VEBT_elim_dead @ T3 @ ( extended_enat2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
            @ TreeList2 ) )
        @ ( vEBT_VEBT_elim_dead @ Summary @ ( extended_enat2 @ ( divide_divide_nat @ L2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.elim_dead.simps(3)
thf(fact_10118_VEBT__internal_Oelim__dead_Oelims,axiom,
    ! [X4: vEBT_VEBT,Xa: extended_enat,Y3: vEBT_VEBT] :
      ( ( ( vEBT_VEBT_elim_dead @ X4 @ Xa )
        = Y3 )
     => ( ! [A5: $o,B5: $o] :
            ( ( X4
              = ( vEBT_Leaf @ A5 @ B5 ) )
           => ( Y3
             != ( vEBT_Leaf @ A5 @ B5 ) ) )
       => ( ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X4
                = ( vEBT_Node @ Info2 @ Deg2 @ TreeList3 @ Summary2 ) )
             => ( ( Xa = extend5688581933313929465d_enat )
               => ( Y3
                 != ( vEBT_Node @ Info2 @ Deg2
                    @ ( map_VE8901447254227204932T_VEBT
                      @ ^ [T3: vEBT_VEBT] : ( vEBT_VEBT_elim_dead @ T3 @ ( extended_enat2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      @ TreeList3 )
                    @ ( vEBT_VEBT_elim_dead @ Summary2 @ extend5688581933313929465d_enat ) ) ) ) )
         => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X4
                  = ( vEBT_Node @ Info2 @ Deg2 @ TreeList3 @ Summary2 ) )
               => ! [L3: nat] :
                    ( ( Xa
                      = ( extended_enat2 @ L3 ) )
                   => ( Y3
                     != ( vEBT_Node @ Info2 @ Deg2
                        @ ( take_VEBT_VEBT @ ( divide_divide_nat @ L3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                          @ ( map_VE8901447254227204932T_VEBT
                            @ ^ [T3: vEBT_VEBT] : ( vEBT_VEBT_elim_dead @ T3 @ ( extended_enat2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                            @ TreeList3 ) )
                        @ ( vEBT_VEBT_elim_dead @ Summary2 @ ( extended_enat2 @ ( divide_divide_nat @ L3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.elim_dead.elims
thf(fact_10119_VEBT__internal_Oelim__dead_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( vEBT_VEBT_elim_dead @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ extend5688581933313929465d_enat )
      = ( vEBT_Node @ Info @ Deg
        @ ( map_VE8901447254227204932T_VEBT
          @ ^ [T3: vEBT_VEBT] : ( vEBT_VEBT_elim_dead @ T3 @ ( extended_enat2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
          @ TreeList2 )
        @ ( vEBT_VEBT_elim_dead @ Summary @ extend5688581933313929465d_enat ) ) ) ).

% VEBT_internal.elim_dead.simps(2)
thf(fact_10120_elimcomplete,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ N )
     => ( ( vEBT_VEBT_elim_dead @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ extend5688581933313929465d_enat )
        = ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) ) ) ).

% elimcomplete
thf(fact_10121_not__infinity__eq,axiom,
    ! [X4: extended_enat] :
      ( ( X4 != extend5688581933313929465d_enat )
      = ( ? [I2: nat] :
            ( X4
            = ( extended_enat2 @ I2 ) ) ) ) ).

% not_infinity_eq
thf(fact_10122_not__enat__eq,axiom,
    ! [X4: extended_enat] :
      ( ( ! [Y: nat] :
            ( X4
           != ( extended_enat2 @ Y ) ) )
      = ( X4 = extend5688581933313929465d_enat ) ) ).

% not_enat_eq
thf(fact_10123_enat__ord__simps_I4_J,axiom,
    ! [Q2: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ Q2 @ extend5688581933313929465d_enat )
      = ( Q2 != extend5688581933313929465d_enat ) ) ).

% enat_ord_simps(4)
thf(fact_10124_enat__ord__simps_I6_J,axiom,
    ! [Q2: extended_enat] :
      ~ ( ord_le72135733267957522d_enat @ extend5688581933313929465d_enat @ Q2 ) ).

% enat_ord_simps(6)
thf(fact_10125_plus__enat__simps_I2_J,axiom,
    ! [Q2: extended_enat] :
      ( ( plus_p3455044024723400733d_enat @ extend5688581933313929465d_enat @ Q2 )
      = extend5688581933313929465d_enat ) ).

% plus_enat_simps(2)
thf(fact_10126_plus__enat__simps_I3_J,axiom,
    ! [Q2: extended_enat] :
      ( ( plus_p3455044024723400733d_enat @ Q2 @ extend5688581933313929465d_enat )
      = extend5688581933313929465d_enat ) ).

% plus_enat_simps(3)
thf(fact_10127_enat__ord__code_I3_J,axiom,
    ! [Q2: extended_enat] : ( ord_le2932123472753598470d_enat @ Q2 @ extend5688581933313929465d_enat ) ).

% enat_ord_code(3)
thf(fact_10128_enat__ord__simps_I5_J,axiom,
    ! [Q2: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ extend5688581933313929465d_enat @ Q2 )
      = ( Q2 = extend5688581933313929465d_enat ) ) ).

% enat_ord_simps(5)
thf(fact_10129_idiff__infinity,axiom,
    ! [N: extended_enat] :
      ( ( minus_3235023915231533773d_enat @ extend5688581933313929465d_enat @ N )
      = extend5688581933313929465d_enat ) ).

% idiff_infinity
thf(fact_10130_times__enat__simps_I2_J,axiom,
    ( ( times_7803423173614009249d_enat @ extend5688581933313929465d_enat @ extend5688581933313929465d_enat )
    = extend5688581933313929465d_enat ) ).

