%------------------------------------------------------------------------------
% File     : ITP208^1 : TPTP v9.0.0. Released v8.1.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem Array_Time 00247_008755
% 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    : 0019_Array_Time_00247_008755 [Des22]

% Status   : Theorem
% Rating   : 0.62 v9.0.0, 0.80 v8.2.0, 0.85 v8.1.0
% Syntax   : Number of formulae    : 10202 (5814 unt; 930 typ;   0 def)
%            Number of atoms       : 21914 (10366 equ;   0 cnn)
%            Maximal formula atoms :   12 (   2 avg)
%            Number of connectives : 77656 (1982   ~; 452   |;1004   &;68006   @)
%                                         (   0 <=>;6212  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   18 (   5 avg)
%            Number of types       :   73 (  72 usr)
%            Number of type conns  : 2931 (2931   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  861 ( 858 usr;  58 con; 0-8 aty)
%            Number of variables   : 19535 (1652   ^;17569   !; 314   ?;19535   :)
% 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
%            2022-02-17 14:11:04.810
%------------------------------------------------------------------------------
% Could-be-implicit typings (72)
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thf(ty_n_t__String__Oliteral,type,
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% Explicit typings (858)
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thf(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_001t__Rat__Orat,type,
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thf(sy_c_Archimedean__Field_Ofrac_001t__Rat__Orat,type,
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thf(sy_c_Archimedean__Field_Oround_001t__Rat__Orat,type,
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thf(sy_c_Array__Time_Oget_001_Eo,type,
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thf(sy_c_Array__Time_Oget_001tf__a,type,
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thf(sy_c_BNF__Cardinal__Order__Relation_OnatLess,type,
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    bNF_re7626690874201225453um_num: ( int > int > $o ) > ( num > num > $o ) > ( int > num ) > ( int > num ) > $o ).

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    bNF_re6250860962936578807nt_int: ( int > int > $o ) > ( product_prod_int_int > product_prod_int_int > $o ) > ( int > product_prod_int_int ) > ( int > product_prod_int_int ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat,type,
    bNF_re2214769303045360666nt_rat: ( int > int > $o ) > ( product_prod_int_int > rat > $o ) > ( int > product_prod_int_int ) > ( int > rat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Code____Numeral__Onatural_001_062_It__Nat__Onat_M_Eo_J_001_062_It__Code____Numeral__Onatural_M_Eo_J,type,
    bNF_re1639080489988575423ural_o: ( nat > code_natural > $o ) > ( ( nat > $o ) > ( code_natural > $o ) > $o ) > ( nat > nat > $o ) > ( code_natural > code_natural > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Code____Numeral__Onatural_001_062_It__Nat__Onat_M_Eo_J_001_062_It__Nat__Onat_M_Eo_J,type,
    bNF_re2895713572246619415_nat_o: ( nat > code_natural > $o ) > ( ( nat > $o ) > ( nat > $o ) > $o ) > ( nat > nat > $o ) > ( code_natural > nat > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Code____Numeral__Onatural_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J,type,
    bNF_re88643428490162567atural: ( nat > code_natural > $o ) > ( ( nat > nat ) > ( code_natural > code_natural ) > $o ) > ( nat > nat > nat ) > ( code_natural > code_natural > code_natural ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Code____Numeral__Onatural_001_Eo_001_Eo,type,
    bNF_re2785088596696291543al_o_o: ( nat > code_natural > $o ) > ( $o > $o > $o ) > ( nat > $o ) > ( code_natural > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Code____Numeral__Onatural_001t__Int__Oint_001t__Code____Numeral__Ointeger,type,
    bNF_re5252274238750452962nteger: ( nat > code_natural > $o ) > ( int > code_integer > $o ) > ( nat > int ) > ( code_natural > code_integer ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Code____Numeral__Onatural_001t__Nat__Onat_001t__Code____Numeral__Onatural,type,
    bNF_re3704215830270325841atural: ( nat > code_natural > $o ) > ( nat > code_natural > $o ) > ( nat > nat ) > ( code_natural > code_natural ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Code____Numeral__Onatural_001t__Nat__Onat_001t__Nat__Onat,type,
    bNF_re7333046803109187065at_nat: ( nat > code_natural > $o ) > ( nat > nat > $o ) > ( nat > nat ) > ( code_natural > nat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001_062_It__Int__Oint_Mt__Int__Oint_J_001_062_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    bNF_re4935368626670024657nteger: ( nat > nat > $o ) > ( ( int > int ) > ( code_integer > code_integer ) > $o ) > ( nat > int > int ) > ( nat > code_integer > code_integer ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001_062_It__Int__Oint_Mt__Int__Oint_J_001_062_It__Int__Oint_Mt__Int__Oint_J,type,
    bNF_re4785983289428654063nt_int: ( nat > nat > $o ) > ( ( int > int ) > ( int > int ) > $o ) > ( nat > int > int ) > ( nat > 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__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J,type,
    bNF_re3692170009638431535atural: ( nat > nat > $o ) > ( ( nat > nat ) > ( code_natural > code_natural ) > $o ) > ( nat > nat > nat ) > ( nat > code_natural > code_natural ) > $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__Int__Oint_001t__Code____Numeral__Ointeger,type,
    bNF_re4153400068438556298nteger: ( nat > nat > $o ) > ( int > code_integer > $o ) > ( nat > int ) > ( nat > code_integer ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__Int__Oint_001t__Int__Oint,type,
    bNF_re6650684261131312217nt_int: ( nat > nat > $o ) > ( int > int > $o ) > ( nat > int ) > ( nat > int ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat_001t__Code____Numeral__Onatural,type,
    bNF_re2605341659958429177atural: ( nat > nat > $o ) > ( nat > code_natural > $o ) > ( nat > nat ) > ( nat > code_natural ) > $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__Nat__Onat_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint,type,
    bNF_re6830278522597306478at_int: ( nat > nat > $o ) > ( product_prod_nat_nat > int > $o ) > ( nat > product_prod_nat_nat ) > ( nat > int ) > $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__Code____Numeral__Ointeger_J,type,
    bNF_re7876454716742015248nteger: ( num > num > $o ) > ( ( num > int ) > ( num > code_integer ) > $o ) > ( num > num > int ) > ( num > num > code_integer ) > $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__Code____Numeral__Ointeger,type,
    bNF_re6501075790457514782nteger: ( num > num > $o ) > ( int > code_integer > $o ) > ( num > int ) > ( num > code_integer ) > $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_BNF__Wellorder__Relation_Owo__rel_001t__Nat__Onat,type,
    bNF_We3818239936649020644el_nat: set_Pr1261947904930325089at_nat > $o ).

thf(sy_c_Basic__BNFs_Opred__prod_001t__Nat__Onat_001t__Nat__Onat,type,
    basic_570253317919918037at_nat: ( nat > $o ) > ( nat > $o ) > product_prod_nat_nat > $o ).

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

thf(sy_c_Binomial_Ogbinomial_001t__Code____Numeral__Ointeger,type,
    gbinom8545251970709558553nteger: code_integer > nat > code_integer ).

thf(sy_c_Binomial_Ogbinomial_001t__Code____Numeral__Onatural,type,
    gbinom7368847122466276068atural: code_natural > nat > code_natural ).

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_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_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_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__Code____Numeral__Onatural,type,
    bit_se2773287842338716102atural: code_natural > code_natural > code_natural ).

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__Code____Numeral__Ointeger,type,
    bit_se3928097537394005634nteger: nat > code_integer > code_integer ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit_001t__Code____Numeral__Onatural,type,
    bit_se2751692689150723149atural: nat > code_natural > code_natural ).

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__Code____Numeral__Onatural,type,
    bit_se168947363167071951atural: nat > code_natural > code_natural ).

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

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thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
    comp_nat_nat_nat: ( nat > nat ) > ( nat > nat ) > nat > nat ).

thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Num__Onum_001t__Int__Oint,type,
    comp_nat_num_int: ( nat > num ) > ( int > nat ) > int > num ).

thf(sy_c_Fun_Oid_001_062_It__Nat__Onat_M_Eo_J,type,
    id_nat_o: ( nat > $o ) > nat > $o ).

thf(sy_c_Fun_Oid_001_Eo,type,
    id_o: $o > $o ).

thf(sy_c_Fun_Oid_001t__Int__Oint,type,
    id_int: int > int ).

thf(sy_c_Fun_Oid_001t__Nat__Onat,type,
    id_nat: nat > nat ).

thf(sy_c_Fun_Oid_001t__Num__Onum,type,
    id_num: num > num ).

thf(sy_c_Fun_Oinj__on_001t__Int__Oint_001t__Nat__Onat,type,
    inj_on_int_nat: ( int > nat ) > set_int > $o ).

thf(sy_c_Fun_Oinj__on_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
    inj_on_list_nat_nat: ( list_nat > nat ) > set_list_nat > $o ).

thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Int__Oint,type,
    inj_on_nat_int: ( nat > int ) > set_nat > $o ).

thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__List__Olist_It__Nat__Onat_J,type,
    inj_on_nat_list_nat: ( nat > list_nat ) > set_nat > $o ).

thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Nat__Onat,type,
    inj_on_nat_nat: ( nat > nat ) > set_nat > $o ).

thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    inj_on5538052773655684606at_nat: ( nat > product_prod_nat_nat ) > set_nat > $o ).

thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__String__Ochar,type,
    inj_on_nat_char: ( nat > char ) > set_nat > $o ).

thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_Fun_Oinj__on_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
    inj_on2178005380612969504at_nat: ( product_prod_nat_nat > nat ) > set_Pr1261947904930325089at_nat > $o ).

thf(sy_c_Fun_Oinj__on_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat,type,
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thf(sy_c_Fun_Oinj__on_001t__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
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thf(sy_c_Fun_Omap__fun_001t__Code____Numeral__Ointeger_001t__Int__Oint_001_062_It__Int__Oint_Mt__Int__Oint_J_001_062_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    map_fu8272188784021352819nteger: ( code_integer > int ) > ( ( int > int ) > code_integer > code_integer ) > ( int > int > int ) > code_integer > code_integer > code_integer ).

thf(sy_c_Fun_Omap__fun_001t__Code____Numeral__Ointeger_001t__Int__Oint_001_062_It__Nat__Onat_M_Eo_J_001_062_It__Nat__Onat_M_Eo_J,type,
    map_fu186220595405829473_nat_o: ( code_integer > int ) > ( ( nat > $o ) > nat > $o ) > ( int > nat > $o ) > code_integer > nat > $o ).

thf(sy_c_Fun_Omap__fun_001t__Code____Numeral__Ointeger_001t__Int__Oint_001t__Int__Oint_001t__Code____Numeral__Ointeger,type,
    map_fu2599414010547811884nteger: ( code_integer > int ) > ( int > code_integer ) > ( int > int ) > code_integer > code_integer ).

thf(sy_c_Fun_Omap__fun_001t__Code____Numeral__Ointeger_001t__Int__Oint_001t__Nat__Onat_001t__Code____Numeral__Onatural,type,
    map_fu1051355602067684763atural: ( code_integer > int ) > ( nat > code_natural ) > ( int > nat ) > code_integer > code_natural ).

thf(sy_c_Fun_Omap__fun_001t__Code____Numeral__Ointeger_001t__Int__Oint_001t__Nat__Onat_001t__Nat__Onat,type,
    map_fu6539832666145259331at_nat: ( code_integer > int ) > ( nat > nat ) > ( int > nat ) > code_integer > nat ).

thf(sy_c_Fun_Omap__fun_001t__Code____Numeral__Ointeger_001t__Int__Oint_001t__Num__Onum_001t__Num__Onum,type,
    map_fu1227494855608507351um_num: ( code_integer > int ) > ( num > num ) > ( int > num ) > code_integer > num ).

thf(sy_c_Fun_Omap__fun_001t__Code____Numeral__Onatural_001t__Nat__Onat_001_062_It__Nat__Onat_M_Eo_J_001_062_It__Nat__Onat_M_Eo_J,type,
    map_fu6244533035418998608_nat_o: ( code_natural > nat ) > ( ( nat > $o ) > nat > $o ) > ( nat > nat > $o ) > code_natural > nat > $o ).

thf(sy_c_Fun_Omap__fun_001t__Code____Numeral__Onatural_001t__Nat__Onat_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J,type,
    map_fu6549440983881763648atural: ( code_natural > nat ) > ( ( nat > nat ) > code_natural > code_natural ) > ( nat > nat > nat ) > code_natural > code_natural > code_natural ).

thf(sy_c_Fun_Omap__fun_001t__Code____Numeral__Onatural_001t__Nat__Onat_001t__Int__Oint_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Fun_Omap__fun_001t__Code____Numeral__Onatural_001t__Nat__Onat_001t__Nat__Onat_001t__Code____Numeral__Onatural,type,
    map_fu1239815594074539274atural: ( code_natural > nat ) > ( nat > code_natural ) > ( nat > nat ) > code_natural > code_natural ).

thf(sy_c_Fun_Omap__fun_001t__Code____Numeral__Onatural_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
    map_fu226235371393106610at_nat: ( code_natural > nat ) > ( nat > nat ) > ( nat > nat ) > code_natural > nat ).

thf(sy_c_Fun_Omap__fun_001t__Int__Oint_001t__Int__Oint_001_062_It__Int__Oint_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001_062_It__Int__Oint_Mt__Rat__Orat_J,type,
    map_fu7831380289885515383nt_rat: ( int > int ) > ( ( int > product_prod_int_int ) > int > rat ) > ( int > int > product_prod_int_int ) > int > int > rat ).

thf(sy_c_Fun_Omap__fun_001t__Int__Oint_001t__Int__Oint_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat,type,
    map_fu3424225382358772769nt_rat: ( int > int ) > ( product_prod_int_int > rat ) > ( int > product_prod_int_int ) > int > rat ).

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,
    map_fu434086159418415080_int_o: ( int > product_prod_nat_nat ) > ( ( product_prod_nat_nat > $o ) > int > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > int > int > $o ).

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_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001_062_It__Int__Oint_Mt__Int__Oint_J,type,
    map_fu4960017516451851995nt_int: ( int > product_prod_nat_nat ) > ( ( product_prod_nat_nat > product_prod_nat_nat ) > int > int ) > ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > int > int > int ).

thf(sy_c_Fun_Omap__fun_001t__Int__Oint_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_Eo_001_Eo,type,
    map_fu4826362097070443709at_o_o: ( int > product_prod_nat_nat ) > ( $o > $o ) > ( product_prod_nat_nat > $o ) > int > $o ).

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 ).

thf(sy_c_Fun_Omap__fun_001t__Int__Oint_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint,type,
    map_fu3667384564859982768at_int: ( int > product_prod_nat_nat ) > ( product_prod_nat_nat > int ) > ( product_prod_nat_nat > product_prod_nat_nat ) > int > int ).

thf(sy_c_Fun_Omap__fun_001t__Nat__Onat_001t__Nat__Onat_001_062_It__Int__Oint_Mt__Int__Oint_J_001_062_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    map_fu8224359078269604234nteger: ( nat > nat ) > ( ( int > int ) > code_integer > code_integer ) > ( nat > int > int ) > nat > code_integer > code_integer ).

thf(sy_c_Fun_Omap__fun_001t__Nat__Onat_001t__Nat__Onat_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J,type,
    map_fu6981160461238011112atural: ( nat > nat ) > ( ( nat > nat ) > code_natural > code_natural ) > ( nat > nat > nat ) > nat > code_natural > code_natural ).

thf(sy_c_Fun_Omap__fun_001t__Nat__Onat_001t__Nat__Onat_001t__Int__Oint_001t__Code____Numeral__Ointeger,type,
    map_fu6290471996055670595nteger: ( nat > nat ) > ( int > code_integer ) > ( nat > int ) > nat > code_integer ).

thf(sy_c_Fun_Omap__fun_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat_001t__Code____Numeral__Onatural,type,
    map_fu4742413587575543474atural: ( nat > nat ) > ( nat > code_natural ) > ( nat > nat ) > nat > code_natural ).

thf(sy_c_Fun_Omap__fun_001t__Num__Onum_001t__Num__Onum_001_062_It__Num__Onum_Mt__Int__Oint_J_001_062_It__Num__Onum_Mt__Code____Numeral__Ointeger_J,type,
    map_fu6891787308814931657nteger: ( num > num ) > ( ( num > int ) > num > code_integer ) > ( num > num > int ) > num > num > code_integer ).

thf(sy_c_Fun_Omap__fun_001t__Num__Onum_001t__Num__Onum_001t__Int__Oint_001t__Code____Numeral__Ointeger,type,
    map_fu8638147718074629079nteger: ( num > num ) > ( int > code_integer ) > ( num > int ) > num > code_integer ).

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,
    map_fu4333342158222067775at_rat: ( rat > product_prod_int_int ) > ( ( product_prod_int_int > product_prod_int_int ) > rat > rat ) > ( product_prod_int_int > product_prod_int_int > product_prod_int_int ) > rat > rat > rat ).

thf(sy_c_Fun_Omap__fun_001t__Rat__Orat_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001_Eo_001_Eo,type,
    map_fu898904425404107465nt_o_o: ( rat > product_prod_int_int ) > ( $o > $o ) > ( product_prod_int_int > $o ) > rat > $o ).

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__Def_Ois__measure_001t__Int__Oint,type,
    fun_is_measure_int: ( int > nat ) > $o ).

thf(sy_c_Fun__Def_Omax__strict,type,
    fun_max_strict: set_Pr4329608150637261639at_nat ).

thf(sy_c_Fun__Def_Omax__weak,type,
    fun_max_weak: set_Pr4329608150637261639at_nat ).

thf(sy_c_Fun__Def_Omin__strict,type,
    fun_min_strict: set_Pr4329608150637261639at_nat ).

thf(sy_c_Fun__Def_Omin__weak,type,
    fun_min_weak: set_Pr4329608150637261639at_nat ).

thf(sy_c_Fun__Def_Opair__leq,type,
    fun_pair_leq: set_Pr8693737435421807431at_nat ).

thf(sy_c_Fun__Def_Opair__less,type,
    fun_pair_less: set_Pr8693737435421807431at_nat ).

thf(sy_c_Fun__Def_Oreduction__pair_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    fun_re2478310338295953701at_nat: produc1319942482725812455at_nat > $o ).

thf(sy_c_GCD_OGcd__class_OGcd_001t__Int__Oint,type,
    gcd_Gcd_int: set_int > int ).

thf(sy_c_GCD_OGcd__class_OGcd_001t__Nat__Onat,type,
    gcd_Gcd_nat: set_nat > nat ).

thf(sy_c_GCD_OGcd__class_OLcm_001t__Int__Oint,type,
    gcd_Lcm_int: set_int > int ).

thf(sy_c_GCD_OGcd__class_OLcm_001t__Nat__Onat,type,
    gcd_Lcm_nat: set_nat > nat ).

thf(sy_c_GCD_Obezw,type,
    bezw: nat > nat > product_prod_int_int ).

thf(sy_c_GCD_Obezw__rel,type,
    bezw_rel: product_prod_nat_nat > product_prod_nat_nat > $o ).

thf(sy_c_GCD_Ogcd__class_Ogcd_001t__Int__Oint,type,
    gcd_gcd_int: int > int > int ).

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__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,
    gcd_nat_rel: product_prod_nat_nat > product_prod_nat_nat > $o ).

thf(sy_c_Groups_Oabs__class_Oabs_001t__Code____Numeral__Ointeger,type,
    abs_abs_Code_integer: code_integer > code_integer ).

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,
    abs_abs_rat: rat > rat ).

thf(sy_c_Groups_Ocomm__monoid_001t__Nat__Onat,type,
    comm_monoid_nat: ( nat > nat > nat ) > nat > $o ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Code____Numeral__Ointeger,type,
    minus_8373710615458151222nteger: code_integer > code_integer > code_integer ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Code____Numeral__Onatural,type,
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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__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,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
    minus_3314409938677909166at_nat: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Product____Type__Ounit_J,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    minus_2163939370556025621et_nat: set_set_nat > set_set_nat > set_set_nat ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
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thf(sy_c_Groups_Omonoid_001t__Nat__Onat,type,
    monoid_nat: ( nat > nat > nat ) > nat > $o ).

thf(sy_c_Groups_Oone__class_Oone_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Groups_Oone__class_Oone_001t__Code____Numeral__Onatural,type,
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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_Oplus__class_Oplus_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Groups_Oplus__class_Oplus_001t__Code____Numeral__Onatural,type,
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thf(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
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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_Osgn__class_Osgn_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Groups_Osgn__class_Osgn_001t__Int__Oint,type,
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thf(sy_c_Groups_Osgn__class_Osgn_001t__Rat__Orat,type,
    sgn_sgn_rat: rat > rat ).

thf(sy_c_Groups_Otimes__class_Otimes_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Groups_Otimes__class_Otimes_001t__Code____Numeral__Onatural,type,
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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_Ouminus__class_Ouminus_001t__Code____Numeral__Ointeger,type,
    uminus1351360451143612070nteger: code_integer > code_integer ).

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

thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Product____Type__Ounit,type,
    uminus2952777764628376836t_unit: product_unit > product_unit ).

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

thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Int__Oint_J,type,
    uminus1532241313380277803et_int: set_int > set_int ).

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

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thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
    uminus935396558254630718at_nat: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat ).

thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    uminus613421341184616069et_nat: set_set_nat > set_set_nat ).

thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    uminus8018730102153342520at_nat: set_se7855581050983116737at_nat > set_se7855581050983116737at_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__Code____Numeral__Onatural,type,
    zero_z2226904508553997617atural: code_natural ).

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,
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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__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__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__List_Ocomm__semiring__0__class_Ohorner__sum_001_Eo_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Groups__List_Ocomm__semiring__0__class_Ohorner__sum_001_Eo_001t__Nat__Onat,type,
    groups9119017779487936845_o_nat: ( $o > nat ) > nat > list_o > nat ).

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    groups8485231416243008693nt_int: ( int > int ) > int > list_int > int ).

thf(sy_c_Groups__List_Ocomm__semiring__0__class_Ohorner__sum_001t__Int__Oint_001t__Nat__Onat,type,
    groups8487721886752058969nt_nat: ( int > nat ) > nat > list_int > nat ).

thf(sy_c_Groups__List_Ocomm__semiring__0__class_Ohorner__sum_001t__Int__Oint_001t__Rat__Orat,type,
    groups7852591826665563233nt_rat: ( int > rat ) > rat > list_int > rat ).

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thf(sy_c_Groups__List_Ocomm__semiring__0__class_Ohorner__sum_001t__Nat__Onat_001t__Int__Oint,type,
    groups7485877704341954137at_int: ( nat > int ) > int > list_nat > int ).

thf(sy_c_Groups__List_Ocomm__semiring__0__class_Ohorner__sum_001t__Nat__Onat_001t__Nat__Onat,type,
    groups7488368174851004413at_nat: ( nat > nat ) > nat > list_nat > nat ).

thf(sy_c_Groups__List_Ocomm__semiring__0__class_Ohorner__sum_001t__Nat__Onat_001t__Rat__Orat,type,
    groups6853238114764508677at_rat: ( nat > rat ) > rat > list_nat > rat ).

thf(sy_c_Groups__List_Omonoid__add__class_Osum__list_001t__Nat__Onat,type,
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thf(sy_c_HOL_ONO__MATCH_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_HOL_ONO__MATCH_001t__Code____Numeral__Ointeger_001t__Int__Oint,type,
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thf(sy_c_HOL_ONO__MATCH_001t__Int__Oint_001t__Nat__Onat,type,
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thf(sy_c_HOL_ONO__MATCH_001t__Int__Oint_001t__Rat__Orat,type,
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thf(sy_c_HOL_ONO__MATCH_001t__Nat__Onat_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_HOL_ONO__MATCH_001t__Nat__Onat_001t__Int__Oint,type,
    nO_MATCH_nat_int: nat > int > $o ).

thf(sy_c_HOL_ONO__MATCH_001t__Nat__Onat_001t__Nat__Onat,type,
    nO_MATCH_nat_nat: nat > nat > $o ).

thf(sy_c_HOL_ONO__MATCH_001t__Nat__Onat_001t__Rat__Orat,type,
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thf(sy_c_HOL_ONO__MATCH_001t__Rat__Orat_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_HOL_ONO__MATCH_001t__Rat__Orat_001t__Rat__Orat,type,
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thf(sy_c_HOL_OThe_001t__Int__Oint,type,
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thf(sy_c_HOL_OThe_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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thf(sy_c_Heap_Oempty,type,
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thf(sy_c_Heap_Oheap_Oheap__ext_001t__Product____Type__Ounit,type,
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thf(sy_c_Heap__Time__Monad_OHeap_OHeap_001t__Product____Type__Ounit,type,
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thf(sy_c_Heap__Time__Monad_Oassert_001t__Nat__Onat,type,
    heap_Time_assert_nat: ( nat > $o ) > nat > heap_Time_Heap_nat ).

thf(sy_c_Heap__Time__Monad_Oeffect_001t__Nat__Onat,type,
    heap_Time_effect_nat: heap_Time_Heap_nat > heap_e7401611519738050253t_unit > heap_e7401611519738050253t_unit > nat > nat > $o ).

thf(sy_c_Heap__Time__Monad_Oreturn_001t__Nat__Onat,type,
    heap_Time_return_nat: nat > heap_Time_Heap_nat ).

thf(sy_c_Heap__Time__Monad_Otap_001t__Nat__Onat,type,
    heap_Time_tap_nat: ( heap_e7401611519738050253t_unit > nat ) > heap_Time_Heap_nat ).

thf(sy_c_Heap__Time__Monad_Oureturn_001t__Nat__Onat,type,
    heap_T6738714419362814988rn_nat: nat > heap_Time_Heap_nat ).

thf(sy_c_Heap__Time__Monad_Owait,type,
    heap_Time_wait: nat > heap_T5738788834812785303t_unit ).

thf(sy_c_If_001_062_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
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thf(sy_c_If_001_062_It__Int__Oint_Mt__Int__Oint_J,type,
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thf(sy_c_If_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_If_001t__Code____Numeral__Onatural,type,
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thf(sy_c_If_001t__Heap____Time____Monad__OHeap_It__Nat__Onat_J,type,
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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__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,
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thf(sy_c_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
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thf(sy_c_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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thf(sy_c_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_If_001t__Rat__Orat,type,
    if_rat: $o > rat > rat > rat ).

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__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_If_001t__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_Int_OAbs__Integ,type,
    abs_Integ: product_prod_nat_nat > int ).

thf(sy_c_Int_ONeg,type,
    neg: num > int ).

thf(sy_c_Int_OPos,type,
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thf(sy_c_Int_ORep__Integ,type,
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thf(sy_c_Int_Ocr__int,type,
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thf(sy_c_Int_Odup,type,
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thf(sy_c_Int_Oint_OAbs__int,type,
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thf(sy_c_Int_Oint_ORep__int,type,
    rep_int: int > set_Pr1261947904930325089at_nat ).

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

thf(sy_c_Int_Ointrel,type,
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thf(sy_c_Int_Onat,type,
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thf(sy_c_Int_Opcr__int,type,
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thf(sy_c_Int_Oring__1__class_OInts_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Int_Oring__1__class_OInts_001t__Int__Oint,type,
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thf(sy_c_Int_Oring__1__class_OInts_001t__Rat__Orat,type,
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thf(sy_c_Int_Oring__1__class_Oof__int_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Int_Oring__1__class_Oof__int_001t__Int__Oint,type,
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thf(sy_c_Int_Oring__1__class_Oof__int_001t__Rat__Orat,type,
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thf(sy_c_Int_Osub,type,
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thf(sy_c_Lattices_Oinf__class_Oinf_001t__Int__Oint,type,
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thf(sy_c_Lattices_Oinf__class_Oinf_001t__Nat__Onat,type,
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thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(sy_c_Lattices_Osemilattice__neutr_001t__Nat__Onat,type,
    semila9081495762789891438tr_nat: ( nat > nat > nat ) > nat > $o ).

thf(sy_c_Lattices_Osemilattice__neutr__order_001t__Nat__Onat,type,
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thf(sy_c_Lattices_Osup__class_Osup_001t__Int__Oint,type,
    sup_sup_int: int > int > int ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Nat__Onat,type,
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thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Lattices__Big_Olinorder__class_OMax_001t__Int__Oint,type,
    lattic8263393255366662781ax_int: set_int > int ).

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

thf(sy_c_Lattices__Big_Osemilattice__neutr__set_OF_001t__Nat__Onat,type,
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thf(sy_c_List_Oappend_001t__Int__Oint,type,
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thf(sy_c_List_Oappend_001t__Nat__Onat,type,
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thf(sy_c_List_Ofold_001t__Int__Oint_001t__Int__Oint,type,
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thf(sy_c_List_Ofold_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_List_Olast_001t__Nat__Onat,type,
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thf(sy_c_List_Olinorder__class_Osorted__list__of__set_001t__Nat__Onat,type,
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thf(sy_c_List_Olist_OCons_001_Eo,type,
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thf(sy_c_List_Olist_OCons_001t__Int__Oint,type,
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thf(sy_c_List_Olist_OCons_001t__Nat__Onat,type,
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thf(sy_c_List_Olist_ONil_001t__Int__Oint,type,
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thf(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
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thf(sy_c_List_Olist_Ohd_001t__Nat__Onat,type,
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thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_List_Olist_Oset_001t__Int__Oint,type,
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thf(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
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thf(sy_c_List_Olist_Osize__list_001t__Typerep__Otyperep,type,
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thf(sy_c_List_Olist_Otl_001t__Nat__Onat,type,
    tl_nat: list_nat > list_nat ).

thf(sy_c_List_Onth_001_Eo,type,
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thf(sy_c_List_Onth_001t__Int__Oint,type,
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thf(sy_c_List_Onth_001t__Nat__Onat,type,
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    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_It__Int__Oint_M_Eo_J,type,
    nth_Pr7514405829937366042_int_o: list_P5087981734274514673_int_o > nat > product_prod_int_o ).

thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    nth_Pr4439495888332055232nt_int: list_P5707943133018811711nt_int > nat > product_prod_int_int ).

thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Int__Oint_Mt__Nat__Onat_J,type,
    nth_Pr8617346907841251940nt_nat: list_P8198026277950538467nt_nat > nat > product_prod_int_nat ).

thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_M_Eo_J,type,
    nth_Pr112076138515278198_nat_o: list_P7333126701944960589_nat_o > nat > product_prod_nat_o ).

thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Int__Oint_J,type,
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thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_List_Onth_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,
    nth_Pr6744343527793145070at_nat: list_P8469869581646625389at_nat > nat > produc859450856879609959at_nat ).

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_001t__Int__Oint_001_Eo,type,
    product_int_o: list_int > list_o > list_P5087981734274514673_int_o ).

thf(sy_c_List_Oproduct_001t__Int__Oint_001t__Int__Oint,type,
    product_int_int: list_int > list_int > list_P5707943133018811711nt_int ).

thf(sy_c_List_Oproduct_001t__Int__Oint_001t__Nat__Onat,type,
    product_int_nat: list_int > list_nat > list_P8198026277950538467nt_nat ).

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__Int__Oint,type,
    product_nat_int: list_nat > list_int > list_P3521021558325789923at_int ).

thf(sy_c_List_Oproduct_001t__Nat__Onat_001t__Nat__Onat,type,
    product_nat_nat: list_nat > list_nat > list_P6011104703257516679at_nat ).

thf(sy_c_List_Oproduct_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    produc3544356994491977349at_nat: list_P6011104703257516679at_nat > list_P6011104703257516679at_nat > list_P8469869581646625389at_nat ).

thf(sy_c_List_Orotate1_001_Eo,type,
    rotate1_o: list_o > list_o ).

thf(sy_c_List_Orotate1_001t__Int__Oint,type,
    rotate1_int: list_int > list_int ).

thf(sy_c_List_Orotate1_001t__Nat__Onat,type,
    rotate1_nat: list_nat > list_nat ).

thf(sy_c_List_Orotate_001_Eo,type,
    rotate_o: nat > list_o > list_o ).

thf(sy_c_List_Orotate_001t__Int__Oint,type,
    rotate_int: nat > list_int > list_int ).

thf(sy_c_List_Orotate_001t__Nat__Onat,type,
    rotate_nat: nat > list_nat > list_nat ).

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_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_List_Ozip_001_Eo_001_Eo,type,
    zip_o_o: list_o > list_o > list_P4002435161011370285od_o_o ).

thf(sy_c_List_Ozip_001_Eo_001t__Int__Oint,type,
    zip_o_int: list_o > list_int > list_P3795440434834930179_o_int ).

thf(sy_c_List_Ozip_001_Eo_001t__Nat__Onat,type,
    zip_o_nat: list_o > list_nat > list_P6285523579766656935_o_nat ).

thf(sy_c_List_Ozip_001t__Int__Oint_001_Eo,type,
    zip_int_o: list_int > list_o > list_P5087981734274514673_int_o ).

thf(sy_c_List_Ozip_001t__Int__Oint_001t__Int__Oint,type,
    zip_int_int: list_int > list_int > list_P5707943133018811711nt_int ).

thf(sy_c_List_Ozip_001t__Int__Oint_001t__Nat__Onat,type,
    zip_int_nat: list_int > list_nat > list_P8198026277950538467nt_nat ).

thf(sy_c_List_Ozip_001t__Nat__Onat_001_Eo,type,
    zip_nat_o: list_nat > list_o > list_P7333126701944960589_nat_o ).

thf(sy_c_List_Ozip_001t__Nat__Onat_001t__Int__Oint,type,
    zip_nat_int: list_nat > list_int > list_P3521021558325789923at_int ).

thf(sy_c_List_Ozip_001t__Nat__Onat_001t__Nat__Onat,type,
    zip_nat_nat: list_nat > list_nat > list_P6011104703257516679at_nat ).

thf(sy_c_List_Ozip_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    zip_Pr4664179122662387191at_nat: list_P6011104703257516679at_nat > list_P6011104703257516679at_nat > list_P8469869581646625389at_nat ).

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_Opred,type,
    pred: nat > nat ).

thf(sy_c_Nat_Osemiring__1__class_ONats_001t__Int__Oint,type,
    semiring_1_Nats_int: set_int ).

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__Code____Numeral__Onatural,type,
    semiri3763490453095760265atural: nat > code_natural ).

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_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__Int__Oint_J,type,
    size_size_list_int: list_int > 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__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_It__Int__Oint_M_Eo_J_J,type,
    size_s4246224855604898693_int_o: list_P5087981734274514673_int_o > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    size_s5157815400016825771nt_int: list_P5707943133018811711nt_int > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Nat__Onat_J_J,type,
    size_s7647898544948552527nt_nat: list_P8198026277950538467nt_nat > 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__Int__Oint_J_J,type,
    size_s2970893825323803983at_int: list_P3521021558325789923at_int > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    size_s5460976970255530739at_nat: list_P6011104703257516679at_nat > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_Itf__a_J,type,
    size_size_list_a: list_a > 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__String__Ochar,type,
    size_size_char: char > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__Typerep__Otyperep,type,
    size_size_typerep: typerep > nat ).

thf(sy_c_Nat__Bijection_Oint__decode,type,
    nat_int_decode: nat > int ).

thf(sy_c_Nat__Bijection_Oint__encode,type,
    nat_int_encode: int > 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_Osum__decode,type,
    nat_sum_decode: nat > sum_sum_nat_nat ).

thf(sy_c_Nat__Bijection_Osum__encode,type,
    nat_sum_encode: sum_sum_nat_nat > nat ).

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

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

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

thf(sy_c_Num_Onat__of__num,type,
    nat_of_num: num > nat ).

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__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__dec_001t__Code____Numeral__Ointeger,type,
    neg_nu7757733837767384882nteger: code_integer > code_integer ).

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__inc_001t__Code____Numeral__Ointeger,type,
    neg_nu5831290666863070958nteger: code_integer > code_integer ).

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_Osub_001t__Code____Numeral__Ointeger,type,
    neg_nu5755505904847501662nteger: num > num > code_integer ).

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

thf(sy_c_Num_Oneg__numeral__class_Osub_001t__Rat__Orat,type,
    neg_numeral_sub_rat: num > num > rat ).

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_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__Code____Numeral__Onatural,type,
    numera5444537566228673987atural: num > code_natural ).

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_Opow,type,
    pow: num > num > num ).

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

thf(sy_c_Num_Oring__1__class_Oiszero_001t__Code____Numeral__Ointeger,type,
    ring_16219924574208605041nteger: code_integer > $o ).

thf(sy_c_Num_Oring__1__class_Oiszero_001t__Int__Oint,type,
    ring_1_iszero_int: int > $o ).

thf(sy_c_Num_Oring__1__class_Oiszero_001t__Rat__Orat,type,
    ring_1_iszero_rat: rat > $o ).

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

thf(sy_c_Option_Ooption_ONone_001t__Num__Onum,type,
    none_num: option_num ).

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__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    some_P1914260805536162275it_nat: produc8664842809031399944it_nat > option8956607266484857688it_nat ).

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_Order__Relation_OunderS_001t__Nat__Onat,type,
    order_underS_nat: set_Pr1261947904930325089at_nat > nat > set_nat ).

thf(sy_c_Order__Relation_Owell__order__on_001t__Nat__Onat,type,
    order_2888998067076097458on_nat: set_nat > set_Pr1261947904930325089at_nat > $o ).

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__Nat__Onat,type,
    bot_bot_nat: nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Code____Numeral__Ointeger_J,type,
    bot_bo3990330152332043303nteger: set_Code_integer ).

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__List__Olist_It__Nat__Onat_J_J,type,
    bot_bot_set_list_nat: set_list_nat ).

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__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    bot_bo2099793752762293965at_nat: set_Pr1261947904930325089at_nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
    bot_bo228742789529271731at_nat: set_Pr4329608150637261639at_nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Ounit_J,type,
    bot_bo3957492148770167129t_unit: set_Product_unit ).

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__Set__Oset_It__Nat__Onat_J_J,type,
    bot_bot_set_set_nat: set_set_nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    bot_bo3083307316010499117at_nat: set_se7855581050983116737at_nat ).

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

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__Code____Numeral__Onatural,type,
    ord_le5570908160329646204atural: code_natural > code_natural > $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,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Num__Onum,type,
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thf(sy_c_Quotient_Oquot__type_Oabs_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint,type,
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thf(sy_c_Quotient_Oquot__type_Orep_001t__Int__Oint_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_Random_Olog,type,
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thf(sy_c_Random_Olog__rel,type,
    log_rel: produc7822875418678951345atural > produc7822875418678951345atural > $o ).

thf(sy_c_Random_Ominus__shift,type,
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thf(sy_c_Random_Orange,type,
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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_Onormalize,type,
    normalize: product_prod_int_int > product_prod_int_int ).

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

thf(sy_c_Rat_Oratrel,type,
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thf(sy_c_Relation_ODomainp_001t__Int__Oint_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Relation_ODomainp_001t__Int__Oint_001t__Int__Oint,type,
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thf(sy_c_Relation_ODomainp_001t__Nat__Onat_001t__Code____Numeral__Onatural,type,
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thf(sy_c_Relation_ODomainp_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_Relation_ODomainp_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat,type,
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thf(sy_c_Relation_ODomainp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint,type,
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thf(sy_c_Relation_OField_001t__Nat__Onat,type,
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thf(sy_c_Relation_OId_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_Rings_Oalgebraic__semidom__class_Ocoprime_001t__Int__Oint,type,
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thf(sy_c_Rings_Oalgebraic__semidom__class_Ocoprime_001t__Nat__Onat,type,
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thf(sy_c_Rings_Odivide__class_Odivide_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Rings_Odivide__class_Odivide_001t__Code____Numeral__Onatural,type,
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thf(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
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thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
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thf(sy_c_Rings_Odivide__class_Odivide_001t__Rat__Orat,type,
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thf(sy_c_Rings_Odvd__class_Odvd_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Rings_Odvd__class_Odvd_001t__Code____Numeral__Onatural,type,
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thf(sy_c_Rings_Odvd__class_Odvd_001t__Int__Oint,type,
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thf(sy_c_Rings_Odvd__class_Odvd_001t__Nat__Onat,type,
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thf(sy_c_Rings_Odvd__class_Odvd_001t__Rat__Orat,type,
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thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Code____Numeral__Onatural,type,
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thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Int__Oint,type,
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thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Nat__Onat,type,
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thf(sy_c_Rings_Onormalization__semidom__class_Onormalize_001t__Int__Oint,type,
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thf(sy_c_Rings_Onormalization__semidom__class_Onormalize_001t__Nat__Onat,type,
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thf(sy_c_Rings_Ounit__factor__class_Ounit__factor_001t__Int__Oint,type,
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thf(sy_c_Rings_Ounit__factor__class_Ounit__factor_001t__Nat__Onat,type,
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thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Code____Numeral__Onatural,type,
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thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Int__Oint,type,
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thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Nat__Onat,type,
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thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Rat__Orat,type,
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thf(sy_c_Set_OCollect_001t__Int__Oint,type,
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thf(sy_c_Set_OCollect_001t__List__Olist_It__Nat__Onat_J,type,
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thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
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thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Set_OCollect_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(sy_c_Set_OPow_001t__Nat__Onat,type,
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thf(sy_c_Set_Oimage_001t__Int__Oint_001t__Int__Oint,type,
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thf(sy_c_Set_Oimage_001t__Int__Oint_001t__Nat__Onat,type,
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thf(sy_c_Set_Oimage_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
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thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Int__Oint,type,
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thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__List__Olist_It__Nat__Onat_J,type,
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thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__String__Ochar,type,
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thf(sy_c_Set_Oinsert_001t__Int__Oint,type,
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thf(sy_c_Set_Oinsert_001t__List__Olist_It__Nat__Onat_J,type,
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thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
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thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
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thf(sy_c_Set_Oinsert_001t__Product____Type__Ounit,type,
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thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(sy_c_Set_Ovimage_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Nat__Onat,type,
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thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Nat__Onat,type,
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thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Num__Onum,type,
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thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Rat__Orat,type,
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thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Set__Oset_It__Nat__Onat_J,type,
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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,
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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_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,
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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,
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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_OlessThan_001t__Nat__Onat,type,
    set_ord_lessThan_nat: nat > set_nat ).