% times_enat_simps(2)
thf(fact_10131_max__enat__simps_I4_J,axiom,
    ! [Q2: extended_enat] :
      ( ( ord_ma741700101516333627d_enat @ Q2 @ extend5688581933313929465d_enat )
      = extend5688581933313929465d_enat ) ).

% max_enat_simps(4)
thf(fact_10132_max__enat__simps_I5_J,axiom,
    ! [Q2: extended_enat] :
      ( ( ord_ma741700101516333627d_enat @ extend5688581933313929465d_enat @ Q2 )
      = extend5688581933313929465d_enat ) ).

% max_enat_simps(5)
thf(fact_10133_min__enat__simps_I4_J,axiom,
    ! [Q2: extended_enat] :
      ( ( ord_mi8085742599997312461d_enat @ Q2 @ extend5688581933313929465d_enat )
      = Q2 ) ).

% min_enat_simps(4)
thf(fact_10134_min__enat__simps_I5_J,axiom,
    ! [Q2: extended_enat] :
      ( ( ord_mi8085742599997312461d_enat @ extend5688581933313929465d_enat @ Q2 )
      = Q2 ) ).

% min_enat_simps(5)
thf(fact_10135_idiff__self,axiom,
    ! [N: extended_enat] :
      ( ( N != extend5688581933313929465d_enat )
     => ( ( minus_3235023915231533773d_enat @ N @ N )
        = zero_z5237406670263579293d_enat ) ) ).

% idiff_self
thf(fact_10136_add__diff__cancel__enat,axiom,
    ! [X4: extended_enat,Y3: extended_enat] :
      ( ( X4 != extend5688581933313929465d_enat )
     => ( ( minus_3235023915231533773d_enat @ ( plus_p3455044024723400733d_enat @ X4 @ Y3 ) @ X4 )
        = Y3 ) ) ).

% add_diff_cancel_enat
thf(fact_10137_idiff__infinity__right,axiom,
    ! [A: nat] :
      ( ( minus_3235023915231533773d_enat @ ( extended_enat2 @ A ) @ extend5688581933313929465d_enat )
      = zero_z5237406670263579293d_enat ) ).

% idiff_infinity_right
thf(fact_10138_times__enat__simps_I3_J,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( times_7803423173614009249d_enat @ extend5688581933313929465d_enat @ ( extended_enat2 @ N ) )
          = zero_z5237406670263579293d_enat ) )
      & ( ( N != zero_zero_nat )
       => ( ( times_7803423173614009249d_enat @ extend5688581933313929465d_enat @ ( extended_enat2 @ N ) )
          = extend5688581933313929465d_enat ) ) ) ).

% times_enat_simps(3)
thf(fact_10139_times__enat__simps_I4_J,axiom,
    ! [M: nat] :
      ( ( ( M = zero_zero_nat )
       => ( ( times_7803423173614009249d_enat @ ( extended_enat2 @ M ) @ extend5688581933313929465d_enat )
          = zero_z5237406670263579293d_enat ) )
      & ( ( M != zero_zero_nat )
       => ( ( times_7803423173614009249d_enat @ ( extended_enat2 @ M ) @ extend5688581933313929465d_enat )
          = extend5688581933313929465d_enat ) ) ) ).

% times_enat_simps(4)
thf(fact_10140_Sup__enat__def,axiom,
    ( comple4398354569131411667d_enat
    = ( ^ [A6: set_Extended_enat] : ( if_Extended_enat @ ( A6 = bot_bo7653980558646680370d_enat ) @ zero_z5237406670263579293d_enat @ ( if_Extended_enat @ ( finite4001608067531595151d_enat @ A6 ) @ ( lattic921264341876707157d_enat @ A6 ) @ extend5688581933313929465d_enat ) ) ) ) ).

% Sup_enat_def
thf(fact_10141_Inf__enat__def,axiom,
    ( comple2295165028678016749d_enat
    = ( ^ [A6: set_Extended_enat] :
          ( if_Extended_enat @ ( A6 = bot_bo7653980558646680370d_enat ) @ extend5688581933313929465d_enat
          @ ( ord_Le1955565732374568822d_enat
            @ ^ [X: extended_enat] : ( member_Extended_enat @ X @ A6 ) ) ) ) ) ).

% Inf_enat_def
thf(fact_10142_imult__is__infinity,axiom,
    ! [A: extended_enat,B2: extended_enat] :
      ( ( ( times_7803423173614009249d_enat @ A @ B2 )
        = extend5688581933313929465d_enat )
      = ( ( ( A = extend5688581933313929465d_enat )
          & ( B2 != zero_z5237406670263579293d_enat ) )
        | ( ( B2 = extend5688581933313929465d_enat )
          & ( A != zero_z5237406670263579293d_enat ) ) ) ) ).

% imult_is_infinity
thf(fact_10143_infinity__ne__i0,axiom,
    extend5688581933313929465d_enat != zero_z5237406670263579293d_enat ).

% infinity_ne_i0
thf(fact_10144_enat__add__left__cancel__less,axiom,
    ! [A: extended_enat,B2: extended_enat,C: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ ( plus_p3455044024723400733d_enat @ A @ B2 ) @ ( plus_p3455044024723400733d_enat @ A @ C ) )
      = ( ( A != extend5688581933313929465d_enat )
        & ( ord_le72135733267957522d_enat @ B2 @ C ) ) ) ).

% enat_add_left_cancel_less
thf(fact_10145_infinity__ne__i1,axiom,
    extend5688581933313929465d_enat != one_on7984719198319812577d_enat ).

% infinity_ne_i1
thf(fact_10146_plus__eq__infty__iff__enat,axiom,
    ! [M: extended_enat,N: extended_enat] :
      ( ( ( plus_p3455044024723400733d_enat @ M @ N )
        = extend5688581933313929465d_enat )
      = ( ( M = extend5688581933313929465d_enat )
        | ( N = extend5688581933313929465d_enat ) ) ) ).

% plus_eq_infty_iff_enat
thf(fact_10147_enat__add__left__cancel,axiom,
    ! [A: extended_enat,B2: extended_enat,C: extended_enat] :
      ( ( ( plus_p3455044024723400733d_enat @ A @ B2 )
        = ( plus_p3455044024723400733d_enat @ A @ C ) )
      = ( ( A = extend5688581933313929465d_enat )
        | ( B2 = C ) ) ) ).

% enat_add_left_cancel
thf(fact_10148_numeral__ne__infinity,axiom,
    ! [K: num] :
      ( ( numera1916890842035813515d_enat @ K )
     != extend5688581933313929465d_enat ) ).

% numeral_ne_infinity
thf(fact_10149_top__enat__def,axiom,
    top_to3028658606643905974d_enat = extend5688581933313929465d_enat ).

% top_enat_def
thf(fact_10150_enat__ord__simps_I3_J,axiom,
    ! [Q2: extended_enat] : ( ord_le2932123472753598470d_enat @ Q2 @ extend5688581933313929465d_enat ) ).

% enat_ord_simps(3)
thf(fact_10151_enat__add__left__cancel__le,axiom,
    ! [A: extended_enat,B2: extended_enat,C: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ ( plus_p3455044024723400733d_enat @ A @ B2 ) @ ( plus_p3455044024723400733d_enat @ A @ C ) )
      = ( ( A = extend5688581933313929465d_enat )
        | ( ord_le2932123472753598470d_enat @ B2 @ C ) ) ) ).

% enat_add_left_cancel_le
thf(fact_10152_enat__ord__code_I5_J,axiom,
    ! [N: nat] :
      ~ ( ord_le2932123472753598470d_enat @ extend5688581933313929465d_enat @ ( extended_enat2 @ N ) ) ).

% enat_ord_code(5)
thf(fact_10153_infinity__ileE,axiom,
    ! [M: nat] :
      ~ ( ord_le2932123472753598470d_enat @ extend5688581933313929465d_enat @ ( extended_enat2 @ M ) ) ).

% infinity_ileE
thf(fact_10154_enat__ex__split,axiom,
    ( ( ^ [P5: extended_enat > $o] :
        ? [X7: extended_enat] : ( P5 @ X7 ) )
    = ( ^ [P6: extended_enat > $o] :
          ( ( P6 @ extend5688581933313929465d_enat )
          | ? [X: nat] : ( P6 @ ( extended_enat2 @ X ) ) ) ) ) ).

% enat_ex_split
thf(fact_10155_enat3__cases,axiom,
    ! [Y3: extended_enat,Ya: extended_enat,Yb: extended_enat] :
      ( ( ? [Nat3: nat] :
            ( Y3
            = ( extended_enat2 @ Nat3 ) )
       => ( ? [Nata: nat] :
              ( Ya
              = ( extended_enat2 @ Nata ) )
         => ! [Natb: nat] :
              ( Yb
             != ( extended_enat2 @ Natb ) ) ) )
     => ( ( ? [Nat3: nat] :
              ( Y3
              = ( extended_enat2 @ Nat3 ) )
         => ( ? [Nata: nat] :
                ( Ya
                = ( extended_enat2 @ Nata ) )
           => ( Yb != extend5688581933313929465d_enat ) ) )
       => ( ( ? [Nat3: nat] :
                ( Y3
                = ( extended_enat2 @ Nat3 ) )
           => ( ( Ya = extend5688581933313929465d_enat )
             => ! [Nata: nat] :
                  ( Yb
                 != ( extended_enat2 @ Nata ) ) ) )
         => ( ( ? [Nat3: nat] :
                  ( Y3
                  = ( extended_enat2 @ Nat3 ) )
             => ( ( Ya = extend5688581933313929465d_enat )
               => ( Yb != extend5688581933313929465d_enat ) ) )
           => ( ( ( Y3 = extend5688581933313929465d_enat )
               => ( ? [Nat3: nat] :
                      ( Ya
                      = ( extended_enat2 @ Nat3 ) )
                 => ! [Nata: nat] :
                      ( Yb
                     != ( extended_enat2 @ Nata ) ) ) )
             => ( ( ( Y3 = extend5688581933313929465d_enat )
                 => ( ? [Nat3: nat] :
                        ( Ya
                        = ( extended_enat2 @ Nat3 ) )
                   => ( Yb != extend5688581933313929465d_enat ) ) )
               => ( ( ( Y3 = extend5688581933313929465d_enat )
                   => ( ( Ya = extend5688581933313929465d_enat )
                     => ! [Nat3: nat] :
                          ( Yb
                         != ( extended_enat2 @ Nat3 ) ) ) )
                 => ~ ( ( Y3 = extend5688581933313929465d_enat )
                     => ( ( Ya = extend5688581933313929465d_enat )
                       => ( Yb != extend5688581933313929465d_enat ) ) ) ) ) ) ) ) ) ) ).

% enat3_cases
thf(fact_10156_enat2__cases,axiom,
    ! [Y3: extended_enat,Ya: extended_enat] :
      ( ( ? [Nat3: nat] :
            ( Y3
            = ( extended_enat2 @ Nat3 ) )
       => ! [Nata: nat] :
            ( Ya
           != ( extended_enat2 @ Nata ) ) )
     => ( ( ? [Nat3: nat] :
              ( Y3
              = ( extended_enat2 @ Nat3 ) )
         => ( Ya != extend5688581933313929465d_enat ) )
       => ( ( ( Y3 = extend5688581933313929465d_enat )
           => ! [Nat3: nat] :
                ( Ya
               != ( extended_enat2 @ Nat3 ) ) )
         => ~ ( ( Y3 = extend5688581933313929465d_enat )
             => ( Ya != extend5688581933313929465d_enat ) ) ) ) ) ).