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

thf(sy_c_String_Ochar_Osize__char,type,
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thf(sy_c_String_Ocomm__semiring__1__class_Oof__char_001t__Nat__Onat,type,
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thf(sy_c_String_Ounique__euclidean__semiring__with__bit__operations__class_Ochar__of_001t__Nat__Onat,type,
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thf(sy_c_Sum__Type_OInl_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_Sum__Type_OInr_001t__Nat__Onat_001t__Nat__Onat,type,
    sum_Inr_nat_nat: nat > sum_sum_nat_nat ).

thf(sy_c_Sum__Type_Osum_Ocase__sum_001t__Nat__Onat_001t__Int__Oint_001t__Nat__Onat,type,
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thf(sy_c_Sum__Type_Osum_Ocase__sum_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
    sum_ca6763686470577984908at_nat: ( nat > nat ) > ( nat > nat ) > sum_sum_nat_nat > nat ).

thf(sy_c_Transfer_Obi__total_001t__Int__Oint_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Transfer_Obi__total_001t__Nat__Onat_001t__Code____Numeral__Onatural,type,
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thf(sy_c_Transfer_Obi__total_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint,type,
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thf(sy_c_Transitive__Closure_Ortrancl_001t__Nat__Onat,type,
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thf(sy_c_Transitive__Closure_Otrancl_001t__Nat__Onat,type,
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thf(sy_c_Transitive__Closure_Otranclp_001t__Nat__Onat,type,
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thf(sy_c_Typedef_Otype__definition_001t__Code____Numeral__Ointeger_001t__Int__Oint,type,
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thf(sy_c_Typerep_Otyperep_OTyperep,type,
    typerep2: literal > list_typerep > typerep ).

thf(sy_c_Typerep_Otyperep_Osize__typerep,type,
    size_typerep: typerep > nat ).

thf(sy_c_Wellfounded_Oaccp_001t__List__Olist_It__Nat__Onat_J,type,
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thf(sy_c_Wellfounded_Oaccp_001t__Nat__Onat,type,
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thf(sy_c_Wellfounded_Oless__than,type,
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thf(sy_c_Wellfounded_Olex__prod_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_Wellfounded_Omin__ext_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    min_ex6901939911449802026at_nat: set_Pr8693737435421807431at_nat > set_Pr4329608150637261639at_nat ).

thf(sy_c_Wellfounded_Opred__nat,type,
    pred_nat: set_Pr1261947904930325089at_nat ).

thf(sy_c_Wellfounded_Owf_001t__Int__Oint,type,
    wf_int: set_Pr958786334691620121nt_int > $o ).

thf(sy_c_Wellfounded_Owf_001t__Nat__Onat,type,
    wf_nat: set_Pr1261947904930325089at_nat > $o ).

thf(sy_c_Wellfounded_Owf_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    wf_Pro7803398752247294826at_nat: set_Pr8693737435421807431at_nat > $o ).

thf(sy_c_member_001t__Code____Numeral__Ointeger,type,
    member_Code_integer: code_integer > set_Code_integer > $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__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    member8757157785044589968at_nat: produc3843707927480180839at_nat > set_Pr4329608150637261639at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Ounit,type,
    member_Product_unit: product_unit > set_Product_unit > $o ).

thf(sy_c_member_001t__Rat__Orat,type,
    member_rat: rat > set_rat > $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__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    member2643936169264416010at_nat: set_Pr1261947904930325089at_nat > set_se7855581050983116737at_nat > $o ).

thf(sy_v_a,type,
    a: array_a ).

thf(sy_v_h,type,
    h: heap_e7401611519738050253t_unit ).

thf(sy_v_h_H,type,
    h2: heap_e7401611519738050253t_unit ).

thf(sy_v_n,type,
    n: nat ).

thf(sy_v_r,type,
    r: nat ).

thf(sy_v_thesis,type,
    thesis: $o ).

% Relevant facts (9231)
thf(fact_0_assms,axiom,
    heap_Time_effect_nat @ ( array_len_a @ a ) @ h @ h2 @ r @ n ).

% assms
thf(fact_1_that,axiom,
    ( ( r
      = ( array_length_a @ h2 @ a ) )
   => ( ( h2 = h )
     => ( ( n = one_one_nat )
       => thesis ) ) ) ).

% that
thf(fact_2_effect__lengthI,axiom,
    ! [H: heap_e7401611519738050253t_unit,H2: heap_e7401611519738050253t_unit,R: nat,A: array_a,N: nat] :
      ( ( H = H2 )
     => ( ( R
          = ( array_length_a @ H2 @ A ) )
       => ( ( N = one_one_nat )
         => ( heap_Time_effect_nat @ ( array_len_a @ A ) @ H2 @ H @ R @ N ) ) ) ) ).

% effect_lengthI
thf(fact_3_one__natural_Orsp,axiom,
    one_one_nat = one_one_nat ).

% one_natural.rsp
thf(fact_4_one__reorient,axiom,
    ! [X: nat] :
      ( ( one_one_nat = X )
      = ( X = one_one_nat ) ) ).

% one_reorient
thf(fact_5_one__reorient,axiom,
    ! [X: int] :
      ( ( one_one_int = X )
      = ( X = one_one_int ) ) ).

% one_reorient
thf(fact_6_one__reorient,axiom,
    ! [X: code_integer] :
      ( ( one_one_Code_integer = X )
      = ( X = one_one_Code_integer ) ) ).

% one_reorient
thf(fact_7_one__reorient,axiom,
    ! [X: rat] :
      ( ( one_one_rat = X )
      = ( X = one_one_rat ) ) ).

% one_reorient
thf(fact_8_one__reorient,axiom,
    ! [X: code_natural] :
      ( ( one_one_Code_natural = X )
      = ( X = one_one_Code_natural ) ) ).

% one_reorient
thf(fact_9_euclidean__size__1,axiom,
    ( ( euclid6377331345833325938nteger @ one_one_Code_integer )
    = one_one_nat ) ).

% euclidean_size_1
thf(fact_10_euclidean__size__1,axiom,
    ( ( euclid4774559944035922753ze_int @ one_one_int )
    = one_one_nat ) ).

% euclidean_size_1
thf(fact_11_euclidean__size__1,axiom,
    ( ( euclid4777050414544973029ze_nat @ one_one_nat )
    = one_one_nat ) ).

% euclidean_size_1
thf(fact_12_euclidean__size__1,axiom,
    ( ( euclid5200926497590043453atural @ one_one_Code_natural )
    = one_one_nat ) ).

% euclidean_size_1
thf(fact_13_fact__1,axiom,
    ( ( semiri1406184849735516958ct_int @ one_one_nat )
    = one_one_int ) ).

% fact_1
thf(fact_14_fact__1,axiom,
    ( ( semiri3624122377584611663nteger @ one_one_nat )
    = one_one_Code_integer ) ).

% fact_1
thf(fact_15_fact__1,axiom,
    ( ( semiri773545260158071498ct_rat @ one_one_nat )
    = one_one_rat ) ).

% fact_1
thf(fact_16_fact__1,axiom,
    ( ( semiri2447717529341329178atural @ one_one_nat )
    = one_one_Code_natural ) ).

% fact_1
thf(fact_17_fact__1,axiom,
    ( ( semiri1408675320244567234ct_nat @ one_one_nat )
    = one_one_nat ) ).

% fact_1
thf(fact_18_of__nat__1,axiom,
    ( ( semiri1314217659103216013at_int @ one_one_nat )
    = one_one_int ) ).

% of_nat_1
thf(fact_19_of__nat__1,axiom,
    ( ( semiri4939895301339042750nteger @ one_one_nat )
    = one_one_Code_integer ) ).

% of_nat_1
thf(fact_20_of__nat__1,axiom,
    ( ( semiri1316708129612266289at_nat @ one_one_nat )
    = one_one_nat ) ).

% of_nat_1
thf(fact_21_of__nat__1,axiom,
    ( ( semiri681578069525770553at_rat @ one_one_nat )
    = one_one_rat ) ).

% of_nat_1
thf(fact_22_of__nat__1,axiom,
    ( ( semiri3763490453095760265atural @ one_one_nat )
    = one_one_Code_natural ) ).

% of_nat_1
thf(fact_23_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_24_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_Code_integer
        = ( semiri4939895301339042750nteger @ N ) )
      = ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_25_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_26_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_27_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_Code_natural
        = ( semiri3763490453095760265atural @ N ) )
      = ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_28_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_29_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( ( semiri4939895301339042750nteger @ N )
        = one_one_Code_integer )
      = ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
thf(fact_30_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_31_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_32_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( ( semiri3763490453095760265atural @ N )
        = one_one_Code_natural )
      = ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
thf(fact_33_dbl__dec__simps_I3_J,axiom,
    ( ( neg_nu3811975205180677377ec_int @ one_one_int )
    = one_one_int ) ).

% dbl_dec_simps(3)
thf(fact_34_dbl__dec__simps_I3_J,axiom,
    ( ( neg_nu7757733837767384882nteger @ one_one_Code_integer )
    = one_one_Code_integer ) ).

% dbl_dec_simps(3)
thf(fact_35_dbl__dec__simps_I3_J,axiom,
    ( ( neg_nu3179335615603231917ec_rat @ one_one_rat )
    = one_one_rat ) ).

% dbl_dec_simps(3)
thf(fact_36_round__1,axiom,
    ( ( archim7778729529865785530nd_rat @ one_one_rat )
    = one_one_int ) ).

% round_1
thf(fact_37_of__nat__eq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ( semiri1314217659103216013at_int @ M )
        = ( semiri1314217659103216013at_int @ N ) )
      = ( M = N ) ) ).

% of_nat_eq_iff
thf(fact_38_of__nat__eq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ( semiri4939895301339042750nteger @ M )
        = ( semiri4939895301339042750nteger @ N ) )
      = ( M = N ) ) ).

% of_nat_eq_iff
thf(fact_39_of__nat__eq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ( semiri1316708129612266289at_nat @ M )
        = ( semiri1316708129612266289at_nat @ N ) )
      = ( M = N ) ) ).

% of_nat_eq_iff
thf(fact_40_of__nat__eq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ( semiri681578069525770553at_rat @ M )
        = ( semiri681578069525770553at_rat @ N ) )
      = ( M = N ) ) ).

% of_nat_eq_iff
thf(fact_41_of__nat__eq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ( semiri3763490453095760265atural @ M )
        = ( semiri3763490453095760265atural @ N ) )
      = ( M = N ) ) ).

% of_nat_eq_iff
thf(fact_42_of__nat__fact,axiom,
    ! [N: nat] :
      ( ( semiri1316708129612266289at_nat @ ( semiri1408675320244567234ct_nat @ N ) )
      = ( semiri1408675320244567234ct_nat @ N ) ) ).

% of_nat_fact
thf(fact_43_of__nat__fact,axiom,
    ! [N: nat] :
      ( ( semiri3763490453095760265atural @ ( semiri1408675320244567234ct_nat @ N ) )
      = ( semiri2447717529341329178atural @ N ) ) ).

% of_nat_fact
thf(fact_44_of__nat__fact,axiom,
    ! [N: nat] :
      ( ( semiri681578069525770553at_rat @ ( semiri1408675320244567234ct_nat @ N ) )
      = ( semiri773545260158071498ct_rat @ N ) ) ).

% of_nat_fact
thf(fact_45_of__nat__fact,axiom,
    ! [N: nat] :
      ( ( semiri4939895301339042750nteger @ ( semiri1408675320244567234ct_nat @ N ) )
      = ( semiri3624122377584611663nteger @ N ) ) ).

% of_nat_fact
thf(fact_46_of__nat__fact,axiom,
    ! [N: nat] :
      ( ( semiri1314217659103216013at_int @ ( semiri1408675320244567234ct_nat @ N ) )
      = ( semiri1406184849735516958ct_int @ N ) ) ).

% of_nat_fact
thf(fact_47_euclidean__size__of__nat,axiom,
    ! [N: nat] :
      ( ( euclid6377331345833325938nteger @ ( semiri4939895301339042750nteger @ N ) )
      = N ) ).

% euclidean_size_of_nat
thf(fact_48_euclidean__size__of__nat,axiom,
    ! [N: nat] :
      ( ( euclid4774559944035922753ze_int @ ( semiri1314217659103216013at_int @ N ) )
      = N ) ).

% euclidean_size_of_nat
thf(fact_49_euclidean__size__of__nat,axiom,
    ! [N: nat] :
      ( ( euclid4777050414544973029ze_nat @ ( semiri1316708129612266289at_nat @ N ) )
      = N ) ).

% euclidean_size_of_nat
thf(fact_50_euclidean__size__of__nat,axiom,
    ! [N: nat] :
      ( ( euclid5200926497590043453atural @ ( semiri3763490453095760265atural @ N ) )
      = N ) ).

% euclidean_size_of_nat
thf(fact_51_round__of__nat,axiom,
    ! [N: nat] :
      ( ( archim7778729529865785530nd_rat @ ( semiri681578069525770553at_rat @ N ) )
      = ( semiri1314217659103216013at_int @ N ) ) ).

% round_of_nat
thf(fact_52_one__integer_Orsp,axiom,
    one_one_int = one_one_int ).

% one_integer.rsp
thf(fact_53_int__ops_I2_J,axiom,
    ( ( semiri1314217659103216013at_int @ one_one_nat )
    = one_one_int ) ).

% int_ops(2)
thf(fact_54_effect__deterministic_I3_J,axiom,
    ! [F: heap_Time_Heap_nat,H2: heap_e7401611519738050253t_unit,H: heap_e7401611519738050253t_unit,A: nat,N: nat,H3: heap_e7401611519738050253t_unit,B: nat,N2: nat] :
      ( ( heap_Time_effect_nat @ F @ H2 @ H @ A @ N )
     => ( ( heap_Time_effect_nat @ F @ H2 @ H3 @ B @ N2 )
       => ( N = N2 ) ) ) ).

% effect_deterministic(3)
thf(fact_55_effect__deterministic_I2_J,axiom,
    ! [F: heap_Time_Heap_nat,H2: heap_e7401611519738050253t_unit,H: heap_e7401611519738050253t_unit,A: nat,N: nat,H3: heap_e7401611519738050253t_unit,B: nat,N2: nat] :
      ( ( heap_Time_effect_nat @ F @ H2 @ H @ A @ N )
     => ( ( heap_Time_effect_nat @ F @ H2 @ H3 @ B @ N2 )
       => ( H = H3 ) ) ) ).

% effect_deterministic(2)
thf(fact_56_effect__deterministic_I1_J,axiom,
    ! [F: heap_Time_Heap_nat,H2: heap_e7401611519738050253t_unit,H: heap_e7401611519738050253t_unit,A: nat,N: nat,H3: heap_e7401611519738050253t_unit,B: nat,N2: nat] :
      ( ( heap_Time_effect_nat @ F @ H2 @ H @ A @ N )
     => ( ( heap_Time_effect_nat @ F @ H2 @ H3 @ B @ N2 )
       => ( A = B ) ) ) ).

% effect_deterministic(1)
thf(fact_57_effect__ifE,axiom,
    ! [C: $o,T: heap_Time_Heap_nat,E: heap_Time_Heap_nat,H2: heap_e7401611519738050253t_unit,H: heap_e7401611519738050253t_unit,R: nat,N: nat] :
      ( ( heap_Time_effect_nat @ ( if_Hea2662716070787841314ap_nat @ C @ T @ E ) @ H2 @ H @ R @ N )
     => ( ( C
         => ~ ( heap_Time_effect_nat @ T @ H2 @ H @ R @ N ) )
       => ~ ( ~ C
           => ~ ( heap_Time_effect_nat @ E @ H2 @ H @ R @ N ) ) ) ) ).

% effect_ifE
thf(fact_58_effect__ifI,axiom,
    ! [C: $o,T: heap_Time_Heap_nat,H2: heap_e7401611519738050253t_unit,H: heap_e7401611519738050253t_unit,R: nat,N: nat,E: heap_Time_Heap_nat] :
      ( ( C
       => ( heap_Time_effect_nat @ T @ H2 @ H @ R @ N ) )
     => ( ( ~ C
         => ( heap_Time_effect_nat @ E @ H2 @ H @ R @ N ) )
       => ( heap_Time_effect_nat @ ( if_Hea2662716070787841314ap_nat @ C @ T @ E ) @ H2 @ H @ R @ N ) ) ) ).

% effect_ifI
thf(fact_59_division__segment__of__nat,axiom,
    ! [N: nat] :
      ( ( euclid3398187327856392827nt_nat @ ( semiri1316708129612266289at_nat @ N ) )
      = one_one_nat ) ).

% division_segment_of_nat
thf(fact_60_division__segment__of__nat,axiom,
    ! [N: nat] :
      ( ( euclid3395696857347342551nt_int @ ( semiri1314217659103216013at_int @ N ) )
      = one_one_int ) ).

% division_segment_of_nat
thf(fact_61_division__segment__of__nat,axiom,
    ! [N: nat] :
      ( ( euclid6289375185220004616nteger @ ( semiri4939895301339042750nteger @ N ) )
      = one_one_Code_integer ) ).

% division_segment_of_nat
thf(fact_62_division__segment__of__nat,axiom,
    ! [N: nat] :
      ( ( euclid5112970336976722131atural @ ( semiri3763490453095760265atural @ N ) )
      = one_one_Code_natural ) ).

% division_segment_of_nat
thf(fact_63_fact__0,axiom,
    ( ( semiri1408675320244567234ct_nat @ zero_zero_nat )
    = one_one_nat ) ).

% fact_0
thf(fact_64_fact__0,axiom,
    ( ( semiri2447717529341329178atural @ zero_zero_nat )
    = one_one_Code_natural ) ).

% fact_0
thf(fact_65_fact__0,axiom,
    ( ( semiri773545260158071498ct_rat @ zero_zero_nat )
    = one_one_rat ) ).

% fact_0
thf(fact_66_fact__0,axiom,
    ( ( semiri3624122377584611663nteger @ zero_zero_nat )
    = one_one_Code_integer ) ).

% fact_0
thf(fact_67_fact__0,axiom,
    ( ( semiri1406184849735516958ct_int @ zero_zero_nat )
    = one_one_int ) ).

% fact_0
thf(fact_68_effect__tapI,axiom,
    ! [H: heap_e7401611519738050253t_unit,H2: heap_e7401611519738050253t_unit,R: nat,F: heap_e7401611519738050253t_unit > nat] :
      ( ( H = H2 )
     => ( ( R
          = ( F @ H2 ) )
       => ( heap_Time_effect_nat @ ( heap_Time_tap_nat @ F ) @ H2 @ H @ R @ one_one_nat ) ) ) ).

% effect_tapI
thf(fact_69_effect__tapE,axiom,
    ! [F: heap_e7401611519738050253t_unit > nat,H2: heap_e7401611519738050253t_unit,H: heap_e7401611519738050253t_unit,R: nat,N: nat] :
      ( ( heap_Time_effect_nat @ ( heap_Time_tap_nat @ F ) @ H2 @ H @ R @ N )
     => ~ ( ( H = H2 )
         => ( ( R
              = ( F @ H2 ) )
           => ( N != one_one_nat ) ) ) ) ).

% effect_tapE
thf(fact_70_division__segment__1,axiom,
    ( ( euclid3398187327856392827nt_nat @ one_one_nat )
    = one_one_nat ) ).

% division_segment_1
thf(fact_71_division__segment__1,axiom,
    ( ( euclid3395696857347342551nt_int @ one_one_int )
    = one_one_int ) ).

% division_segment_1
thf(fact_72_division__segment__1,axiom,
    ( ( euclid6289375185220004616nteger @ one_one_Code_integer )
    = one_one_Code_integer ) ).

% division_segment_1
thf(fact_73_division__segment__1,axiom,
    ( ( euclid5112970336976722131atural @ one_one_Code_natural )
    = one_one_Code_natural ) ).

% division_segment_1
thf(fact_74_round__0,axiom,
    ( ( archim7778729529865785530nd_rat @ zero_zero_rat )
    = zero_zero_int ) ).

% round_0
thf(fact_75_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_76_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( ( semiri4939895301339042750nteger @ M )
        = zero_z3403309356797280102nteger )
      = ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
thf(fact_77_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_78_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_79_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( ( semiri3763490453095760265atural @ M )
        = zero_z2226904508553997617atural )
      = ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
thf(fact_80_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_81_of__nat__0__eq__iff,axiom,
    ! [N: nat] :
      ( ( zero_z3403309356797280102nteger
        = ( semiri4939895301339042750nteger @ N ) )
      = ( zero_zero_nat = N ) ) ).

% of_nat_0_eq_iff
thf(fact_82_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_83_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_84_of__nat__0__eq__iff,axiom,
    ! [N: nat] :
      ( ( zero_z2226904508553997617atural
        = ( semiri3763490453095760265atural @ N ) )
      = ( zero_zero_nat = N ) ) ).

% of_nat_0_eq_iff
thf(fact_85_of__nat__0,axiom,
    ( ( semiri1314217659103216013at_int @ zero_zero_nat )
    = zero_zero_int ) ).

% of_nat_0
thf(fact_86_of__nat__0,axiom,
    ( ( semiri4939895301339042750nteger @ zero_zero_nat )
    = zero_z3403309356797280102nteger ) ).

% of_nat_0
thf(fact_87_of__nat__0,axiom,
    ( ( semiri1316708129612266289at_nat @ zero_zero_nat )
    = zero_zero_nat ) ).

% of_nat_0
thf(fact_88_of__nat__0,axiom,
    ( ( semiri681578069525770553at_rat @ zero_zero_nat )
    = zero_zero_rat ) ).

% of_nat_0
thf(fact_89_of__nat__0,axiom,
    ( ( semiri3763490453095760265atural @ zero_zero_nat )
    = zero_z2226904508553997617atural ) ).

% of_nat_0
thf(fact_90_euclidean__size__eq__0__iff,axiom,
    ! [B: code_integer] :
      ( ( ( euclid6377331345833325938nteger @ B )
        = zero_zero_nat )
      = ( B = zero_z3403309356797280102nteger ) ) ).

% euclidean_size_eq_0_iff
thf(fact_91_euclidean__size__eq__0__iff,axiom,
    ! [B: int] :
      ( ( ( euclid4774559944035922753ze_int @ B )
        = zero_zero_nat )
      = ( B = zero_zero_int ) ) ).

% euclidean_size_eq_0_iff
thf(fact_92_euclidean__size__eq__0__iff,axiom,
    ! [B: nat] :
      ( ( ( euclid4777050414544973029ze_nat @ B )
        = zero_zero_nat )
      = ( B = zero_zero_nat ) ) ).

% euclidean_size_eq_0_iff
thf(fact_93_euclidean__size__eq__0__iff,axiom,
    ! [B: code_natural] :
      ( ( ( euclid5200926497590043453atural @ B )
        = zero_zero_nat )
      = ( B = zero_z2226904508553997617atural ) ) ).

% euclidean_size_eq_0_iff
thf(fact_94_size__0,axiom,
    ( ( euclid6377331345833325938nteger @ zero_z3403309356797280102nteger )
    = zero_zero_nat ) ).

% size_0
thf(fact_95_size__0,axiom,
    ( ( euclid4774559944035922753ze_int @ zero_zero_int )
    = zero_zero_nat ) ).

% size_0
thf(fact_96_size__0,axiom,
    ( ( euclid4777050414544973029ze_nat @ zero_zero_nat )
    = zero_zero_nat ) ).

% size_0
thf(fact_97_size__0,axiom,
    ( ( euclid5200926497590043453atural @ zero_z2226904508553997617atural )
    = zero_zero_nat ) ).

% size_0
thf(fact_98_nat__int__comparison_I1_J,axiom,
    ( ( ^ [Y: nat,Z: nat] : ( Y = Z ) )
    = ( ^ [A2: nat,B2: nat] :
          ( ( semiri1314217659103216013at_int @ A2 )
          = ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).

% nat_int_comparison(1)
thf(fact_99_zero__natural_Orsp,axiom,
    zero_zero_nat = zero_zero_nat ).

% zero_natural.rsp
thf(fact_100_division__segment__not__0,axiom,
    ! [A: nat] :
      ( ( euclid3398187327856392827nt_nat @ A )
     != zero_zero_nat ) ).

% division_segment_not_0
thf(fact_101_division__segment__not__0,axiom,
    ! [A: int] :
      ( ( euclid3395696857347342551nt_int @ A )
     != zero_zero_int ) ).

% division_segment_not_0
thf(fact_102_division__segment__not__0,axiom,
    ! [A: code_integer] :
      ( ( euclid6289375185220004616nteger @ A )
     != zero_z3403309356797280102nteger ) ).

% division_segment_not_0
thf(fact_103_division__segment__not__0,axiom,
    ! [A: code_natural] :
      ( ( euclid5112970336976722131atural @ A )
     != zero_z2226904508553997617atural ) ).

% division_segment_not_0
thf(fact_104_mem__Collect__eq,axiom,
    ! [A: product_prod_nat_nat,P: product_prod_nat_nat > $o] :
      ( ( member8440522571783428010at_nat @ A @ ( collec3392354462482085612at_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_105_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_106_mem__Collect__eq,axiom,
    ! [A: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
      ( ( member2643936169264416010at_nat @ A @ ( collec5514110066124741708at_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_107_mem__Collect__eq,axiom,
    ! [A: set_nat,P: set_nat > $o] :
      ( ( member_set_nat @ A @ ( collect_set_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_108_mem__Collect__eq,axiom,
    ! [A: nat,P: nat > $o] :
      ( ( member_nat @ A @ ( collect_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_109_mem__Collect__eq,axiom,
    ! [A: int,P: int > $o] :
      ( ( member_int @ A @ ( collect_int @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_110_Collect__mem__eq,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ( ( collec3392354462482085612at_nat
        @ ^ [X2: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X2 @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_111_Collect__mem__eq,axiom,
    ! [A3: set_list_nat] :
      ( ( collect_list_nat
        @ ^ [X2: list_nat] : ( member_list_nat @ X2 @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_112_Collect__mem__eq,axiom,
    ! [A3: set_se7855581050983116737at_nat] :
      ( ( collec5514110066124741708at_nat
        @ ^ [X2: set_Pr1261947904930325089at_nat] : ( member2643936169264416010at_nat @ X2 @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_113_Collect__mem__eq,axiom,
    ! [A3: set_set_nat] :
      ( ( collect_set_nat
        @ ^ [X2: set_nat] : ( member_set_nat @ X2 @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_114_Collect__mem__eq,axiom,
    ! [A3: set_nat] :
      ( ( collect_nat
        @ ^ [X2: nat] : ( member_nat @ X2 @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_115_Collect__mem__eq,axiom,
    ! [A3: set_int] :
      ( ( collect_int
        @ ^ [X2: int] : ( member_int @ X2 @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_116_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_117_Collect__cong,axiom,
    ! [P: set_Pr1261947904930325089at_nat > $o,Q: set_Pr1261947904930325089at_nat > $o] :
      ( ! [X3: set_Pr1261947904930325089at_nat] :
          ( ( P @ X3 )
          = ( Q @ X3 ) )
     => ( ( collec5514110066124741708at_nat @ P )
        = ( collec5514110066124741708at_nat @ Q ) ) ) ).

% Collect_cong
thf(fact_118_Collect__cong,axiom,
    ! [P: set_nat > $o,Q: set_nat > $o] :
      ( ! [X3: set_nat] :
          ( ( P @ X3 )
          = ( Q @ X3 ) )
     => ( ( collect_set_nat @ P )
        = ( collect_set_nat @ Q ) ) ) ).

% Collect_cong
thf(fact_119_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_120_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_121_int__if,axiom,
    ! [P: $o,A: nat,B: nat] :
      ( ( P
       => ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B ) )
          = ( semiri1314217659103216013at_int @ A ) ) )
      & ( ~ P
       => ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B ) )
          = ( semiri1314217659103216013at_int @ B ) ) ) ) ).

% int_if
thf(fact_122_zero__reorient,axiom,
    ! [X: nat] :
      ( ( zero_zero_nat = X )
      = ( X = zero_zero_nat ) ) ).

% zero_reorient
thf(fact_123_zero__reorient,axiom,
    ! [X: int] :
      ( ( zero_zero_int = X )
      = ( X = zero_zero_int ) ) ).

% zero_reorient
thf(fact_124_zero__reorient,axiom,
    ! [X: code_integer] :
      ( ( zero_z3403309356797280102nteger = X )
      = ( X = zero_z3403309356797280102nteger ) ) ).

% zero_reorient
thf(fact_125_zero__reorient,axiom,
    ! [X: rat] :
      ( ( zero_zero_rat = X )
      = ( X = zero_zero_rat ) ) ).

% zero_reorient
thf(fact_126_zero__reorient,axiom,
    ! [X: code_natural] :
      ( ( zero_z2226904508553997617atural = X )
      = ( X = zero_z2226904508553997617atural ) ) ).

% zero_reorient
thf(fact_127_fact__nonzero,axiom,
    ! [N: nat] :
      ( ( semiri1408675320244567234ct_nat @ N )
     != zero_zero_nat ) ).

% fact_nonzero
thf(fact_128_fact__nonzero,axiom,
    ! [N: nat] :
      ( ( semiri2447717529341329178atural @ N )
     != zero_z2226904508553997617atural ) ).

% fact_nonzero
thf(fact_129_fact__nonzero,axiom,
    ! [N: nat] :
      ( ( semiri773545260158071498ct_rat @ N )
     != zero_zero_rat ) ).

% fact_nonzero
thf(fact_130_fact__nonzero,axiom,
    ! [N: nat] :
      ( ( semiri3624122377584611663nteger @ N )
     != zero_z3403309356797280102nteger ) ).

% fact_nonzero
thf(fact_131_fact__nonzero,axiom,
    ! [N: nat] :
      ( ( semiri1406184849735516958ct_int @ N )
     != zero_zero_int ) ).

% fact_nonzero
thf(fact_132_division__segment__nat__def,axiom,
    ( euclid3398187327856392827nt_nat
    = ( ^ [N3: nat] : one_one_nat ) ) ).

% division_segment_nat_def
thf(fact_133_division__segment__eq__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( ( euclid3398187327856392827nt_nat @ A )
        = ( euclid3398187327856392827nt_nat @ B ) )
     => ( ( ( euclid4777050414544973029ze_nat @ A )
          = ( euclid4777050414544973029ze_nat @ B ) )
       => ( A = B ) ) ) ).

% division_segment_eq_iff
thf(fact_134_division__segment__eq__iff,axiom,
    ! [A: int,B: int] :
      ( ( ( euclid3395696857347342551nt_int @ A )
        = ( euclid3395696857347342551nt_int @ B ) )
     => ( ( ( euclid4774559944035922753ze_int @ A )
          = ( euclid4774559944035922753ze_int @ B ) )
       => ( A = B ) ) ) ).

% division_segment_eq_iff
thf(fact_135_division__segment__eq__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( euclid6289375185220004616nteger @ A )
        = ( euclid6289375185220004616nteger @ B ) )
     => ( ( ( euclid6377331345833325938nteger @ A )
          = ( euclid6377331345833325938nteger @ B ) )
       => ( A = B ) ) ) ).

% division_segment_eq_iff
thf(fact_136_division__segment__eq__iff,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( ( euclid5112970336976722131atural @ A )
        = ( euclid5112970336976722131atural @ B ) )
     => ( ( ( euclid5200926497590043453atural @ A )
          = ( euclid5200926497590043453atural @ B ) )
       => ( A = B ) ) ) ).

% division_segment_eq_iff
thf(fact_137_zero__neq__one,axiom,
    zero_zero_nat != one_one_nat ).

% zero_neq_one
thf(fact_138_zero__neq__one,axiom,
    zero_zero_int != one_one_int ).

% zero_neq_one
thf(fact_139_zero__neq__one,axiom,
    zero_z3403309356797280102nteger != one_one_Code_integer ).

% zero_neq_one
thf(fact_140_zero__neq__one,axiom,
    zero_zero_rat != one_one_rat ).

% zero_neq_one
thf(fact_141_zero__neq__one,axiom,
    zero_z2226904508553997617atural != one_one_Code_natural ).

% zero_neq_one
thf(fact_142_dbl__inc__simps_I2_J,axiom,
    ( ( neg_nu5851722552734809277nc_int @ zero_zero_int )
    = one_one_int ) ).

% dbl_inc_simps(2)
thf(fact_143_dbl__inc__simps_I2_J,axiom,
    ( ( neg_nu5831290666863070958nteger @ zero_z3403309356797280102nteger )
    = one_one_Code_integer ) ).

% dbl_inc_simps(2)
thf(fact_144_dbl__inc__simps_I2_J,axiom,
    ( ( neg_nu5219082963157363817nc_rat @ zero_zero_rat )
    = one_one_rat ) ).

% dbl_inc_simps(2)
thf(fact_145_effect__ureturnI,axiom,
    ! [H2: heap_e7401611519738050253t_unit,H: heap_e7401611519738050253t_unit,X: nat] :
      ( ( H2 = H )
     => ( heap_Time_effect_nat @ ( heap_T6738714419362814988rn_nat @ X ) @ H2 @ H @ X @ zero_zero_nat ) ) ).

% effect_ureturnI
thf(fact_146_effect__ureturnE,axiom,
    ! [X: nat,H2: heap_e7401611519738050253t_unit,H: heap_e7401611519738050253t_unit,R: nat,N: nat] :
      ( ( heap_Time_effect_nat @ ( heap_T6738714419362814988rn_nat @ X ) @ H2 @ H @ R @ N )
     => ~ ( ( R = X )
         => ( ( H = H2 )
           => ( N != zero_zero_nat ) ) ) ) ).

% effect_ureturnE
thf(fact_147_effect__assertE,axiom,
    ! [P: nat > $o,X: nat,H2: heap_e7401611519738050253t_unit,H: heap_e7401611519738050253t_unit,R: nat,N: nat] :
      ( ( heap_Time_effect_nat @ ( heap_Time_assert_nat @ P @ X ) @ H2 @ H @ R @ N )
     => ~ ( ( P @ X )
         => ( ( R = X )
           => ( ( H = H2 )
             => ( N != one_one_nat ) ) ) ) ) ).

% effect_assertE
thf(fact_148_effect__assertI,axiom,
    ! [P: nat > $o,X: nat,H: heap_e7401611519738050253t_unit,H2: heap_e7401611519738050253t_unit,R: nat,N: nat] :
      ( ( P @ X )
     => ( ( H = H2 )
       => ( ( R = X )
         => ( ( N = one_one_nat )
           => ( heap_Time_effect_nat @ ( heap_Time_assert_nat @ P @ X ) @ H2 @ H @ R @ N ) ) ) ) ) ).

% effect_assertI
thf(fact_149_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu3811975205180677377ec_int @ zero_zero_int )
    = ( uminus_uminus_int @ one_one_int ) ) ).

% dbl_dec_simps(2)
thf(fact_150_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu7757733837767384882nteger @ zero_z3403309356797280102nteger )
    = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% dbl_dec_simps(2)
thf(fact_151_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu3179335615603231917ec_rat @ zero_zero_rat )
    = ( uminus_uminus_rat @ one_one_rat ) ) ).

% dbl_dec_simps(2)
thf(fact_152_division__segment__euclidean__size,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ ( euclid3398187327856392827nt_nat @ A ) @ ( semiri1316708129612266289at_nat @ ( euclid4777050414544973029ze_nat @ A ) ) )
      = A ) ).

% division_segment_euclidean_size
thf(fact_153_division__segment__euclidean__size,axiom,
    ! [A: int] :
      ( ( times_times_int @ ( euclid3395696857347342551nt_int @ A ) @ ( semiri1314217659103216013at_int @ ( euclid4774559944035922753ze_int @ A ) ) )
      = A ) ).

% division_segment_euclidean_size
thf(fact_154_division__segment__euclidean__size,axiom,
    ! [A: code_integer] :
      ( ( times_3573771949741848930nteger @ ( euclid6289375185220004616nteger @ A ) @ ( semiri4939895301339042750nteger @ ( euclid6377331345833325938nteger @ A ) ) )
      = A ) ).

% division_segment_euclidean_size
thf(fact_155_division__segment__euclidean__size,axiom,
    ! [A: code_natural] :
      ( ( times_2397367101498566445atural @ ( euclid5112970336976722131atural @ A ) @ ( semiri3763490453095760265atural @ ( euclid5200926497590043453atural @ A ) ) )
      = A ) ).

% division_segment_euclidean_size
thf(fact_156_euclidean__size__greater__0__iff,axiom,
    ! [B: code_integer] :
      ( ( ord_less_nat @ zero_zero_nat @ ( euclid6377331345833325938nteger @ B ) )
      = ( B != zero_z3403309356797280102nteger ) ) ).

% euclidean_size_greater_0_iff
thf(fact_157_euclidean__size__greater__0__iff,axiom,
    ! [B: int] :
      ( ( ord_less_nat @ zero_zero_nat @ ( euclid4774559944035922753ze_int @ B ) )
      = ( B != zero_zero_int ) ) ).

% euclidean_size_greater_0_iff
thf(fact_158_euclidean__size__greater__0__iff,axiom,
    ! [B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( euclid4777050414544973029ze_nat @ B ) )
      = ( B != zero_zero_nat ) ) ).

% euclidean_size_greater_0_iff
thf(fact_159_euclidean__size__greater__0__iff,axiom,
    ! [B: code_natural] :
      ( ( ord_less_nat @ zero_zero_nat @ ( euclid5200926497590043453atural @ B ) )
      = ( B != zero_z2226904508553997617atural ) ) ).

% euclidean_size_greater_0_iff
thf(fact_160_fact__Suc__0,axiom,
    ( ( semiri1408675320244567234ct_nat @ ( suc @ zero_zero_nat ) )
    = one_one_nat ) ).

% fact_Suc_0
thf(fact_161_fact__Suc__0,axiom,
    ( ( semiri2447717529341329178atural @ ( suc @ zero_zero_nat ) )
    = one_one_Code_natural ) ).

% fact_Suc_0
thf(fact_162_fact__Suc__0,axiom,
    ( ( semiri773545260158071498ct_rat @ ( suc @ zero_zero_nat ) )
    = one_one_rat ) ).

% fact_Suc_0
thf(fact_163_fact__Suc__0,axiom,
    ( ( semiri3624122377584611663nteger @ ( suc @ zero_zero_nat ) )
    = one_one_Code_integer ) ).

% fact_Suc_0
thf(fact_164_fact__Suc__0,axiom,
    ( ( semiri1406184849735516958ct_int @ ( suc @ zero_zero_nat ) )
    = one_one_int ) ).

% fact_Suc_0
thf(fact_165_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( ( ord_le3102999989581377725nteger @ ( semiri4939895301339042750nteger @ M ) @ zero_z3403309356797280102nteger )
      = ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
thf(fact_166_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( ( ord_le1926595141338095240atural @ ( semiri3763490453095760265atural @ M ) @ zero_z2226904508553997617atural )
      = ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
thf(fact_167_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_168_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_169_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_170_effect__returnE,axiom,
    ! [X: nat,H2: heap_e7401611519738050253t_unit,H: heap_e7401611519738050253t_unit,R: nat,N: nat] :
      ( ( heap_Time_effect_nat @ ( heap_Time_return_nat @ X ) @ H2 @ H @ R @ N )
     => ~ ( ( R = X )
         => ( ( H = H2 )
           => ( N != one_one_nat ) ) ) ) ).