% enat2_cases
thf(fact_10157_enat_Oexhaust,axiom,
    ! [Y3: extended_enat] :
      ( ! [Nat3: nat] :
          ( Y3
         != ( extended_enat2 @ Nat3 ) )
     => ( Y3 = extend5688581933313929465d_enat ) ) ).

% enat.exhaust
thf(fact_10158_enat_Odistinct_I1_J,axiom,
    ! [Nat: nat] :
      ( ( extended_enat2 @ Nat )
     != extend5688581933313929465d_enat ) ).

% enat.distinct(1)
thf(fact_10159_enat_Odistinct_I2_J,axiom,
    ! [Nat: nat] :
      ( extend5688581933313929465d_enat
     != ( extended_enat2 @ Nat ) ) ).

% enat.distinct(2)
thf(fact_10160_infinity__ilessE,axiom,
    ! [M: nat] :
      ~ ( ord_le72135733267957522d_enat @ extend5688581933313929465d_enat @ ( extended_enat2 @ M ) ) ).

% infinity_ilessE
thf(fact_10161_less__infinityE,axiom,
    ! [N: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ N @ extend5688581933313929465d_enat )
     => ~ ! [K2: nat] :
            ( N
           != ( extended_enat2 @ K2 ) ) ) ).

% less_infinityE
thf(fact_10162_enat__ord__code_I4_J,axiom,
    ! [M: nat] : ( ord_le72135733267957522d_enat @ ( extended_enat2 @ M ) @ extend5688581933313929465d_enat ) ).

% enat_ord_code(4)
thf(fact_10163_imult__infinity,axiom,
    ! [N: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N )
     => ( ( times_7803423173614009249d_enat @ extend5688581933313929465d_enat @ N )
        = extend5688581933313929465d_enat ) ) ).

% imult_infinity
thf(fact_10164_imult__infinity__right,axiom,
    ! [N: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N )
     => ( ( times_7803423173614009249d_enat @ N @ extend5688581933313929465d_enat )
        = extend5688581933313929465d_enat ) ) ).

% imult_infinity_right
thf(fact_10165_plus__enat__def,axiom,
    ( plus_p3455044024723400733d_enat
    = ( ^ [M6: extended_enat,N2: extended_enat] :
          ( extend3600170679010898289d_enat
          @ ^ [O: nat] :
              ( extend3600170679010898289d_enat
              @ ^ [P3: nat] : ( extended_enat2 @ ( plus_plus_nat @ O @ P3 ) )
              @ extend5688581933313929465d_enat
              @ N2 )
          @ extend5688581933313929465d_enat
          @ M6 ) ) ) ).

% plus_enat_def
thf(fact_10166_diff__enat__def,axiom,
    ( minus_3235023915231533773d_enat
    = ( ^ [A4: extended_enat,B3: extended_enat] :
          ( extend3600170679010898289d_enat
          @ ^ [X: nat] :
              ( extend3600170679010898289d_enat
              @ ^ [Y: nat] : ( extended_enat2 @ ( minus_minus_nat @ X @ Y ) )
              @ zero_z5237406670263579293d_enat
              @ B3 )
          @ extend5688581933313929465d_enat
          @ A4 ) ) ) ).

% diff_enat_def
thf(fact_10167_times__enat__def,axiom,
    ( times_7803423173614009249d_enat
    = ( ^ [M6: extended_enat,N2: extended_enat] :
          ( extend3600170679010898289d_enat
          @ ^ [O: nat] :
              ( extend3600170679010898289d_enat
              @ ^ [P3: nat] : ( extended_enat2 @ ( times_times_nat @ O @ P3 ) )
              @ ( if_Extended_enat @ ( O = zero_zero_nat ) @ zero_z5237406670263579293d_enat @ extend5688581933313929465d_enat )
              @ N2 )
          @ ( if_Extended_enat @ ( N2 = zero_z5237406670263579293d_enat ) @ zero_z5237406670263579293d_enat @ extend5688581933313929465d_enat )
          @ M6 ) ) ) ).