% effect_returnE
thf(fact_171_verit__minus__simplify_I4_J,axiom,
    ! [B: int] :
      ( ( uminus_uminus_int @ ( uminus_uminus_int @ B ) )
      = B ) ).

% verit_minus_simplify(4)
thf(fact_172_verit__minus__simplify_I4_J,axiom,
    ! [B: code_integer] :
      ( ( uminus1351360451143612070nteger @ ( uminus1351360451143612070nteger @ B ) )
      = B ) ).

% verit_minus_simplify(4)
thf(fact_173_verit__minus__simplify_I4_J,axiom,
    ! [B: rat] :
      ( ( uminus_uminus_rat @ ( uminus_uminus_rat @ B ) )
      = B ) ).

% verit_minus_simplify(4)
thf(fact_174_neg__equal__iff__equal,axiom,
    ! [A: int,B: int] :
      ( ( ( uminus_uminus_int @ A )
        = ( uminus_uminus_int @ B ) )
      = ( A = B ) ) ).

% neg_equal_iff_equal
thf(fact_175_neg__equal__iff__equal,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( uminus1351360451143612070nteger @ A )
        = ( uminus1351360451143612070nteger @ B ) )
      = ( A = B ) ) ).

% neg_equal_iff_equal
thf(fact_176_neg__equal__iff__equal,axiom,
    ! [A: rat,B: rat] :
      ( ( ( uminus_uminus_rat @ A )
        = ( uminus_uminus_rat @ B ) )
      = ( A = B ) ) ).

% neg_equal_iff_equal
thf(fact_177_add_Oinverse__inverse,axiom,
    ! [A: int] :
      ( ( uminus_uminus_int @ ( uminus_uminus_int @ A ) )
      = A ) ).

% add.inverse_inverse
thf(fact_178_add_Oinverse__inverse,axiom,
    ! [A: code_integer] :
      ( ( uminus1351360451143612070nteger @ ( uminus1351360451143612070nteger @ A ) )
      = A ) ).

% add.inverse_inverse
thf(fact_179_add_Oinverse__inverse,axiom,
    ! [A: rat] :
      ( ( uminus_uminus_rat @ ( uminus_uminus_rat @ A ) )
      = A ) ).

% add.inverse_inverse
thf(fact_180_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_181_old_Onat_Oinject,axiom,
    ! [Nat: nat,Nat2: nat] :
      ( ( ( suc @ Nat )
        = ( suc @ Nat2 ) )
      = ( Nat = Nat2 ) ) ).

% old.nat.inject
thf(fact_182_nat_Oinject,axiom,
    ! [X22: nat,Y2: nat] :
      ( ( ( suc @ X22 )
        = ( suc @ Y2 ) )
      = ( X22 = Y2 ) ) ).

% nat.inject
thf(fact_183_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_184_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_185_mult__0__right,axiom,
    ! [M: nat] :
      ( ( times_times_nat @ M @ zero_zero_nat )
      = zero_zero_nat ) ).

% mult_0_right
thf(fact_186_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_187_le0,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).

% le0
thf(fact_188_bot__nat__0_Oextremum,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).

% bot_nat_0.extremum
thf(fact_189_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_190_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_191_le__zero__eq,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ N @ zero_zero_nat )
      = ( N = zero_zero_nat ) ) ).

% le_zero_eq
thf(fact_192_not__gr__zero,axiom,
    ! [N: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
      = ( N = zero_zero_nat ) ) ).

% not_gr_zero
thf(fact_193_mult__zero__left,axiom,
    ! [A: code_natural] :
      ( ( times_2397367101498566445atural @ zero_z2226904508553997617atural @ A )
      = zero_z2226904508553997617atural ) ).

% mult_zero_left
thf(fact_194_mult__zero__left,axiom,
    ! [A: code_integer] :
      ( ( times_3573771949741848930nteger @ zero_z3403309356797280102nteger @ A )
      = zero_z3403309356797280102nteger ) ).

% mult_zero_left
thf(fact_195_mult__zero__left,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ zero_zero_rat @ A )
      = zero_zero_rat ) ).

% mult_zero_left
thf(fact_196_mult__zero__left,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% mult_zero_left
thf(fact_197_mult__zero__left,axiom,
    ! [A: int] :
      ( ( times_times_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% mult_zero_left
thf(fact_198_mult__zero__right,axiom,
    ! [A: code_natural] :
      ( ( times_2397367101498566445atural @ A @ zero_z2226904508553997617atural )
      = zero_z2226904508553997617atural ) ).

% mult_zero_right
thf(fact_199_mult__zero__right,axiom,
    ! [A: code_integer] :
      ( ( times_3573771949741848930nteger @ A @ zero_z3403309356797280102nteger )
      = zero_z3403309356797280102nteger ) ).

% mult_zero_right
thf(fact_200_mult__zero__right,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ A @ zero_zero_rat )
      = zero_zero_rat ) ).

% mult_zero_right
thf(fact_201_mult__zero__right,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ zero_zero_nat )
      = zero_zero_nat ) ).

% mult_zero_right
thf(fact_202_mult__zero__right,axiom,
    ! [A: int] :
      ( ( times_times_int @ A @ zero_zero_int )
      = zero_zero_int ) ).

% mult_zero_right
thf(fact_203_mult__eq__0__iff,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( ( times_2397367101498566445atural @ A @ B )
        = zero_z2226904508553997617atural )
      = ( ( A = zero_z2226904508553997617atural )
        | ( B = zero_z2226904508553997617atural ) ) ) ).

% mult_eq_0_iff
thf(fact_204_mult__eq__0__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( times_3573771949741848930nteger @ A @ B )
        = zero_z3403309356797280102nteger )
      = ( ( A = zero_z3403309356797280102nteger )
        | ( B = zero_z3403309356797280102nteger ) ) ) ).

% mult_eq_0_iff
thf(fact_205_mult__eq__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ( times_times_rat @ A @ B )
        = zero_zero_rat )
      = ( ( A = zero_zero_rat )
        | ( B = zero_zero_rat ) ) ) ).

% mult_eq_0_iff
thf(fact_206_mult__eq__0__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( ( times_times_nat @ A @ B )
        = zero_zero_nat )
      = ( ( A = zero_zero_nat )
        | ( B = zero_zero_nat ) ) ) ).

% mult_eq_0_iff
thf(fact_207_mult__eq__0__iff,axiom,
    ! [A: int,B: int] :
      ( ( ( times_times_int @ A @ B )
        = zero_zero_int )
      = ( ( A = zero_zero_int )
        | ( B = zero_zero_int ) ) ) ).

% mult_eq_0_iff
thf(fact_208_mult__cancel__left,axiom,
    ! [C: code_natural,A: code_natural,B: code_natural] :
      ( ( ( times_2397367101498566445atural @ C @ A )
        = ( times_2397367101498566445atural @ C @ B ) )
      = ( ( C = zero_z2226904508553997617atural )
        | ( A = B ) ) ) ).

% mult_cancel_left
thf(fact_209_mult__cancel__left,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( ( times_3573771949741848930nteger @ C @ A )
        = ( times_3573771949741848930nteger @ C @ B ) )
      = ( ( C = zero_z3403309356797280102nteger )
        | ( A = B ) ) ) ).

% mult_cancel_left
thf(fact_210_mult__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ( times_times_rat @ C @ A )
        = ( times_times_rat @ C @ B ) )
      = ( ( C = zero_zero_rat )
        | ( A = B ) ) ) ).

% mult_cancel_left
thf(fact_211_mult__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ( times_times_nat @ C @ A )
        = ( times_times_nat @ C @ B ) )
      = ( ( C = zero_zero_nat )
        | ( A = B ) ) ) ).

% mult_cancel_left
thf(fact_212_mult__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ( times_times_int @ C @ A )
        = ( times_times_int @ C @ B ) )
      = ( ( C = zero_zero_int )
        | ( A = B ) ) ) ).

% mult_cancel_left
thf(fact_213_mult__cancel__right,axiom,
    ! [A: code_natural,C: code_natural,B: code_natural] :
      ( ( ( times_2397367101498566445atural @ A @ C )
        = ( times_2397367101498566445atural @ B @ C ) )
      = ( ( C = zero_z2226904508553997617atural )
        | ( A = B ) ) ) ).

% mult_cancel_right
thf(fact_214_mult__cancel__right,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( ( times_3573771949741848930nteger @ A @ C )
        = ( times_3573771949741848930nteger @ B @ C ) )
      = ( ( C = zero_z3403309356797280102nteger )
        | ( A = B ) ) ) ).

% mult_cancel_right
thf(fact_215_mult__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ( times_times_rat @ A @ C )
        = ( times_times_rat @ B @ C ) )
      = ( ( C = zero_zero_rat )
        | ( A = B ) ) ) ).

% mult_cancel_right
thf(fact_216_mult__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ( times_times_nat @ A @ C )
        = ( times_times_nat @ B @ C ) )
      = ( ( C = zero_zero_nat )
        | ( A = B ) ) ) ).

% mult_cancel_right
thf(fact_217_mult__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ( times_times_int @ A @ C )
        = ( times_times_int @ B @ C ) )
      = ( ( C = zero_zero_int )
        | ( A = B ) ) ) ).

% mult_cancel_right
thf(fact_218_neg__le__iff__le,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) )
      = ( ord_le3102999989581377725nteger @ A @ B ) ) ).

% neg_le_iff_le
thf(fact_219_neg__le__iff__le,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) )
      = ( ord_less_eq_rat @ A @ B ) ) ).

% neg_le_iff_le
thf(fact_220_neg__le__iff__le,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
      = ( ord_less_eq_int @ A @ B ) ) ).

% neg_le_iff_le
thf(fact_221_neg__equal__zero,axiom,
    ! [A: int] :
      ( ( ( uminus_uminus_int @ A )
        = A )
      = ( A = zero_zero_int ) ) ).

% neg_equal_zero
thf(fact_222_neg__equal__zero,axiom,
    ! [A: code_integer] :
      ( ( ( uminus1351360451143612070nteger @ A )
        = A )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% neg_equal_zero
thf(fact_223_neg__equal__zero,axiom,
    ! [A: rat] :
      ( ( ( uminus_uminus_rat @ A )
        = A )
      = ( A = zero_zero_rat ) ) ).

% neg_equal_zero
thf(fact_224_equal__neg__zero,axiom,
    ! [A: int] :
      ( ( A
        = ( uminus_uminus_int @ A ) )
      = ( A = zero_zero_int ) ) ).

% equal_neg_zero
thf(fact_225_equal__neg__zero,axiom,
    ! [A: code_integer] :
      ( ( A
        = ( uminus1351360451143612070nteger @ A ) )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% equal_neg_zero
thf(fact_226_equal__neg__zero,axiom,
    ! [A: rat] :
      ( ( A
        = ( uminus_uminus_rat @ A ) )
      = ( A = zero_zero_rat ) ) ).

% equal_neg_zero
thf(fact_227_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_228_neg__equal__0__iff__equal,axiom,
    ! [A: code_integer] :
      ( ( ( uminus1351360451143612070nteger @ A )
        = zero_z3403309356797280102nteger )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% neg_equal_0_iff_equal
thf(fact_229_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_230_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_231_neg__0__equal__iff__equal,axiom,
    ! [A: code_integer] :
      ( ( zero_z3403309356797280102nteger
        = ( uminus1351360451143612070nteger @ A ) )
      = ( zero_z3403309356797280102nteger = A ) ) ).

% neg_0_equal_iff_equal
thf(fact_232_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_233_add_Oinverse__neutral,axiom,
    ( ( uminus_uminus_int @ zero_zero_int )
    = zero_zero_int ) ).

% add.inverse_neutral
thf(fact_234_add_Oinverse__neutral,axiom,
    ( ( uminus1351360451143612070nteger @ zero_z3403309356797280102nteger )
    = zero_z3403309356797280102nteger ) ).

% add.inverse_neutral
thf(fact_235_add_Oinverse__neutral,axiom,
    ( ( uminus_uminus_rat @ zero_zero_rat )
    = zero_zero_rat ) ).

% add.inverse_neutral
thf(fact_236_mult__1,axiom,
    ! [A: code_natural] :
      ( ( times_2397367101498566445atural @ one_one_Code_natural @ A )
      = A ) ).

% mult_1
thf(fact_237_mult__1,axiom,
    ! [A: code_integer] :
      ( ( times_3573771949741848930nteger @ one_one_Code_integer @ A )
      = A ) ).

% mult_1
thf(fact_238_mult__1,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ one_one_rat @ A )
      = A ) ).

% mult_1
thf(fact_239_mult__1,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ one_one_nat @ A )
      = A ) ).

% mult_1
thf(fact_240_mult__1,axiom,
    ! [A: int] :
      ( ( times_times_int @ one_one_int @ A )
      = A ) ).

% mult_1
thf(fact_241_mult_Oright__neutral,axiom,
    ! [A: code_natural] :
      ( ( times_2397367101498566445atural @ A @ one_one_Code_natural )
      = A ) ).

% mult.right_neutral
thf(fact_242_mult_Oright__neutral,axiom,
    ! [A: code_integer] :
      ( ( times_3573771949741848930nteger @ A @ one_one_Code_integer )
      = A ) ).

% mult.right_neutral
thf(fact_243_mult_Oright__neutral,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ A @ one_one_rat )
      = A ) ).

% mult.right_neutral
thf(fact_244_mult_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ one_one_nat )
      = A ) ).

% mult.right_neutral
thf(fact_245_mult_Oright__neutral,axiom,
    ! [A: int] :
      ( ( times_times_int @ A @ one_one_int )
      = A ) ).

% mult.right_neutral
thf(fact_246_neg__less__iff__less,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
      = ( ord_less_int @ A @ B ) ) ).

% neg_less_iff_less
thf(fact_247_neg__less__iff__less,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) )
      = ( ord_le6747313008572928689nteger @ A @ B ) ) ).

% neg_less_iff_less
thf(fact_248_neg__less__iff__less,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) )
      = ( ord_less_rat @ A @ B ) ) ).

% neg_less_iff_less
thf(fact_249_mult__minus__left,axiom,
    ! [A: int,B: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ A ) @ B )
      = ( uminus_uminus_int @ ( times_times_int @ A @ B ) ) ) ).

% mult_minus_left
thf(fact_250_mult__minus__left,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
      = ( uminus1351360451143612070nteger @ ( times_3573771949741848930nteger @ A @ B ) ) ) ).

% mult_minus_left
thf(fact_251_mult__minus__left,axiom,
    ! [A: rat,B: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ A ) @ B )
      = ( uminus_uminus_rat @ ( times_times_rat @ A @ B ) ) ) ).

% mult_minus_left
thf(fact_252_minus__mult__minus,axiom,
    ! [A: int,B: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
      = ( times_times_int @ A @ B ) ) ).

% minus_mult_minus
thf(fact_253_minus__mult__minus,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) )
      = ( times_3573771949741848930nteger @ A @ B ) ) ).

% minus_mult_minus
thf(fact_254_minus__mult__minus,axiom,
    ! [A: rat,B: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) )
      = ( times_times_rat @ A @ B ) ) ).

% minus_mult_minus
thf(fact_255_mult__minus__right,axiom,
    ! [A: int,B: int] :
      ( ( times_times_int @ A @ ( uminus_uminus_int @ B ) )
      = ( uminus_uminus_int @ ( times_times_int @ A @ B ) ) ) ).

% mult_minus_right
thf(fact_256_mult__minus__right,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( times_3573771949741848930nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
      = ( uminus1351360451143612070nteger @ ( times_3573771949741848930nteger @ A @ B ) ) ) ).

% mult_minus_right
thf(fact_257_mult__minus__right,axiom,
    ! [A: rat,B: rat] :
      ( ( times_times_rat @ A @ ( uminus_uminus_rat @ B ) )
      = ( uminus_uminus_rat @ ( times_times_rat @ A @ B ) ) ) ).

% mult_minus_right
thf(fact_258_of__nat__le__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_le3102999989581377725nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% of_nat_le_iff
thf(fact_259_of__nat__le__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_le1926595141338095240atural @ ( semiri3763490453095760265atural @ M ) @ ( semiri3763490453095760265atural @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% of_nat_le_iff
thf(fact_260_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_261_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_262_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_263_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_264_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_265_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_266_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_267_of__nat__mult,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri4939895301339042750nteger @ ( times_times_nat @ M @ N ) )
      = ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) ) ) ).

% of_nat_mult
thf(fact_268_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_269_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_270_of__nat__mult,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri3763490453095760265atural @ ( times_times_nat @ M @ N ) )
      = ( times_2397367101498566445atural @ ( semiri3763490453095760265atural @ M ) @ ( semiri3763490453095760265atural @ N ) ) ) ).

% of_nat_mult
thf(fact_271_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_272_Suc__mono,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).

% Suc_mono
thf(fact_273_lessI,axiom,
    ! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).

% lessI
thf(fact_274_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_275_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_276_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_277_less__nat__zero__code,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% less_nat_zero_code
thf(fact_278_neq0__conv,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
      = ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% neq0_conv
thf(fact_279_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_280_neg__less__eq__nonneg,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
      = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).

% neg_less_eq_nonneg
thf(fact_281_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_282_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_283_less__eq__neg__nonpos,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
      = ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) ) ).

% less_eq_neg_nonpos
thf(fact_284_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_285_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_286_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_287_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_288_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_289_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_290_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_291_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_292_mult__cancel__right2,axiom,
    ! [A: code_integer,C: code_integer] :
      ( ( ( times_3573771949741848930nteger @ A @ C )
        = C )
      = ( ( C = zero_z3403309356797280102nteger )
        | ( A = one_one_Code_integer ) ) ) ).

% mult_cancel_right2
thf(fact_293_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_294_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_295_mult__cancel__right1,axiom,
    ! [C: code_integer,B: code_integer] :
      ( ( C
        = ( times_3573771949741848930nteger @ B @ C ) )
      = ( ( C = zero_z3403309356797280102nteger )
        | ( B = one_one_Code_integer ) ) ) ).

% mult_cancel_right1
thf(fact_296_mult__cancel__right1,axiom,
    ! [C: rat,B: rat] :
      ( ( C
        = ( times_times_rat @ B @ C ) )
      = ( ( C = zero_zero_rat )
        | ( B = one_one_rat ) ) ) ).

% mult_cancel_right1
thf(fact_297_mult__cancel__right1,axiom,
    ! [C: int,B: int] :
      ( ( C
        = ( times_times_int @ B @ C ) )
      = ( ( C = zero_zero_int )
        | ( B = one_one_int ) ) ) ).

% mult_cancel_right1
thf(fact_298_mult__cancel__left2,axiom,
    ! [C: code_integer,A: code_integer] :
      ( ( ( times_3573771949741848930nteger @ C @ A )
        = C )
      = ( ( C = zero_z3403309356797280102nteger )
        | ( A = one_one_Code_integer ) ) ) ).

% mult_cancel_left2
thf(fact_299_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_300_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_301_mult__cancel__left1,axiom,
    ! [C: code_integer,B: code_integer] :
      ( ( C
        = ( times_3573771949741848930nteger @ C @ B ) )
      = ( ( C = zero_z3403309356797280102nteger )
        | ( B = one_one_Code_integer ) ) ) ).

% mult_cancel_left1
thf(fact_302_mult__cancel__left1,axiom,
    ! [C: rat,B: rat] :
      ( ( C
        = ( times_times_rat @ C @ B ) )
      = ( ( C = zero_zero_rat )
        | ( B = one_one_rat ) ) ) ).

% mult_cancel_left1
thf(fact_303_mult__cancel__left1,axiom,
    ! [C: int,B: int] :
      ( ( C
        = ( times_times_int @ C @ B ) )
      = ( ( C = zero_zero_int )
        | ( B = one_one_int ) ) ) ).

% mult_cancel_left1
thf(fact_304_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_305_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_306_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_307_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_308_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_309_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_310_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_311_neg__less__pos,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
      = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).

% neg_less_pos
thf(fact_312_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_313_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_314_less__neg__neg,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
      = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).

% less_neg_neg
thf(fact_315_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_316_mult__minus1,axiom,
    ! [Z2: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ one_one_int ) @ Z2 )
      = ( uminus_uminus_int @ Z2 ) ) ).

% mult_minus1
thf(fact_317_mult__minus1,axiom,
    ! [Z2: code_integer] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ Z2 )
      = ( uminus1351360451143612070nteger @ Z2 ) ) ).

% mult_minus1
thf(fact_318_mult__minus1,axiom,
    ! [Z2: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ one_one_rat ) @ Z2 )
      = ( uminus_uminus_rat @ Z2 ) ) ).

% mult_minus1
thf(fact_319_mult__minus1__right,axiom,
    ! [Z2: int] :
      ( ( times_times_int @ Z2 @ ( uminus_uminus_int @ one_one_int ) )
      = ( uminus_uminus_int @ Z2 ) ) ).

% mult_minus1_right
thf(fact_320_mult__minus1__right,axiom,
    ! [Z2: code_integer] :
      ( ( times_3573771949741848930nteger @ Z2 @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( uminus1351360451143612070nteger @ Z2 ) ) ).

% mult_minus1_right
thf(fact_321_mult__minus1__right,axiom,
    ! [Z2: rat] :
      ( ( times_times_rat @ Z2 @ ( uminus_uminus_rat @ one_one_rat ) )
      = ( uminus_uminus_rat @ Z2 ) ) ).

% mult_minus1_right
thf(fact_322_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_323_of__nat__less__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_le6747313008572928689nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% of_nat_less_iff
thf(fact_324_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_325_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_326_of__nat__less__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_le5570908160329646204atural @ ( semiri3763490453095760265atural @ M ) @ ( semiri3763490453095760265atural @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% of_nat_less_iff
thf(fact_327_zero__less__Suc,axiom,
    ! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).

% zero_less_Suc
thf(fact_328_less__Suc0,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
      = ( N = zero_zero_nat ) ) ).

% less_Suc0
thf(fact_329_less__one,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ N @ one_one_nat )
      = ( N = zero_zero_nat ) ) ).

% less_one
thf(fact_330_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_331_dbl__inc__simps_I4_J,axiom,
    ( ( neg_nu5831290666863070958nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% dbl_inc_simps(4)
thf(fact_332_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_333_fact__Suc,axiom,
    ! [N: nat] :
      ( ( semiri1408675320244567234ct_nat @ ( suc @ N ) )
      = ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( suc @ N ) ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).

% fact_Suc
thf(fact_334_fact__Suc,axiom,
    ! [N: nat] :
      ( ( semiri2447717529341329178atural @ ( suc @ N ) )
      = ( times_2397367101498566445atural @ ( semiri3763490453095760265atural @ ( suc @ N ) ) @ ( semiri2447717529341329178atural @ N ) ) ) ).

% fact_Suc
thf(fact_335_fact__Suc,axiom,
    ! [N: nat] :
      ( ( semiri773545260158071498ct_rat @ ( suc @ N ) )
      = ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ N ) ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).

% fact_Suc
thf(fact_336_fact__Suc,axiom,
    ! [N: nat] :
      ( ( semiri3624122377584611663nteger @ ( suc @ N ) )
      = ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ ( suc @ N ) ) @ ( semiri3624122377584611663nteger @ N ) ) ) ).

% fact_Suc
thf(fact_337_fact__Suc,axiom,
    ! [N: nat] :
      ( ( semiri1406184849735516958ct_int @ ( suc @ N ) )
      = ( times_times_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ).

% fact_Suc
thf(fact_338_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_339_of__nat__0__less__iff,axiom,
    ! [N: nat] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( semiri4939895301339042750nteger @ N ) )
      = ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% of_nat_0_less_iff
thf(fact_340_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_341_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_342_of__nat__0__less__iff,axiom,
    ! [N: nat] :
      ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ ( semiri3763490453095760265atural @ N ) )
      = ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% of_nat_0_less_iff
thf(fact_343_zero__integer_Orsp,axiom,
    zero_zero_int = zero_zero_int ).

% zero_integer.rsp
thf(fact_344_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_345_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_346_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_347_ex__least__nat__less,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ N )
     => ( ~ ( P @ zero_zero_nat )
       => ? [K2: nat] :
            ( ( ord_less_nat @ K2 @ N )
            & ! [I: nat] :
                ( ( ord_less_eq_nat @ I @ K2 )
               => ~ ( P @ I ) )
            & ( P @ ( suc @ K2 ) ) ) ) ) ).

% ex_least_nat_less
thf(fact_348_mult__le__cancel__iff1,axiom,
    ! [Z2: code_integer,X: code_integer,Y3: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ Z2 )
     => ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ X @ Z2 ) @ ( times_3573771949741848930nteger @ Y3 @ Z2 ) )
        = ( ord_le3102999989581377725nteger @ X @ Y3 ) ) ) ).

% mult_le_cancel_iff1
thf(fact_349_mult__le__cancel__iff1,axiom,
    ! [Z2: rat,X: rat,Y3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Z2 )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ X @ Z2 ) @ ( times_times_rat @ Y3 @ Z2 ) )
        = ( ord_less_eq_rat @ X @ Y3 ) ) ) ).

% mult_le_cancel_iff1
thf(fact_350_mult__le__cancel__iff1,axiom,
    ! [Z2: int,X: int,Y3: int] :
      ( ( ord_less_int @ zero_zero_int @ Z2 )
     => ( ( ord_less_eq_int @ ( times_times_int @ X @ Z2 ) @ ( times_times_int @ Y3 @ Z2 ) )
        = ( ord_less_eq_int @ X @ Y3 ) ) ) ).

% mult_le_cancel_iff1
thf(fact_351_mult__le__cancel__iff2,axiom,
    ! [Z2: code_integer,X: code_integer,Y3: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ Z2 )
     => ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ Z2 @ X ) @ ( times_3573771949741848930nteger @ Z2 @ Y3 ) )
        = ( ord_le3102999989581377725nteger @ X @ Y3 ) ) ) ).

% mult_le_cancel_iff2
thf(fact_352_mult__le__cancel__iff2,axiom,
    ! [Z2: rat,X: rat,Y3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Z2 )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ Z2 @ X ) @ ( times_times_rat @ Z2 @ Y3 ) )
        = ( ord_less_eq_rat @ X @ Y3 ) ) ) ).

% mult_le_cancel_iff2
thf(fact_353_mult__le__cancel__iff2,axiom,
    ! [Z2: int,X: int,Y3: int] :
      ( ( ord_less_int @ zero_zero_int @ Z2 )
     => ( ( ord_less_eq_int @ ( times_times_int @ Z2 @ X ) @ ( times_times_int @ Z2 @ Y3 ) )
        = ( ord_less_eq_int @ X @ Y3 ) ) ) ).

% mult_le_cancel_iff2
thf(fact_354_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_355_verit__la__disequality,axiom,
    ! [A: rat,B: rat] :
      ( ( A = B )
      | ~ ( ord_less_eq_rat @ A @ B )
      | ~ ( ord_less_eq_rat @ B @ A ) ) ).

% verit_la_disequality
thf(fact_356_verit__la__disequality,axiom,
    ! [A: num,B: num] :
      ( ( A = B )
      | ~ ( ord_less_eq_num @ A @ B )
      | ~ ( ord_less_eq_num @ B @ A ) ) ).

% verit_la_disequality
thf(fact_357_verit__la__disequality,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
      | ~ ( ord_less_eq_nat @ A @ B )
      | ~ ( ord_less_eq_nat @ B @ A ) ) ).

% verit_la_disequality
thf(fact_358_verit__la__disequality,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
      | ~ ( ord_less_eq_int @ A @ B )
      | ~ ( ord_less_eq_int @ B @ A ) ) ).

% verit_la_disequality
thf(fact_359_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( ( ord_le5570908160329646204atural @ A @ B )
     => ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ C )
       => ( ord_le5570908160329646204atural @ ( times_2397367101498566445atural @ C @ A ) @ ( times_2397367101498566445atural @ C @ B ) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_360_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ B )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
       => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_361_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_362_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ zero_zero_nat @ C )
       => ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_363_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ zero_zero_int @ C )
       => ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_364_mult__less__cancel__right__disj,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) )
      = ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
          & ( ord_le6747313008572928689nteger @ A @ B ) )
        | ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
          & ( ord_le6747313008572928689nteger @ B @ A ) ) ) ) ).

% mult_less_cancel_right_disj
thf(fact_365_mult__less__cancel__right__disj,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
          & ( ord_less_rat @ A @ B ) )
        | ( ( ord_less_rat @ C @ zero_zero_rat )
          & ( ord_less_rat @ B @ A ) ) ) ) ).

% mult_less_cancel_right_disj
thf(fact_366_mult__less__cancel__right__disj,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
          & ( ord_less_int @ A @ B ) )
        | ( ( ord_less_int @ C @ zero_zero_int )
          & ( ord_less_int @ B @ A ) ) ) ) ).

% mult_less_cancel_right_disj
thf(fact_367_mult__strict__right__mono,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( ( ord_le5570908160329646204atural @ A @ B )
     => ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ C )
       => ( ord_le5570908160329646204atural @ ( times_2397367101498566445atural @ A @ C ) @ ( times_2397367101498566445atural @ B @ C ) ) ) ) ).

% mult_strict_right_mono
thf(fact_368_mult__strict__right__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ B )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
       => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) ) ) ) ).

% mult_strict_right_mono
thf(fact_369_mult__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).

% mult_strict_right_mono
thf(fact_370_mult__strict__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ zero_zero_nat @ C )
       => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).

% mult_strict_right_mono
thf(fact_371_mult__strict__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ zero_zero_int @ C )
       => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).

% mult_strict_right_mono
thf(fact_372_mult__strict__right__mono__neg,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( ord_le6747313008572928689nteger @ B @ A )
     => ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
       => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) ) ) ) ).

% mult_strict_right_mono_neg
thf(fact_373_mult__strict__right__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( ord_less_rat @ C @ zero_zero_rat )
       => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).

% mult_strict_right_mono_neg
thf(fact_374_mult__strict__right__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( ord_less_int @ C @ zero_zero_int )
       => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).

% mult_strict_right_mono_neg
thf(fact_375_mult__less__cancel__left__disj,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) )
      = ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
          & ( ord_le6747313008572928689nteger @ A @ B ) )
        | ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
          & ( ord_le6747313008572928689nteger @ B @ A ) ) ) ) ).

% mult_less_cancel_left_disj
thf(fact_376_mult__less__cancel__left__disj,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
          & ( ord_less_rat @ A @ B ) )
        | ( ( ord_less_rat @ C @ zero_zero_rat )
          & ( ord_less_rat @ B @ A ) ) ) ) ).

% mult_less_cancel_left_disj
thf(fact_377_mult__less__cancel__left__disj,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
          & ( ord_less_int @ A @ B ) )
        | ( ( ord_less_int @ C @ zero_zero_int )
          & ( ord_less_int @ B @ A ) ) ) ) ).

% mult_less_cancel_left_disj
thf(fact_378_mult__strict__left__mono,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( ( ord_le5570908160329646204atural @ A @ B )
     => ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ C )
       => ( ord_le5570908160329646204atural @ ( times_2397367101498566445atural @ C @ A ) @ ( times_2397367101498566445atural @ C @ B ) ) ) ) ).

% mult_strict_left_mono
thf(fact_379_mult__strict__left__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ B )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
       => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) ) ) ) ).

% mult_strict_left_mono
thf(fact_380_mult__strict__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).

% mult_strict_left_mono
thf(fact_381_mult__strict__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ zero_zero_nat @ C )
       => ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).

% mult_strict_left_mono
thf(fact_382_mult__strict__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ zero_zero_int @ C )
       => ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).

% mult_strict_left_mono
thf(fact_383_mult__less__le__imp__less,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural,D: code_natural] :
      ( ( ord_le5570908160329646204atural @ A @ B )
     => ( ( ord_le1926595141338095240atural @ C @ D )
       => ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ A )
         => ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ C )
           => ( ord_le5570908160329646204atural @ ( times_2397367101498566445atural @ A @ C ) @ ( times_2397367101498566445atural @ B @ D ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_384_mult__less__le__imp__less,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ B )
     => ( ( ord_le3102999989581377725nteger @ C @ D )
       => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
         => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
           => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ D ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_385_mult__less__le__imp__less,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( 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 @ B @ D ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_386_mult__less__le__imp__less,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( 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 @ B @ D ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_387_mult__less__le__imp__less,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( 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 @ B @ D ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_388_mult__le__less__imp__less,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural,D: code_natural] :
      ( ( ord_le1926595141338095240atural @ A @ B )
     => ( ( ord_le5570908160329646204atural @ C @ D )
       => ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ A )
         => ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ C )
           => ( ord_le5570908160329646204atural @ ( times_2397367101498566445atural @ A @ C ) @ ( times_2397367101498566445atural @ B @ D ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_389_mult__le__less__imp__less,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ B )
     => ( ( ord_le6747313008572928689nteger @ C @ D )
       => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
         => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
           => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ D ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_390_mult__le__less__imp__less,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( 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 @ B @ D ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_391_mult__le__less__imp__less,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( 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 @ B @ D ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_392_mult__le__less__imp__less,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( 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 @ B @ D ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_393_mult__strict__left__mono__neg,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( ord_le6747313008572928689nteger @ B @ A )
     => ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
       => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) ) ) ) ).

% mult_strict_left_mono_neg
thf(fact_394_mult__strict__left__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( ord_less_rat @ C @ zero_zero_rat )
       => ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).

% mult_strict_left_mono_neg
thf(fact_395_mult__strict__left__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( ord_less_int @ C @ zero_zero_int )
       => ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).

% mult_strict_left_mono_neg
thf(fact_396_mult__less__cancel__left__pos,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
     => ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) )
        = ( ord_le6747313008572928689nteger @ A @ B ) ) ) ).

% mult_less_cancel_left_pos
thf(fact_397_mult__less__cancel__left__pos,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
        = ( ord_less_rat @ A @ B ) ) ) ).

% mult_less_cancel_left_pos
thf(fact_398_mult__less__cancel__left__pos,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ C )
     => ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
        = ( ord_less_int @ A @ B ) ) ) ).

% mult_less_cancel_left_pos
thf(fact_399_mult__less__cancel__left__neg,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
     => ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) )
        = ( ord_le6747313008572928689nteger @ B @ A ) ) ) ).

% mult_less_cancel_left_neg
thf(fact_400_mult__less__cancel__left__neg,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
        = ( ord_less_rat @ B @ A ) ) ) ).

% mult_less_cancel_left_neg
thf(fact_401_mult__less__cancel__left__neg,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ C @ zero_zero_int )
     => ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
        = ( ord_less_int @ B @ A ) ) ) ).

% mult_less_cancel_left_neg
thf(fact_402_mult__right__le__imp__le,axiom,
    ! [A: code_natural,C: code_natural,B: code_natural] :
      ( ( ord_le1926595141338095240atural @ ( times_2397367101498566445atural @ A @ C ) @ ( times_2397367101498566445atural @ B @ C ) )
     => ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ C )
       => ( ord_le1926595141338095240atural @ A @ B ) ) ) ).

% mult_right_le_imp_le
thf(fact_403_mult__right__le__imp__le,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
       => ( ord_le3102999989581377725nteger @ A @ B ) ) ) ).

% mult_right_le_imp_le
thf(fact_404_mult__right__le__imp__le,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ord_less_eq_rat @ A @ B ) ) ) ).

% mult_right_le_imp_le
thf(fact_405_mult__right__le__imp__le,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
     => ( ( ord_less_nat @ zero_zero_nat @ C )
       => ( ord_less_eq_nat @ A @ B ) ) ) ).

% mult_right_le_imp_le
thf(fact_406_mult__right__le__imp__le,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
     => ( ( ord_less_int @ zero_zero_int @ C )
       => ( ord_less_eq_int @ A @ B ) ) ) ).

% mult_right_le_imp_le
thf(fact_407_zero__less__mult__pos2,axiom,
    ! [B: code_natural,A: code_natural] :
      ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ ( times_2397367101498566445atural @ B @ A ) )
     => ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ A )
       => ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ B ) ) ) ).

% zero_less_mult_pos2
thf(fact_408_zero__less__mult__pos2,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( times_3573771949741848930nteger @ B @ A ) )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
       => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B ) ) ) ).

% zero_less_mult_pos2
thf(fact_409_zero__less__mult__pos2,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ B @ A ) )
     => ( ( ord_less_rat @ zero_zero_rat @ A )
       => ( ord_less_rat @ zero_zero_rat @ B ) ) ) ).

% zero_less_mult_pos2
thf(fact_410_zero__less__mult__pos2,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B @ A ) )
     => ( ( ord_less_nat @ zero_zero_nat @ A )
       => ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).

% zero_less_mult_pos2
thf(fact_411_zero__less__mult__pos2,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B @ A ) )
     => ( ( ord_less_int @ zero_zero_int @ A )
       => ( ord_less_int @ zero_zero_int @ B ) ) ) ).

% zero_less_mult_pos2
thf(fact_412_mult__left__le__imp__le,axiom,
    ! [C: code_natural,A: code_natural,B: code_natural] :
      ( ( ord_le1926595141338095240atural @ ( times_2397367101498566445atural @ C @ A ) @ ( times_2397367101498566445atural @ C @ B ) )
     => ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ C )
       => ( ord_le1926595141338095240atural @ A @ B ) ) ) ).

% mult_left_le_imp_le
thf(fact_413_mult__left__le__imp__le,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
       => ( ord_le3102999989581377725nteger @ A @ B ) ) ) ).

% mult_left_le_imp_le
thf(fact_414_mult__left__le__imp__le,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ord_less_eq_rat @ A @ B ) ) ) ).

% mult_left_le_imp_le
thf(fact_415_mult__left__le__imp__le,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
     => ( ( ord_less_nat @ zero_zero_nat @ C )
       => ( ord_less_eq_nat @ A @ B ) ) ) ).

% mult_left_le_imp_le
thf(fact_416_mult__left__le__imp__le,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
     => ( ( ord_less_int @ zero_zero_int @ C )
       => ( ord_less_eq_int @ A @ B ) ) ) ).

% mult_left_le_imp_le
thf(fact_417_mult__le__cancel__left__pos,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
     => ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) )
        = ( ord_le3102999989581377725nteger @ A @ B ) ) ) ).

% mult_le_cancel_left_pos
thf(fact_418_mult__le__cancel__left__pos,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
        = ( ord_less_eq_rat @ A @ B ) ) ) ).

% mult_le_cancel_left_pos
thf(fact_419_mult__le__cancel__left__pos,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ C )
     => ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
        = ( ord_less_eq_int @ A @ B ) ) ) ).

% mult_le_cancel_left_pos
thf(fact_420_mult__le__cancel__left__neg,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
     => ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) )
        = ( ord_le3102999989581377725nteger @ B @ A ) ) ) ).

% mult_le_cancel_left_neg
thf(fact_421_mult__le__cancel__left__neg,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
        = ( ord_less_eq_rat @ B @ A ) ) ) ).

% mult_le_cancel_left_neg
thf(fact_422_mult__le__cancel__left__neg,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ C @ zero_zero_int )
     => ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
        = ( ord_less_eq_int @ B @ A ) ) ) ).

% mult_le_cancel_left_neg
thf(fact_423_zero__less__mult__pos,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ ( times_2397367101498566445atural @ A @ B ) )
     => ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ A )
       => ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ B ) ) ) ).

% zero_less_mult_pos
thf(fact_424_zero__less__mult__pos,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( times_3573771949741848930nteger @ A @ B ) )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
       => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B ) ) ) ).

% zero_less_mult_pos
thf(fact_425_zero__less__mult__pos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
     => ( ( ord_less_rat @ zero_zero_rat @ A )
       => ( ord_less_rat @ zero_zero_rat @ B ) ) ) ).

% zero_less_mult_pos
thf(fact_426_zero__less__mult__pos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) )
     => ( ( ord_less_nat @ zero_zero_nat @ A )
       => ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).

% zero_less_mult_pos
thf(fact_427_zero__less__mult__pos,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
     => ( ( ord_less_int @ zero_zero_int @ A )
       => ( ord_less_int @ zero_zero_int @ B ) ) ) ).

% zero_less_mult_pos
thf(fact_428_mult__less__cancel__right,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) )
      = ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
         => ( ord_le6747313008572928689nteger @ A @ B ) )
        & ( ( ord_le3102999989581377725nteger @ C @ zero_z3403309356797280102nteger )
         => ( ord_le6747313008572928689nteger @ B @ A ) ) ) ) ).

% mult_less_cancel_right
thf(fact_429_mult__less__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ A @ B ) )
        & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ B @ A ) ) ) ) ).

% mult_less_cancel_right
thf(fact_430_mult__less__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ A @ B ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ B @ A ) ) ) ) ).