% times_enat_def
thf(fact_10168_VEBT__internal_Oelim__dead_Opelims,axiom,
    ! [X4: vEBT_VEBT,Xa: extended_enat,Y3: vEBT_VEBT] :
      ( ( ( vEBT_VEBT_elim_dead @ X4 @ Xa )
        = Y3 )
     => ( ( accp_P6183159247885693666d_enat @ vEBT_V312737461966249ad_rel @ ( produc581526299967858633d_enat @ X4 @ Xa ) )
       => ( ! [A5: $o,B5: $o] :
              ( ( X4
                = ( vEBT_Leaf @ A5 @ B5 ) )
             => ( ( Y3
                  = ( vEBT_Leaf @ A5 @ B5 ) )
               => ~ ( accp_P6183159247885693666d_enat @ vEBT_V312737461966249ad_rel @ ( produc581526299967858633d_enat @ ( vEBT_Leaf @ A5 @ B5 ) @ Xa ) ) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X4
                  = ( vEBT_Node @ Info2 @ Deg2 @ TreeList3 @ Summary2 ) )
               => ( ( Xa = extend5688581933313929465d_enat )
                 => ( ( Y3
                      = ( vEBT_Node @ Info2 @ Deg2
                        @ ( map_VE8901447254227204932T_VEBT
                          @ ^ [T3: vEBT_VEBT] : ( vEBT_VEBT_elim_dead @ T3 @ ( extended_enat2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                          @ TreeList3 )
                        @ ( vEBT_VEBT_elim_dead @ Summary2 @ extend5688581933313929465d_enat ) ) )
                   => ~ ( accp_P6183159247885693666d_enat @ vEBT_V312737461966249ad_rel @ ( produc581526299967858633d_enat @ ( vEBT_Node @ Info2 @ Deg2 @ TreeList3 @ Summary2 ) @ extend5688581933313929465d_enat ) ) ) ) )
           => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                  ( ( X4
                    = ( vEBT_Node @ Info2 @ Deg2 @ TreeList3 @ Summary2 ) )
                 => ! [L3: nat] :
                      ( ( Xa
                        = ( extended_enat2 @ L3 ) )
                     => ( ( Y3
                          = ( vEBT_Node @ Info2 @ Deg2
                            @ ( take_VEBT_VEBT @ ( divide_divide_nat @ L3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                              @ ( map_VE8901447254227204932T_VEBT
                                @ ^ [T3: vEBT_VEBT] : ( vEBT_VEBT_elim_dead @ T3 @ ( extended_enat2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                @ TreeList3 ) )
                            @ ( vEBT_VEBT_elim_dead @ Summary2 @ ( extended_enat2 @ ( divide_divide_nat @ L3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
                       => ~ ( accp_P6183159247885693666d_enat @ vEBT_V312737461966249ad_rel @ ( produc581526299967858633d_enat @ ( vEBT_Node @ Info2 @ Deg2 @ TreeList3 @ Summary2 ) @ ( extended_enat2 @ L3 ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.elim_dead.pelims
thf(fact_10169_the__enat_Osimps,axiom,
    ! [N: nat] :
      ( ( extended_the_enat @ ( extended_enat2 @ N ) )
      = N ) ).

% the_enat.simps
thf(fact_10170_eSuc__Max,axiom,
    ! [A3: set_Extended_enat] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ( A3 != bot_bo7653980558646680370d_enat )
       => ( ( extended_eSuc @ ( lattic921264341876707157d_enat @ A3 ) )
          = ( lattic921264341876707157d_enat @ ( image_80655429650038917d_enat @ extended_eSuc @ A3 ) ) ) ) ) ).

% eSuc_Max
thf(fact_10171_eSuc__inject,axiom,
    ! [M: extended_enat,N: extended_enat] :
      ( ( ( extended_eSuc @ M )
        = ( extended_eSuc @ N ) )
      = ( M = N ) ) ).

% eSuc_inject
thf(fact_10172_eSuc__infinity,axiom,
    ( ( extended_eSuc @ extend5688581933313929465d_enat )
    = extend5688581933313929465d_enat ) ).

% eSuc_infinity
thf(fact_10173_eSuc__mono,axiom,
    ! [N: extended_enat,M: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ ( extended_eSuc @ N ) @ ( extended_eSuc @ M ) )
      = ( ord_le72135733267957522d_enat @ N @ M ) ) ).

% eSuc_mono
thf(fact_10174_eSuc__ile__mono,axiom,
    ! [N: extended_enat,M: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ ( extended_eSuc @ N ) @ ( extended_eSuc @ M ) )
      = ( ord_le2932123472753598470d_enat @ N @ M ) ) ).

% eSuc_ile_mono
thf(fact_10175_eSuc__minus__eSuc,axiom,
    ! [N: extended_enat,M: extended_enat] :
      ( ( minus_3235023915231533773d_enat @ ( extended_eSuc @ N ) @ ( extended_eSuc @ M ) )
      = ( minus_3235023915231533773d_enat @ N @ M ) ) ).

% eSuc_minus_eSuc
thf(fact_10176_iless__eSuc0,axiom,
    ! [N: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ N @ ( extended_eSuc @ zero_z5237406670263579293d_enat ) )
      = ( N = zero_z5237406670263579293d_enat ) ) ).

% iless_eSuc0
thf(fact_10177_eSuc__minus__1,axiom,
    ! [N: extended_enat] :
      ( ( minus_3235023915231533773d_enat @ ( extended_eSuc @ N ) @ one_on7984719198319812577d_enat )
      = N ) ).

% eSuc_minus_1
thf(fact_10178_eSuc__numeral,axiom,
    ! [K: num] :
      ( ( extended_eSuc @ ( numera1916890842035813515d_enat @ K ) )
      = ( numera1916890842035813515d_enat @ ( plus_plus_num @ K @ one ) ) ) ).

% eSuc_numeral
thf(fact_10179_iless__Suc__eq,axiom,
    ! [M: nat,N: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ ( extended_enat2 @ M ) @ ( extended_eSuc @ N ) )
      = ( ord_le2932123472753598470d_enat @ ( extended_enat2 @ M ) @ N ) ) ).

% iless_Suc_eq
thf(fact_10180_one__eSuc,axiom,
    ( one_on7984719198319812577d_enat
    = ( extended_eSuc @ zero_z5237406670263579293d_enat ) ) ).

% one_eSuc
thf(fact_10181_i0__iless__eSuc,axiom,
    ! [N: extended_enat] : ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( extended_eSuc @ N ) ) ).

% i0_iless_eSuc
thf(fact_10182_zero__ne__eSuc,axiom,
    ! [N: extended_enat] :
      ( zero_z5237406670263579293d_enat
     != ( extended_eSuc @ N ) ) ).

% zero_ne_eSuc
thf(fact_10183_eSuc__plus__1,axiom,
    ( extended_eSuc
    = ( ^ [N2: extended_enat] : ( plus_p3455044024723400733d_enat @ N2 @ one_on7984719198319812577d_enat ) ) ) ).

% eSuc_plus_1
thf(fact_10184_plus__1__eSuc_I1_J,axiom,
    ! [Q2: extended_enat] :
      ( ( plus_p3455044024723400733d_enat @ one_on7984719198319812577d_enat @ Q2 )
      = ( extended_eSuc @ Q2 ) ) ).

% plus_1_eSuc(1)
thf(fact_10185_plus__1__eSuc_I2_J,axiom,
    ! [Q2: extended_enat] :
      ( ( plus_p3455044024723400733d_enat @ Q2 @ one_on7984719198319812577d_enat )
      = ( extended_eSuc @ Q2 ) ) ).

% plus_1_eSuc(2)
thf(fact_10186_mono__eSuc,axiom,
    order_4130057895858720880d_enat @ extended_eSuc ).

% mono_eSuc
thf(fact_10187_iadd__Suc__right,axiom,
    ! [M: extended_enat,N: extended_enat] :
      ( ( plus_p3455044024723400733d_enat @ M @ ( extended_eSuc @ N ) )
      = ( extended_eSuc @ ( plus_p3455044024723400733d_enat @ M @ N ) ) ) ).

% iadd_Suc_right
thf(fact_10188_iadd__Suc,axiom,
    ! [M: extended_enat,N: extended_enat] :
      ( ( plus_p3455044024723400733d_enat @ ( extended_eSuc @ M ) @ N )
      = ( extended_eSuc @ ( plus_p3455044024723400733d_enat @ M @ N ) ) ) ).

% iadd_Suc
thf(fact_10189_eSuc__max,axiom,
    ! [X4: extended_enat,Y3: extended_enat] :
      ( ( extended_eSuc @ ( ord_ma741700101516333627d_enat @ X4 @ Y3 ) )
      = ( ord_ma741700101516333627d_enat @ ( extended_eSuc @ X4 ) @ ( extended_eSuc @ Y3 ) ) ) ).

% eSuc_max
thf(fact_10190_mult__eSuc__right,axiom,
    ! [M: extended_enat,N: extended_enat] :
      ( ( times_7803423173614009249d_enat @ M @ ( extended_eSuc @ N ) )
      = ( plus_p3455044024723400733d_enat @ M @ ( times_7803423173614009249d_enat @ M @ N ) ) ) ).

% mult_eSuc_right
thf(fact_10191_mult__eSuc,axiom,
    ! [M: extended_enat,N: extended_enat] :
      ( ( times_7803423173614009249d_enat @ ( extended_eSuc @ M ) @ N )
      = ( plus_p3455044024723400733d_enat @ N @ ( times_7803423173614009249d_enat @ M @ N ) ) ) ).

% mult_eSuc
thf(fact_10192_ileI1,axiom,
    ! [M: extended_enat,N: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ M @ N )
     => ( ord_le2932123472753598470d_enat @ ( extended_eSuc @ M ) @ N ) ) ).

% ileI1
thf(fact_10193_ile__eSuc,axiom,
    ! [N: extended_enat] : ( ord_le2932123472753598470d_enat @ N @ ( extended_eSuc @ N ) ) ).

% ile_eSuc
thf(fact_10194_not__eSuc__ilei0,axiom,
    ! [N: extended_enat] :
      ~ ( ord_le2932123472753598470d_enat @ ( extended_eSuc @ N ) @ zero_z5237406670263579293d_enat ) ).

% not_eSuc_ilei0
thf(fact_10195_eSuc__enat,axiom,
    ! [N: nat] :
      ( ( extended_eSuc @ ( extended_enat2 @ N ) )
      = ( extended_enat2 @ ( suc @ N ) ) ) ).

% eSuc_enat
thf(fact_10196_eSuc__enat__iff,axiom,
    ! [X4: extended_enat,Y3: nat] :
      ( ( ( extended_eSuc @ X4 )
        = ( extended_enat2 @ Y3 ) )
      = ( ? [N2: nat] :
            ( ( Y3
              = ( suc @ N2 ) )
            & ( X4
              = ( extended_enat2 @ N2 ) ) ) ) ) ).

% eSuc_enat_iff
thf(fact_10197_enat__eSuc__iff,axiom,
    ! [Y3: nat,X4: extended_enat] :
      ( ( ( extended_enat2 @ Y3 )
        = ( extended_eSuc @ X4 ) )
      = ( ? [N2: nat] :
            ( ( Y3
              = ( suc @ N2 ) )
            & ( ( extended_enat2 @ N2 )
              = X4 ) ) ) ) ).

% enat_eSuc_iff
thf(fact_10198_eSuc__Sup,axiom,
    ! [A3: set_Extended_enat] :
      ( ( A3 != bot_bo7653980558646680370d_enat )
     => ( ( extended_eSuc @ ( comple4398354569131411667d_enat @ A3 ) )
        = ( comple4398354569131411667d_enat @ ( image_80655429650038917d_enat @ extended_eSuc @ A3 ) ) ) ) ).

% eSuc_Sup
thf(fact_10199_eSuc__def,axiom,
    ( extended_eSuc
    = ( extend3600170679010898289d_enat
      @ ^ [N2: nat] : ( extended_enat2 @ ( suc @ N2 ) )
      @ extend5688581933313929465d_enat ) ) ).

% eSuc_def
thf(fact_10200_less__than__iff,axiom,
    ! [X4: nat,Y3: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y3 ) @ less_than )
      = ( ord_less_nat @ X4 @ Y3 ) ) ).

% less_than_iff
thf(fact_10201_Quotient__real,axiom,
    quotie3684837364556693515t_real @ realrel @ real2 @ rep_real @ cr_real ).