% mult_less_cancel_right
thf(fact_431_mult__strict__mono_H,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural,D: code_natural] :
      ( ( ord_le5570908160329646204atural @ A @ B )
     => ( ( ord_le5570908160329646204atural @ C @ D )
       => ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ A )
         => ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ C )
           => ( ord_le5570908160329646204atural @ ( times_2397367101498566445atural @ A @ C ) @ ( times_2397367101498566445atural @ B @ D ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_432_mult__strict__mono_H,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ B )
     => ( ( ord_le6747313008572928689nteger @ C @ D )
       => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
         => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
           => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ D ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_433_mult__strict__mono_H,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( 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 @ B @ D ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_434_mult__strict__mono_H,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( 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 @ B @ D ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_435_mult__strict__mono_H,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( 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 @ B @ D ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_436_mult__right__less__imp__less,axiom,
    ! [A: code_natural,C: code_natural,B: code_natural] :
      ( ( ord_le5570908160329646204atural @ ( times_2397367101498566445atural @ A @ C ) @ ( times_2397367101498566445atural @ B @ C ) )
     => ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ C )
       => ( ord_le5570908160329646204atural @ A @ B ) ) ) ).

% mult_right_less_imp_less
thf(fact_437_mult__right__less__imp__less,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
       => ( ord_le6747313008572928689nteger @ A @ B ) ) ) ).

% mult_right_less_imp_less
thf(fact_438_mult__right__less__imp__less,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ord_less_rat @ A @ B ) ) ) ).

% mult_right_less_imp_less
thf(fact_439_mult__right__less__imp__less,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
       => ( ord_less_nat @ A @ B ) ) ) ).

% mult_right_less_imp_less
thf(fact_440_mult__right__less__imp__less,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ C )
       => ( ord_less_int @ A @ B ) ) ) ).

% mult_right_less_imp_less
thf(fact_441_mult__less__cancel__left,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) )
      = ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
         => ( ord_le6747313008572928689nteger @ A @ B ) )
        & ( ( ord_le3102999989581377725nteger @ C @ zero_z3403309356797280102nteger )
         => ( ord_le6747313008572928689nteger @ B @ A ) ) ) ) ).

% mult_less_cancel_left
thf(fact_442_mult__less__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ A @ B ) )
        & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ B @ A ) ) ) ) ).

% mult_less_cancel_left
thf(fact_443_mult__less__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ A @ B ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ B @ A ) ) ) ) ).

% mult_less_cancel_left
thf(fact_444_mult__strict__mono,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural,D: code_natural] :
      ( ( ord_le5570908160329646204atural @ A @ B )
     => ( ( ord_le5570908160329646204atural @ C @ D )
       => ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ B )
         => ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ C )
           => ( ord_le5570908160329646204atural @ ( times_2397367101498566445atural @ A @ C ) @ ( times_2397367101498566445atural @ B @ D ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_445_mult__strict__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ B )
     => ( ( ord_le6747313008572928689nteger @ C @ D )
       => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
         => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
           => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ D ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_446_mult__strict__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ C @ D )
       => ( ( ord_less_rat @ zero_zero_rat @ B )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
           => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_447_mult__strict__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ C @ D )
       => ( ( ord_less_nat @ zero_zero_nat @ B )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
           => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_448_mult__strict__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ C @ D )
       => ( ( ord_less_int @ zero_zero_int @ B )
         => ( ( ord_less_eq_int @ zero_zero_int @ C )
           => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_449_mult__left__less__imp__less,axiom,
    ! [C: code_natural,A: code_natural,B: code_natural] :
      ( ( ord_le5570908160329646204atural @ ( times_2397367101498566445atural @ C @ A ) @ ( times_2397367101498566445atural @ C @ B ) )
     => ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ C )
       => ( ord_le5570908160329646204atural @ A @ B ) ) ) ).

% mult_left_less_imp_less
thf(fact_450_mult__left__less__imp__less,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
       => ( ord_le6747313008572928689nteger @ A @ B ) ) ) ).

% mult_left_less_imp_less
thf(fact_451_mult__left__less__imp__less,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ord_less_rat @ A @ B ) ) ) ).

% mult_left_less_imp_less
thf(fact_452_mult__left__less__imp__less,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
       => ( ord_less_nat @ A @ B ) ) ) ).

% mult_left_less_imp_less
thf(fact_453_mult__left__less__imp__less,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ C )
       => ( ord_less_int @ A @ B ) ) ) ).

% mult_left_less_imp_less
thf(fact_454_mult__le__cancel__right,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) )
      = ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
         => ( ord_le3102999989581377725nteger @ A @ B ) )
        & ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
         => ( ord_le3102999989581377725nteger @ B @ A ) ) ) ) ).

% mult_le_cancel_right
thf(fact_455_mult__le__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ A @ B ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ B @ A ) ) ) ) ).

% mult_le_cancel_right
thf(fact_456_mult__le__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ A @ B ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ B @ A ) ) ) ) ).

% mult_le_cancel_right
thf(fact_457_mult__le__cancel__left,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) )
      = ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
         => ( ord_le3102999989581377725nteger @ A @ B ) )
        & ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
         => ( ord_le3102999989581377725nteger @ B @ A ) ) ) ) ).

% mult_le_cancel_left
thf(fact_458_mult__le__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ A @ B ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ B @ A ) ) ) ) ).

% mult_le_cancel_left
thf(fact_459_mult__le__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ A @ B ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ B @ A ) ) ) ) ).

% mult_le_cancel_left
thf(fact_460_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( ( ord_le1926595141338095240atural @ A @ B )
     => ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ C )
       => ( ord_le1926595141338095240atural @ ( times_2397367101498566445atural @ C @ A ) @ ( times_2397367101498566445atural @ C @ B ) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_461_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ B )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
       => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_462_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_463_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
       => ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_464_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ zero_zero_int @ C )
       => ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_465_zero__less__mult__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( times_3573771949741848930nteger @ A @ B ) )
      = ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
          & ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B ) )
        | ( ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger )
          & ( ord_le6747313008572928689nteger @ B @ zero_z3403309356797280102nteger ) ) ) ) ).

% zero_less_mult_iff
thf(fact_466_zero__less__mult__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A )
          & ( ord_less_rat @ zero_zero_rat @ B ) )
        | ( ( ord_less_rat @ A @ zero_zero_rat )
          & ( ord_less_rat @ B @ zero_zero_rat ) ) ) ) ).

% zero_less_mult_iff
thf(fact_467_zero__less__mult__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
      = ( ( ( ord_less_int @ zero_zero_int @ A )
          & ( ord_less_int @ zero_zero_int @ B ) )
        | ( ( ord_less_int @ A @ zero_zero_int )
          & ( ord_less_int @ B @ zero_zero_int ) ) ) ) ).

% zero_less_mult_iff
thf(fact_468_mult__pos__neg2,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ A )
     => ( ( ord_le5570908160329646204atural @ B @ zero_z2226904508553997617atural )
       => ( ord_le5570908160329646204atural @ ( times_2397367101498566445atural @ B @ A ) @ zero_z2226904508553997617atural ) ) ) ).

% mult_pos_neg2
thf(fact_469_mult__pos__neg2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le6747313008572928689nteger @ B @ zero_z3403309356797280102nteger )
       => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ B @ A ) @ zero_z3403309356797280102nteger ) ) ) ).

% mult_pos_neg2
thf(fact_470_mult__pos__neg2,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ B @ zero_zero_rat )
       => ( ord_less_rat @ ( times_times_rat @ B @ A ) @ zero_zero_rat ) ) ) ).

% mult_pos_neg2
thf(fact_471_mult__pos__neg2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ B @ zero_zero_nat )
       => ( ord_less_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).

% mult_pos_neg2
thf(fact_472_mult__pos__neg2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ B @ zero_zero_int )
       => ( ord_less_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).

% mult_pos_neg2
thf(fact_473_mult__pos__pos,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ A )
     => ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ B )
       => ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ ( times_2397367101498566445atural @ A @ B ) ) ) ) ).

% mult_pos_pos
thf(fact_474_mult__pos__pos,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
       => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( times_3573771949741848930nteger @ A @ B ) ) ) ) ).

% mult_pos_pos
thf(fact_475_mult__pos__pos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ zero_zero_rat @ B )
       => ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).

% mult_pos_pos
thf(fact_476_mult__pos__pos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ B )
       => ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).

% mult_pos_pos
thf(fact_477_mult__pos__pos,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).

% mult_pos_pos
thf(fact_478_mult__pos__neg,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ A )
     => ( ( ord_le5570908160329646204atural @ B @ zero_z2226904508553997617atural )
       => ( ord_le5570908160329646204atural @ ( times_2397367101498566445atural @ A @ B ) @ zero_z2226904508553997617atural ) ) ) ).

% mult_pos_neg
thf(fact_479_mult__pos__neg,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le6747313008572928689nteger @ B @ zero_z3403309356797280102nteger )
       => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A @ B ) @ zero_z3403309356797280102nteger ) ) ) ).

% mult_pos_neg
thf(fact_480_mult__pos__neg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ B @ zero_zero_rat )
       => ( ord_less_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).

% mult_pos_neg
thf(fact_481_mult__pos__neg,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ B @ zero_zero_nat )
       => ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% mult_pos_neg
thf(fact_482_mult__pos__neg,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ B @ zero_zero_int )
       => ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).

% mult_pos_neg
thf(fact_483_mult__neg__pos,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( ord_le5570908160329646204atural @ A @ zero_z2226904508553997617atural )
     => ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ B )
       => ( ord_le5570908160329646204atural @ ( times_2397367101498566445atural @ A @ B ) @ zero_z2226904508553997617atural ) ) ) ).

% mult_neg_pos
thf(fact_484_mult__neg__pos,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
       => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A @ B ) @ zero_z3403309356797280102nteger ) ) ) ).

% mult_neg_pos
thf(fact_485_mult__neg__pos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( ord_less_rat @ zero_zero_rat @ B )
       => ( ord_less_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).

% mult_neg_pos
thf(fact_486_mult__neg__pos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ zero_zero_nat )
     => ( ( ord_less_nat @ zero_zero_nat @ B )
       => ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% mult_neg_pos
thf(fact_487_mult__neg__pos,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).

% mult_neg_pos
thf(fact_488_zero__le__mult__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( times_3573771949741848930nteger @ A @ B ) )
      = ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
          & ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B ) )
        | ( ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger )
          & ( ord_le3102999989581377725nteger @ B @ zero_z3403309356797280102nteger ) ) ) ) ).

% zero_le_mult_iff
thf(fact_489_zero__le__mult__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
          & ( ord_less_eq_rat @ zero_zero_rat @ B ) )
        | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
          & ( ord_less_eq_rat @ B @ zero_zero_rat ) ) ) ) ).

% zero_le_mult_iff
thf(fact_490_zero__le__mult__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ A )
          & ( ord_less_eq_int @ zero_zero_int @ B ) )
        | ( ( ord_less_eq_int @ A @ zero_zero_int )
          & ( ord_less_eq_int @ B @ zero_zero_int ) ) ) ) ).

% zero_le_mult_iff
thf(fact_491_mult__nonneg__nonpos2,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ A )
     => ( ( ord_le1926595141338095240atural @ B @ zero_z2226904508553997617atural )
       => ( ord_le1926595141338095240atural @ ( times_2397367101498566445atural @ B @ A ) @ zero_z2226904508553997617atural ) ) ) ).

% mult_nonneg_nonpos2
thf(fact_492_mult__nonneg__nonpos2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le3102999989581377725nteger @ B @ zero_z3403309356797280102nteger )
       => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ B @ A ) @ zero_z3403309356797280102nteger ) ) ) ).

% mult_nonneg_nonpos2
thf(fact_493_mult__nonneg__nonpos2,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( times_times_rat @ B @ A ) @ zero_zero_rat ) ) ) ).

% mult_nonneg_nonpos2
thf(fact_494_mult__nonneg__nonpos2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
       => ( ord_less_eq_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).

% mult_nonneg_nonpos2
thf(fact_495_mult__nonneg__nonpos2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ B @ zero_zero_int )
       => ( ord_less_eq_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).

% mult_nonneg_nonpos2
thf(fact_496_mult__less__0__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A @ B ) @ zero_z3403309356797280102nteger )
      = ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
          & ( ord_le6747313008572928689nteger @ B @ zero_z3403309356797280102nteger ) )
        | ( ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger )
          & ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B ) ) ) ) ).

% mult_less_0_iff
thf(fact_497_mult__less__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A )
          & ( ord_less_rat @ B @ zero_zero_rat ) )
        | ( ( ord_less_rat @ A @ zero_zero_rat )
          & ( ord_less_rat @ zero_zero_rat @ B ) ) ) ) ).

% mult_less_0_iff
thf(fact_498_mult__less__0__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
      = ( ( ( ord_less_int @ zero_zero_int @ A )
          & ( ord_less_int @ B @ zero_zero_int ) )
        | ( ( ord_less_int @ A @ zero_zero_int )
          & ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).

% mult_less_0_iff
thf(fact_499_mult__nonpos__nonneg,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( ord_le1926595141338095240atural @ A @ zero_z2226904508553997617atural )
     => ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ B )
       => ( ord_le1926595141338095240atural @ ( times_2397367101498566445atural @ A @ B ) @ zero_z2226904508553997617atural ) ) ) ).

% mult_nonpos_nonneg
thf(fact_500_mult__nonpos__nonneg,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
       => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A @ B ) @ zero_z3403309356797280102nteger ) ) ) ).

% mult_nonpos_nonneg
thf(fact_501_mult__nonpos__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
       => ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).

% mult_nonpos_nonneg
thf(fact_502_mult__nonpos__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% mult_nonpos_nonneg
thf(fact_503_mult__nonpos__nonneg,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( ord_less_eq_int @ zero_zero_int @ B )
       => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).

% mult_nonpos_nonneg
thf(fact_504_mult__nonneg__nonpos,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ A )
     => ( ( ord_le1926595141338095240atural @ B @ zero_z2226904508553997617atural )
       => ( ord_le1926595141338095240atural @ ( times_2397367101498566445atural @ A @ B ) @ zero_z2226904508553997617atural ) ) ) ).

% mult_nonneg_nonpos
thf(fact_505_mult__nonneg__nonpos,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le3102999989581377725nteger @ B @ zero_z3403309356797280102nteger )
       => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A @ B ) @ zero_z3403309356797280102nteger ) ) ) ).

% mult_nonneg_nonpos
thf(fact_506_mult__nonneg__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).

% mult_nonneg_nonpos
thf(fact_507_mult__nonneg__nonpos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
       => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% mult_nonneg_nonpos
thf(fact_508_mult__nonneg__nonpos,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ B @ zero_zero_int )
       => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).

% mult_nonneg_nonpos
thf(fact_509_mult__nonneg__nonneg,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ A )
     => ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ B )
       => ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ ( times_2397367101498566445atural @ A @ B ) ) ) ) ).

% mult_nonneg_nonneg
thf(fact_510_mult__nonneg__nonneg,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
       => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( times_3573771949741848930nteger @ A @ B ) ) ) ) ).

% mult_nonneg_nonneg
thf(fact_511_mult__nonneg__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).

% mult_nonneg_nonneg
thf(fact_512_mult__nonneg__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).

% mult_nonneg_nonneg
thf(fact_513_mult__nonneg__nonneg,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ zero_zero_int @ B )
       => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).

% mult_nonneg_nonneg
thf(fact_514_split__mult__neg__le,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ A )
          & ( ord_le1926595141338095240atural @ B @ zero_z2226904508553997617atural ) )
        | ( ( ord_le1926595141338095240atural @ A @ zero_z2226904508553997617atural )
          & ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ B ) ) )
     => ( ord_le1926595141338095240atural @ ( times_2397367101498566445atural @ A @ B ) @ zero_z2226904508553997617atural ) ) ).

% split_mult_neg_le
thf(fact_515_split__mult__neg__le,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
          & ( ord_le3102999989581377725nteger @ B @ zero_z3403309356797280102nteger ) )
        | ( ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger )
          & ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B ) ) )
     => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A @ B ) @ zero_z3403309356797280102nteger ) ) ).

% split_mult_neg_le
thf(fact_516_split__mult__neg__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
          & ( ord_less_eq_rat @ B @ zero_zero_rat ) )
        | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
          & ( ord_less_eq_rat @ zero_zero_rat @ B ) ) )
     => ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ).

% split_mult_neg_le
thf(fact_517_split__mult__neg__le,axiom,
    ! [A: nat,B: nat] :
      ( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A )
          & ( ord_less_eq_nat @ B @ zero_zero_nat ) )
        | ( ( ord_less_eq_nat @ A @ zero_zero_nat )
          & ( ord_less_eq_nat @ zero_zero_nat @ B ) ) )
     => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ).

% split_mult_neg_le
thf(fact_518_split__mult__neg__le,axiom,
    ! [A: int,B: int] :
      ( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
          & ( ord_less_eq_int @ B @ zero_zero_int ) )
        | ( ( ord_less_eq_int @ A @ zero_zero_int )
          & ( ord_less_eq_int @ zero_zero_int @ B ) ) )
     => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ).

% split_mult_neg_le
thf(fact_519_mult__le__0__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A @ B ) @ zero_z3403309356797280102nteger )
      = ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
          & ( ord_le3102999989581377725nteger @ B @ zero_z3403309356797280102nteger ) )
        | ( ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger )
          & ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B ) ) ) ) ).

% mult_le_0_iff
thf(fact_520_mult__le__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
          & ( ord_less_eq_rat @ B @ zero_zero_rat ) )
        | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
          & ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ) ) ).

% mult_le_0_iff
thf(fact_521_mult__le__0__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ A )
          & ( ord_less_eq_int @ B @ zero_zero_int ) )
        | ( ( ord_less_eq_int @ A @ zero_zero_int )
          & ( ord_less_eq_int @ zero_zero_int @ B ) ) ) ) ).

% mult_le_0_iff
thf(fact_522_not__square__less__zero,axiom,
    ! [A: code_integer] :
      ~ ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A @ A ) @ zero_z3403309356797280102nteger ) ).

% not_square_less_zero
thf(fact_523_not__square__less__zero,axiom,
    ! [A: rat] :
      ~ ( ord_less_rat @ ( times_times_rat @ A @ A ) @ zero_zero_rat ) ).

% not_square_less_zero
thf(fact_524_not__square__less__zero,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ ( times_times_int @ A @ A ) @ zero_zero_int ) ).

% not_square_less_zero
thf(fact_525_mult__neg__neg,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger )
     => ( ( ord_le6747313008572928689nteger @ B @ zero_z3403309356797280102nteger )
       => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( times_3573771949741848930nteger @ A @ B ) ) ) ) ).

% mult_neg_neg
thf(fact_526_mult__neg__neg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( ord_less_rat @ B @ zero_zero_rat )
       => ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).

% mult_neg_neg
thf(fact_527_mult__neg__neg,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ B @ zero_zero_int )
       => ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).

% mult_neg_neg
thf(fact_528_mult__right__mono,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( ( ord_le1926595141338095240atural @ A @ B )
     => ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ C )
       => ( ord_le1926595141338095240atural @ ( times_2397367101498566445atural @ A @ C ) @ ( times_2397367101498566445atural @ B @ C ) ) ) ) ).

% mult_right_mono
thf(fact_529_mult__right__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ B )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
       => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) ) ) ) ).

% mult_right_mono
thf(fact_530_mult__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).

% mult_right_mono
thf(fact_531_mult__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
       => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).

% mult_right_mono
thf(fact_532_mult__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ zero_zero_int @ C )
       => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).

% mult_right_mono
thf(fact_533_mult__right__mono__neg,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( ord_le3102999989581377725nteger @ B @ A )
     => ( ( ord_le3102999989581377725nteger @ C @ zero_z3403309356797280102nteger )
       => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) ) ) ) ).

% mult_right_mono_neg
thf(fact_534_mult__right__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).

% mult_right_mono_neg
thf(fact_535_mult__right__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_less_eq_int @ C @ zero_zero_int )
       => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).

% mult_right_mono_neg
thf(fact_536_mult__left__mono,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( ( ord_le1926595141338095240atural @ A @ B )
     => ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ C )
       => ( ord_le1926595141338095240atural @ ( times_2397367101498566445atural @ C @ A ) @ ( times_2397367101498566445atural @ C @ B ) ) ) ) ).

% mult_left_mono
thf(fact_537_mult__left__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ B )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
       => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) ) ) ) ).

% mult_left_mono
thf(fact_538_mult__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).

% mult_left_mono
thf(fact_539_mult__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
       => ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).

% mult_left_mono
thf(fact_540_mult__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ zero_zero_int @ C )
       => ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).

% mult_left_mono
thf(fact_541_mult__nonpos__nonpos,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger )
     => ( ( ord_le3102999989581377725nteger @ B @ zero_z3403309356797280102nteger )
       => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( times_3573771949741848930nteger @ A @ B ) ) ) ) ).

% mult_nonpos_nonpos
thf(fact_542_mult__nonpos__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).

% mult_nonpos_nonpos
thf(fact_543_mult__nonpos__nonpos,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( ord_less_eq_int @ B @ zero_zero_int )
       => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).

% mult_nonpos_nonpos
thf(fact_544_mult__left__mono__neg,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( ord_le3102999989581377725nteger @ B @ A )
     => ( ( ord_le3102999989581377725nteger @ C @ zero_z3403309356797280102nteger )
       => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) ) ) ) ).

% mult_left_mono_neg
thf(fact_545_mult__left__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).

% mult_left_mono_neg
thf(fact_546_mult__left__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_less_eq_int @ C @ zero_zero_int )
       => ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).

% mult_left_mono_neg
thf(fact_547_split__mult__pos__le,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
          & ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B ) )
        | ( ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger )
          & ( ord_le3102999989581377725nteger @ B @ zero_z3403309356797280102nteger ) ) )
     => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( times_3573771949741848930nteger @ A @ B ) ) ) ).

% split_mult_pos_le
thf(fact_548_split__mult__pos__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
          & ( ord_less_eq_rat @ zero_zero_rat @ B ) )
        | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
          & ( ord_less_eq_rat @ B @ zero_zero_rat ) ) )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ).

% split_mult_pos_le
thf(fact_549_split__mult__pos__le,axiom,
    ! [A: int,B: int] :
      ( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
          & ( ord_less_eq_int @ zero_zero_int @ B ) )
        | ( ( ord_less_eq_int @ A @ zero_zero_int )
          & ( ord_less_eq_int @ B @ zero_zero_int ) ) )
     => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ).

% split_mult_pos_le
thf(fact_550_zero__le__square,axiom,
    ! [A: code_integer] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( times_3573771949741848930nteger @ A @ A ) ) ).

% zero_le_square
thf(fact_551_zero__le__square,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ A ) ) ).

% zero_le_square
thf(fact_552_zero__le__square,axiom,
    ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ A ) ) ).

% zero_le_square
thf(fact_553_mult__mono_H,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural,D: code_natural] :
      ( ( ord_le1926595141338095240atural @ A @ B )
     => ( ( ord_le1926595141338095240atural @ C @ D )
       => ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ A )
         => ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ C )
           => ( ord_le1926595141338095240atural @ ( times_2397367101498566445atural @ A @ C ) @ ( times_2397367101498566445atural @ B @ D ) ) ) ) ) ) ).

% mult_mono'
thf(fact_554_mult__mono_H,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ B )
     => ( ( ord_le3102999989581377725nteger @ C @ D )
       => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
         => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
           => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ D ) ) ) ) ) ) ).

% mult_mono'
thf(fact_555_mult__mono_H,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( 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 @ B @ D ) ) ) ) ) ) ).

% mult_mono'
thf(fact_556_mult__mono_H,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( 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 @ B @ D ) ) ) ) ) ) ).

% mult_mono'
thf(fact_557_mult__mono_H,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( 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 @ B @ D ) ) ) ) ) ) ).

% mult_mono'
thf(fact_558_mult__mono,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural,D: code_natural] :
      ( ( ord_le1926595141338095240atural @ A @ B )
     => ( ( ord_le1926595141338095240atural @ C @ D )
       => ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ B )
         => ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ C )
           => ( ord_le1926595141338095240atural @ ( times_2397367101498566445atural @ A @ C ) @ ( times_2397367101498566445atural @ B @ D ) ) ) ) ) ) ).

% mult_mono
thf(fact_559_mult__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ B )
     => ( ( ord_le3102999989581377725nteger @ C @ D )
       => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
         => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
           => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ D ) ) ) ) ) ) ).

% mult_mono
thf(fact_560_mult__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ C @ D )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
           => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).

% mult_mono
thf(fact_561_mult__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ D )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
           => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).

% mult_mono
thf(fact_562_mult__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ C @ D )
       => ( ( ord_less_eq_int @ zero_zero_int @ B )
         => ( ( ord_less_eq_int @ zero_zero_int @ C )
           => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).

% mult_mono
thf(fact_563_verit__negate__coefficient_I2_J,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).

% verit_negate_coefficient(2)
thf(fact_564_verit__negate__coefficient_I2_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ B )
     => ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).

% verit_negate_coefficient(2)
thf(fact_565_verit__negate__coefficient_I2_J,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).

% verit_negate_coefficient(2)
thf(fact_566_verit__negate__coefficient_I3_J,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
     => ( ( uminus_uminus_int @ A )
        = ( uminus_uminus_int @ B ) ) ) ).

% verit_negate_coefficient(3)
thf(fact_567_verit__negate__coefficient_I3_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A = B )
     => ( ( uminus1351360451143612070nteger @ A )
        = ( uminus1351360451143612070nteger @ B ) ) ) ).

% verit_negate_coefficient(3)
thf(fact_568_verit__negate__coefficient_I3_J,axiom,
    ! [A: rat,B: rat] :
      ( ( A = B )
     => ( ( uminus_uminus_rat @ A )
        = ( uminus_uminus_rat @ B ) ) ) ).

% verit_negate_coefficient(3)
thf(fact_569_verit__comp__simplify1_I1_J,axiom,
    ! [A: code_integer] :
      ~ ( ord_le6747313008572928689nteger @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_570_verit__comp__simplify1_I1_J,axiom,
    ! [A: rat] :
      ~ ( ord_less_rat @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_571_verit__comp__simplify1_I1_J,axiom,
    ! [A: num] :
      ~ ( ord_less_num @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_572_verit__comp__simplify1_I1_J,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_573_verit__comp__simplify1_I1_J,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_574_verit__comp__simplify1_I2_J,axiom,
    ! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_575_verit__comp__simplify1_I2_J,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_576_verit__comp__simplify1_I2_J,axiom,
    ! [A: num] : ( ord_less_eq_num @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_577_verit__comp__simplify1_I2_J,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_578_verit__comp__simplify1_I2_J,axiom,
    ! [A: int] : ( ord_less_eq_int @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_579_verit__comp__simplify1_I3_J,axiom,
    ! [B3: code_integer,A4: code_integer] :
      ( ( ~ ( ord_le3102999989581377725nteger @ B3 @ A4 ) )
      = ( ord_le6747313008572928689nteger @ A4 @ B3 ) ) ).

% verit_comp_simplify1(3)
thf(fact_580_verit__comp__simplify1_I3_J,axiom,
    ! [B3: rat,A4: rat] :
      ( ( ~ ( ord_less_eq_rat @ B3 @ A4 ) )
      = ( ord_less_rat @ A4 @ B3 ) ) ).

% verit_comp_simplify1(3)
thf(fact_581_verit__comp__simplify1_I3_J,axiom,
    ! [B3: num,A4: num] :
      ( ( ~ ( ord_less_eq_num @ B3 @ A4 ) )
      = ( ord_less_num @ A4 @ B3 ) ) ).

% verit_comp_simplify1(3)
thf(fact_582_verit__comp__simplify1_I3_J,axiom,
    ! [B3: nat,A4: nat] :
      ( ( ~ ( ord_less_eq_nat @ B3 @ A4 ) )
      = ( ord_less_nat @ A4 @ B3 ) ) ).

% verit_comp_simplify1(3)
thf(fact_583_verit__comp__simplify1_I3_J,axiom,
    ! [B3: int,A4: int] :
      ( ( ~ ( ord_less_eq_int @ B3 @ A4 ) )
      = ( ord_less_int @ A4 @ B3 ) ) ).

% verit_comp_simplify1(3)
thf(fact_584_square__eq__iff,axiom,
    ! [A: int,B: int] :
      ( ( ( times_times_int @ A @ A )
        = ( times_times_int @ B @ B ) )
      = ( ( A = B )
        | ( A
          = ( uminus_uminus_int @ B ) ) ) ) ).

% square_eq_iff
thf(fact_585_square__eq__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( times_3573771949741848930nteger @ A @ A )
        = ( times_3573771949741848930nteger @ B @ B ) )
      = ( ( A = B )
        | ( A
          = ( uminus1351360451143612070nteger @ B ) ) ) ) ).

% square_eq_iff
thf(fact_586_square__eq__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ( times_times_rat @ A @ A )
        = ( times_times_rat @ B @ B ) )
      = ( ( A = B )
        | ( A
          = ( uminus_uminus_rat @ B ) ) ) ) ).

% square_eq_iff
thf(fact_587_minus__mult__commute,axiom,
    ! [A: int,B: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ A ) @ B )
      = ( times_times_int @ A @ ( uminus_uminus_int @ B ) ) ) ).

% minus_mult_commute
thf(fact_588_minus__mult__commute,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
      = ( times_3573771949741848930nteger @ A @ ( uminus1351360451143612070nteger @ B ) ) ) ).

% minus_mult_commute
thf(fact_589_minus__mult__commute,axiom,
    ! [A: rat,B: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ A ) @ B )
      = ( times_times_rat @ A @ ( uminus_uminus_rat @ B ) ) ) ).

% minus_mult_commute
thf(fact_590_linorder__neqE__linordered__idom,axiom,
    ! [X: code_integer,Y3: code_integer] :
      ( ( X != Y3 )
     => ( ~ ( ord_le6747313008572928689nteger @ X @ Y3 )
       => ( ord_le6747313008572928689nteger @ Y3 @ X ) ) ) ).

% linorder_neqE_linordered_idom
thf(fact_591_linorder__neqE__linordered__idom,axiom,
    ! [X: rat,Y3: rat] :
      ( ( X != Y3 )
     => ( ~ ( ord_less_rat @ X @ Y3 )
       => ( ord_less_rat @ Y3 @ X ) ) ) ).

% linorder_neqE_linordered_idom
thf(fact_592_linorder__neqE__linordered__idom,axiom,
    ! [X: int,Y3: int] :
      ( ( X != Y3 )
     => ( ~ ( ord_less_int @ X @ Y3 )
       => ( ord_less_int @ Y3 @ X ) ) ) ).

% linorder_neqE_linordered_idom
thf(fact_593_transitive__stepwise__le,axiom,
    ! [M: nat,N: nat,R2: nat > nat > $o] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ! [X3: nat] : ( R2 @ X3 @ X3 )
       => ( ! [X3: nat,Y4: nat,Z3: nat] :
              ( ( R2 @ X3 @ Y4 )
             => ( ( R2 @ Y4 @ Z3 )
               => ( R2 @ X3 @ Z3 ) ) )
         => ( ! [N4: nat] : ( R2 @ N4 @ ( suc @ N4 ) )
           => ( R2 @ M @ N ) ) ) ) ) ).

% transitive_stepwise_le
thf(fact_594_less__mono__imp__le__mono,axiom,
    ! [F: nat > nat,I2: nat,J: nat] :
      ( ! [I3: nat,J2: nat] :
          ( ( ord_less_nat @ I3 @ J2 )
         => ( ord_less_nat @ ( F @ I3 ) @ ( F @ J2 ) ) )
     => ( ( ord_less_eq_nat @ I2 @ J )
       => ( ord_less_eq_nat @ ( F @ I2 ) @ ( F @ J ) ) ) ) ).

% less_mono_imp_le_mono
thf(fact_595_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_596_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_597_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_598_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_599_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_600_strict__inc__induct,axiom,
    ! [I2: nat,J: nat,P: nat > $o] :
      ( ( ord_less_nat @ I2 @ J )
     => ( ! [I3: nat] :
            ( ( J
              = ( suc @ I3 ) )
           => ( P @ I3 ) )
       => ( ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ J )
             => ( ( P @ ( suc @ I3 ) )
               => ( P @ I3 ) ) )
         => ( P @ I2 ) ) ) ) ).

% strict_inc_induct
thf(fact_601_linorder__neqE__nat,axiom,
    ! [X: nat,Y3: nat] :
      ( ( X != Y3 )
     => ( ~ ( ord_less_nat @ X @ Y3 )
       => ( ord_less_nat @ Y3 @ X ) ) ) ).

% linorder_neqE_nat
thf(fact_602_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_603_le__eq__less__or__eq,axiom,
    ( ord_less_eq_nat
    = ( ^ [M2: nat,N3: nat] :
          ( ( ord_less_nat @ M2 @ N3 )
          | ( M2 = N3 ) ) ) ) ).

% le_eq_less_or_eq
thf(fact_604_infinite__descent,axiom,
    ! [P: nat > $o,N: nat] :
      ( ! [N4: nat] :
          ( ~ ( P @ N4 )
         => ? [M3: nat] :
              ( ( ord_less_nat @ M3 @ N4 )
              & ~ ( P @ M3 ) ) )
     => ( P @ N ) ) ).

% infinite_descent
thf(fact_605_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_606_nat__less__induct,axiom,
    ! [P: nat > $o,N: nat] :
      ( ! [N4: nat] :
          ( ! [M3: nat] :
              ( ( ord_less_nat @ M3 @ N4 )
             => ( P @ M3 ) )
         => ( P @ N4 ) )
     => ( P @ N ) ) ).

% nat_less_induct
thf(fact_607_less__irrefl__nat,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ N ) ).

% less_irrefl_nat
thf(fact_608_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_609_less__Suc__induct,axiom,
    ! [I2: nat,J: nat,P: nat > nat > $o] :
      ( ( ord_less_nat @ I2 @ 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 @ I2 @ J ) ) ) ) ).

% less_Suc_induct
thf(fact_610_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_611_full__nat__induct,axiom,
    ! [P: nat > $o,N: nat] :
      ( ! [N4: nat] :
          ( ! [M3: nat] :
              ( ( ord_less_eq_nat @ ( suc @ M3 ) @ N4 )
             => ( P @ M3 ) )
         => ( P @ N4 ) )
     => ( P @ N ) ) ).

% full_nat_induct
thf(fact_612_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_613_less__trans__Suc,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ I2 @ J )
     => ( ( ord_less_nat @ J @ K )
       => ( ord_less_nat @ ( suc @ I2 ) @ K ) ) ) ).

% less_trans_Suc
thf(fact_614_less__not__refl3,axiom,
    ! [S: nat,T: nat] :
      ( ( ord_less_nat @ S @ T )
     => ( S != T ) ) ).

% less_not_refl3
thf(fact_615_less__not__refl2,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ N @ M )
     => ( M != N ) ) ).

% less_not_refl2
thf(fact_616_less__eq__Suc__le,axiom,
    ( ord_less_nat
    = ( ^ [N3: nat] : ( ord_less_eq_nat @ ( suc @ N3 ) ) ) ) ).

% less_eq_Suc_le
thf(fact_617_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_618_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_619_Suc__n__not__le__n,axiom,
    ! [N: nat] :
      ~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).

% Suc_n_not_le_n
thf(fact_620_less__not__refl,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ N ) ).

% less_not_refl
thf(fact_621_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_622_less__antisym,axiom,
    ! [N: nat,M: nat] :
      ( ~ ( ord_less_nat @ N @ M )
     => ( ( ord_less_nat @ N @ ( suc @ M ) )
       => ( M = N ) ) ) ).

% less_antisym
thf(fact_623_Suc__less__eq2,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ ( suc @ N ) @ M )
      = ( ? [M4: nat] :
            ( ( M
              = ( suc @ M4 ) )
            & ( ord_less_nat @ N @ M4 ) ) ) ) ).

% Suc_less_eq2
thf(fact_624_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_625_All__less__Suc,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( suc @ N ) )
           => ( P @ I4 ) ) )
      = ( ( P @ N )
        & ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ N )
           => ( P @ I4 ) ) ) ) ).

% All_less_Suc
thf(fact_626_not__less__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ~ ( ord_less_nat @ M @ N ) )
      = ( ord_less_nat @ N @ ( suc @ M ) ) ) ).

% not_less_eq
thf(fact_627_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_628_nat__less__le,axiom,
    ( ord_less_nat
    = ( ^ [M2: nat,N3: nat] :
          ( ( ord_less_eq_nat @ M2 @ N3 )
          & ( M2 != N3 ) ) ) ) ).

% nat_less_le
thf(fact_629_n__not__Suc__n,axiom,
    ! [N: nat] :
      ( N
     != ( suc @ N ) ) ).

% n_not_Suc_n
thf(fact_630_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_631_Ex__less__Suc,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( suc @ N ) )
            & ( P @ I4 ) ) )
      = ( ( P @ N )
        | ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ N )
            & ( P @ I4 ) ) ) ) ).

% Ex_less_Suc
thf(fact_632_inc__induct,axiom,
    ! [I2: nat,J: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ I2 @ J )
     => ( ( P @ J )
       => ( ! [N4: nat] :
              ( ( ord_less_eq_nat @ I2 @ N4 )
             => ( ( ord_less_nat @ N4 @ J )
               => ( ( P @ ( suc @ N4 ) )
                 => ( P @ N4 ) ) ) )
         => ( P @ I2 ) ) ) ) ).

% inc_induct
thf(fact_633_dec__induct,axiom,
    ! [I2: nat,J: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ I2 @ J )
     => ( ( P @ I2 )
       => ( ! [N4: nat] :
              ( ( ord_less_eq_nat @ I2 @ N4 )
             => ( ( ord_less_nat @ N4 @ J )
               => ( ( P @ N4 )
                 => ( P @ ( suc @ N4 ) ) ) ) )
         => ( P @ J ) ) ) ) ).

% dec_induct
thf(fact_634_Suc__inject,axiom,
    ! [X: nat,Y3: nat] :
      ( ( ( suc @ X )
        = ( suc @ Y3 ) )
     => ( X = Y3 ) ) ).

% Suc_inject
thf(fact_635_less__SucI,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_nat @ M @ ( suc @ N ) ) ) ).

% less_SucI
thf(fact_636_less__SucE,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N ) )
     => ( ~ ( ord_less_nat @ M @ N )
       => ( M = N ) ) ) ).

% less_SucE
thf(fact_637_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_638_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_639_Suc__lessE,axiom,
    ! [I2: nat,K: nat] :
      ( ( ord_less_nat @ ( suc @ I2 ) @ K )
     => ~ ! [J2: nat] :
            ( ( ord_less_nat @ I2 @ J2 )
           => ( K
             != ( suc @ J2 ) ) ) ) ).

% Suc_lessE
thf(fact_640_Suc__lessD,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ M ) @ N )
     => ( ord_less_nat @ M @ N ) ) ).

% Suc_lessD
thf(fact_641_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_642_Suc__le__D,axiom,
    ! [N: nat,M5: nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ M5 )
     => ? [M6: nat] :
          ( M5
          = ( suc @ M6 ) ) ) ).

% Suc_le_D
thf(fact_643_le__SucI,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).

% le_SucI
thf(fact_644_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_645_Suc__leI,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).

% Suc_leI
thf(fact_646_Suc__leD,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% Suc_leD
thf(fact_647_Nat_OlessE,axiom,
    ! [I2: nat,K: nat] :
      ( ( ord_less_nat @ I2 @ K )
     => ( ( K
         != ( suc @ I2 ) )
       => ~ ! [J2: nat] :
              ( ( ord_less_nat @ I2 @ J2 )
             => ( K
               != ( suc @ J2 ) ) ) ) ) ).

% Nat.lessE
thf(fact_648_square__eq__1__iff,axiom,
    ! [X: int] :
      ( ( ( times_times_int @ X @ X )
        = one_one_int )
      = ( ( X = one_one_int )
        | ( X
          = ( uminus_uminus_int @ one_one_int ) ) ) ) ).

% square_eq_1_iff
thf(fact_649_square__eq__1__iff,axiom,
    ! [X: code_integer] :
      ( ( ( times_3573771949741848930nteger @ X @ X )
        = one_one_Code_integer )
      = ( ( X = one_one_Code_integer )
        | ( X
          = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).

% square_eq_1_iff
thf(fact_650_square__eq__1__iff,axiom,
    ! [X: rat] :
      ( ( ( times_times_rat @ X @ X )
        = one_one_rat )
      = ( ( X = one_one_rat )
        | ( X
          = ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).

% square_eq_1_iff
thf(fact_651_minus__less__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ ( uminus_uminus_int @ A ) @ B )
      = ( ord_less_int @ ( uminus_uminus_int @ B ) @ A ) ) ).

% minus_less_iff
thf(fact_652_minus__less__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
      = ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B ) @ A ) ) ).

% minus_less_iff
thf(fact_653_minus__less__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ B )
      = ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ A ) ) ).

% minus_less_iff
thf(fact_654_less__minus__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ ( uminus_uminus_int @ B ) )
      = ( ord_less_int @ B @ ( uminus_uminus_int @ A ) ) ) ).

% less_minus_iff
thf(fact_655_less__minus__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
      = ( ord_le6747313008572928689nteger @ B @ ( uminus1351360451143612070nteger @ A ) ) ) ).

% less_minus_iff
thf(fact_656_less__minus__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ B ) )
      = ( ord_less_rat @ B @ ( uminus_uminus_rat @ A ) ) ) ).

% less_minus_iff
thf(fact_657_mult_Oleft__commute,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( times_3573771949741848930nteger @ B @ ( times_3573771949741848930nteger @ A @ C ) )
      = ( times_3573771949741848930nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) ) ) ).

% mult.left_commute
thf(fact_658_mult_Oleft__commute,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( times_times_rat @ B @ ( times_times_rat @ A @ C ) )
      = ( times_times_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).

% mult.left_commute
thf(fact_659_mult_Oleft__commute,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( times_times_nat @ B @ ( times_times_nat @ A @ C ) )
      = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).

% mult.left_commute
thf(fact_660_mult_Oleft__commute,axiom,
    ! [B: int,A: int,C: int] :
      ( ( times_times_int @ B @ ( times_times_int @ A @ C ) )
      = ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).

% mult.left_commute
thf(fact_661_le__imp__neg__le,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ B )
     => ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).

% le_imp_neg_le
thf(fact_662_le__imp__neg__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).

% le_imp_neg_le
thf(fact_663_le__imp__neg__le,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).

% le_imp_neg_le
thf(fact_664_minus__le__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
      = ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B ) @ A ) ) ).

% minus_le_iff
thf(fact_665_minus__le__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B )
      = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ A ) ) ).

% minus_le_iff
thf(fact_666_minus__le__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B )
      = ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ A ) ) ).

% minus_le_iff
thf(fact_667_le__minus__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
      = ( ord_le3102999989581377725nteger @ B @ ( uminus1351360451143612070nteger @ A ) ) ) ).

% le_minus_iff
thf(fact_668_le__minus__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ B ) )
      = ( ord_less_eq_rat @ B @ ( uminus_uminus_rat @ A ) ) ) ).

% le_minus_iff
thf(fact_669_le__minus__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ B ) )
      = ( ord_less_eq_int @ B @ ( uminus_uminus_int @ A ) ) ) ).

% le_minus_iff
thf(fact_670_mult_Ocommute,axiom,
    ( times_3573771949741848930nteger
    = ( ^ [A2: code_integer,B2: code_integer] : ( times_3573771949741848930nteger @ B2 @ A2 ) ) ) ).

% mult.commute
thf(fact_671_mult_Ocommute,axiom,
    ( times_times_rat
    = ( ^ [A2: rat,B2: rat] : ( times_times_rat @ B2 @ A2 ) ) ) ).

% mult.commute
thf(fact_672_mult_Ocommute,axiom,
    ( times_times_nat
    = ( ^ [A2: nat,B2: nat] : ( times_times_nat @ B2 @ A2 ) ) ) ).

% mult.commute
thf(fact_673_mult_Ocommute,axiom,
    ( times_times_int
    = ( ^ [A2: int,B2: int] : ( times_times_int @ B2 @ A2 ) ) ) ).

% mult.commute
thf(fact_674_less__1__mult,axiom,
    ! [M: code_natural,N: code_natural] :
      ( ( ord_le5570908160329646204atural @ one_one_Code_natural @ M )
     => ( ( ord_le5570908160329646204atural @ one_one_Code_natural @ N )
       => ( ord_le5570908160329646204atural @ one_one_Code_natural @ ( times_2397367101498566445atural @ M @ N ) ) ) ) ).

% less_1_mult
thf(fact_675_less__1__mult,axiom,
    ! [M: code_integer,N: code_integer] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ M )
     => ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ N )
       => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( times_3573771949741848930nteger @ M @ N ) ) ) ) ).

% less_1_mult
thf(fact_676_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_677_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_678_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_679_minus__equation__iff,axiom,
    ! [A: int,B: int] :
      ( ( ( uminus_uminus_int @ A )
        = B )
      = ( ( uminus_uminus_int @ B )
        = A ) ) ).

% minus_equation_iff
thf(fact_680_minus__equation__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( uminus1351360451143612070nteger @ A )
        = B )
      = ( ( uminus1351360451143612070nteger @ B )
        = A ) ) ).

% minus_equation_iff
thf(fact_681_minus__equation__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ( uminus_uminus_rat @ A )
        = B )
      = ( ( uminus_uminus_rat @ B )
        = A ) ) ).

% minus_equation_iff
thf(fact_682_equation__minus__iff,axiom,
    ! [A: int,B: int] :
      ( ( A
        = ( uminus_uminus_int @ B ) )
      = ( B
        = ( uminus_uminus_int @ A ) ) ) ).

% equation_minus_iff
thf(fact_683_equation__minus__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A
        = ( uminus1351360451143612070nteger @ B ) )
      = ( B
        = ( uminus1351360451143612070nteger @ A ) ) ) ).

% equation_minus_iff
thf(fact_684_equation__minus__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( A
        = ( uminus_uminus_rat @ B ) )
      = ( B
        = ( uminus_uminus_rat @ A ) ) ) ).

% equation_minus_iff
thf(fact_685_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > code_integer,N: nat,M: nat] :
      ( ! [N4: nat] : ( ord_le6747313008572928689nteger @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
     => ( ( ord_le6747313008572928689nteger @ ( F @ N ) @ ( F @ M ) )
        = ( ord_less_nat @ N @ M ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_686_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_687_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_688_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_689_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_690_mult_Oassoc,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( times_3573771949741848930nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C )
      = ( times_3573771949741848930nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) ) ) ).

% mult.assoc
thf(fact_691_mult_Oassoc,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ ( times_times_rat @ A @ B ) @ C )
      = ( times_times_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).

% mult.assoc
thf(fact_692_mult_Oassoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
      = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).

% mult.assoc
thf(fact_693_mult_Oassoc,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
      = ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).

% mult.assoc
thf(fact_694_lift__Suc__antimono__le,axiom,
    ! [F: nat > set_nat,N: nat,N2: nat] :
      ( ! [N4: nat] : ( ord_less_eq_set_nat @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
     => ( ( ord_less_eq_nat @ N @ N2 )
       => ( ord_less_eq_set_nat @ ( F @ N2 ) @ ( F @ N ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_695_lift__Suc__antimono__le,axiom,
    ! [F: nat > rat,N: nat,N2: nat] :
      ( ! [N4: nat] : ( ord_less_eq_rat @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
     => ( ( ord_less_eq_nat @ N @ N2 )
       => ( ord_less_eq_rat @ ( F @ N2 ) @ ( F @ N ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_696_lift__Suc__antimono__le,axiom,
    ! [F: nat > num,N: nat,N2: nat] :
      ( ! [N4: nat] : ( ord_less_eq_num @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
     => ( ( ord_less_eq_nat @ N @ N2 )
       => ( ord_less_eq_num @ ( F @ N2 ) @ ( F @ N ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_697_lift__Suc__antimono__le,axiom,
    ! [F: nat > nat,N: nat,N2: nat] :
      ( ! [N4: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
     => ( ( ord_less_eq_nat @ N @ N2 )
       => ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ N ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_698_lift__Suc__antimono__le,axiom,
    ! [F: nat > int,N: nat,N2: nat] :
      ( ! [N4: nat] : ( ord_less_eq_int @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
     => ( ( ord_less_eq_nat @ N @ N2 )
       => ( ord_less_eq_int @ ( F @ N2 ) @ ( F @ N ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_699_lift__Suc__mono__less,axiom,
    ! [F: nat > code_integer,N: nat,N2: nat] :
      ( ! [N4: nat] : ( ord_le6747313008572928689nteger @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
     => ( ( ord_less_nat @ N @ N2 )
       => ( ord_le6747313008572928689nteger @ ( F @ N ) @ ( F @ N2 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_700_lift__Suc__mono__less,axiom,
    ! [F: nat > rat,N: nat,N2: nat] :
      ( ! [N4: nat] : ( ord_less_rat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
     => ( ( ord_less_nat @ N @ N2 )
       => ( ord_less_rat @ ( F @ N ) @ ( F @ N2 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_701_lift__Suc__mono__less,axiom,
    ! [F: nat > num,N: nat,N2: nat] :
      ( ! [N4: nat] : ( ord_less_num @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
     => ( ( ord_less_nat @ N @ N2 )
       => ( ord_less_num @ ( F @ N ) @ ( F @ N2 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_702_lift__Suc__mono__less,axiom,
    ! [F: nat > nat,N: nat,N2: nat] :
      ( ! [N4: nat] : ( ord_less_nat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
     => ( ( ord_less_nat @ N @ N2 )
       => ( ord_less_nat @ ( F @ N ) @ ( F @ N2 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_703_lift__Suc__mono__less,axiom,
    ! [F: nat > int,N: nat,N2: nat] :
      ( ! [N4: nat] : ( ord_less_int @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
     => ( ( ord_less_nat @ N @ N2 )
       => ( ord_less_int @ ( F @ N ) @ ( F @ N2 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_704_lift__Suc__mono__le,axiom,
    ! [F: nat > set_nat,N: nat,N2: nat] :
      ( ! [N4: nat] : ( ord_less_eq_set_nat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
     => ( ( ord_less_eq_nat @ N @ N2 )
       => ( ord_less_eq_set_nat @ ( F @ N ) @ ( F @ N2 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_705_lift__Suc__mono__le,axiom,
    ! [F: nat > rat,N: nat,N2: nat] :
      ( ! [N4: nat] : ( ord_less_eq_rat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
     => ( ( ord_less_eq_nat @ N @ N2 )
       => ( ord_less_eq_rat @ ( F @ N ) @ ( F @ N2 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_706_lift__Suc__mono__le,axiom,
    ! [F: nat > num,N: nat,N2: nat] :
      ( ! [N4: nat] : ( ord_less_eq_num @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
     => ( ( ord_less_eq_nat @ N @ N2 )
       => ( ord_less_eq_num @ ( F @ N ) @ ( F @ N2 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_707_lift__Suc__mono__le,axiom,
    ! [F: nat > nat,N: nat,N2: nat] :
      ( ! [N4: nat] : ( ord_less_eq_nat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
     => ( ( ord_less_eq_nat @ N @ N2 )
       => ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N2 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_708_lift__Suc__mono__le,axiom,
    ! [F: nat > int,N: nat,N2: nat] :
      ( ! [N4: nat] : ( ord_less_eq_int @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
     => ( ( ord_less_eq_nat @ N @ N2 )
       => ( ord_less_eq_int @ ( F @ N ) @ ( F @ N2 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_709_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( times_3573771949741848930nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C )
      = ( times_3573771949741848930nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) ) ) ).

% ab_semigroup_mult_class.mult_ac(1)
thf(fact_710_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ ( times_times_rat @ A @ B ) @ C )
      = ( times_times_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).

% ab_semigroup_mult_class.mult_ac(1)
thf(fact_711_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
      = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).

% ab_semigroup_mult_class.mult_ac(1)
thf(fact_712_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
      = ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).

% ab_semigroup_mult_class.mult_ac(1)
thf(fact_713_mult__less__cancel__right2,axiom,
    ! [A: code_integer,C: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A @ C ) @ C )
      = ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
         => ( ord_le6747313008572928689nteger @ A @ one_one_Code_integer ) )
        & ( ( ord_le3102999989581377725nteger @ C @ zero_z3403309356797280102nteger )
         => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A ) ) ) ) ).

% mult_less_cancel_right2
thf(fact_714_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_715_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_716_mult__less__cancel__right1,axiom,
    ! [C: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ C @ ( times_3573771949741848930nteger @ B @ C ) )
      = ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
         => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ B ) )
        & ( ( ord_le3102999989581377725nteger @ C @ zero_z3403309356797280102nteger )
         => ( ord_le6747313008572928689nteger @ B @ one_one_Code_integer ) ) ) ) ).

% mult_less_cancel_right1
thf(fact_717_mult__less__cancel__right1,axiom,
    ! [C: rat,B: rat] :
      ( ( ord_less_rat @ C @ ( times_times_rat @ B @ C ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ one_one_rat @ B ) )
        & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ B @ one_one_rat ) ) ) ) ).

% mult_less_cancel_right1
thf(fact_718_mult__less__cancel__right1,axiom,
    ! [C: int,B: int] :
      ( ( ord_less_int @ C @ ( times_times_int @ B @ C ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ one_one_int @ B ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ B @ one_one_int ) ) ) ) ).

% mult_less_cancel_right1
thf(fact_719_mult__less__cancel__left2,axiom,
    ! [C: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ C @ A ) @ C )
      = ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
         => ( ord_le6747313008572928689nteger @ A @ one_one_Code_integer ) )
        & ( ( ord_le3102999989581377725nteger @ C @ zero_z3403309356797280102nteger )
         => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A ) ) ) ) ).

% mult_less_cancel_left2
thf(fact_720_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_721_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_722_mult__less__cancel__left1,axiom,
    ! [C: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ C @ ( times_3573771949741848930nteger @ C @ B ) )
      = ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
         => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ B ) )
        & ( ( ord_le3102999989581377725nteger @ C @ zero_z3403309356797280102nteger )
         => ( ord_le6747313008572928689nteger @ B @ one_one_Code_integer ) ) ) ) ).

% mult_less_cancel_left1
thf(fact_723_mult__less__cancel__left1,axiom,
    ! [C: rat,B: rat] :
      ( ( ord_less_rat @ C @ ( times_times_rat @ C @ B ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ one_one_rat @ B ) )
        & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ B @ one_one_rat ) ) ) ) ).

% mult_less_cancel_left1
thf(fact_724_mult__less__cancel__left1,axiom,
    ! [C: int,B: int] :
      ( ( ord_less_int @ C @ ( times_times_int @ C @ B ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ one_one_int @ B ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ B @ one_one_int ) ) ) ) ).

% mult_less_cancel_left1
thf(fact_725_mult__le__cancel__right2,axiom,
    ! [A: code_integer,C: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A @ C ) @ C )
      = ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
         => ( ord_le3102999989581377725nteger @ A @ one_one_Code_integer ) )
        & ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
         => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ A ) ) ) ) ).

% mult_le_cancel_right2
thf(fact_726_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_727_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_728_mult__le__cancel__right1,axiom,
    ! [C: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ C @ ( times_3573771949741848930nteger @ B @ C ) )
      = ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
         => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ B ) )
        & ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
         => ( ord_le3102999989581377725nteger @ B @ one_one_Code_integer ) ) ) ) ).

% mult_le_cancel_right1
thf(fact_729_mult__le__cancel__right1,axiom,
    ! [C: rat,B: rat] :
      ( ( ord_less_eq_rat @ C @ ( times_times_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ one_one_rat @ B ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ B @ one_one_rat ) ) ) ) ).

% mult_le_cancel_right1
thf(fact_730_mult__le__cancel__right1,axiom,
    ! [C: int,B: int] :
      ( ( ord_less_eq_int @ C @ ( times_times_int @ B @ C ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ one_one_int @ B ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).

% mult_le_cancel_right1
thf(fact_731_mult__le__cancel__left2,axiom,
    ! [C: code_integer,A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ C @ A ) @ C )
      = ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
         => ( ord_le3102999989581377725nteger @ A @ one_one_Code_integer ) )
        & ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
         => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ A ) ) ) ) ).

% mult_le_cancel_left2
thf(fact_732_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_733_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_734_mult__le__cancel__left1,axiom,
    ! [C: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ C @ ( times_3573771949741848930nteger @ C @ B ) )
      = ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
         => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ B ) )
        & ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
         => ( ord_le3102999989581377725nteger @ B @ one_one_Code_integer ) ) ) ) ).

% mult_le_cancel_left1
thf(fact_735_mult__le__cancel__left1,axiom,
    ! [C: rat,B: rat] :
      ( ( ord_less_eq_rat @ C @ ( times_times_rat @ C @ B ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ one_one_rat @ B ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ B @ one_one_rat ) ) ) ) ).

% mult_le_cancel_left1
thf(fact_736_mult__le__cancel__left1,axiom,
    ! [C: int,B: int] :
      ( ( ord_less_eq_int @ C @ ( times_times_int @ C @ B ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ one_one_int @ B ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).

% mult_le_cancel_left1
thf(fact_737_mult__less__iff1,axiom,
    ! [Z2: code_integer,X: code_integer,Y3: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ Z2 )
     => ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ X @ Z2 ) @ ( times_3573771949741848930nteger @ Y3 @ Z2 ) )
        = ( ord_le6747313008572928689nteger @ X @ Y3 ) ) ) ).

% mult_less_iff1
thf(fact_738_mult__less__iff1,axiom,
    ! [Z2: rat,X: rat,Y3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Z2 )
     => ( ( ord_less_rat @ ( times_times_rat @ X @ Z2 ) @ ( times_times_rat @ Y3 @ Z2 ) )
        = ( ord_less_rat @ X @ Y3 ) ) ) ).

% mult_less_iff1
thf(fact_739_mult__less__iff1,axiom,
    ! [Z2: int,X: int,Y3: int] :
      ( ( ord_less_int @ zero_zero_int @ Z2 )
     => ( ( ord_less_int @ ( times_times_int @ X @ Z2 ) @ ( times_times_int @ Y3 @ Z2 ) )
        = ( ord_less_int @ X @ Y3 ) ) ) ).

% mult_less_iff1
thf(fact_740_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_741_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_742_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_743_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_744_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_745_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_746_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_747_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_748_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_749_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_750_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_751_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_752_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_753_gr0__implies__Suc,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ? [M6: nat] :
          ( N
          = ( suc @ M6 ) ) ) ).

% gr0_implies_Suc
thf(fact_754_All__less__Suc2,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( suc @ N ) )
           => ( P @ I4 ) ) )
      = ( ( P @ zero_zero_nat )
        & ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ N )
           => ( P @ ( suc @ I4 ) ) ) ) ) ).

% All_less_Suc2
thf(fact_755_gr0__conv__Suc,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
      = ( ? [M2: nat] :
            ( N
            = ( suc @ M2 ) ) ) ) ).

% gr0_conv_Suc
thf(fact_756_Ex__less__Suc2,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( suc @ N ) )
            & ( P @ I4 ) ) )
      = ( ( P @ zero_zero_nat )
        | ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ N )
            & ( P @ ( suc @ I4 ) ) ) ) ) ).

% Ex_less_Suc2
thf(fact_757_mult__less__mono2,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ I2 @ J )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ord_less_nat @ ( times_times_nat @ K @ I2 ) @ ( times_times_nat @ K @ J ) ) ) ) ).

% mult_less_mono2
thf(fact_758_mult__less__mono1,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ I2 @ J )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ord_less_nat @ ( times_times_nat @ I2 @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).

% mult_less_mono1
thf(fact_759_ex__least__nat__le,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ N )
     => ( ~ ( P @ zero_zero_nat )
       => ? [K2: nat] :
            ( ( ord_less_eq_nat @ K2 @ N )
            & ! [I: nat] :
                ( ( ord_less_nat @ I @ K2 )
               => ~ ( P @ I ) )
            & ( P @ K2 ) ) ) ) ).

% ex_least_nat_le
thf(fact_760_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_761_less__imp__of__nat__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_le6747313008572928689nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) ) ) ).

% less_imp_of_nat_less
thf(fact_762_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_763_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_764_less__imp__of__nat__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_le5570908160329646204atural @ ( semiri3763490453095760265atural @ M ) @ ( semiri3763490453095760265atural @ N ) ) ) ).

% less_imp_of_nat_less
thf(fact_765_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_766_of__nat__less__imp__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_le6747313008572928689nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) )
     => ( ord_less_nat @ M @ N ) ) ).

% of_nat_less_imp_less
thf(fact_767_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_768_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_769_of__nat__less__imp__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_le5570908160329646204atural @ ( semiri3763490453095760265atural @ M ) @ ( semiri3763490453095760265atural @ N ) )
     => ( ord_less_nat @ M @ N ) ) ).

% of_nat_less_imp_less
thf(fact_770_of__nat__mono,axiom,
    ! [I2: nat,J: nat] :
      ( ( ord_less_eq_nat @ I2 @ J )
     => ( ord_le3102999989581377725nteger @ ( semiri4939895301339042750nteger @ I2 ) @ ( semiri4939895301339042750nteger @ J ) ) ) ).

% of_nat_mono
thf(fact_771_of__nat__mono,axiom,
    ! [I2: nat,J: nat] :
      ( ( ord_less_eq_nat @ I2 @ J )
     => ( ord_le1926595141338095240atural @ ( semiri3763490453095760265atural @ I2 ) @ ( semiri3763490453095760265atural @ J ) ) ) ).

% of_nat_mono
thf(fact_772_of__nat__mono,axiom,
    ! [I2: nat,J: nat] :
      ( ( ord_less_eq_nat @ I2 @ J )
     => ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ I2 ) @ ( semiri681578069525770553at_rat @ J ) ) ) ).

% of_nat_mono
thf(fact_773_of__nat__mono,axiom,
    ! [I2: nat,J: nat] :
      ( ( ord_less_eq_nat @ I2 @ J )
     => ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I2 ) @ ( semiri1316708129612266289at_nat @ J ) ) ) ).

% of_nat_mono
thf(fact_774_of__nat__mono,axiom,
    ! [I2: nat,J: nat] :
      ( ( ord_less_eq_nat @ I2 @ J )
     => ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I2 ) @ ( semiri1314217659103216013at_int @ J ) ) ) ).

% of_nat_mono
thf(fact_775_int__ops_I7_J,axiom,
    ! [A: nat,B: nat] :
      ( ( semiri1314217659103216013at_int @ ( times_times_nat @ A @ B ) )
      = ( times_times_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).

% int_ops(7)
thf(fact_776_nat__int__comparison_I3_J,axiom,
    ( ord_less_eq_nat
    = ( ^ [A2: nat,B2: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).

% nat_int_comparison(3)
thf(fact_777_mult__not__zero,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( ( times_2397367101498566445atural @ A @ B )
       != zero_z2226904508553997617atural )
     => ( ( A != zero_z2226904508553997617atural )
        & ( B != zero_z2226904508553997617atural ) ) ) ).

% mult_not_zero
thf(fact_778_mult__not__zero,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( times_3573771949741848930nteger @ A @ B )
       != zero_z3403309356797280102nteger )
     => ( ( A != zero_z3403309356797280102nteger )
        & ( B != zero_z3403309356797280102nteger ) ) ) ).

% mult_not_zero
thf(fact_779_mult__not__zero,axiom,
    ! [A: rat,B: rat] :
      ( ( ( times_times_rat @ A @ B )
       != zero_zero_rat )
     => ( ( A != zero_zero_rat )
        & ( B != zero_zero_rat ) ) ) ).

% mult_not_zero
thf(fact_780_mult__not__zero,axiom,
    ! [A: nat,B: nat] :
      ( ( ( times_times_nat @ A @ B )
       != zero_zero_nat )
     => ( ( A != zero_zero_nat )
        & ( B != zero_zero_nat ) ) ) ).

% mult_not_zero
thf(fact_781_mult__not__zero,axiom,
    ! [A: int,B: int] :
      ( ( ( times_times_int @ A @ B )
       != zero_zero_int )
     => ( ( A != zero_zero_int )
        & ( B != zero_zero_int ) ) ) ).

% mult_not_zero
thf(fact_782_divisors__zero,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( ( times_2397367101498566445atural @ A @ B )
        = zero_z2226904508553997617atural )
     => ( ( A = zero_z2226904508553997617atural )
        | ( B = zero_z2226904508553997617atural ) ) ) ).

% divisors_zero
thf(fact_783_divisors__zero,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( times_3573771949741848930nteger @ A @ B )
        = zero_z3403309356797280102nteger )
     => ( ( A = zero_z3403309356797280102nteger )
        | ( B = zero_z3403309356797280102nteger ) ) ) ).

% divisors_zero
thf(fact_784_divisors__zero,axiom,
    ! [A: rat,B: rat] :
      ( ( ( times_times_rat @ A @ B )
        = zero_zero_rat )
     => ( ( A = zero_zero_rat )
        | ( B = zero_zero_rat ) ) ) ).

% divisors_zero
thf(fact_785_divisors__zero,axiom,
    ! [A: nat,B: nat] :
      ( ( ( times_times_nat @ A @ B )
        = zero_zero_nat )
     => ( ( A = zero_zero_nat )
        | ( B = zero_zero_nat ) ) ) ).

% divisors_zero
thf(fact_786_divisors__zero,axiom,
    ! [A: int,B: int] :
      ( ( ( times_times_int @ A @ B )
        = zero_zero_int )
     => ( ( A = zero_zero_int )
        | ( B = zero_zero_int ) ) ) ).

% divisors_zero
thf(fact_787_no__zero__divisors,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( A != zero_z2226904508553997617atural )
     => ( ( B != zero_z2226904508553997617atural )
       => ( ( times_2397367101498566445atural @ A @ B )
         != zero_z2226904508553997617atural ) ) ) ).

% no_zero_divisors
thf(fact_788_no__zero__divisors,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( B != zero_z3403309356797280102nteger )
       => ( ( times_3573771949741848930nteger @ A @ B )
         != zero_z3403309356797280102nteger ) ) ) ).

% no_zero_divisors
thf(fact_789_no__zero__divisors,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B != zero_zero_rat )
       => ( ( times_times_rat @ A @ B )
         != zero_zero_rat ) ) ) ).

% no_zero_divisors
thf(fact_790_no__zero__divisors,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( ( B != zero_zero_nat )
       => ( ( times_times_nat @ A @ B )
         != zero_zero_nat ) ) ) ).

% no_zero_divisors
thf(fact_791_no__zero__divisors,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( ( B != zero_zero_int )
       => ( ( times_times_int @ A @ B )
         != zero_zero_int ) ) ) ).

% no_zero_divisors
thf(fact_792_mult__left__cancel,axiom,
    ! [C: code_natural,A: code_natural,B: code_natural] :
      ( ( C != zero_z2226904508553997617atural )
     => ( ( ( times_2397367101498566445atural @ C @ A )
          = ( times_2397367101498566445atural @ C @ B ) )
        = ( A = B ) ) ) ).

% mult_left_cancel
thf(fact_793_mult__left__cancel,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( C != zero_z3403309356797280102nteger )
     => ( ( ( times_3573771949741848930nteger @ C @ A )
          = ( times_3573771949741848930nteger @ C @ B ) )
        = ( A = B ) ) ) ).

% mult_left_cancel
thf(fact_794_mult__left__cancel,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( ( times_times_rat @ C @ A )
          = ( times_times_rat @ C @ B ) )
        = ( A = B ) ) ) ).

% mult_left_cancel
thf(fact_795_mult__left__cancel,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( C != zero_zero_nat )
     => ( ( ( times_times_nat @ C @ A )
          = ( times_times_nat @ C @ B ) )
        = ( A = B ) ) ) ).

% mult_left_cancel
thf(fact_796_mult__left__cancel,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( ( ( times_times_int @ C @ A )
          = ( times_times_int @ C @ B ) )
        = ( A = B ) ) ) ).

% mult_left_cancel
thf(fact_797_mult__right__cancel,axiom,
    ! [C: code_natural,A: code_natural,B: code_natural] :
      ( ( C != zero_z2226904508553997617atural )
     => ( ( ( times_2397367101498566445atural @ A @ C )
          = ( times_2397367101498566445atural @ B @ C ) )
        = ( A = B ) ) ) ).

% mult_right_cancel
thf(fact_798_mult__right__cancel,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( C != zero_z3403309356797280102nteger )
     => ( ( ( times_3573771949741848930nteger @ A @ C )
          = ( times_3573771949741848930nteger @ B @ C ) )
        = ( A = B ) ) ) ).

% mult_right_cancel
thf(fact_799_mult__right__cancel,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( ( times_times_rat @ A @ C )
          = ( times_times_rat @ B @ C ) )
        = ( A = B ) ) ) ).

% mult_right_cancel
thf(fact_800_mult__right__cancel,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( C != zero_zero_nat )
     => ( ( ( times_times_nat @ A @ C )
          = ( times_times_nat @ B @ C ) )
        = ( A = B ) ) ) ).

% mult_right_cancel
thf(fact_801_mult__right__cancel,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( ( ( times_times_int @ A @ C )
          = ( times_times_int @ B @ C ) )
        = ( A = B ) ) ) ).

% mult_right_cancel
thf(fact_802_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_803_fact__mono,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_le1926595141338095240atural @ ( semiri2447717529341329178atural @ M ) @ ( semiri2447717529341329178atural @ N ) ) ) ).

% fact_mono
thf(fact_804_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_805_fact__mono,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_le3102999989581377725nteger @ ( semiri3624122377584611663nteger @ M ) @ ( semiri3624122377584611663nteger @ N ) ) ) ).

% fact_mono
thf(fact_806_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_807_euclidean__size__mult,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( euclid6377331345833325938nteger @ ( times_3573771949741848930nteger @ A @ B ) )
      = ( times_times_nat @ ( euclid6377331345833325938nteger @ A ) @ ( euclid6377331345833325938nteger @ B ) ) ) ).

% euclidean_size_mult
thf(fact_808_euclidean__size__mult,axiom,
    ! [A: int,B: int] :
      ( ( euclid4774559944035922753ze_int @ ( times_times_int @ A @ B ) )
      = ( times_times_nat @ ( euclid4774559944035922753ze_int @ A ) @ ( euclid4774559944035922753ze_int @ B ) ) ) ).

% euclidean_size_mult
thf(fact_809_euclidean__size__mult,axiom,
    ! [A: nat,B: nat] :
      ( ( euclid4777050414544973029ze_nat @ ( times_times_nat @ A @ B ) )
      = ( times_times_nat @ ( euclid4777050414544973029ze_nat @ A ) @ ( euclid4777050414544973029ze_nat @ B ) ) ) ).

% euclidean_size_mult
thf(fact_810_euclidean__size__mult,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( euclid5200926497590043453atural @ ( times_2397367101498566445atural @ A @ B ) )
      = ( times_times_nat @ ( euclid5200926497590043453atural @ A ) @ ( euclid5200926497590043453atural @ B ) ) ) ).

% euclidean_size_mult
thf(fact_811_round__mono,axiom,
    ! [X: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ X @ Y3 )
     => ( ord_less_eq_int @ ( archim7778729529865785530nd_rat @ X ) @ ( archim7778729529865785530nd_rat @ Y3 ) ) ) ).

% round_mono
thf(fact_812_mult__left__le,axiom,
    ! [C: code_natural,A: code_natural] :
      ( ( ord_le1926595141338095240atural @ C @ one_one_Code_natural )
     => ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ A )
       => ( ord_le1926595141338095240atural @ ( times_2397367101498566445atural @ A @ C ) @ A ) ) ) ).

% mult_left_le
thf(fact_813_mult__left__le,axiom,
    ! [C: code_integer,A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ C @ one_one_Code_integer )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
       => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A @ C ) @ A ) ) ) ).

% mult_left_le
thf(fact_814_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_815_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_816_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_817_mult__le__one,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( ord_le1926595141338095240atural @ A @ one_one_Code_natural )
     => ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ B )
       => ( ( ord_le1926595141338095240atural @ B @ one_one_Code_natural )
         => ( ord_le1926595141338095240atural @ ( times_2397367101498566445atural @ A @ B ) @ one_one_Code_natural ) ) ) ) ).

% mult_le_one
thf(fact_818_mult__le__one,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ one_one_Code_integer )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
       => ( ( ord_le3102999989581377725nteger @ B @ one_one_Code_integer )
         => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A @ B ) @ one_one_Code_integer ) ) ) ) ).

% mult_le_one
thf(fact_819_mult__le__one,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ one_one_rat )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
       => ( ( ord_less_eq_rat @ B @ one_one_rat )
         => ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ one_one_rat ) ) ) ) ).

% mult_le_one
thf(fact_820_mult__le__one,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ one_one_nat )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ( ord_less_eq_nat @ B @ one_one_nat )
         => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ) ).

% mult_le_one
thf(fact_821_mult__le__one,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ one_one_int )
     => ( ( ord_less_eq_int @ zero_zero_int @ B )
       => ( ( ord_less_eq_int @ B @ one_one_int )
         => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ one_one_int ) ) ) ) ).

% mult_le_one
thf(fact_822_mult__right__le__one__le,axiom,
    ! [X: code_integer,Y3: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ Y3 )
       => ( ( ord_le3102999989581377725nteger @ Y3 @ one_one_Code_integer )
         => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ X @ Y3 ) @ X ) ) ) ) ).

% mult_right_le_one_le
thf(fact_823_mult__right__le__one__le,axiom,
    ! [X: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
       => ( ( ord_less_eq_rat @ Y3 @ one_one_rat )
         => ( ord_less_eq_rat @ ( times_times_rat @ X @ Y3 ) @ X ) ) ) ) ).

% mult_right_le_one_le
thf(fact_824_mult__right__le__one__le,axiom,
    ! [X: int,Y3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
       => ( ( ord_less_eq_int @ Y3 @ one_one_int )
         => ( ord_less_eq_int @ ( times_times_int @ X @ Y3 ) @ X ) ) ) ) ).

% mult_right_le_one_le
thf(fact_825_mult__left__le__one__le,axiom,
    ! [X: code_integer,Y3: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ Y3 )
       => ( ( ord_le3102999989581377725nteger @ Y3 @ one_one_Code_integer )
         => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ Y3 @ X ) @ X ) ) ) ) ).

% mult_left_le_one_le
thf(fact_826_mult__left__le__one__le,axiom,
    ! [X: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
       => ( ( ord_less_eq_rat @ Y3 @ one_one_rat )
         => ( ord_less_eq_rat @ ( times_times_rat @ Y3 @ X ) @ X ) ) ) ) ).

% mult_left_le_one_le
thf(fact_827_mult__left__le__one__le,axiom,
    ! [X: int,Y3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
       => ( ( ord_less_eq_int @ Y3 @ one_one_int )
         => ( ord_less_eq_int @ ( times_times_int @ Y3 @ X ) @ X ) ) ) ) ).

% mult_left_le_one_le
thf(fact_828_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_829_less__minus__one__simps_I1_J,axiom,
    ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).

% less_minus_one_simps(1)
thf(fact_830_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_831_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_832_less__minus__one__simps_I3_J,axiom,
    ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% less_minus_one_simps(3)
thf(fact_833_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_834_le__minus__one__simps_I1_J,axiom,
    ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).

% le_minus_one_simps(1)
thf(fact_835_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_836_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_837_le__minus__one__simps_I3_J,axiom,
    ~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% le_minus_one_simps(3)
thf(fact_838_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_839_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_840_ex__less__of__nat__mult,axiom,
    ! [X: rat,Y3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ X )
     => ? [N4: nat] : ( ord_less_rat @ Y3 @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N4 ) @ X ) ) ) ).

% ex_less_of_nat_mult
thf(fact_841_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_842_size__mult__mono_H,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( ord_less_eq_nat @ ( euclid6377331345833325938nteger @ A ) @ ( euclid6377331345833325938nteger @ ( times_3573771949741848930nteger @ B @ A ) ) ) ) ).

% size_mult_mono'
thf(fact_843_size__mult__mono_H,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( ord_less_eq_nat @ ( euclid4774559944035922753ze_int @ A ) @ ( euclid4774559944035922753ze_int @ ( times_times_int @ B @ A ) ) ) ) ).

% size_mult_mono'
thf(fact_844_size__mult__mono_H,axiom,
    ! [B: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( ord_less_eq_nat @ ( euclid4777050414544973029ze_nat @ A ) @ ( euclid4777050414544973029ze_nat @ ( times_times_nat @ B @ A ) ) ) ) ).

% size_mult_mono'
thf(fact_845_size__mult__mono_H,axiom,
    ! [B: code_natural,A: code_natural] :
      ( ( B != zero_z2226904508553997617atural )
     => ( ord_less_eq_nat @ ( euclid5200926497590043453atural @ A ) @ ( euclid5200926497590043453atural @ ( times_2397367101498566445atural @ B @ A ) ) ) ) ).

% size_mult_mono'
thf(fact_846_size__mult__mono,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( ord_less_eq_nat @ ( euclid6377331345833325938nteger @ A ) @ ( euclid6377331345833325938nteger @ ( times_3573771949741848930nteger @ A @ B ) ) ) ) ).

% size_mult_mono
thf(fact_847_size__mult__mono,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( ord_less_eq_nat @ ( euclid4774559944035922753ze_int @ A ) @ ( euclid4774559944035922753ze_int @ ( times_times_int @ A @ B ) ) ) ) ).

% size_mult_mono
thf(fact_848_size__mult__mono,axiom,
    ! [B: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( ord_less_eq_nat @ ( euclid4777050414544973029ze_nat @ A ) @ ( euclid4777050414544973029ze_nat @ ( times_times_nat @ A @ B ) ) ) ) ).

% size_mult_mono
thf(fact_849_size__mult__mono,axiom,
    ! [B: code_natural,A: code_natural] :
      ( ( B != zero_z2226904508553997617atural )
     => ( ord_less_eq_nat @ ( euclid5200926497590043453atural @ A ) @ ( euclid5200926497590043453atural @ ( times_2397367101498566445atural @ A @ B ) ) ) ) ).

% size_mult_mono
thf(fact_850_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_851_fact__less__mono,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_nat @ M @ N )
       => ( ord_le5570908160329646204atural @ ( semiri2447717529341329178atural @ M ) @ ( semiri2447717529341329178atural @ N ) ) ) ) ).

% fact_less_mono
thf(fact_852_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_853_fact__less__mono,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_nat @ M @ N )
       => ( ord_le6747313008572928689nteger @ ( semiri3624122377584611663nteger @ M ) @ ( semiri3624122377584611663nteger @ N ) ) ) ) ).

% fact_less_mono
thf(fact_854_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_855_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_856_one__neq__neg__one,axiom,
    ( one_one_int
   != ( uminus_uminus_int @ one_one_int ) ) ).

% one_neq_neg_one
thf(fact_857_one__neq__neg__one,axiom,
    ( one_one_Code_integer
   != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% one_neq_neg_one
thf(fact_858_one__neq__neg__one,axiom,
    ( one_one_rat
   != ( uminus_uminus_rat @ one_one_rat ) ) ).

% one_neq_neg_one
thf(fact_859_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_860_gr__implies__not__zero,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( N != zero_zero_nat ) ) ).

% gr_implies_not_zero
thf(fact_861_not__less__zero,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% not_less_zero
thf(fact_862_gr__zeroI,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% gr_zeroI
thf(fact_863_less__numeral__extra_I3_J,axiom,
    ~ ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ zero_z2226904508553997617atural ) ).

% less_numeral_extra(3)
thf(fact_864_less__numeral__extra_I3_J,axiom,
    ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger ) ).

% less_numeral_extra(3)
thf(fact_865_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_rat @ zero_zero_rat @ zero_zero_rat ) ).

% less_numeral_extra(3)
thf(fact_866_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).

% less_numeral_extra(3)
thf(fact_867_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).

% less_numeral_extra(3)
thf(fact_868_mult_Ocomm__neutral,axiom,
    ! [A: code_natural] :
      ( ( times_2397367101498566445atural @ A @ one_one_Code_natural )
      = A ) ).

% mult.comm_neutral
thf(fact_869_mult_Ocomm__neutral,axiom,
    ! [A: code_integer] :
      ( ( times_3573771949741848930nteger @ A @ one_one_Code_integer )
      = A ) ).

% mult.comm_neutral
thf(fact_870_mult_Ocomm__neutral,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ A @ one_one_rat )
      = A ) ).

% mult.comm_neutral
thf(fact_871_mult_Ocomm__neutral,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ one_one_nat )
      = A ) ).

% mult.comm_neutral
thf(fact_872_mult_Ocomm__neutral,axiom,
    ! [A: int] :
      ( ( times_times_int @ A @ one_one_int )
      = A ) ).

% mult.comm_neutral
thf(fact_873_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: code_natural] :
      ( ( times_2397367101498566445atural @ one_one_Code_natural @ A )
      = A ) ).

% comm_monoid_mult_class.mult_1
thf(fact_874_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: code_integer] :
      ( ( times_3573771949741848930nteger @ one_one_Code_integer @ A )
      = A ) ).

% comm_monoid_mult_class.mult_1
thf(fact_875_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_876_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_877_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_878_zero__le,axiom,
    ! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).

% zero_le
thf(fact_879_le__numeral__extra_I3_J,axiom,
    ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger ).