% Quotient_real
thf(fact_10202_lcm__code__integer,axiom,
    ( gcd_lcm_Code_integer
    = ( ^ [A4: code_integer,B3: code_integer] : ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A4 ) @ ( abs_abs_Code_integer @ B3 ) ) @ ( gcd_gcd_Code_integer @ A4 @ B3 ) ) ) ) ).

% lcm_code_integer
thf(fact_10203_lcm__1__iff__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( ( gcd_lcm_nat @ M @ N )
        = ( suc @ zero_zero_nat ) )
      = ( ( M
          = ( suc @ zero_zero_nat ) )
        & ( N
          = ( suc @ zero_zero_nat ) ) ) ) ).

% lcm_1_iff_nat
thf(fact_10204_prod__gcd__lcm__int,axiom,
    ! [M: int,N: int] :
      ( ( times_times_int @ ( abs_abs_int @ M ) @ ( abs_abs_int @ N ) )
      = ( times_times_int @ ( gcd_gcd_int @ M @ N ) @ ( gcd_lcm_int @ M @ N ) ) ) ).

% prod_gcd_lcm_int
thf(fact_10205_prod__gcd__lcm__nat,axiom,
    ( times_times_nat
    = ( ^ [M6: nat,N2: nat] : ( times_times_nat @ ( gcd_gcd_nat @ M6 @ N2 ) @ ( gcd_lcm_nat @ M6 @ N2 ) ) ) ) ).

% prod_gcd_lcm_nat
thf(fact_10206_lcm__nat__def,axiom,
    ( gcd_lcm_nat
    = ( ^ [X: nat,Y: nat] : ( divide_divide_nat @ ( times_times_nat @ X @ Y ) @ ( gcd_gcd_nat @ X @ Y ) ) ) ) ).

% lcm_nat_def
thf(fact_10207_lcm__pos__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_nat @ zero_zero_nat @ ( gcd_lcm_nat @ M @ N ) ) ) ) ).

% lcm_pos_nat
thf(fact_10208_lcm__pos__int,axiom,
    ! [M: int,N: int] :
      ( ( M != zero_zero_int )
     => ( ( N != zero_zero_int )
       => ( ord_less_int @ zero_zero_int @ ( gcd_lcm_int @ M @ N ) ) ) ) ).

% lcm_pos_int
thf(fact_10209_lcm__altdef__int,axiom,
    ( gcd_lcm_int
    = ( ^ [A4: int,B3: int] : ( divide_divide_int @ ( times_times_int @ ( abs_abs_int @ A4 ) @ ( abs_abs_int @ B3 ) ) @ ( gcd_gcd_int @ A4 @ B3 ) ) ) ) ).

% lcm_altdef_int

% Helper facts (46)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
    ! [X4: int,Y3: int] :
      ( ( if_int @ $false @ X4 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
    ! [X4: int,Y3: int] :
      ( ( if_int @ $true @ X4 @ Y3 )
      = X4 ) ).

thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
    ! [X4: nat,Y3: nat] :
      ( ( if_nat @ $false @ X4 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
    ! [X4: nat,Y3: nat] :
      ( ( if_nat @ $true @ X4 @ Y3 )
      = X4 ) ).

thf(help_If_2_1_If_001t__Num__Onum_T,axiom,
    ! [X4: num,Y3: num] :
      ( ( if_num @ $false @ X4 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Num__Onum_T,axiom,
    ! [X4: num,Y3: num] :
      ( ( if_num @ $true @ X4 @ Y3 )
      = X4 ) ).

thf(help_If_2_1_If_001t__Rat__Orat_T,axiom,
    ! [X4: rat,Y3: rat] :
      ( ( if_rat @ $false @ X4 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Rat__Orat_T,axiom,
    ! [X4: rat,Y3: rat] :
      ( ( if_rat @ $true @ X4 @ Y3 )
      = X4 ) ).

thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
    ! [X4: real,Y3: real] :
      ( ( if_real @ $false @ X4 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
    ! [X4: real,Y3: real] :
      ( ( if_real @ $true @ X4 @ Y3 )
      = X4 ) ).

thf(help_fChoice_1_1_fChoice_001t__Real__Oreal_T,axiom,
    ! [P: real > $o] :
      ( ( P @ ( fChoice_real @ P ) )
      = ( ? [X6: real] : ( P @ X6 ) ) ) ).

thf(help_If_2_1_If_001t__Complex__Ocomplex_T,axiom,
    ! [X4: complex,Y3: complex] :
      ( ( if_complex @ $false @ X4 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Complex__Ocomplex_T,axiom,
    ! [X4: complex,Y3: complex] :
      ( ( if_complex @ $true @ X4 @ Y3 )
      = X4 ) ).

thf(help_If_2_1_If_001t__Extended____Nat__Oenat_T,axiom,
    ! [X4: extended_enat,Y3: extended_enat] :
      ( ( if_Extended_enat @ $false @ X4 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Extended____Nat__Oenat_T,axiom,
    ! [X4: extended_enat,Y3: extended_enat] :
      ( ( if_Extended_enat @ $true @ X4 @ Y3 )
      = X4 ) ).

thf(help_If_2_1_If_001t__Code____Numeral__Ointeger_T,axiom,
    ! [X4: code_integer,Y3: code_integer] :
      ( ( if_Code_integer @ $false @ X4 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Code____Numeral__Ointeger_T,axiom,
    ! [X4: code_integer,Y3: code_integer] :
      ( ( if_Code_integer @ $true @ X4 @ Y3 )
      = X4 ) ).