% le_numeral_extra(3)
thf(fact_880_le__numeral__extra_I3_J,axiom,
    ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ zero_z2226904508553997617atural ).

% le_numeral_extra(3)
thf(fact_881_le__numeral__extra_I3_J,axiom,
    ord_less_eq_rat @ zero_zero_rat @ zero_zero_rat ).

% le_numeral_extra(3)
thf(fact_882_le__numeral__extra_I3_J,axiom,
    ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).

% le_numeral_extra(3)
thf(fact_883_le__numeral__extra_I3_J,axiom,
    ord_less_eq_int @ zero_zero_int @ zero_zero_int ).

% le_numeral_extra(3)
thf(fact_884_less__numeral__extra_I4_J,axiom,
    ~ ( ord_le5570908160329646204atural @ one_one_Code_natural @ one_one_Code_natural ) ).

% less_numeral_extra(4)
thf(fact_885_less__numeral__extra_I4_J,axiom,
    ~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ one_one_Code_integer ) ).

% less_numeral_extra(4)
thf(fact_886_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_rat @ one_one_rat @ one_one_rat ) ).

% less_numeral_extra(4)
thf(fact_887_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).

% less_numeral_extra(4)
thf(fact_888_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_int @ one_one_int @ one_one_int ) ).

% less_numeral_extra(4)
thf(fact_889_exists__least__lemma,axiom,
    ! [P: nat > $o] :
      ( ~ ( P @ zero_zero_nat )
     => ( ? [X_1: nat] : ( P @ X_1 )
       => ? [N4: nat] :
            ( ~ ( P @ N4 )
            & ( P @ ( suc @ N4 ) ) ) ) ) ).

% exists_least_lemma
thf(fact_890_not0__implies__Suc,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ? [M6: nat] :
          ( N
          = ( suc @ M6 ) ) ) ).

% not0_implies_Suc
thf(fact_891_Zero__not__Suc,axiom,
    ! [M: nat] :
      ( zero_zero_nat
     != ( suc @ M ) ) ).

% Zero_not_Suc
thf(fact_892_Zero__neq__Suc,axiom,
    ! [M: nat] :
      ( zero_zero_nat
     != ( suc @ M ) ) ).

% Zero_neq_Suc
thf(fact_893_Suc__neq__Zero,axiom,
    ! [M: nat] :
      ( ( suc @ M )
     != zero_zero_nat ) ).

% Suc_neq_Zero
thf(fact_894_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_895_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_896_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_897_old_Onat_Oexhaust,axiom,
    ! [Y3: nat] :
      ( ( Y3 != zero_zero_nat )
     => ~ ! [Nat3: nat] :
            ( Y3
           != ( suc @ Nat3 ) ) ) ).

% old.nat.exhaust
thf(fact_898_nat_OdiscI,axiom,
    ! [Nat: nat,X22: nat] :
      ( ( Nat
        = ( suc @ X22 ) )
     => ( Nat != zero_zero_nat ) ) ).

% nat.discI
thf(fact_899_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] :
      ( zero_zero_nat
     != ( suc @ Nat2 ) ) ).

% old.nat.distinct(1)
thf(fact_900_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] :
      ( ( suc @ Nat2 )
     != zero_zero_nat ) ).

% old.nat.distinct(2)
thf(fact_901_nat_Odistinct_I1_J,axiom,
    ! [X22: nat] :
      ( zero_zero_nat
     != ( suc @ X22 ) ) ).

% nat.distinct(1)
thf(fact_902_le__numeral__extra_I4_J,axiom,
    ord_le3102999989581377725nteger @ one_one_Code_integer @ one_one_Code_integer ).

% le_numeral_extra(4)
thf(fact_903_le__numeral__extra_I4_J,axiom,
    ord_le1926595141338095240atural @ one_one_Code_natural @ one_one_Code_natural ).

% le_numeral_extra(4)
thf(fact_904_le__numeral__extra_I4_J,axiom,
    ord_less_eq_rat @ one_one_rat @ one_one_rat ).

% le_numeral_extra(4)
thf(fact_905_le__numeral__extra_I4_J,axiom,
    ord_less_eq_nat @ one_one_nat @ one_one_nat ).

% le_numeral_extra(4)
thf(fact_906_le__numeral__extra_I4_J,axiom,
    ord_less_eq_int @ one_one_int @ one_one_int ).

% le_numeral_extra(4)
thf(fact_907_mult__0,axiom,
    ! [N: nat] :
      ( ( times_times_nat @ zero_zero_nat @ N )
      = zero_zero_nat ) ).

% mult_0
thf(fact_908_mult__of__nat__commute,axiom,
    ! [X: nat,Y3: int] :
      ( ( times_times_int @ ( semiri1314217659103216013at_int @ X ) @ Y3 )
      = ( times_times_int @ Y3 @ ( semiri1314217659103216013at_int @ X ) ) ) ).

% mult_of_nat_commute
thf(fact_909_mult__of__nat__commute,axiom,
    ! [X: nat,Y3: code_integer] :
      ( ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ X ) @ Y3 )
      = ( times_3573771949741848930nteger @ Y3 @ ( semiri4939895301339042750nteger @ X ) ) ) ).

% mult_of_nat_commute
thf(fact_910_mult__of__nat__commute,axiom,
    ! [X: nat,Y3: nat] :
      ( ( times_times_nat @ ( semiri1316708129612266289at_nat @ X ) @ Y3 )
      = ( times_times_nat @ Y3 @ ( semiri1316708129612266289at_nat @ X ) ) ) ).

% mult_of_nat_commute
thf(fact_911_mult__of__nat__commute,axiom,
    ! [X: nat,Y3: rat] :
      ( ( times_times_rat @ ( semiri681578069525770553at_rat @ X ) @ Y3 )
      = ( times_times_rat @ Y3 @ ( semiri681578069525770553at_rat @ X ) ) ) ).

% mult_of_nat_commute
thf(fact_912_mult__of__nat__commute,axiom,
    ! [X: nat,Y3: code_natural] :
      ( ( times_2397367101498566445atural @ ( semiri3763490453095760265atural @ X ) @ Y3 )
      = ( times_2397367101498566445atural @ Y3 @ ( semiri3763490453095760265atural @ X ) ) ) ).

% mult_of_nat_commute
thf(fact_913_infinite__descent0,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N4: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N4 )
           => ( ~ ( P @ N4 )
             => ? [M3: nat] :
                  ( ( ord_less_nat @ M3 @ N4 )
                  & ~ ( P @ M3 ) ) ) )
       => ( P @ N ) ) ) ).

% infinite_descent0
thf(fact_914_gr__implies__not0,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( N != zero_zero_nat ) ) ).

% gr_implies_not0
thf(fact_915_less__zeroE,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% less_zeroE
thf(fact_916_not__less0,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% not_less0
thf(fact_917_not__gr0,axiom,
    ! [N: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
      = ( N = zero_zero_nat ) ) ).

% not_gr0
thf(fact_918_gr0I,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% gr0I
thf(fact_919_bot__nat__0_Oextremum__strict,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ zero_zero_nat ) ).

% bot_nat_0.extremum_strict
thf(fact_920_reals__Archimedean2,axiom,
    ! [X: rat] :
    ? [N4: nat] : ( ord_less_rat @ X @ ( semiri681578069525770553at_rat @ N4 ) ) ).

% reals_Archimedean2
thf(fact_921_le__0__eq,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ N @ zero_zero_nat )
      = ( N = zero_zero_nat ) ) ).

% le_0_eq
thf(fact_922_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_923_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_924_less__eq__nat_Osimps_I1_J,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).

% less_eq_nat.simps(1)
thf(fact_925_real__arch__simple,axiom,
    ! [X: rat] :
    ? [N4: nat] : ( ord_less_eq_rat @ X @ ( semiri681578069525770553at_rat @ N4 ) ) ).

% real_arch_simple
thf(fact_926_nat__mult__1__right,axiom,
    ! [N: nat] :
      ( ( times_times_nat @ N @ one_one_nat )
      = N ) ).

% nat_mult_1_right
thf(fact_927_nat__mult__1,axiom,
    ! [N: nat] :
      ( ( times_times_nat @ one_one_nat @ N )
      = N ) ).

% nat_mult_1
thf(fact_928_not__one__less__zero,axiom,
    ~ ( ord_le5570908160329646204atural @ one_one_Code_natural @ zero_z2226904508553997617atural ) ).

% not_one_less_zero
thf(fact_929_not__one__less__zero,axiom,
    ~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ zero_z3403309356797280102nteger ) ).

% not_one_less_zero
thf(fact_930_not__one__less__zero,axiom,
    ~ ( ord_less_rat @ one_one_rat @ zero_zero_rat ) ).

% not_one_less_zero
thf(fact_931_not__one__less__zero,axiom,
    ~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).

% not_one_less_zero
thf(fact_932_not__one__less__zero,axiom,
    ~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).

% not_one_less_zero
thf(fact_933_zero__less__one,axiom,
    ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ one_one_Code_natural ).

% zero_less_one
thf(fact_934_zero__less__one,axiom,
    ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ one_one_Code_integer ).

% zero_less_one
thf(fact_935_zero__less__one,axiom,
    ord_less_rat @ zero_zero_rat @ one_one_rat ).

% zero_less_one
thf(fact_936_zero__less__one,axiom,
    ord_less_nat @ zero_zero_nat @ one_one_nat ).

% zero_less_one
thf(fact_937_zero__less__one,axiom,
    ord_less_int @ zero_zero_int @ one_one_int ).

% zero_less_one
thf(fact_938_not__one__le__zero,axiom,
    ~ ( ord_le3102999989581377725nteger @ one_one_Code_integer @ zero_z3403309356797280102nteger ) ).

% not_one_le_zero
thf(fact_939_not__one__le__zero,axiom,
    ~ ( ord_le1926595141338095240atural @ one_one_Code_natural @ zero_z2226904508553997617atural ) ).

% not_one_le_zero
thf(fact_940_not__one__le__zero,axiom,
    ~ ( ord_less_eq_rat @ one_one_rat @ zero_zero_rat ) ).

% not_one_le_zero
thf(fact_941_not__one__le__zero,axiom,
    ~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).

% not_one_le_zero
thf(fact_942_not__one__le__zero,axiom,
    ~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).

% not_one_le_zero
thf(fact_943_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ one_one_Code_integer ).

% linordered_nonzero_semiring_class.zero_le_one
thf(fact_944_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ one_one_Code_natural ).

% linordered_nonzero_semiring_class.zero_le_one
thf(fact_945_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_946_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_947_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_948_zero__less__one__class_Ozero__le__one,axiom,
    ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ one_one_Code_integer ).

% zero_less_one_class.zero_le_one
thf(fact_949_zero__less__one__class_Ozero__le__one,axiom,
    ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ one_one_Code_natural ).

% zero_less_one_class.zero_le_one
thf(fact_950_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_951_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_952_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_953_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_954_fact__ge__self,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ N @ ( semiri1408675320244567234ct_nat @ N ) ) ).

% fact_ge_self
thf(fact_955_zero__neq__neg__one,axiom,
    ( zero_zero_int
   != ( uminus_uminus_int @ one_one_int ) ) ).

% zero_neq_neg_one
thf(fact_956_zero__neq__neg__one,axiom,
    ( zero_z3403309356797280102nteger
   != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% zero_neq_neg_one
thf(fact_957_zero__neq__neg__one,axiom,
    ( zero_zero_rat
   != ( uminus_uminus_rat @ one_one_rat ) ) ).

% zero_neq_neg_one
thf(fact_958_less__numeral__extra_I1_J,axiom,
    ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ one_one_Code_natural ).

% less_numeral_extra(1)
thf(fact_959_less__numeral__extra_I1_J,axiom,
    ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ one_one_Code_integer ).

% less_numeral_extra(1)
thf(fact_960_less__numeral__extra_I1_J,axiom,
    ord_less_rat @ zero_zero_rat @ one_one_rat ).

% less_numeral_extra(1)
thf(fact_961_less__numeral__extra_I1_J,axiom,
    ord_less_nat @ zero_zero_nat @ one_one_nat ).

% less_numeral_extra(1)
thf(fact_962_less__numeral__extra_I1_J,axiom,
    ord_less_int @ zero_zero_int @ one_one_int ).

% less_numeral_extra(1)
thf(fact_963_of__nat__neq__0,axiom,
    ! [N: nat] :
      ( ( semiri1314217659103216013at_int @ ( suc @ N ) )
     != zero_zero_int ) ).

% of_nat_neq_0
thf(fact_964_of__nat__neq__0,axiom,
    ! [N: nat] :
      ( ( semiri4939895301339042750nteger @ ( suc @ N ) )
     != zero_z3403309356797280102nteger ) ).

% of_nat_neq_0
thf(fact_965_of__nat__neq__0,axiom,
    ! [N: nat] :
      ( ( semiri1316708129612266289at_nat @ ( suc @ N ) )
     != zero_zero_nat ) ).

% of_nat_neq_0
thf(fact_966_of__nat__neq__0,axiom,
    ! [N: nat] :
      ( ( semiri681578069525770553at_rat @ ( suc @ N ) )
     != zero_zero_rat ) ).

% of_nat_neq_0
thf(fact_967_of__nat__neq__0,axiom,
    ! [N: nat] :
      ( ( semiri3763490453095760265atural @ ( suc @ N ) )
     != zero_z2226904508553997617atural ) ).

% of_nat_neq_0
thf(fact_968_of__nat__less__0__iff,axiom,
    ! [M: nat] :
      ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int ) ).

% of_nat_less_0_iff
thf(fact_969_of__nat__less__0__iff,axiom,
    ! [M: nat] :
      ~ ( ord_le6747313008572928689nteger @ ( semiri4939895301339042750nteger @ M ) @ zero_z3403309356797280102nteger ) ).

% of_nat_less_0_iff
thf(fact_970_of__nat__less__0__iff,axiom,
    ! [M: nat] :
      ~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat ) ).

% of_nat_less_0_iff
thf(fact_971_of__nat__less__0__iff,axiom,
    ! [M: nat] :
      ~ ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ zero_zero_rat ) ).

% of_nat_less_0_iff
thf(fact_972_of__nat__less__0__iff,axiom,
    ! [M: nat] :
      ~ ( ord_le5570908160329646204atural @ ( semiri3763490453095760265atural @ M ) @ zero_z2226904508553997617atural ) ).

% of_nat_less_0_iff
thf(fact_973_of__nat__0__le__iff,axiom,
    ! [N: nat] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( semiri4939895301339042750nteger @ N ) ) ).

% of_nat_0_le_iff
thf(fact_974_of__nat__0__le__iff,axiom,
    ! [N: nat] : ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ ( semiri3763490453095760265atural @ N ) ) ).

% of_nat_0_le_iff
thf(fact_975_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_976_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_977_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_978_One__nat__def,axiom,
    ( one_one_nat
    = ( suc @ zero_zero_nat ) ) ).

% One_nat_def
thf(fact_979_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_980_fact__gt__zero,axiom,
    ! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( semiri1408675320244567234ct_nat @ N ) ) ).

% fact_gt_zero
thf(fact_981_fact__gt__zero,axiom,
    ! [N: nat] : ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ ( semiri2447717529341329178atural @ N ) ) ).

% fact_gt_zero
thf(fact_982_fact__gt__zero,axiom,
    ! [N: nat] : ( ord_less_rat @ zero_zero_rat @ ( semiri773545260158071498ct_rat @ N ) ) ).

% fact_gt_zero
thf(fact_983_fact__gt__zero,axiom,
    ! [N: nat] : ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( semiri3624122377584611663nteger @ N ) ) ).

% fact_gt_zero
thf(fact_984_fact__gt__zero,axiom,
    ! [N: nat] : ( ord_less_int @ zero_zero_int @ ( semiri1406184849735516958ct_int @ N ) ) ).

% fact_gt_zero
thf(fact_985_fact__not__neg,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ N ) @ zero_zero_nat ) ).

% fact_not_neg
thf(fact_986_fact__not__neg,axiom,
    ! [N: nat] :
      ~ ( ord_le5570908160329646204atural @ ( semiri2447717529341329178atural @ N ) @ zero_z2226904508553997617atural ) ).

% fact_not_neg
thf(fact_987_fact__not__neg,axiom,
    ! [N: nat] :
      ~ ( ord_less_rat @ ( semiri773545260158071498ct_rat @ N ) @ zero_zero_rat ) ).

% fact_not_neg
thf(fact_988_fact__not__neg,axiom,
    ! [N: nat] :
      ~ ( ord_le6747313008572928689nteger @ ( semiri3624122377584611663nteger @ N ) @ zero_z3403309356797280102nteger ) ).

% fact_not_neg
thf(fact_989_fact__not__neg,axiom,
    ! [N: nat] :
      ~ ( ord_less_int @ ( semiri1406184849735516958ct_int @ N ) @ zero_zero_int ) ).

% fact_not_neg
thf(fact_990_fact__ge__zero,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1408675320244567234ct_nat @ N ) ) ).

% fact_ge_zero
thf(fact_991_fact__ge__zero,axiom,
    ! [N: nat] : ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ ( semiri2447717529341329178atural @ N ) ) ).

% fact_ge_zero
thf(fact_992_fact__ge__zero,axiom,
    ! [N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( semiri773545260158071498ct_rat @ N ) ) ).

% fact_ge_zero
thf(fact_993_fact__ge__zero,axiom,
    ! [N: nat] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( semiri3624122377584611663nteger @ N ) ) ).

% fact_ge_zero
thf(fact_994_fact__ge__zero,axiom,
    ! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1406184849735516958ct_int @ N ) ) ).

% fact_ge_zero
thf(fact_995_division__segment__mult,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( ( B != zero_zero_nat )
       => ( ( euclid3398187327856392827nt_nat @ ( times_times_nat @ A @ B ) )
          = ( times_times_nat @ ( euclid3398187327856392827nt_nat @ A ) @ ( euclid3398187327856392827nt_nat @ B ) ) ) ) ) ).

% division_segment_mult
thf(fact_996_division__segment__mult,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( ( B != zero_zero_int )
       => ( ( euclid3395696857347342551nt_int @ ( times_times_int @ A @ B ) )
          = ( times_times_int @ ( euclid3395696857347342551nt_int @ A ) @ ( euclid3395696857347342551nt_int @ B ) ) ) ) ) ).

% division_segment_mult
thf(fact_997_division__segment__mult,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( B != zero_z3403309356797280102nteger )
       => ( ( euclid6289375185220004616nteger @ ( times_3573771949741848930nteger @ A @ B ) )
          = ( times_3573771949741848930nteger @ ( euclid6289375185220004616nteger @ A ) @ ( euclid6289375185220004616nteger @ B ) ) ) ) ) ).

% division_segment_mult
thf(fact_998_division__segment__mult,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( A != zero_z2226904508553997617atural )
     => ( ( B != zero_z2226904508553997617atural )
       => ( ( euclid5112970336976722131atural @ ( times_2397367101498566445atural @ A @ B ) )
          = ( times_2397367101498566445atural @ ( euclid5112970336976722131atural @ A ) @ ( euclid5112970336976722131atural @ B ) ) ) ) ) ).

% division_segment_mult
thf(fact_999_fact__ge__1,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ one_one_nat @ ( semiri1408675320244567234ct_nat @ N ) ) ).

% fact_ge_1
thf(fact_1000_fact__ge__1,axiom,
    ! [N: nat] : ( ord_le1926595141338095240atural @ one_one_Code_natural @ ( semiri2447717529341329178atural @ N ) ) ).

% fact_ge_1
thf(fact_1001_fact__ge__1,axiom,
    ! [N: nat] : ( ord_less_eq_rat @ one_one_rat @ ( semiri773545260158071498ct_rat @ N ) ) ).

% fact_ge_1
thf(fact_1002_fact__ge__1,axiom,
    ! [N: nat] : ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( semiri3624122377584611663nteger @ N ) ) ).

% fact_ge_1
thf(fact_1003_fact__ge__1,axiom,
    ! [N: nat] : ( ord_less_eq_int @ one_one_int @ ( semiri1406184849735516958ct_int @ N ) ) ).

% fact_ge_1
thf(fact_1004_int__ops_I1_J,axiom,
    ( ( semiri1314217659103216013at_int @ zero_zero_nat )
    = zero_zero_int ) ).

% int_ops(1)
thf(fact_1005_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_1006_effect__returnI,axiom,
    ! [H2: heap_e7401611519738050253t_unit,H: heap_e7401611519738050253t_unit,X: nat] :
      ( ( H2 = H )
     => ( heap_Time_effect_nat @ ( heap_Time_return_nat @ X ) @ H2 @ H @ X @ one_one_nat ) ) ).

% effect_returnI
thf(fact_1007_negative__eq__positive,axiom,
    ! [N: nat,M: nat] :
      ( ( ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) )
        = ( semiri1314217659103216013at_int @ M ) )
      = ( ( N = zero_zero_nat )
        & ( M = zero_zero_nat ) ) ) ).

% negative_eq_positive
thf(fact_1008_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_1009_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_1010_compl__less__compl__iff,axiom,
    ! [X: product_unit,Y3: product_unit] :
      ( ( ord_le361264281704409273t_unit @ ( uminus2952777764628376836t_unit @ X ) @ ( uminus2952777764628376836t_unit @ Y3 ) )
      = ( ord_le361264281704409273t_unit @ Y3 @ X ) ) ).

% compl_less_compl_iff
thf(fact_1011_compl__le__compl__iff,axiom,
    ! [X: product_unit,Y3: product_unit] :
      ( ( ord_le3221252021190050221t_unit @ ( uminus2952777764628376836t_unit @ X ) @ ( uminus2952777764628376836t_unit @ Y3 ) )
      = ( ord_le3221252021190050221t_unit @ Y3 @ X ) ) ).

% compl_le_compl_iff
thf(fact_1012_compl__le__compl__iff,axiom,
    ! [X: set_nat,Y3: set_nat] :
      ( ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ ( uminus5710092332889474511et_nat @ Y3 ) )
      = ( ord_less_eq_set_nat @ Y3 @ X ) ) ).

% compl_le_compl_iff
thf(fact_1013_field__le__mult__one__interval,axiom,
    ! [X: rat,Y3: rat] :
      ( ! [Z3: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ Z3 )
         => ( ( ord_less_rat @ Z3 @ one_one_rat )
           => ( ord_less_eq_rat @ ( times_times_rat @ Z3 @ X ) @ Y3 ) ) )
     => ( ord_less_eq_rat @ X @ Y3 ) ) ).

% field_le_mult_one_interval
thf(fact_1014_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_1015_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_1016_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_1017_negative__zle,axiom,
    ! [N: nat,M: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).

% negative_zle
thf(fact_1018_boolean__algebra__class_Oboolean__algebra_Odouble__compl,axiom,
    ! [X: product_unit] :
      ( ( uminus2952777764628376836t_unit @ ( uminus2952777764628376836t_unit @ X ) )
      = X ) ).

% boolean_algebra_class.boolean_algebra.double_compl
thf(fact_1019_boolean__algebra__class_Oboolean__algebra_Ocompl__eq__compl__iff,axiom,
    ! [X: product_unit,Y3: product_unit] :
      ( ( ( uminus2952777764628376836t_unit @ X )
        = ( uminus2952777764628376836t_unit @ Y3 ) )
      = ( X = Y3 ) ) ).

% boolean_algebra_class.boolean_algebra.compl_eq_compl_iff
thf(fact_1020_negative__zless,axiom,
    ! [N: nat,M: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).

% negative_zless
thf(fact_1021_verit__la__generic,axiom,
    ! [A: int,X: int] :
      ( ( ord_less_eq_int @ A @ X )
      | ( A = X )
      | ( ord_less_eq_int @ X @ A ) ) ).

% verit_la_generic
thf(fact_1022_Nat_Oex__has__greatest__nat,axiom,
    ! [P: nat > $o,K: nat,B: nat] :
      ( ( P @ K )
     => ( ! [Y4: nat] :
            ( ( P @ Y4 )
           => ( ord_less_eq_nat @ Y4 @ B ) )
       => ? [X3: nat] :
            ( ( P @ X3 )
            & ! [Y5: nat] :
                ( ( P @ Y5 )
               => ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ) ).

% Nat.ex_has_greatest_nat
thf(fact_1023_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_1024_mult__le__mono2,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I2 @ J )
     => ( ord_less_eq_nat @ ( times_times_nat @ K @ I2 ) @ ( times_times_nat @ K @ J ) ) ) ).

% mult_le_mono2
thf(fact_1025_mult__le__mono1,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I2 @ J )
     => ( ord_less_eq_nat @ ( times_times_nat @ I2 @ K ) @ ( times_times_nat @ J @ K ) ) ) ).

% mult_le_mono1
thf(fact_1026_mult__le__mono,axiom,
    ! [I2: nat,J: nat,K: nat,L: nat] :
      ( ( ord_less_eq_nat @ I2 @ J )
     => ( ( ord_less_eq_nat @ K @ L )
       => ( ord_less_eq_nat @ ( times_times_nat @ I2 @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).

% mult_le_mono
thf(fact_1027_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_1028_le__square,axiom,
    ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).

% le_square
thf(fact_1029_eq__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( M = N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% eq_imp_le
thf(fact_1030_le__trans,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I2 @ J )
     => ( ( ord_less_eq_nat @ J @ K )
       => ( ord_less_eq_nat @ I2 @ K ) ) ) ).

% le_trans
thf(fact_1031_le__refl,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).

% le_refl
thf(fact_1032_le__cube,axiom,
    ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).

% le_cube
thf(fact_1033_less__int__code_I1_J,axiom,
    ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).

% less_int_code(1)
thf(fact_1034_zmult__zless__mono2,axiom,
    ! [I2: int,J: int,K: int] :
      ( ( ord_less_int @ I2 @ J )
     => ( ( ord_less_int @ zero_zero_int @ K )
       => ( ord_less_int @ ( times_times_int @ K @ I2 ) @ ( times_times_int @ K @ J ) ) ) ) ).

% zmult_zless_mono2
thf(fact_1035_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_1036_int__one__le__iff__zero__less,axiom,
    ! [Z2: int] :
      ( ( ord_less_eq_int @ one_one_int @ Z2 )
      = ( ord_less_int @ zero_zero_int @ Z2 ) ) ).

% int_one_le_iff_zero_less
thf(fact_1037_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_1038_zmult__zless__mono2__lemma,axiom,
    ! [I2: int,J: int,K: nat] :
      ( ( ord_less_int @ I2 @ J )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ord_less_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ I2 ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ J ) ) ) ) ).

% zmult_zless_mono2_lemma
thf(fact_1039_negative__zless__0,axiom,
    ! [N: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ zero_zero_int ) ).

% negative_zless_0
thf(fact_1040_negD,axiom,
    ! [X: int] :
      ( ( ord_less_int @ X @ zero_zero_int )
     => ? [N4: nat] :
          ( X
          = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N4 ) ) ) ) ) ).

% negD
thf(fact_1041_nat__int__comparison_I2_J,axiom,
    ( ord_less_nat
    = ( ^ [A2: nat,B2: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).

% nat_int_comparison(2)
thf(fact_1042_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_1043_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_1044_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_1045_linordered__field__no__ub,axiom,
    ! [X4: rat] :
    ? [X_12: rat] : ( ord_less_rat @ X4 @ X_12 ) ).

% linordered_field_no_ub
thf(fact_1046_linordered__field__no__lb,axiom,
    ! [X4: rat] :
    ? [Y4: rat] : ( ord_less_rat @ Y4 @ X4 ) ).

% linordered_field_no_lb
thf(fact_1047_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_1048_times__int__code_I2_J,axiom,
    ! [L: int] :
      ( ( times_times_int @ zero_zero_int @ L )
      = zero_zero_int ) ).

% times_int_code(2)
thf(fact_1049_times__int__code_I1_J,axiom,
    ! [K: int] :
      ( ( times_times_int @ K @ zero_zero_int )
      = zero_zero_int ) ).

% times_int_code(1)
thf(fact_1050_uminus__int__code_I1_J,axiom,
    ( ( uminus_uminus_int @ zero_zero_int )
    = zero_zero_int ) ).

% uminus_int_code(1)
thf(fact_1051_less__eq__int__code_I1_J,axiom,
    ord_less_eq_int @ zero_zero_int @ zero_zero_int ).

% less_eq_int_code(1)
thf(fact_1052_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_1053_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_1054_int__int__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ( semiri1314217659103216013at_int @ M )
        = ( semiri1314217659103216013at_int @ N ) )
      = ( M = N ) ) ).

% int_int_eq
thf(fact_1055_int__cases2,axiom,
    ! [Z2: int] :
      ( ! [N4: nat] :
          ( Z2
         != ( semiri1314217659103216013at_int @ N4 ) )
     => ~ ! [N4: nat] :
            ( Z2
           != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N4 ) ) ) ) ).

% int_cases2
thf(fact_1056_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_1057_compl__le__swap2,axiom,
    ! [Y3: product_unit,X: product_unit] :
      ( ( ord_le3221252021190050221t_unit @ ( uminus2952777764628376836t_unit @ Y3 ) @ X )
     => ( ord_le3221252021190050221t_unit @ ( uminus2952777764628376836t_unit @ X ) @ Y3 ) ) ).

% compl_le_swap2
thf(fact_1058_compl__le__swap2,axiom,
    ! [Y3: set_nat,X: set_nat] :
      ( ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ Y3 ) @ X )
     => ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ Y3 ) ) ).

% compl_le_swap2
thf(fact_1059_compl__le__swap1,axiom,
    ! [Y3: product_unit,X: product_unit] :
      ( ( ord_le3221252021190050221t_unit @ Y3 @ ( uminus2952777764628376836t_unit @ X ) )
     => ( ord_le3221252021190050221t_unit @ X @ ( uminus2952777764628376836t_unit @ Y3 ) ) ) ).

% compl_le_swap1
thf(fact_1060_compl__le__swap1,axiom,
    ! [Y3: set_nat,X: set_nat] :
      ( ( ord_less_eq_set_nat @ Y3 @ ( uminus5710092332889474511et_nat @ X ) )
     => ( ord_less_eq_set_nat @ X @ ( uminus5710092332889474511et_nat @ Y3 ) ) ) ).

% compl_le_swap1
thf(fact_1061_compl__mono,axiom,
    ! [X: product_unit,Y3: product_unit] :
      ( ( ord_le3221252021190050221t_unit @ X @ Y3 )
     => ( ord_le3221252021190050221t_unit @ ( uminus2952777764628376836t_unit @ Y3 ) @ ( uminus2952777764628376836t_unit @ X ) ) ) ).

% compl_mono
thf(fact_1062_compl__mono,axiom,
    ! [X: set_nat,Y3: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ Y3 )
     => ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ Y3 ) @ ( uminus5710092332889474511et_nat @ X ) ) ) ).

% compl_mono
thf(fact_1063_compl__less__swap2,axiom,
    ! [Y3: product_unit,X: product_unit] :
      ( ( ord_le361264281704409273t_unit @ ( uminus2952777764628376836t_unit @ Y3 ) @ X )
     => ( ord_le361264281704409273t_unit @ ( uminus2952777764628376836t_unit @ X ) @ Y3 ) ) ).

% compl_less_swap2
thf(fact_1064_compl__less__swap1,axiom,
    ! [Y3: product_unit,X: product_unit] :
      ( ( ord_le361264281704409273t_unit @ Y3 @ ( uminus2952777764628376836t_unit @ X ) )
     => ( ord_le361264281704409273t_unit @ X @ ( uminus2952777764628376836t_unit @ Y3 ) ) ) ).

% compl_less_swap1
thf(fact_1065_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_1066_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_1067_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_1068_int__of__nat__induct,axiom,
    ! [P: int > $o,Z2: int] :
      ( ! [N4: nat] : ( P @ ( semiri1314217659103216013at_int @ N4 ) )
     => ( ! [N4: nat] : ( P @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N4 ) ) ) )
       => ( P @ Z2 ) ) ) ).

% int_of_nat_induct
thf(fact_1069_int__cases,axiom,
    ! [Z2: int] :
      ( ! [N4: nat] :
          ( Z2
         != ( semiri1314217659103216013at_int @ N4 ) )
     => ~ ! [N4: nat] :
            ( Z2
           != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N4 ) ) ) ) ) ).

% int_cases
thf(fact_1070_int__zle__neg,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) )
      = ( ( N = zero_zero_nat )
        & ( M = zero_zero_nat ) ) ) ).

% int_zle_neg
thf(fact_1071_zero__le__imp__eq__int,axiom,
    ! [K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ? [N4: nat] :
          ( K
          = ( semiri1314217659103216013at_int @ N4 ) ) ) ).

% zero_le_imp_eq_int
thf(fact_1072_nonpos__int__cases,axiom,
    ! [K: int] :
      ( ( ord_less_eq_int @ K @ zero_zero_int )
     => ~ ! [N4: nat] :
            ( K
           != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N4 ) ) ) ) ).

% nonpos_int_cases
thf(fact_1073_nonneg__int__cases,axiom,
    ! [K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ~ ! [N4: nat] :
            ( K
           != ( semiri1314217659103216013at_int @ N4 ) ) ) ).

% nonneg_int_cases
thf(fact_1074_negative__zle__0,axiom,
    ! [N: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ zero_zero_int ) ).

% negative_zle_0
thf(fact_1075_dual__order_Orefl,axiom,
    ! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).

% dual_order.refl
thf(fact_1076_dual__order_Orefl,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ A @ A ) ).

% dual_order.refl
thf(fact_1077_dual__order_Orefl,axiom,
    ! [A: num] : ( ord_less_eq_num @ A @ A ) ).

% dual_order.refl
thf(fact_1078_dual__order_Orefl,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).

% dual_order.refl
thf(fact_1079_dual__order_Orefl,axiom,
    ! [A: int] : ( ord_less_eq_int @ A @ A ) ).

% dual_order.refl
thf(fact_1080_order__refl,axiom,
    ! [X: set_nat] : ( ord_less_eq_set_nat @ X @ X ) ).

% order_refl
thf(fact_1081_order__refl,axiom,
    ! [X: rat] : ( ord_less_eq_rat @ X @ X ) ).

% order_refl
thf(fact_1082_order__refl,axiom,
    ! [X: num] : ( ord_less_eq_num @ X @ X ) ).

% order_refl
thf(fact_1083_order__refl,axiom,
    ! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).

% order_refl
thf(fact_1084_order__refl,axiom,
    ! [X: int] : ( ord_less_eq_int @ X @ X ) ).

% order_refl
thf(fact_1085_imp__le__cong,axiom,
    ! [X: int,X5: int,P: $o,P2: $o] :
      ( ( X = X5 )
     => ( ( ( ord_less_eq_int @ zero_zero_int @ X5 )
         => ( P = P2 ) )
       => ( ( ( ord_less_eq_int @ zero_zero_int @ X )
           => P )
          = ( ( ord_less_eq_int @ zero_zero_int @ X5 )
           => P2 ) ) ) ) ).

% imp_le_cong
thf(fact_1086_conj__le__cong,axiom,
    ! [X: int,X5: int,P: $o,P2: $o] :
      ( ( X = X5 )
     => ( ( ( ord_less_eq_int @ zero_zero_int @ X5 )
         => ( P = P2 ) )
       => ( ( ( ord_less_eq_int @ zero_zero_int @ X )
            & P )
          = ( ( ord_less_eq_int @ zero_zero_int @ X5 )
            & P2 ) ) ) ) ).

% conj_le_cong
thf(fact_1087_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 )
           => ( ! [I: nat] :
                  ( ( ord_less_nat @ K2 @ I )
                 => ( P @ I ) )
             => ( P @ K2 ) ) )
       => ( P @ M ) ) ) ).

% nat_descend_induct
thf(fact_1088_list__decode_Ocases,axiom,
    ! [X: nat] :
      ( ( X != zero_zero_nat )
     => ~ ! [N4: nat] :
            ( X
           != ( suc @ N4 ) ) ) ).

% list_decode.cases
thf(fact_1089_order__le__imp__less__or__eq,axiom,
    ! [X: code_integer,Y3: code_integer] :
      ( ( ord_le3102999989581377725nteger @ X @ Y3 )
     => ( ( ord_le6747313008572928689nteger @ X @ Y3 )
        | ( X = Y3 ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_1090_order__le__imp__less__or__eq,axiom,
    ! [X: set_nat,Y3: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ Y3 )
     => ( ( ord_less_set_nat @ X @ Y3 )
        | ( X = Y3 ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_1091_order__le__imp__less__or__eq,axiom,
    ! [X: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ X @ Y3 )
     => ( ( ord_less_rat @ X @ Y3 )
        | ( X = Y3 ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_1092_order__le__imp__less__or__eq,axiom,
    ! [X: num,Y3: num] :
      ( ( ord_less_eq_num @ X @ Y3 )
     => ( ( ord_less_num @ X @ Y3 )
        | ( X = Y3 ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_1093_order__le__imp__less__or__eq,axiom,
    ! [X: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ X @ Y3 )
     => ( ( ord_less_nat @ X @ Y3 )
        | ( X = Y3 ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_1094_order__le__imp__less__or__eq,axiom,
    ! [X: int,Y3: int] :
      ( ( ord_less_eq_int @ X @ Y3 )
     => ( ( ord_less_int @ X @ Y3 )
        | ( X = Y3 ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_1095_nle__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ~ ( ord_less_eq_rat @ A @ B ) )
      = ( ( ord_less_eq_rat @ B @ A )
        & ( B != A ) ) ) ).

% nle_le
thf(fact_1096_nle__le,axiom,
    ! [A: num,B: num] :
      ( ( ~ ( ord_less_eq_num @ A @ B ) )
      = ( ( ord_less_eq_num @ B @ A )
        & ( B != A ) ) ) ).

% nle_le
thf(fact_1097_nle__le,axiom,
    ! [A: nat,B: nat] :
      ( ( ~ ( ord_less_eq_nat @ A @ B ) )
      = ( ( ord_less_eq_nat @ B @ A )
        & ( B != A ) ) ) ).

% nle_le
thf(fact_1098_nle__le,axiom,
    ! [A: int,B: int] :
      ( ( ~ ( ord_less_eq_int @ A @ B ) )
      = ( ( ord_less_eq_int @ B @ A )
        & ( B != A ) ) ) ).

% nle_le
thf(fact_1099_le__cases3,axiom,
    ! [X: rat,Y3: rat,Z2: rat] :
      ( ( ( ord_less_eq_rat @ X @ Y3 )
       => ~ ( ord_less_eq_rat @ Y3 @ Z2 ) )
     => ( ( ( ord_less_eq_rat @ Y3 @ X )
         => ~ ( ord_less_eq_rat @ X @ Z2 ) )
       => ( ( ( ord_less_eq_rat @ X @ Z2 )
           => ~ ( ord_less_eq_rat @ Z2 @ Y3 ) )
         => ( ( ( ord_less_eq_rat @ Z2 @ Y3 )
             => ~ ( ord_less_eq_rat @ Y3 @ X ) )
           => ( ( ( ord_less_eq_rat @ Y3 @ Z2 )
               => ~ ( ord_less_eq_rat @ Z2 @ X ) )
             => ~ ( ( ord_less_eq_rat @ Z2 @ X )
                 => ~ ( ord_less_eq_rat @ X @ Y3 ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_1100_le__cases3,axiom,
    ! [X: num,Y3: num,Z2: num] :
      ( ( ( ord_less_eq_num @ X @ Y3 )
       => ~ ( ord_less_eq_num @ Y3 @ Z2 ) )
     => ( ( ( ord_less_eq_num @ Y3 @ X )
         => ~ ( ord_less_eq_num @ X @ Z2 ) )
       => ( ( ( ord_less_eq_num @ X @ Z2 )
           => ~ ( ord_less_eq_num @ Z2 @ Y3 ) )
         => ( ( ( ord_less_eq_num @ Z2 @ Y3 )
             => ~ ( ord_less_eq_num @ Y3 @ X ) )
           => ( ( ( ord_less_eq_num @ Y3 @ Z2 )
               => ~ ( ord_less_eq_num @ Z2 @ X ) )
             => ~ ( ( ord_less_eq_num @ Z2 @ X )
                 => ~ ( ord_less_eq_num @ X @ Y3 ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_1101_le__cases3,axiom,
    ! [X: nat,Y3: nat,Z2: nat] :
      ( ( ( ord_less_eq_nat @ X @ Y3 )
       => ~ ( ord_less_eq_nat @ Y3 @ Z2 ) )
     => ( ( ( ord_less_eq_nat @ Y3 @ X )
         => ~ ( ord_less_eq_nat @ X @ Z2 ) )
       => ( ( ( ord_less_eq_nat @ X @ Z2 )
           => ~ ( ord_less_eq_nat @ Z2 @ Y3 ) )
         => ( ( ( ord_less_eq_nat @ Z2 @ Y3 )
             => ~ ( ord_less_eq_nat @ Y3 @ X ) )
           => ( ( ( ord_less_eq_nat @ Y3 @ Z2 )
               => ~ ( ord_less_eq_nat @ Z2 @ X ) )
             => ~ ( ( ord_less_eq_nat @ Z2 @ X )
                 => ~ ( ord_less_eq_nat @ X @ Y3 ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_1102_le__cases3,axiom,
    ! [X: int,Y3: int,Z2: int] :
      ( ( ( ord_less_eq_int @ X @ Y3 )
       => ~ ( ord_less_eq_int @ Y3 @ Z2 ) )
     => ( ( ( ord_less_eq_int @ Y3 @ X )
         => ~ ( ord_less_eq_int @ X @ Z2 ) )
       => ( ( ( ord_less_eq_int @ X @ Z2 )
           => ~ ( ord_less_eq_int @ Z2 @ Y3 ) )
         => ( ( ( ord_less_eq_int @ Z2 @ Y3 )
             => ~ ( ord_less_eq_int @ Y3 @ X ) )
           => ( ( ( ord_less_eq_int @ Y3 @ Z2 )
               => ~ ( ord_less_eq_int @ Z2 @ X ) )
             => ~ ( ( ord_less_eq_int @ Z2 @ X )
                 => ~ ( ord_less_eq_int @ X @ Y3 ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_1103_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y: set_nat,Z: set_nat] : ( Y = Z ) )
    = ( ^ [X2: set_nat,Y6: set_nat] :
          ( ( ord_less_eq_set_nat @ X2 @ Y6 )
          & ( ord_less_eq_set_nat @ Y6 @ X2 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_1104_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y: rat,Z: rat] : ( Y = Z ) )
    = ( ^ [X2: rat,Y6: rat] :
          ( ( ord_less_eq_rat @ X2 @ Y6 )
          & ( ord_less_eq_rat @ Y6 @ X2 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_1105_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y: num,Z: num] : ( Y = Z ) )
    = ( ^ [X2: num,Y6: num] :
          ( ( ord_less_eq_num @ X2 @ Y6 )
          & ( ord_less_eq_num @ Y6 @ X2 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_1106_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y: nat,Z: nat] : ( Y = Z ) )
    = ( ^ [X2: nat,Y6: nat] :
          ( ( ord_less_eq_nat @ X2 @ Y6 )
          & ( ord_less_eq_nat @ Y6 @ X2 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_1107_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y: int,Z: int] : ( Y = Z ) )
    = ( ^ [X2: int,Y6: int] :
          ( ( ord_less_eq_int @ X2 @ Y6 )
          & ( ord_less_eq_int @ Y6 @ X2 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_1108_ord__eq__le__trans,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( ( A = B )
     => ( ( ord_less_eq_set_nat @ B @ C )
       => ( ord_less_eq_set_nat @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_1109_ord__eq__le__trans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( A = B )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ord_less_eq_rat @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_1110_ord__eq__le__trans,axiom,
    ! [A: num,B: num,C: num] :
      ( ( A = B )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ord_less_eq_num @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_1111_ord__eq__le__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A = B )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ord_less_eq_nat @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_1112_ord__eq__le__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A = B )
     => ( ( ord_less_eq_int @ B @ C )
       => ( ord_less_eq_int @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_1113_ord__le__eq__trans,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( B = C )
       => ( ord_less_eq_set_nat @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_1114_ord__le__eq__trans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( B = C )
       => ( ord_less_eq_rat @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_1115_ord__le__eq__trans,axiom,
    ! [A: num,B: num,C: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( B = C )
       => ( ord_less_eq_num @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_1116_ord__le__eq__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( B = C )
       => ( ord_less_eq_nat @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_1117_ord__le__eq__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( B = C )
       => ( ord_less_eq_int @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_1118_order__antisym,axiom,
    ! [X: set_nat,Y3: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ Y3 )
     => ( ( ord_less_eq_set_nat @ Y3 @ X )
       => ( X = Y3 ) ) ) ).

% order_antisym
thf(fact_1119_order__antisym,axiom,
    ! [X: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ X @ Y3 )
     => ( ( ord_less_eq_rat @ Y3 @ X )
       => ( X = Y3 ) ) ) ).

% order_antisym
thf(fact_1120_order__antisym,axiom,
    ! [X: num,Y3: num] :
      ( ( ord_less_eq_num @ X @ Y3 )
     => ( ( ord_less_eq_num @ Y3 @ X )
       => ( X = Y3 ) ) ) ).

% order_antisym
thf(fact_1121_order__antisym,axiom,
    ! [X: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ X @ Y3 )
     => ( ( ord_less_eq_nat @ Y3 @ X )
       => ( X = Y3 ) ) ) ).

% order_antisym
thf(fact_1122_order__antisym,axiom,
    ! [X: int,Y3: int] :
      ( ( ord_less_eq_int @ X @ Y3 )
     => ( ( ord_less_eq_int @ Y3 @ X )
       => ( X = Y3 ) ) ) ).

% order_antisym
thf(fact_1123_order_Otrans,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( ord_less_eq_set_nat @ B @ C )
       => ( ord_less_eq_set_nat @ A @ C ) ) ) ).

% order.trans
thf(fact_1124_order_Otrans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ord_less_eq_rat @ A @ C ) ) ) ).

% order.trans
thf(fact_1125_order_Otrans,axiom,
    ! [A: num,B: num,C: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ord_less_eq_num @ A @ C ) ) ) ).

% order.trans
thf(fact_1126_order_Otrans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ord_less_eq_nat @ A @ C ) ) ) ).

% order.trans
thf(fact_1127_order_Otrans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ B @ C )
       => ( ord_less_eq_int @ A @ C ) ) ) ).

% order.trans
thf(fact_1128_order__trans,axiom,
    ! [X: set_nat,Y3: set_nat,Z2: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ Y3 )
     => ( ( ord_less_eq_set_nat @ Y3 @ Z2 )
       => ( ord_less_eq_set_nat @ X @ Z2 ) ) ) ).

% order_trans
thf(fact_1129_order__trans,axiom,
    ! [X: rat,Y3: rat,Z2: rat] :
      ( ( ord_less_eq_rat @ X @ Y3 )
     => ( ( ord_less_eq_rat @ Y3 @ Z2 )
       => ( ord_less_eq_rat @ X @ Z2 ) ) ) ).

% order_trans
thf(fact_1130_order__trans,axiom,
    ! [X: num,Y3: num,Z2: num] :
      ( ( ord_less_eq_num @ X @ Y3 )
     => ( ( ord_less_eq_num @ Y3 @ Z2 )
       => ( ord_less_eq_num @ X @ Z2 ) ) ) ).

% order_trans
thf(fact_1131_order__trans,axiom,
    ! [X: nat,Y3: nat,Z2: nat] :
      ( ( ord_less_eq_nat @ X @ Y3 )
     => ( ( ord_less_eq_nat @ Y3 @ Z2 )
       => ( ord_less_eq_nat @ X @ Z2 ) ) ) ).

% order_trans
thf(fact_1132_order__trans,axiom,
    ! [X: int,Y3: int,Z2: int] :
      ( ( ord_less_eq_int @ X @ Y3 )
     => ( ( ord_less_eq_int @ Y3 @ Z2 )
       => ( ord_less_eq_int @ X @ Z2 ) ) ) ).

% order_trans
thf(fact_1133_linorder__wlog,axiom,
    ! [P: rat > rat > $o,A: rat,B: rat] :
      ( ! [A5: rat,B4: rat] :
          ( ( ord_less_eq_rat @ A5 @ B4 )
         => ( P @ A5 @ B4 ) )
     => ( ! [A5: rat,B4: rat] :
            ( ( P @ B4 @ A5 )
           => ( P @ A5 @ B4 ) )
       => ( P @ A @ B ) ) ) ).

% linorder_wlog
thf(fact_1134_linorder__wlog,axiom,
    ! [P: num > num > $o,A: num,B: num] :
      ( ! [A5: num,B4: num] :
          ( ( ord_less_eq_num @ A5 @ B4 )
         => ( P @ A5 @ B4 ) )
     => ( ! [A5: num,B4: num] :
            ( ( P @ B4 @ A5 )
           => ( P @ A5 @ B4 ) )
       => ( P @ A @ B ) ) ) ).

% linorder_wlog
thf(fact_1135_linorder__wlog,axiom,
    ! [P: nat > nat > $o,A: nat,B: nat] :
      ( ! [A5: nat,B4: nat] :
          ( ( ord_less_eq_nat @ A5 @ B4 )
         => ( P @ A5 @ B4 ) )
     => ( ! [A5: nat,B4: nat] :
            ( ( P @ B4 @ A5 )
           => ( P @ A5 @ B4 ) )
       => ( P @ A @ B ) ) ) ).

% linorder_wlog
thf(fact_1136_linorder__wlog,axiom,
    ! [P: int > int > $o,A: int,B: int] :
      ( ! [A5: int,B4: int] :
          ( ( ord_less_eq_int @ A5 @ B4 )
         => ( P @ A5 @ B4 ) )
     => ( ! [A5: int,B4: int] :
            ( ( P @ B4 @ A5 )
           => ( P @ A5 @ B4 ) )
       => ( P @ A @ B ) ) ) ).

% linorder_wlog
thf(fact_1137_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y: set_nat,Z: set_nat] : ( Y = Z ) )
    = ( ^ [A2: set_nat,B2: set_nat] :
          ( ( ord_less_eq_set_nat @ B2 @ A2 )
          & ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_1138_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y: rat,Z: rat] : ( Y = Z ) )
    = ( ^ [A2: rat,B2: rat] :
          ( ( ord_less_eq_rat @ B2 @ A2 )
          & ( ord_less_eq_rat @ A2 @ B2 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_1139_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y: num,Z: num] : ( Y = Z ) )
    = ( ^ [A2: num,B2: num] :
          ( ( ord_less_eq_num @ B2 @ A2 )
          & ( ord_less_eq_num @ A2 @ B2 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_1140_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y: nat,Z: nat] : ( Y = Z ) )
    = ( ^ [A2: nat,B2: nat] :
          ( ( ord_less_eq_nat @ B2 @ A2 )
          & ( ord_less_eq_nat @ A2 @ B2 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_1141_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y: int,Z: int] : ( Y = Z ) )
    = ( ^ [A2: int,B2: int] :
          ( ( ord_less_eq_int @ B2 @ A2 )
          & ( ord_less_eq_int @ A2 @ B2 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_1142_dual__order_Oantisym,axiom,
    ! [B: set_nat,A: set_nat] :
      ( ( ord_less_eq_set_nat @ B @ A )
     => ( ( ord_less_eq_set_nat @ A @ B )
       => ( A = B ) ) ) ).

% dual_order.antisym
thf(fact_1143_dual__order_Oantisym,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( ord_less_eq_rat @ A @ B )
       => ( A = B ) ) ) ).

% dual_order.antisym
thf(fact_1144_dual__order_Oantisym,axiom,
    ! [B: num,A: num] :
      ( ( ord_less_eq_num @ B @ A )
     => ( ( ord_less_eq_num @ A @ B )
       => ( A = B ) ) ) ).

% dual_order.antisym
thf(fact_1145_dual__order_Oantisym,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_eq_nat @ A @ B )
       => ( A = B ) ) ) ).

% dual_order.antisym
thf(fact_1146_dual__order_Oantisym,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_less_eq_int @ A @ B )
       => ( A = B ) ) ) ).

% dual_order.antisym
thf(fact_1147_dual__order_Otrans,axiom,
    ! [B: set_nat,A: set_nat,C: set_nat] :
      ( ( ord_less_eq_set_nat @ B @ A )
     => ( ( ord_less_eq_set_nat @ C @ B )
       => ( ord_less_eq_set_nat @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_1148_dual__order_Otrans,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( ord_less_eq_rat @ C @ B )
       => ( ord_less_eq_rat @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_1149_dual__order_Otrans,axiom,
    ! [B: num,A: num,C: num] :
      ( ( ord_less_eq_num @ B @ A )
     => ( ( ord_less_eq_num @ C @ B )
       => ( ord_less_eq_num @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_1150_dual__order_Otrans,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_eq_nat @ C @ B )
       => ( ord_less_eq_nat @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_1151_dual__order_Otrans,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_less_eq_int @ C @ B )
       => ( ord_less_eq_int @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_1152_antisym,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( ord_less_eq_set_nat @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_1153_antisym,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_1154_antisym,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_eq_num @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_1155_antisym,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_1156_antisym,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_1157_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y: set_nat,Z: set_nat] : ( Y = Z ) )
    = ( ^ [A2: set_nat,B2: set_nat] :
          ( ( ord_less_eq_set_nat @ A2 @ B2 )
          & ( ord_less_eq_set_nat @ B2 @ A2 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_1158_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y: rat,Z: rat] : ( Y = Z ) )
    = ( ^ [A2: rat,B2: rat] :
          ( ( ord_less_eq_rat @ A2 @ B2 )
          & ( ord_less_eq_rat @ B2 @ A2 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_1159_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y: num,Z: num] : ( Y = Z ) )
    = ( ^ [A2: num,B2: num] :
          ( ( ord_less_eq_num @ A2 @ B2 )
          & ( ord_less_eq_num @ B2 @ A2 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_1160_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y: nat,Z: nat] : ( Y = Z ) )
    = ( ^ [A2: nat,B2: nat] :
          ( ( ord_less_eq_nat @ A2 @ B2 )
          & ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_1161_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y: int,Z: int] : ( Y = Z ) )
    = ( ^ [A2: int,B2: int] :
          ( ( ord_less_eq_int @ A2 @ B2 )
          & ( ord_less_eq_int @ B2 @ A2 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_1162_order__subst1,axiom,
    ! [A: rat,F: rat > rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ 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_1163_order__subst1,axiom,
    ! [A: rat,F: num > rat,B: num,C: num] :
      ( ( ord_less_eq_rat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ 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_1164_order__subst1,axiom,
    ! [A: rat,F: nat > rat,B: nat,C: nat] :
      ( ( ord_less_eq_rat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ 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_1165_order__subst1,axiom,
    ! [A: rat,F: int > rat,B: int,C: int] :
      ( ( ord_less_eq_rat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_int @ B @ 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_1166_order__subst1,axiom,
    ! [A: num,F: rat > num,B: rat,C: rat] :
      ( ( ord_less_eq_num @ A @ ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ 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_1167_order__subst1,axiom,
    ! [A: num,F: num > num,B: num,C: num] :
      ( ( ord_less_eq_num @ A @ ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ 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_1168_order__subst1,axiom,
    ! [A: num,F: nat > num,B: nat,C: nat] :
      ( ( ord_less_eq_num @ A @ ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ 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_1169_order__subst1,axiom,
    ! [A: num,F: int > num,B: int,C: int] :
      ( ( ord_less_eq_num @ A @ ( F @ B ) )
     => ( ( ord_less_eq_int @ B @ 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_1170_order__subst1,axiom,
    ! [A: nat,F: rat > nat,B: rat,C: rat] :
      ( ( ord_less_eq_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ 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_1171_order__subst1,axiom,
    ! [A: nat,F: num > nat,B: num,C: num] :
      ( ( ord_less_eq_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ 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_1172_order__subst2,axiom,
    ! [A: rat,B: rat,F: rat > rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ ( F @ B ) @ 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_1173_order__subst2,axiom,
    ! [A: rat,B: rat,F: rat > num,C: num] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_num @ ( F @ B ) @ 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_1174_order__subst2,axiom,
    ! [A: rat,B: rat,F: rat > nat,C: nat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_nat @ ( F @ B ) @ 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_1175_order__subst2,axiom,
    ! [A: rat,B: rat,F: rat > int,C: int] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_int @ ( F @ B ) @ 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_1176_order__subst2,axiom,
    ! [A: num,B: num,F: num > rat,C: rat] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_eq_rat @ ( F @ B ) @ 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_1177_order__subst2,axiom,
    ! [A: num,B: num,F: num > num,C: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_eq_num @ ( F @ B ) @ 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_1178_order__subst2,axiom,
    ! [A: num,B: num,F: num > nat,C: nat] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_eq_nat @ ( F @ B ) @ 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_1179_order__subst2,axiom,
    ! [A: num,B: num,F: num > int,C: int] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_eq_int @ ( F @ B ) @ 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_1180_order__subst2,axiom,
    ! [A: nat,B: nat,F: nat > rat,C: rat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_rat @ ( F @ B ) @ 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_1181_order__subst2,axiom,
    ! [A: nat,B: nat,F: nat > num,C: num] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_num @ ( F @ B ) @ 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_1182_order__eq__refl,axiom,
    ! [X: set_nat,Y3: set_nat] :
      ( ( X = Y3 )
     => ( ord_less_eq_set_nat @ X @ Y3 ) ) ).

% order_eq_refl
thf(fact_1183_order__eq__refl,axiom,
    ! [X: rat,Y3: rat] :
      ( ( X = Y3 )
     => ( ord_less_eq_rat @ X @ Y3 ) ) ).

% order_eq_refl
thf(fact_1184_order__eq__refl,axiom,
    ! [X: num,Y3: num] :
      ( ( X = Y3 )
     => ( ord_less_eq_num @ X @ Y3 ) ) ).

% order_eq_refl
thf(fact_1185_order__eq__refl,axiom,
    ! [X: nat,Y3: nat] :
      ( ( X = Y3 )
     => ( ord_less_eq_nat @ X @ Y3 ) ) ).

% order_eq_refl
thf(fact_1186_order__eq__refl,axiom,
    ! [X: int,Y3: int] :
      ( ( X = Y3 )
     => ( ord_less_eq_int @ X @ Y3 ) ) ).

% order_eq_refl
thf(fact_1187_linorder__linear,axiom,
    ! [X: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ X @ Y3 )
      | ( ord_less_eq_rat @ Y3 @ X ) ) ).

% linorder_linear
thf(fact_1188_linorder__linear,axiom,
    ! [X: num,Y3: num] :
      ( ( ord_less_eq_num @ X @ Y3 )
      | ( ord_less_eq_num @ Y3 @ X ) ) ).

% linorder_linear
thf(fact_1189_linorder__linear,axiom,
    ! [X: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ X @ Y3 )
      | ( ord_less_eq_nat @ Y3 @ X ) ) ).

% linorder_linear
thf(fact_1190_linorder__linear,axiom,
    ! [X: int,Y3: int] :
      ( ( ord_less_eq_int @ X @ Y3 )
      | ( ord_less_eq_int @ Y3 @ X ) ) ).

% linorder_linear
thf(fact_1191_ord__eq__le__subst,axiom,
    ! [A: rat,F: rat > rat,B: rat,C: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ 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_1192_ord__eq__le__subst,axiom,
    ! [A: num,F: rat > num,B: rat,C: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ 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_1193_ord__eq__le__subst,axiom,
    ! [A: nat,F: rat > nat,B: rat,C: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ 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_1194_ord__eq__le__subst,axiom,
    ! [A: int,F: rat > int,B: rat,C: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ 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_1195_ord__eq__le__subst,axiom,
    ! [A: rat,F: num > rat,B: num,C: num] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ 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_1196_ord__eq__le__subst,axiom,
    ! [A: num,F: num > num,B: num,C: num] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ 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_1197_ord__eq__le__subst,axiom,
    ! [A: nat,F: num > nat,B: num,C: num] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ 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_1198_ord__eq__le__subst,axiom,
    ! [A: int,F: num > int,B: num,C: num] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ 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_1199_ord__eq__le__subst,axiom,
    ! [A: rat,F: nat > rat,B: nat,C: nat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ 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_1200_ord__eq__le__subst,axiom,
    ! [A: num,F: nat > num,B: nat,C: nat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ 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_1201_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ( F @ B )
          = 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_1202_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > num,C: num] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ( F @ B )
          = 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_1203_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > nat,C: nat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ( F @ B )
          = 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_1204_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > int,C: int] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ( F @ B )
          = 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_1205_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: num > rat,C: rat] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ( F @ B )
          = 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_1206_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: num > num,C: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ( F @ B )
          = 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_1207_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: num > nat,C: nat] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ( F @ B )
          = 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_1208_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: num > int,C: int] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ( F @ B )
          = 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_1209_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F: nat > rat,C: rat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ( F @ B )
          = 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_1210_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F: nat > num,C: num] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ( F @ B )
          = 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_1211_linorder__le__cases,axiom,
    ! [X: rat,Y3: rat] :
      ( ~ ( ord_less_eq_rat @ X @ Y3 )
     => ( ord_less_eq_rat @ Y3 @ X ) ) ).

% linorder_le_cases
thf(fact_1212_linorder__le__cases,axiom,
    ! [X: num,Y3: num] :
      ( ~ ( ord_less_eq_num @ X @ Y3 )
     => ( ord_less_eq_num @ Y3 @ X ) ) ).

% linorder_le_cases
thf(fact_1213_linorder__le__cases,axiom,
    ! [X: nat,Y3: nat] :
      ( ~ ( ord_less_eq_nat @ X @ Y3 )
     => ( ord_less_eq_nat @ Y3 @ X ) ) ).

% linorder_le_cases
thf(fact_1214_linorder__le__cases,axiom,
    ! [X: int,Y3: int] :
      ( ~ ( ord_less_eq_int @ X @ Y3 )
     => ( ord_less_eq_int @ Y3 @ X ) ) ).

% linorder_le_cases
thf(fact_1215_order__antisym__conv,axiom,
    ! [Y3: set_nat,X: set_nat] :
      ( ( ord_less_eq_set_nat @ Y3 @ X )
     => ( ( ord_less_eq_set_nat @ X @ Y3 )
        = ( X = Y3 ) ) ) ).

% order_antisym_conv
thf(fact_1216_order__antisym__conv,axiom,
    ! [Y3: rat,X: rat] :
      ( ( ord_less_eq_rat @ Y3 @ X )
     => ( ( ord_less_eq_rat @ X @ Y3 )
        = ( X = Y3 ) ) ) ).

% order_antisym_conv
thf(fact_1217_order__antisym__conv,axiom,
    ! [Y3: num,X: num] :
      ( ( ord_less_eq_num @ Y3 @ X )
     => ( ( ord_less_eq_num @ X @ Y3 )
        = ( X = Y3 ) ) ) ).

% order_antisym_conv
thf(fact_1218_order__antisym__conv,axiom,
    ! [Y3: nat,X: nat] :
      ( ( ord_less_eq_nat @ Y3 @ X )
     => ( ( ord_less_eq_nat @ X @ Y3 )
        = ( X = Y3 ) ) ) ).

% order_antisym_conv
thf(fact_1219_order__antisym__conv,axiom,
    ! [Y3: int,X: int] :
      ( ( ord_less_eq_int @ Y3 @ X )
     => ( ( ord_less_eq_int @ X @ Y3 )
        = ( X = Y3 ) ) ) ).

% order_antisym_conv
thf(fact_1220_minf_I7_J,axiom,
    ! [T: code_integer] :
    ? [Z3: code_integer] :
    ! [X4: code_integer] :
      ( ( ord_le6747313008572928689nteger @ X4 @ Z3 )
     => ~ ( ord_le6747313008572928689nteger @ T @ X4 ) ) ).

% minf(7)
thf(fact_1221_minf_I7_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X4: rat] :
      ( ( ord_less_rat @ X4 @ Z3 )
     => ~ ( ord_less_rat @ T @ X4 ) ) ).

% minf(7)
thf(fact_1222_minf_I7_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X4: num] :
      ( ( ord_less_num @ X4 @ Z3 )
     => ~ ( ord_less_num @ T @ X4 ) ) ).

% minf(7)
thf(fact_1223_minf_I7_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ X4 @ Z3 )
     => ~ ( ord_less_nat @ T @ X4 ) ) ).

% minf(7)
thf(fact_1224_minf_I7_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X4: int] :
      ( ( ord_less_int @ X4 @ Z3 )
     => ~ ( ord_less_int @ T @ X4 ) ) ).

% minf(7)
thf(fact_1225_minf_I5_J,axiom,
    ! [T: code_integer] :
    ? [Z3: code_integer] :
    ! [X4: code_integer] :
      ( ( ord_le6747313008572928689nteger @ X4 @ Z3 )
     => ( ord_le6747313008572928689nteger @ X4 @ T ) ) ).

% minf(5)
thf(fact_1226_minf_I5_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X4: rat] :
      ( ( ord_less_rat @ X4 @ Z3 )
     => ( ord_less_rat @ X4 @ T ) ) ).

% minf(5)
thf(fact_1227_minf_I5_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X4: num] :
      ( ( ord_less_num @ X4 @ Z3 )
     => ( ord_less_num @ X4 @ T ) ) ).

% minf(5)
thf(fact_1228_minf_I5_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ X4 @ Z3 )
     => ( ord_less_nat @ X4 @ T ) ) ).

% minf(5)
thf(fact_1229_minf_I5_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X4: int] :
      ( ( ord_less_int @ X4 @ Z3 )
     => ( ord_less_int @ X4 @ T ) ) ).

% minf(5)
thf(fact_1230_minf_I4_J,axiom,
    ! [T: code_integer] :
    ? [Z3: code_integer] :
    ! [X4: code_integer] :
      ( ( ord_le6747313008572928689nteger @ X4 @ Z3 )
     => ( X4 != T ) ) ).

% minf(4)
thf(fact_1231_minf_I4_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X4: rat] :
      ( ( ord_less_rat @ X4 @ Z3 )
     => ( X4 != T ) ) ).

% minf(4)
thf(fact_1232_minf_I4_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X4: num] :
      ( ( ord_less_num @ X4 @ Z3 )
     => ( X4 != T ) ) ).

% minf(4)
thf(fact_1233_minf_I4_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ X4 @ Z3 )
     => ( X4 != T ) ) ).

% minf(4)
thf(fact_1234_minf_I4_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X4: int] :
      ( ( ord_less_int @ X4 @ Z3 )
     => ( X4 != T ) ) ).

% minf(4)
thf(fact_1235_minf_I3_J,axiom,
    ! [T: code_integer] :
    ? [Z3: code_integer] :
    ! [X4: code_integer] :
      ( ( ord_le6747313008572928689nteger @ X4 @ Z3 )
     => ( X4 != T ) ) ).

% minf(3)
thf(fact_1236_minf_I3_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X4: rat] :
      ( ( ord_less_rat @ X4 @ Z3 )
     => ( X4 != T ) ) ).

% minf(3)
thf(fact_1237_minf_I3_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X4: num] :
      ( ( ord_less_num @ X4 @ Z3 )
     => ( X4 != T ) ) ).

% minf(3)
thf(fact_1238_minf_I3_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ X4 @ Z3 )
     => ( X4 != T ) ) ).

% minf(3)
thf(fact_1239_minf_I3_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X4: int] :
      ( ( ord_less_int @ X4 @ Z3 )
     => ( X4 != T ) ) ).

% minf(3)
thf(fact_1240_minf_I2_J,axiom,
    ! [P: code_integer > $o,P2: code_integer > $o,Q: code_integer > $o,Q2: code_integer > $o] :
      ( ? [Z4: code_integer] :
        ! [X3: code_integer] :
          ( ( ord_le6747313008572928689nteger @ X3 @ Z4 )
         => ( ( P @ X3 )
            = ( P2 @ X3 ) ) )
     => ( ? [Z4: code_integer] :
          ! [X3: code_integer] :
            ( ( ord_le6747313008572928689nteger @ X3 @ Z4 )
           => ( ( Q @ X3 )
              = ( Q2 @ X3 ) ) )
       => ? [Z3: code_integer] :
          ! [X4: code_integer] :
            ( ( ord_le6747313008572928689nteger @ X4 @ Z3 )
           => ( ( ( P @ X4 )
                | ( Q @ X4 ) )
              = ( ( P2 @ X4 )
                | ( Q2 @ X4 ) ) ) ) ) ) ).

% minf(2)
thf(fact_1241_minf_I2_J,axiom,
    ! [P: rat > $o,P2: rat > $o,Q: rat > $o,Q2: rat > $o] :
      ( ? [Z4: rat] :
        ! [X3: rat] :
          ( ( ord_less_rat @ X3 @ Z4 )
         => ( ( P @ X3 )
            = ( P2 @ X3 ) ) )
     => ( ? [Z4: rat] :
          ! [X3: rat] :
            ( ( ord_less_rat @ X3 @ Z4 )
           => ( ( Q @ X3 )
              = ( Q2 @ X3 ) ) )
       => ? [Z3: rat] :
          ! [X4: rat] :
            ( ( ord_less_rat @ X4 @ Z3 )
           => ( ( ( P @ X4 )
                | ( Q @ X4 ) )
              = ( ( P2 @ X4 )
                | ( Q2 @ X4 ) ) ) ) ) ) ).

% minf(2)
thf(fact_1242_minf_I2_J,axiom,
    ! [P: num > $o,P2: num > $o,Q: num > $o,Q2: num > $o] :
      ( ? [Z4: num] :
        ! [X3: num] :
          ( ( ord_less_num @ X3 @ Z4 )
         => ( ( P @ X3 )
            = ( P2 @ X3 ) ) )
     => ( ? [Z4: num] :
          ! [X3: num] :
            ( ( ord_less_num @ X3 @ Z4 )
           => ( ( Q @ X3 )
              = ( Q2 @ X3 ) ) )
       => ? [Z3: num] :
          ! [X4: num] :
            ( ( ord_less_num @ X4 @ Z3 )
           => ( ( ( P @ X4 )
                | ( Q @ X4 ) )
              = ( ( P2 @ X4 )
                | ( Q2 @ X4 ) ) ) ) ) ) ).

% minf(2)
thf(fact_1243_minf_I2_J,axiom,
    ! [P: nat > $o,P2: nat > $o,Q: nat > $o,Q2: nat > $o] :
      ( ? [Z4: nat] :
        ! [X3: nat] :
          ( ( ord_less_nat @ X3 @ Z4 )
         => ( ( P @ X3 )
            = ( P2 @ X3 ) ) )
     => ( ? [Z4: nat] :
          ! [X3: nat] :
            ( ( ord_less_nat @ X3 @ Z4 )
           => ( ( Q @ X3 )
              = ( Q2 @ X3 ) ) )
       => ? [Z3: nat] :
          ! [X4: nat] :
            ( ( ord_less_nat @ X4 @ Z3 )
           => ( ( ( P @ X4 )
                | ( Q @ X4 ) )
              = ( ( P2 @ X4 )
                | ( Q2 @ X4 ) ) ) ) ) ) ).

% minf(2)
thf(fact_1244_minf_I2_J,axiom,
    ! [P: int > $o,P2: int > $o,Q: int > $o,Q2: int > $o] :
      ( ? [Z4: int] :
        ! [X3: int] :
          ( ( ord_less_int @ X3 @ Z4 )
         => ( ( P @ X3 )
            = ( P2 @ X3 ) ) )
     => ( ? [Z4: int] :
          ! [X3: int] :
            ( ( ord_less_int @ X3 @ Z4 )
           => ( ( Q @ X3 )
              = ( Q2 @ X3 ) ) )
       => ? [Z3: int] :
          ! [X4: int] :
            ( ( ord_less_int @ X4 @ Z3 )
           => ( ( ( P @ X4 )
                | ( Q @ X4 ) )
              = ( ( P2 @ X4 )
                | ( Q2 @ X4 ) ) ) ) ) ) ).

% minf(2)
thf(fact_1245_minf_I1_J,axiom,
    ! [P: code_integer > $o,P2: code_integer > $o,Q: code_integer > $o,Q2: code_integer > $o] :
      ( ? [Z4: code_integer] :
        ! [X3: code_integer] :
          ( ( ord_le6747313008572928689nteger @ X3 @ Z4 )
         => ( ( P @ X3 )
            = ( P2 @ X3 ) ) )
     => ( ? [Z4: code_integer] :
          ! [X3: code_integer] :
            ( ( ord_le6747313008572928689nteger @ X3 @ Z4 )
           => ( ( Q @ X3 )
              = ( Q2 @ X3 ) ) )
       => ? [Z3: code_integer] :
          ! [X4: code_integer] :
            ( ( ord_le6747313008572928689nteger @ X4 @ Z3 )
           => ( ( ( P @ X4 )
                & ( Q @ X4 ) )
              = ( ( P2 @ X4 )
                & ( Q2 @ X4 ) ) ) ) ) ) ).

% minf(1)
thf(fact_1246_minf_I1_J,axiom,
    ! [P: rat > $o,P2: rat > $o,Q: rat > $o,Q2: rat > $o] :
      ( ? [Z4: rat] :
        ! [X3: rat] :
          ( ( ord_less_rat @ X3 @ Z4 )
         => ( ( P @ X3 )
            = ( P2 @ X3 ) ) )
     => ( ? [Z4: rat] :
          ! [X3: rat] :
            ( ( ord_less_rat @ X3 @ Z4 )
           => ( ( Q @ X3 )
              = ( Q2 @ X3 ) ) )
       => ? [Z3: rat] :
          ! [X4: rat] :
            ( ( ord_less_rat @ X4 @ Z3 )
           => ( ( ( P @ X4 )
                & ( Q @ X4 ) )
              = ( ( P2 @ X4 )
                & ( Q2 @ X4 ) ) ) ) ) ) ).

% minf(1)
thf(fact_1247_minf_I1_J,axiom,
    ! [P: num > $o,P2: num > $o,Q: num > $o,Q2: num > $o] :
      ( ? [Z4: num] :
        ! [X3: num] :
          ( ( ord_less_num @ X3 @ Z4 )
         => ( ( P @ X3 )
            = ( P2 @ X3 ) ) )
     => ( ? [Z4: num] :
          ! [X3: num] :
            ( ( ord_less_num @ X3 @ Z4 )
           => ( ( Q @ X3 )
              = ( Q2 @ X3 ) ) )
       => ? [Z3: num] :
          ! [X4: num] :
            ( ( ord_less_num @ X4 @ Z3 )
           => ( ( ( P @ X4 )
                & ( Q @ X4 ) )
              = ( ( P2 @ X4 )
                & ( Q2 @ X4 ) ) ) ) ) ) ).

% minf(1)
thf(fact_1248_minf_I1_J,axiom,
    ! [P: nat > $o,P2: nat > $o,Q: nat > $o,Q2: nat > $o] :
      ( ? [Z4: nat] :
        ! [X3: nat] :
          ( ( ord_less_nat @ X3 @ Z4 )
         => ( ( P @ X3 )
            = ( P2 @ X3 ) ) )
     => ( ? [Z4: nat] :
          ! [X3: nat] :
            ( ( ord_less_nat @ X3 @ Z4 )
           => ( ( Q @ X3 )
              = ( Q2 @ X3 ) ) )
       => ? [Z3: nat] :
          ! [X4: nat] :
            ( ( ord_less_nat @ X4 @ Z3 )
           => ( ( ( P @ X4 )
                & ( Q @ X4 ) )
              = ( ( P2 @ X4 )
                & ( Q2 @ X4 ) ) ) ) ) ) ).

% minf(1)
thf(fact_1249_minf_I1_J,axiom,
    ! [P: int > $o,P2: int > $o,Q: int > $o,Q2: int > $o] :
      ( ? [Z4: int] :
        ! [X3: int] :
          ( ( ord_less_int @ X3 @ Z4 )
         => ( ( P @ X3 )
            = ( P2 @ X3 ) ) )
     => ( ? [Z4: int] :
          ! [X3: int] :
            ( ( ord_less_int @ X3 @ Z4 )
           => ( ( Q @ X3 )
              = ( Q2 @ X3 ) ) )
       => ? [Z3: int] :
          ! [X4: int] :
            ( ( ord_less_int @ X4 @ Z3 )
           => ( ( ( P @ X4 )
                & ( Q @ X4 ) )
              = ( ( P2 @ X4 )
                & ( Q2 @ X4 ) ) ) ) ) ) ).

% minf(1)
thf(fact_1250_pinf_I7_J,axiom,
    ! [T: code_integer] :
    ? [Z3: code_integer] :
    ! [X4: code_integer] :
      ( ( ord_le6747313008572928689nteger @ Z3 @ X4 )
     => ( ord_le6747313008572928689nteger @ T @ X4 ) ) ).

% pinf(7)
thf(fact_1251_pinf_I7_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X4: rat] :
      ( ( ord_less_rat @ Z3 @ X4 )
     => ( ord_less_rat @ T @ X4 ) ) ).

% pinf(7)
thf(fact_1252_pinf_I7_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X4: num] :
      ( ( ord_less_num @ Z3 @ X4 )
     => ( ord_less_num @ T @ X4 ) ) ).

% pinf(7)
thf(fact_1253_pinf_I7_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ Z3 @ X4 )
     => ( ord_less_nat @ T @ X4 ) ) ).

% pinf(7)
thf(fact_1254_pinf_I7_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X4: int] :
      ( ( ord_less_int @ Z3 @ X4 )
     => ( ord_less_int @ T @ X4 ) ) ).

% pinf(7)
thf(fact_1255_pinf_I5_J,axiom,
    ! [T: code_integer] :
    ? [Z3: code_integer] :
    ! [X4: code_integer] :
      ( ( ord_le6747313008572928689nteger @ Z3 @ X4 )
     => ~ ( ord_le6747313008572928689nteger @ X4 @ T ) ) ).

% pinf(5)
thf(fact_1256_pinf_I5_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X4: rat] :
      ( ( ord_less_rat @ Z3 @ X4 )
     => ~ ( ord_less_rat @ X4 @ T ) ) ).

% pinf(5)
thf(fact_1257_pinf_I5_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X4: num] :
      ( ( ord_less_num @ Z3 @ X4 )
     => ~ ( ord_less_num @ X4 @ T ) ) ).

% pinf(5)
thf(fact_1258_pinf_I5_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ Z3 @ X4 )
     => ~ ( ord_less_nat @ X4 @ T ) ) ).

% pinf(5)
thf(fact_1259_pinf_I5_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X4: int] :
      ( ( ord_less_int @ Z3 @ X4 )
     => ~ ( ord_less_int @ X4 @ T ) ) ).

% pinf(5)
thf(fact_1260_pinf_I4_J,axiom,
    ! [T: code_integer] :
    ? [Z3: code_integer] :
    ! [X4: code_integer] :
      ( ( ord_le6747313008572928689nteger @ Z3 @ X4 )
     => ( X4 != T ) ) ).

% pinf(4)
thf(fact_1261_pinf_I4_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X4: rat] :
      ( ( ord_less_rat @ Z3 @ X4 )
     => ( X4 != T ) ) ).

% pinf(4)
thf(fact_1262_pinf_I4_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X4: num] :
      ( ( ord_less_num @ Z3 @ X4 )
     => ( X4 != T ) ) ).

% pinf(4)
thf(fact_1263_pinf_I4_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ Z3 @ X4 )
     => ( X4 != T ) ) ).

% pinf(4)
thf(fact_1264_pinf_I4_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X4: int] :
      ( ( ord_less_int @ Z3 @ X4 )
     => ( X4 != T ) ) ).

% pinf(4)
thf(fact_1265_pinf_I3_J,axiom,
    ! [T: code_integer] :
    ? [Z3: code_integer] :
    ! [X4: code_integer] :
      ( ( ord_le6747313008572928689nteger @ Z3 @ X4 )
     => ( X4 != T ) ) ).

% pinf(3)
thf(fact_1266_pinf_I3_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X4: rat] :
      ( ( ord_less_rat @ Z3 @ X4 )
     => ( X4 != T ) ) ).

% pinf(3)
thf(fact_1267_pinf_I3_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X4: num] :
      ( ( ord_less_num @ Z3 @ X4 )
     => ( X4 != T ) ) ).

% pinf(3)
thf(fact_1268_pinf_I3_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ Z3 @ X4 )
     => ( X4 != T ) ) ).

% pinf(3)
thf(fact_1269_pinf_I3_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X4: int] :
      ( ( ord_less_int @ Z3 @ X4 )
     => ( X4 != T ) ) ).