thf(help_If_2_1_If_001t__Set__Oset_It__Int__Oint_J_T,axiom,
    ! [X4: set_int,Y3: set_int] :
      ( ( if_set_int @ $false @ X4 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Set__Oset_It__Int__Oint_J_T,axiom,
    ! [X4: set_int,Y3: set_int] :
      ( ( if_set_int @ $true @ X4 @ Y3 )
      = X4 ) ).

thf(help_If_2_1_If_001t__VEBT____Definitions__OVEBT_T,axiom,
    ! [X4: vEBT_VEBT,Y3: vEBT_VEBT] :
      ( ( if_VEBT_VEBT @ $false @ X4 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__VEBT____Definitions__OVEBT_T,axiom,
    ! [X4: vEBT_VEBT,Y3: vEBT_VEBT] :
      ( ( if_VEBT_VEBT @ $true @ X4 @ Y3 )
      = X4 ) ).

thf(help_If_2_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
    ! [X4: list_int,Y3: list_int] :
      ( ( if_list_int @ $false @ X4 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
    ! [X4: list_int,Y3: list_int] :
      ( ( if_list_int @ $true @ X4 @ Y3 )
      = X4 ) ).

thf(help_If_2_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [X4: list_nat,Y3: list_nat] :
      ( ( if_list_nat @ $false @ X4 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [X4: list_nat,Y3: list_nat] :
      ( ( if_list_nat @ $true @ X4 @ Y3 )
      = X4 ) ).

thf(help_If_2_1_If_001_062_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X4: int > int,Y3: int > int] :
      ( ( if_int_int @ $false @ X4 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001_062_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X4: int > int,Y3: int > int] :
      ( ( if_int_int @ $true @ X4 @ Y3 )
      = X4 ) ).

thf(help_If_2_1_If_001_062_It__Nat__Onat_Mt__Rat__Orat_J_T,axiom,
    ! [X4: nat > rat,Y3: nat > rat] :
      ( ( if_nat_rat @ $false @ X4 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001_062_It__Nat__Onat_Mt__Rat__Orat_J_T,axiom,
    ! [X4: nat > rat,Y3: nat > rat] :
      ( ( if_nat_rat @ $true @ X4 @ Y3 )
      = X4 ) ).

thf(help_If_2_1_If_001t__Option__Ooption_It__Nat__Onat_J_T,axiom,
    ! [X4: option_nat,Y3: option_nat] :
      ( ( if_option_nat @ $false @ X4 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Option__Ooption_It__Nat__Onat_J_T,axiom,
    ! [X4: option_nat,Y3: option_nat] :
      ( ( if_option_nat @ $true @ X4 @ Y3 )
      = X4 ) ).

thf(help_If_2_1_If_001t__Option__Ooption_It__Num__Onum_J_T,axiom,
    ! [X4: option_num,Y3: option_num] :
      ( ( if_option_num @ $false @ X4 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Option__Ooption_It__Num__Onum_J_T,axiom,
    ! [X4: option_num,Y3: option_num] :
      ( ( if_option_num @ $true @ X4 @ Y3 )
      = X4 ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X4: product_prod_int_int,Y3: product_prod_int_int] :
      ( ( if_Pro3027730157355071871nt_int @ $false @ X4 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X4: product_prod_int_int,Y3: product_prod_int_int] :
      ( ( if_Pro3027730157355071871nt_int @ $true @ X4 @ Y3 )
      = X4 ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X4: product_prod_nat_nat,Y3: product_prod_nat_nat] :
      ( ( if_Pro6206227464963214023at_nat @ $false @ X4 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X4: product_prod_nat_nat,Y3: product_prod_nat_nat] :
      ( ( if_Pro6206227464963214023at_nat @ $true @ X4 @ Y3 )
      = X4 ) ).

thf(help_If_2_1_If_001_062_It__Nat__Onat_M_062_It__Int__Oint_Mt__Int__Oint_J_J_T,axiom,
    ! [X4: nat > int > int,Y3: nat > int > int] :
      ( ( if_nat_int_int @ $false @ X4 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001_062_It__Nat__Onat_M_062_It__Int__Oint_Mt__Int__Oint_J_J_T,axiom,
    ! [X4: nat > int > int,Y3: nat > int > int] :
      ( ( if_nat_int_int @ $true @ X4 @ Y3 )
      = X4 ) ).

thf(help_If_2_1_If_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_T,axiom,
    ! [X4: nat > nat > nat,Y3: nat > nat > nat] :
      ( ( if_nat_nat_nat @ $false @ X4 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_T,axiom,
    ! [X4: nat > nat > nat,Y3: nat > nat > nat] :
      ( ( if_nat_nat_nat @ $true @ X4 @ Y3 )
      = X4 ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_T,axiom,
    ! [X4: produc6271795597528267376eger_o,Y3: produc6271795597528267376eger_o] :
      ( ( if_Pro5737122678794959658eger_o @ $false @ X4 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_T,axiom,
    ! [X4: produc6271795597528267376eger_o,Y3: produc6271795597528267376eger_o] :
      ( ( if_Pro5737122678794959658eger_o @ $true @ X4 @ Y3 )
      = X4 ) ).

thf(help_If_3_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
    ! [P: $o] :
      ( ( P = $true )
      | ( P = $false ) ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
    ! [X4: produc8923325533196201883nteger,Y3: produc8923325533196201883nteger] :
      ( ( if_Pro6119634080678213985nteger @ $false @ X4 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
    ! [X4: produc8923325533196201883nteger,Y3: produc8923325533196201883nteger] :
      ( ( if_Pro6119634080678213985nteger @ $true @ X4 @ Y3 )
      = X4 ) ).

% Conjectures (1)
thf(conj_0,conjecture,
    ( ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
    = ( vEBT_VEBT_high @ za @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

%------------------------------------------------------------------------------