% pinf(3)
thf(fact_1270_pinf_I2_J,axiom,
    ! [P: code_integer > $o,P2: code_integer > $o,Q: code_integer > $o,Q2: code_integer > $o] :
      ( ? [Z4: code_integer] :
        ! [X3: code_integer] :
          ( ( ord_le6747313008572928689nteger @ Z4 @ X3 )
         => ( ( P @ X3 )
            = ( P2 @ X3 ) ) )
     => ( ? [Z4: code_integer] :
          ! [X3: code_integer] :
            ( ( ord_le6747313008572928689nteger @ Z4 @ X3 )
           => ( ( Q @ X3 )
              = ( Q2 @ X3 ) ) )
       => ? [Z3: code_integer] :
          ! [X4: code_integer] :
            ( ( ord_le6747313008572928689nteger @ Z3 @ X4 )
           => ( ( ( P @ X4 )
                | ( Q @ X4 ) )
              = ( ( P2 @ X4 )
                | ( Q2 @ X4 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_1271_pinf_I2_J,axiom,
    ! [P: rat > $o,P2: rat > $o,Q: rat > $o,Q2: rat > $o] :
      ( ? [Z4: rat] :
        ! [X3: rat] :
          ( ( ord_less_rat @ Z4 @ X3 )
         => ( ( P @ X3 )
            = ( P2 @ X3 ) ) )
     => ( ? [Z4: rat] :
          ! [X3: rat] :
            ( ( ord_less_rat @ Z4 @ X3 )
           => ( ( Q @ X3 )
              = ( Q2 @ X3 ) ) )
       => ? [Z3: rat] :
          ! [X4: rat] :
            ( ( ord_less_rat @ Z3 @ X4 )
           => ( ( ( P @ X4 )
                | ( Q @ X4 ) )
              = ( ( P2 @ X4 )
                | ( Q2 @ X4 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_1272_pinf_I2_J,axiom,
    ! [P: num > $o,P2: num > $o,Q: num > $o,Q2: num > $o] :
      ( ? [Z4: num] :
        ! [X3: num] :
          ( ( ord_less_num @ Z4 @ X3 )
         => ( ( P @ X3 )
            = ( P2 @ X3 ) ) )
     => ( ? [Z4: num] :
          ! [X3: num] :
            ( ( ord_less_num @ Z4 @ X3 )
           => ( ( Q @ X3 )
              = ( Q2 @ X3 ) ) )
       => ? [Z3: num] :
          ! [X4: num] :
            ( ( ord_less_num @ Z3 @ X4 )
           => ( ( ( P @ X4 )
                | ( Q @ X4 ) )
              = ( ( P2 @ X4 )
                | ( Q2 @ X4 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_1273_pinf_I2_J,axiom,
    ! [P: nat > $o,P2: nat > $o,Q: nat > $o,Q2: nat > $o] :
      ( ? [Z4: nat] :
        ! [X3: nat] :
          ( ( ord_less_nat @ Z4 @ X3 )
         => ( ( P @ X3 )
            = ( P2 @ X3 ) ) )
     => ( ? [Z4: nat] :
          ! [X3: nat] :
            ( ( ord_less_nat @ Z4 @ X3 )
           => ( ( Q @ X3 )
              = ( Q2 @ X3 ) ) )
       => ? [Z3: nat] :
          ! [X4: nat] :
            ( ( ord_less_nat @ Z3 @ X4 )
           => ( ( ( P @ X4 )
                | ( Q @ X4 ) )
              = ( ( P2 @ X4 )
                | ( Q2 @ X4 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_1274_pinf_I2_J,axiom,
    ! [P: int > $o,P2: int > $o,Q: int > $o,Q2: int > $o] :
      ( ? [Z4: int] :
        ! [X3: int] :
          ( ( ord_less_int @ Z4 @ X3 )
         => ( ( P @ X3 )
            = ( P2 @ X3 ) ) )
     => ( ? [Z4: int] :
          ! [X3: int] :
            ( ( ord_less_int @ Z4 @ X3 )
           => ( ( Q @ X3 )
              = ( Q2 @ X3 ) ) )
       => ? [Z3: int] :
          ! [X4: int] :
            ( ( ord_less_int @ Z3 @ X4 )
           => ( ( ( P @ X4 )
                | ( Q @ X4 ) )
              = ( ( P2 @ X4 )
                | ( Q2 @ X4 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_1275_pinf_I1_J,axiom,
    ! [P: code_integer > $o,P2: code_integer > $o,Q: code_integer > $o,Q2: code_integer > $o] :
      ( ? [Z4: code_integer] :
        ! [X3: code_integer] :
          ( ( ord_le6747313008572928689nteger @ Z4 @ X3 )
         => ( ( P @ X3 )
            = ( P2 @ X3 ) ) )
     => ( ? [Z4: code_integer] :
          ! [X3: code_integer] :
            ( ( ord_le6747313008572928689nteger @ Z4 @ X3 )
           => ( ( Q @ X3 )
              = ( Q2 @ X3 ) ) )
       => ? [Z3: code_integer] :
          ! [X4: code_integer] :
            ( ( ord_le6747313008572928689nteger @ Z3 @ X4 )
           => ( ( ( P @ X4 )
                & ( Q @ X4 ) )
              = ( ( P2 @ X4 )
                & ( Q2 @ X4 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_1276_pinf_I1_J,axiom,
    ! [P: rat > $o,P2: rat > $o,Q: rat > $o,Q2: rat > $o] :
      ( ? [Z4: rat] :
        ! [X3: rat] :
          ( ( ord_less_rat @ Z4 @ X3 )
         => ( ( P @ X3 )
            = ( P2 @ X3 ) ) )
     => ( ? [Z4: rat] :
          ! [X3: rat] :
            ( ( ord_less_rat @ Z4 @ X3 )
           => ( ( Q @ X3 )
              = ( Q2 @ X3 ) ) )
       => ? [Z3: rat] :
          ! [X4: rat] :
            ( ( ord_less_rat @ Z3 @ X4 )
           => ( ( ( P @ X4 )
                & ( Q @ X4 ) )
              = ( ( P2 @ X4 )
                & ( Q2 @ X4 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_1277_pinf_I1_J,axiom,
    ! [P: num > $o,P2: num > $o,Q: num > $o,Q2: num > $o] :
      ( ? [Z4: num] :
        ! [X3: num] :
          ( ( ord_less_num @ Z4 @ X3 )
         => ( ( P @ X3 )
            = ( P2 @ X3 ) ) )
     => ( ? [Z4: num] :
          ! [X3: num] :
            ( ( ord_less_num @ Z4 @ X3 )
           => ( ( Q @ X3 )
              = ( Q2 @ X3 ) ) )
       => ? [Z3: num] :
          ! [X4: num] :
            ( ( ord_less_num @ Z3 @ X4 )
           => ( ( ( P @ X4 )
                & ( Q @ X4 ) )
              = ( ( P2 @ X4 )
                & ( Q2 @ X4 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_1278_pinf_I1_J,axiom,
    ! [P: nat > $o,P2: nat > $o,Q: nat > $o,Q2: nat > $o] :
      ( ? [Z4: nat] :
        ! [X3: nat] :
          ( ( ord_less_nat @ Z4 @ X3 )
         => ( ( P @ X3 )
            = ( P2 @ X3 ) ) )
     => ( ? [Z4: nat] :
          ! [X3: nat] :
            ( ( ord_less_nat @ Z4 @ X3 )
           => ( ( Q @ X3 )
              = ( Q2 @ X3 ) ) )
       => ? [Z3: nat] :
          ! [X4: nat] :
            ( ( ord_less_nat @ Z3 @ X4 )
           => ( ( ( P @ X4 )
                & ( Q @ X4 ) )
              = ( ( P2 @ X4 )
                & ( Q2 @ X4 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_1279_pinf_I1_J,axiom,
    ! [P: int > $o,P2: int > $o,Q: int > $o,Q2: int > $o] :
      ( ? [Z4: int] :
        ! [X3: int] :
          ( ( ord_less_int @ Z4 @ X3 )
         => ( ( P @ X3 )
            = ( P2 @ X3 ) ) )
     => ( ? [Z4: int] :
          ! [X3: int] :
            ( ( ord_less_int @ Z4 @ X3 )
           => ( ( Q @ X3 )
              = ( Q2 @ X3 ) ) )
       => ? [Z3: int] :
          ! [X4: int] :
            ( ( ord_less_int @ Z3 @ X4 )
           => ( ( ( P @ X4 )
                & ( Q @ X4 ) )
              = ( ( P2 @ X4 )
                & ( Q2 @ X4 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_1280_lt__ex,axiom,
    ! [X: rat] :
    ? [Y4: rat] : ( ord_less_rat @ Y4 @ X ) ).

% lt_ex
thf(fact_1281_lt__ex,axiom,
    ! [X: int] :
    ? [Y4: int] : ( ord_less_int @ Y4 @ X ) ).

% lt_ex
thf(fact_1282_gt__ex,axiom,
    ! [X: rat] :
    ? [X_12: rat] : ( ord_less_rat @ X @ X_12 ) ).

% gt_ex
thf(fact_1283_gt__ex,axiom,
    ! [X: nat] :
    ? [X_12: nat] : ( ord_less_nat @ X @ X_12 ) ).

% gt_ex
thf(fact_1284_gt__ex,axiom,
    ! [X: int] :
    ? [X_12: int] : ( ord_less_int @ X @ X_12 ) ).

% gt_ex
thf(fact_1285_dense,axiom,
    ! [X: rat,Y3: rat] :
      ( ( ord_less_rat @ X @ Y3 )
     => ? [Z3: rat] :
          ( ( ord_less_rat @ X @ Z3 )
          & ( ord_less_rat @ Z3 @ Y3 ) ) ) ).

% dense
thf(fact_1286_less__imp__neq,axiom,
    ! [X: code_integer,Y3: code_integer] :
      ( ( ord_le6747313008572928689nteger @ X @ Y3 )
     => ( X != Y3 ) ) ).

% less_imp_neq
thf(fact_1287_less__imp__neq,axiom,
    ! [X: rat,Y3: rat] :
      ( ( ord_less_rat @ X @ Y3 )
     => ( X != Y3 ) ) ).

% less_imp_neq
thf(fact_1288_less__imp__neq,axiom,
    ! [X: num,Y3: num] :
      ( ( ord_less_num @ X @ Y3 )
     => ( X != Y3 ) ) ).

% less_imp_neq
thf(fact_1289_less__imp__neq,axiom,
    ! [X: nat,Y3: nat] :
      ( ( ord_less_nat @ X @ Y3 )
     => ( X != Y3 ) ) ).

% less_imp_neq
thf(fact_1290_less__imp__neq,axiom,
    ! [X: int,Y3: int] :
      ( ( ord_less_int @ X @ Y3 )
     => ( X != Y3 ) ) ).

% less_imp_neq
thf(fact_1291_order_Oasym,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ B )
     => ~ ( ord_le6747313008572928689nteger @ B @ A ) ) ).

% order.asym
thf(fact_1292_order_Oasym,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ~ ( ord_less_rat @ B @ A ) ) ).

% order.asym
thf(fact_1293_order_Oasym,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_num @ A @ B )
     => ~ ( ord_less_num @ B @ A ) ) ).

% order.asym
thf(fact_1294_order_Oasym,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ( ord_less_nat @ B @ A ) ) ).

% order.asym
thf(fact_1295_order_Oasym,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ~ ( ord_less_int @ B @ A ) ) ).

% order.asym
thf(fact_1296_ord__eq__less__trans,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( A = B )
     => ( ( ord_le6747313008572928689nteger @ B @ C )
       => ( ord_le6747313008572928689nteger @ A @ C ) ) ) ).

% ord_eq_less_trans
thf(fact_1297_ord__eq__less__trans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( A = B )
     => ( ( ord_less_rat @ B @ C )
       => ( ord_less_rat @ A @ C ) ) ) ).

% ord_eq_less_trans
thf(fact_1298_ord__eq__less__trans,axiom,
    ! [A: num,B: num,C: num] :
      ( ( A = B )
     => ( ( ord_less_num @ B @ C )
       => ( ord_less_num @ A @ C ) ) ) ).

% ord_eq_less_trans
thf(fact_1299_ord__eq__less__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A = B )
     => ( ( ord_less_nat @ B @ C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% ord_eq_less_trans
thf(fact_1300_ord__eq__less__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A = B )
     => ( ( ord_less_int @ B @ C )
       => ( ord_less_int @ A @ C ) ) ) ).

% ord_eq_less_trans
thf(fact_1301_ord__less__eq__trans,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ B )
     => ( ( B = C )
       => ( ord_le6747313008572928689nteger @ A @ C ) ) ) ).

% ord_less_eq_trans
thf(fact_1302_ord__less__eq__trans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( B = C )
       => ( ord_less_rat @ A @ C ) ) ) ).

% ord_less_eq_trans
thf(fact_1303_ord__less__eq__trans,axiom,
    ! [A: num,B: num,C: num] :
      ( ( ord_less_num @ A @ B )
     => ( ( B = C )
       => ( ord_less_num @ A @ C ) ) ) ).

% ord_less_eq_trans
thf(fact_1304_ord__less__eq__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( B = C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% ord_less_eq_trans
thf(fact_1305_ord__less__eq__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( B = C )
       => ( ord_less_int @ A @ C ) ) ) ).

% ord_less_eq_trans
thf(fact_1306_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_1307_antisym__conv3,axiom,
    ! [Y3: code_integer,X: code_integer] :
      ( ~ ( ord_le6747313008572928689nteger @ Y3 @ X )
     => ( ( ~ ( ord_le6747313008572928689nteger @ X @ Y3 ) )
        = ( X = Y3 ) ) ) ).

% antisym_conv3
thf(fact_1308_antisym__conv3,axiom,
    ! [Y3: rat,X: rat] :
      ( ~ ( ord_less_rat @ Y3 @ X )
     => ( ( ~ ( ord_less_rat @ X @ Y3 ) )
        = ( X = Y3 ) ) ) ).

% antisym_conv3
thf(fact_1309_antisym__conv3,axiom,
    ! [Y3: num,X: num] :
      ( ~ ( ord_less_num @ Y3 @ X )
     => ( ( ~ ( ord_less_num @ X @ Y3 ) )
        = ( X = Y3 ) ) ) ).

% antisym_conv3
thf(fact_1310_antisym__conv3,axiom,
    ! [Y3: nat,X: nat] :
      ( ~ ( ord_less_nat @ Y3 @ X )
     => ( ( ~ ( ord_less_nat @ X @ Y3 ) )
        = ( X = Y3 ) ) ) ).

% antisym_conv3
thf(fact_1311_antisym__conv3,axiom,
    ! [Y3: int,X: int] :
      ( ~ ( ord_less_int @ Y3 @ X )
     => ( ( ~ ( ord_less_int @ X @ Y3 ) )
        = ( X = Y3 ) ) ) ).

% antisym_conv3
thf(fact_1312_linorder__cases,axiom,
    ! [X: code_integer,Y3: code_integer] :
      ( ~ ( ord_le6747313008572928689nteger @ X @ Y3 )
     => ( ( X != Y3 )
       => ( ord_le6747313008572928689nteger @ Y3 @ X ) ) ) ).

% linorder_cases
thf(fact_1313_linorder__cases,axiom,
    ! [X: rat,Y3: rat] :
      ( ~ ( ord_less_rat @ X @ Y3 )
     => ( ( X != Y3 )
       => ( ord_less_rat @ Y3 @ X ) ) ) ).

% linorder_cases
thf(fact_1314_linorder__cases,axiom,
    ! [X: num,Y3: num] :
      ( ~ ( ord_less_num @ X @ Y3 )
     => ( ( X != Y3 )
       => ( ord_less_num @ Y3 @ X ) ) ) ).

% linorder_cases
thf(fact_1315_linorder__cases,axiom,
    ! [X: nat,Y3: nat] :
      ( ~ ( ord_less_nat @ X @ Y3 )
     => ( ( X != Y3 )
       => ( ord_less_nat @ Y3 @ X ) ) ) ).

% linorder_cases
thf(fact_1316_linorder__cases,axiom,
    ! [X: int,Y3: int] :
      ( ~ ( ord_less_int @ X @ Y3 )
     => ( ( X != Y3 )
       => ( ord_less_int @ Y3 @ X ) ) ) ).

% linorder_cases
thf(fact_1317_dual__order_Oasym,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ B @ A )
     => ~ ( ord_le6747313008572928689nteger @ A @ B ) ) ).

% dual_order.asym
thf(fact_1318_dual__order_Oasym,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ B @ A )
     => ~ ( ord_less_rat @ A @ B ) ) ).

% dual_order.asym
thf(fact_1319_dual__order_Oasym,axiom,
    ! [B: num,A: num] :
      ( ( ord_less_num @ B @ A )
     => ~ ( ord_less_num @ A @ B ) ) ).

% dual_order.asym
thf(fact_1320_dual__order_Oasym,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ~ ( ord_less_nat @ A @ B ) ) ).

% dual_order.asym
thf(fact_1321_dual__order_Oasym,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ A )
     => ~ ( ord_less_int @ A @ B ) ) ).

% dual_order.asym
thf(fact_1322_dual__order_Oirrefl,axiom,
    ! [A: code_integer] :
      ~ ( ord_le6747313008572928689nteger @ A @ A ) ).

% dual_order.irrefl
thf(fact_1323_dual__order_Oirrefl,axiom,
    ! [A: rat] :
      ~ ( ord_less_rat @ A @ A ) ).

% dual_order.irrefl
thf(fact_1324_dual__order_Oirrefl,axiom,
    ! [A: num] :
      ~ ( ord_less_num @ A @ A ) ).

% dual_order.irrefl
thf(fact_1325_dual__order_Oirrefl,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ A ) ).

% dual_order.irrefl
thf(fact_1326_dual__order_Oirrefl,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ A @ A ) ).

% dual_order.irrefl
thf(fact_1327_exists__least__iff,axiom,
    ( ( ^ [P3: nat > $o] :
        ? [X6: nat] : ( P3 @ X6 ) )
    = ( ^ [P4: nat > $o] :
        ? [N3: nat] :
          ( ( P4 @ N3 )
          & ! [M2: nat] :
              ( ( ord_less_nat @ M2 @ N3 )
             => ~ ( P4 @ M2 ) ) ) ) ) ).

% exists_least_iff
thf(fact_1328_linorder__less__wlog,axiom,
    ! [P: code_integer > code_integer > $o,A: code_integer,B: code_integer] :
      ( ! [A5: code_integer,B4: code_integer] :
          ( ( ord_le6747313008572928689nteger @ A5 @ B4 )
         => ( P @ A5 @ B4 ) )
     => ( ! [A5: code_integer] : ( P @ A5 @ A5 )
       => ( ! [A5: code_integer,B4: code_integer] :
              ( ( P @ B4 @ A5 )
             => ( P @ A5 @ B4 ) )
         => ( P @ A @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_1329_linorder__less__wlog,axiom,
    ! [P: rat > rat > $o,A: rat,B: rat] :
      ( ! [A5: rat,B4: rat] :
          ( ( ord_less_rat @ A5 @ B4 )
         => ( P @ A5 @ B4 ) )
     => ( ! [A5: rat] : ( P @ A5 @ A5 )
       => ( ! [A5: rat,B4: rat] :
              ( ( P @ B4 @ A5 )
             => ( P @ A5 @ B4 ) )
         => ( P @ A @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_1330_linorder__less__wlog,axiom,
    ! [P: num > num > $o,A: num,B: num] :
      ( ! [A5: num,B4: num] :
          ( ( ord_less_num @ A5 @ B4 )
         => ( P @ A5 @ B4 ) )
     => ( ! [A5: num] : ( P @ A5 @ A5 )
       => ( ! [A5: num,B4: num] :
              ( ( P @ B4 @ A5 )
             => ( P @ A5 @ B4 ) )
         => ( P @ A @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_1331_linorder__less__wlog,axiom,
    ! [P: nat > nat > $o,A: nat,B: nat] :
      ( ! [A5: nat,B4: nat] :
          ( ( ord_less_nat @ A5 @ B4 )
         => ( P @ A5 @ B4 ) )
     => ( ! [A5: nat] : ( P @ A5 @ A5 )
       => ( ! [A5: nat,B4: nat] :
              ( ( P @ B4 @ A5 )
             => ( P @ A5 @ B4 ) )
         => ( P @ A @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_1332_linorder__less__wlog,axiom,
    ! [P: int > int > $o,A: int,B: int] :
      ( ! [A5: int,B4: int] :
          ( ( ord_less_int @ A5 @ B4 )
         => ( P @ A5 @ B4 ) )
     => ( ! [A5: int] : ( P @ A5 @ A5 )
       => ( ! [A5: int,B4: int] :
              ( ( P @ B4 @ A5 )
             => ( P @ A5 @ B4 ) )
         => ( P @ A @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_1333_order_Ostrict__trans,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ B )
     => ( ( ord_le6747313008572928689nteger @ B @ C )
       => ( ord_le6747313008572928689nteger @ A @ C ) ) ) ).

% order.strict_trans
thf(fact_1334_order_Ostrict__trans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ B @ C )
       => ( ord_less_rat @ A @ C ) ) ) ).

% order.strict_trans
thf(fact_1335_order_Ostrict__trans,axiom,
    ! [A: num,B: num,C: num] :
      ( ( ord_less_num @ A @ B )
     => ( ( ord_less_num @ B @ C )
       => ( ord_less_num @ A @ C ) ) ) ).

% order.strict_trans
thf(fact_1336_order_Ostrict__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ B @ C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% order.strict_trans
thf(fact_1337_order_Ostrict__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ B @ C )
       => ( ord_less_int @ A @ C ) ) ) ).

% order.strict_trans
thf(fact_1338_not__less__iff__gr__or__eq,axiom,
    ! [X: code_integer,Y3: code_integer] :
      ( ( ~ ( ord_le6747313008572928689nteger @ X @ Y3 ) )
      = ( ( ord_le6747313008572928689nteger @ Y3 @ X )
        | ( X = Y3 ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_1339_not__less__iff__gr__or__eq,axiom,
    ! [X: rat,Y3: rat] :
      ( ( ~ ( ord_less_rat @ X @ Y3 ) )
      = ( ( ord_less_rat @ Y3 @ X )
        | ( X = Y3 ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_1340_not__less__iff__gr__or__eq,axiom,
    ! [X: num,Y3: num] :
      ( ( ~ ( ord_less_num @ X @ Y3 ) )
      = ( ( ord_less_num @ Y3 @ X )
        | ( X = Y3 ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_1341_not__less__iff__gr__or__eq,axiom,
    ! [X: nat,Y3: nat] :
      ( ( ~ ( ord_less_nat @ X @ Y3 ) )
      = ( ( ord_less_nat @ Y3 @ X )
        | ( X = Y3 ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_1342_not__less__iff__gr__or__eq,axiom,
    ! [X: int,Y3: int] :
      ( ( ~ ( ord_less_int @ X @ Y3 ) )
      = ( ( ord_less_int @ Y3 @ X )
        | ( X = Y3 ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_1343_dual__order_Ostrict__trans,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( ord_le6747313008572928689nteger @ B @ A )
     => ( ( ord_le6747313008572928689nteger @ C @ B )
       => ( ord_le6747313008572928689nteger @ C @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_1344_dual__order_Ostrict__trans,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( ord_less_rat @ C @ B )
       => ( ord_less_rat @ C @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_1345_dual__order_Ostrict__trans,axiom,
    ! [B: num,A: num,C: num] :
      ( ( ord_less_num @ B @ A )
     => ( ( ord_less_num @ C @ B )
       => ( ord_less_num @ C @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_1346_dual__order_Ostrict__trans,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( ord_less_nat @ C @ B )
       => ( ord_less_nat @ C @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_1347_dual__order_Ostrict__trans,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( ord_less_int @ C @ B )
       => ( ord_less_int @ C @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_1348_order_Ostrict__implies__not__eq,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_1349_order_Ostrict__implies__not__eq,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_1350_order_Ostrict__implies__not__eq,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_num @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_1351_order_Ostrict__implies__not__eq,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_1352_order_Ostrict__implies__not__eq,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_1353_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ B @ A )
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_1354_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_1355_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: num,A: num] :
      ( ( ord_less_num @ B @ A )
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_1356_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_1357_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ A )
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_1358_linorder__neqE,axiom,
    ! [X: code_integer,Y3: code_integer] :
      ( ( X != Y3 )
     => ( ~ ( ord_le6747313008572928689nteger @ X @ Y3 )
       => ( ord_le6747313008572928689nteger @ Y3 @ X ) ) ) ).

% linorder_neqE
thf(fact_1359_linorder__neqE,axiom,
    ! [X: rat,Y3: rat] :
      ( ( X != Y3 )
     => ( ~ ( ord_less_rat @ X @ Y3 )
       => ( ord_less_rat @ Y3 @ X ) ) ) ).

% linorder_neqE
thf(fact_1360_linorder__neqE,axiom,
    ! [X: num,Y3: num] :
      ( ( X != Y3 )
     => ( ~ ( ord_less_num @ X @ Y3 )
       => ( ord_less_num @ Y3 @ X ) ) ) ).

% linorder_neqE
thf(fact_1361_linorder__neqE,axiom,
    ! [X: nat,Y3: nat] :
      ( ( X != Y3 )
     => ( ~ ( ord_less_nat @ X @ Y3 )
       => ( ord_less_nat @ Y3 @ X ) ) ) ).

% linorder_neqE
thf(fact_1362_linorder__neqE,axiom,
    ! [X: int,Y3: int] :
      ( ( X != Y3 )
     => ( ~ ( ord_less_int @ X @ Y3 )
       => ( ord_less_int @ Y3 @ X ) ) ) ).

% linorder_neqE
thf(fact_1363_order__less__asym,axiom,
    ! [X: code_integer,Y3: code_integer] :
      ( ( ord_le6747313008572928689nteger @ X @ Y3 )
     => ~ ( ord_le6747313008572928689nteger @ Y3 @ X ) ) ).

% order_less_asym
thf(fact_1364_order__less__asym,axiom,
    ! [X: rat,Y3: rat] :
      ( ( ord_less_rat @ X @ Y3 )
     => ~ ( ord_less_rat @ Y3 @ X ) ) ).

% order_less_asym
thf(fact_1365_order__less__asym,axiom,
    ! [X: num,Y3: num] :
      ( ( ord_less_num @ X @ Y3 )
     => ~ ( ord_less_num @ Y3 @ X ) ) ).

% order_less_asym
thf(fact_1366_order__less__asym,axiom,
    ! [X: nat,Y3: nat] :
      ( ( ord_less_nat @ X @ Y3 )
     => ~ ( ord_less_nat @ Y3 @ X ) ) ).

% order_less_asym
thf(fact_1367_order__less__asym,axiom,
    ! [X: int,Y3: int] :
      ( ( ord_less_int @ X @ Y3 )
     => ~ ( ord_less_int @ Y3 @ X ) ) ).

% order_less_asym
thf(fact_1368_linorder__neq__iff,axiom,
    ! [X: code_integer,Y3: code_integer] :
      ( ( X != Y3 )
      = ( ( ord_le6747313008572928689nteger @ X @ Y3 )
        | ( ord_le6747313008572928689nteger @ Y3 @ X ) ) ) ).

% linorder_neq_iff
thf(fact_1369_linorder__neq__iff,axiom,
    ! [X: rat,Y3: rat] :
      ( ( X != Y3 )
      = ( ( ord_less_rat @ X @ Y3 )
        | ( ord_less_rat @ Y3 @ X ) ) ) ).

% linorder_neq_iff
thf(fact_1370_linorder__neq__iff,axiom,
    ! [X: num,Y3: num] :
      ( ( X != Y3 )
      = ( ( ord_less_num @ X @ Y3 )
        | ( ord_less_num @ Y3 @ X ) ) ) ).

% linorder_neq_iff
thf(fact_1371_linorder__neq__iff,axiom,
    ! [X: nat,Y3: nat] :
      ( ( X != Y3 )
      = ( ( ord_less_nat @ X @ Y3 )
        | ( ord_less_nat @ Y3 @ X ) ) ) ).

% linorder_neq_iff
thf(fact_1372_linorder__neq__iff,axiom,
    ! [X: int,Y3: int] :
      ( ( X != Y3 )
      = ( ( ord_less_int @ X @ Y3 )
        | ( ord_less_int @ Y3 @ X ) ) ) ).

% linorder_neq_iff
thf(fact_1373_order__less__asym_H,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ B )
     => ~ ( ord_le6747313008572928689nteger @ B @ A ) ) ).

% order_less_asym'
thf(fact_1374_order__less__asym_H,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ~ ( ord_less_rat @ B @ A ) ) ).

% order_less_asym'
thf(fact_1375_order__less__asym_H,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_num @ A @ B )
     => ~ ( ord_less_num @ B @ A ) ) ).

% order_less_asym'
thf(fact_1376_order__less__asym_H,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ( ord_less_nat @ B @ A ) ) ).

% order_less_asym'
thf(fact_1377_order__less__asym_H,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ~ ( ord_less_int @ B @ A ) ) ).

% order_less_asym'
thf(fact_1378_order__less__trans,axiom,
    ! [X: code_integer,Y3: code_integer,Z2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ X @ Y3 )
     => ( ( ord_le6747313008572928689nteger @ Y3 @ Z2 )
       => ( ord_le6747313008572928689nteger @ X @ Z2 ) ) ) ).

% order_less_trans
thf(fact_1379_order__less__trans,axiom,
    ! [X: rat,Y3: rat,Z2: rat] :
      ( ( ord_less_rat @ X @ Y3 )
     => ( ( ord_less_rat @ Y3 @ Z2 )
       => ( ord_less_rat @ X @ Z2 ) ) ) ).

% order_less_trans
thf(fact_1380_order__less__trans,axiom,
    ! [X: num,Y3: num,Z2: num] :
      ( ( ord_less_num @ X @ Y3 )
     => ( ( ord_less_num @ Y3 @ Z2 )
       => ( ord_less_num @ X @ Z2 ) ) ) ).

% order_less_trans
thf(fact_1381_order__less__trans,axiom,
    ! [X: nat,Y3: nat,Z2: nat] :
      ( ( ord_less_nat @ X @ Y3 )
     => ( ( ord_less_nat @ Y3 @ Z2 )
       => ( ord_less_nat @ X @ Z2 ) ) ) ).

% order_less_trans
thf(fact_1382_order__less__trans,axiom,
    ! [X: int,Y3: int,Z2: int] :
      ( ( ord_less_int @ X @ Y3 )
     => ( ( ord_less_int @ Y3 @ Z2 )
       => ( ord_less_int @ X @ Z2 ) ) ) ).

% order_less_trans
thf(fact_1383_ord__eq__less__subst,axiom,
    ! [A: code_integer,F: code_integer > code_integer,B: code_integer,C: code_integer] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_le6747313008572928689nteger @ B @ C )
       => ( ! [X3: code_integer,Y4: code_integer] :
              ( ( ord_le6747313008572928689nteger @ X3 @ Y4 )
             => ( ord_le6747313008572928689nteger @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_le6747313008572928689nteger @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_1384_ord__eq__less__subst,axiom,
    ! [A: rat,F: code_integer > rat,B: code_integer,C: code_integer] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_le6747313008572928689nteger @ B @ C )
       => ( ! [X3: code_integer,Y4: code_integer] :
              ( ( ord_le6747313008572928689nteger @ X3 @ Y4 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_1385_ord__eq__less__subst,axiom,
    ! [A: num,F: code_integer > num,B: code_integer,C: code_integer] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_le6747313008572928689nteger @ B @ C )
       => ( ! [X3: code_integer,Y4: code_integer] :
              ( ( ord_le6747313008572928689nteger @ X3 @ Y4 )
             => ( ord_less_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_1386_ord__eq__less__subst,axiom,
    ! [A: nat,F: code_integer > nat,B: code_integer,C: code_integer] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_le6747313008572928689nteger @ B @ C )
       => ( ! [X3: code_integer,Y4: code_integer] :
              ( ( ord_le6747313008572928689nteger @ X3 @ Y4 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_1387_ord__eq__less__subst,axiom,
    ! [A: int,F: code_integer > int,B: code_integer,C: code_integer] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_le6747313008572928689nteger @ B @ C )
       => ( ! [X3: code_integer,Y4: code_integer] :
              ( ( ord_le6747313008572928689nteger @ X3 @ Y4 )
             => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_1388_ord__eq__less__subst,axiom,
    ! [A: code_integer,F: rat > code_integer,B: rat,C: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_rat @ B @ C )
       => ( ! [X3: rat,Y4: rat] :
              ( ( ord_less_rat @ X3 @ Y4 )
             => ( ord_le6747313008572928689nteger @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_le6747313008572928689nteger @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_1389_ord__eq__less__subst,axiom,
    ! [A: rat,F: rat > rat,B: rat,C: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_rat @ B @ 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_1390_ord__eq__less__subst,axiom,
    ! [A: num,F: rat > num,B: rat,C: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_rat @ B @ 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_1391_ord__eq__less__subst,axiom,
    ! [A: nat,F: rat > nat,B: rat,C: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_rat @ B @ 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_1392_ord__eq__less__subst,axiom,
    ! [A: int,F: rat > int,B: rat,C: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_rat @ B @ 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_1393_ord__less__eq__subst,axiom,
    ! [A: code_integer,B: code_integer,F: code_integer > code_integer,C: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: code_integer,Y4: code_integer] :
              ( ( ord_le6747313008572928689nteger @ X3 @ Y4 )
             => ( ord_le6747313008572928689nteger @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_le6747313008572928689nteger @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_1394_ord__less__eq__subst,axiom,
    ! [A: code_integer,B: code_integer,F: code_integer > rat,C: rat] :
      ( ( ord_le6747313008572928689nteger @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: code_integer,Y4: code_integer] :
              ( ( ord_le6747313008572928689nteger @ X3 @ Y4 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_1395_ord__less__eq__subst,axiom,
    ! [A: code_integer,B: code_integer,F: code_integer > num,C: num] :
      ( ( ord_le6747313008572928689nteger @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: code_integer,Y4: code_integer] :
              ( ( ord_le6747313008572928689nteger @ X3 @ Y4 )
             => ( ord_less_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_1396_ord__less__eq__subst,axiom,
    ! [A: code_integer,B: code_integer,F: code_integer > nat,C: nat] :
      ( ( ord_le6747313008572928689nteger @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: code_integer,Y4: code_integer] :
              ( ( ord_le6747313008572928689nteger @ X3 @ Y4 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_1397_ord__less__eq__subst,axiom,
    ! [A: code_integer,B: code_integer,F: code_integer > int,C: int] :
      ( ( ord_le6747313008572928689nteger @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: code_integer,Y4: code_integer] :
              ( ( ord_le6747313008572928689nteger @ X3 @ Y4 )
             => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_1398_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > code_integer,C: code_integer] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: rat,Y4: rat] :
              ( ( ord_less_rat @ X3 @ Y4 )
             => ( ord_le6747313008572928689nteger @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_le6747313008572928689nteger @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_1399_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ( F @ B )
          = 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_1400_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > num,C: num] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ( F @ B )
          = 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_1401_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > nat,C: nat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ( F @ B )
          = 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_1402_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > int,C: int] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ( F @ B )
          = 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_1403_order__less__irrefl,axiom,
    ! [X: code_integer] :
      ~ ( ord_le6747313008572928689nteger @ X @ X ) ).

% order_less_irrefl
thf(fact_1404_order__less__irrefl,axiom,
    ! [X: rat] :
      ~ ( ord_less_rat @ X @ X ) ).

% order_less_irrefl
thf(fact_1405_order__less__irrefl,axiom,
    ! [X: num] :
      ~ ( ord_less_num @ X @ X ) ).

% order_less_irrefl
thf(fact_1406_order__less__irrefl,axiom,
    ! [X: nat] :
      ~ ( ord_less_nat @ X @ X ) ).

% order_less_irrefl
thf(fact_1407_order__less__irrefl,axiom,
    ! [X: int] :
      ~ ( ord_less_int @ X @ X ) ).

% order_less_irrefl
thf(fact_1408_order__less__subst1,axiom,
    ! [A: code_integer,F: code_integer > code_integer,B: code_integer,C: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ ( F @ B ) )
     => ( ( ord_le6747313008572928689nteger @ B @ C )
       => ( ! [X3: code_integer,Y4: code_integer] :
              ( ( ord_le6747313008572928689nteger @ X3 @ Y4 )
             => ( ord_le6747313008572928689nteger @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_le6747313008572928689nteger @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_1409_order__less__subst1,axiom,
    ! [A: code_integer,F: rat > code_integer,B: rat,C: rat] :
      ( ( ord_le6747313008572928689nteger @ A @ ( F @ B ) )
     => ( ( ord_less_rat @ B @ C )
       => ( ! [X3: rat,Y4: rat] :
              ( ( ord_less_rat @ X3 @ Y4 )
             => ( ord_le6747313008572928689nteger @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_le6747313008572928689nteger @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_1410_order__less__subst1,axiom,
    ! [A: code_integer,F: num > code_integer,B: num,C: num] :
      ( ( ord_le6747313008572928689nteger @ A @ ( F @ B ) )
     => ( ( ord_less_num @ B @ C )
       => ( ! [X3: num,Y4: num] :
              ( ( ord_less_num @ X3 @ Y4 )
             => ( ord_le6747313008572928689nteger @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_le6747313008572928689nteger @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_1411_order__less__subst1,axiom,
    ! [A: code_integer,F: nat > code_integer,B: nat,C: nat] :
      ( ( ord_le6747313008572928689nteger @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_nat @ X3 @ Y4 )
             => ( ord_le6747313008572928689nteger @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_le6747313008572928689nteger @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_1412_order__less__subst1,axiom,
    ! [A: code_integer,F: int > code_integer,B: int,C: int] :
      ( ( ord_le6747313008572928689nteger @ A @ ( F @ B ) )
     => ( ( ord_less_int @ B @ C )
       => ( ! [X3: int,Y4: int] :
              ( ( ord_less_int @ X3 @ Y4 )
             => ( ord_le6747313008572928689nteger @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_le6747313008572928689nteger @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_1413_order__less__subst1,axiom,
    ! [A: rat,F: code_integer > rat,B: code_integer,C: code_integer] :
      ( ( ord_less_rat @ A @ ( F @ B ) )
     => ( ( ord_le6747313008572928689nteger @ B @ C )
       => ( ! [X3: code_integer,Y4: code_integer] :
              ( ( ord_le6747313008572928689nteger @ X3 @ Y4 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_1414_order__less__subst1,axiom,
    ! [A: rat,F: rat > rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ ( F @ B ) )
     => ( ( ord_less_rat @ B @ 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_1415_order__less__subst1,axiom,
    ! [A: rat,F: num > rat,B: num,C: num] :
      ( ( ord_less_rat @ A @ ( F @ B ) )
     => ( ( ord_less_num @ B @ 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_1416_order__less__subst1,axiom,
    ! [A: rat,F: nat > rat,B: nat,C: nat] :
      ( ( ord_less_rat @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ 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_1417_order__less__subst1,axiom,
    ! [A: rat,F: int > rat,B: int,C: int] :
      ( ( ord_less_rat @ A @ ( F @ B ) )
     => ( ( ord_less_int @ B @ 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_1418_order__less__subst2,axiom,
    ! [A: code_integer,B: code_integer,F: code_integer > code_integer,C: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ B )
     => ( ( ord_le6747313008572928689nteger @ ( F @ B ) @ C )
       => ( ! [X3: code_integer,Y4: code_integer] :
              ( ( ord_le6747313008572928689nteger @ X3 @ Y4 )
             => ( ord_le6747313008572928689nteger @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_le6747313008572928689nteger @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_1419_order__less__subst2,axiom,
    ! [A: code_integer,B: code_integer,F: code_integer > rat,C: rat] :
      ( ( ord_le6747313008572928689nteger @ A @ B )
     => ( ( ord_less_rat @ ( F @ B ) @ C )
       => ( ! [X3: code_integer,Y4: code_integer] :
              ( ( ord_le6747313008572928689nteger @ X3 @ Y4 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_1420_order__less__subst2,axiom,
    ! [A: code_integer,B: code_integer,F: code_integer > num,C: num] :
      ( ( ord_le6747313008572928689nteger @ A @ B )
     => ( ( ord_less_num @ ( F @ B ) @ C )
       => ( ! [X3: code_integer,Y4: code_integer] :
              ( ( ord_le6747313008572928689nteger @ X3 @ Y4 )
             => ( ord_less_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_1421_order__less__subst2,axiom,
    ! [A: code_integer,B: code_integer,F: code_integer > nat,C: nat] :
      ( ( ord_le6747313008572928689nteger @ A @ B )
     => ( ( ord_less_nat @ ( F @ B ) @ C )
       => ( ! [X3: code_integer,Y4: code_integer] :
              ( ( ord_le6747313008572928689nteger @ X3 @ Y4 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_1422_order__less__subst2,axiom,
    ! [A: code_integer,B: code_integer,F: code_integer > int,C: int] :
      ( ( ord_le6747313008572928689nteger @ A @ B )
     => ( ( ord_less_int @ ( F @ B ) @ C )
       => ( ! [X3: code_integer,Y4: code_integer] :
              ( ( ord_le6747313008572928689nteger @ X3 @ Y4 )
             => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_1423_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > code_integer,C: code_integer] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_le6747313008572928689nteger @ ( F @ B ) @ C )
       => ( ! [X3: rat,Y4: rat] :
              ( ( ord_less_rat @ X3 @ Y4 )
             => ( ord_le6747313008572928689nteger @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_le6747313008572928689nteger @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_1424_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ ( F @ B ) @ 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_1425_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > num,C: num] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_num @ ( F @ B ) @ 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_1426_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > nat,C: nat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_nat @ ( F @ B ) @ 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_1427_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > int,C: int] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_int @ ( F @ B ) @ 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_1428_order__less__not__sym,axiom,
    ! [X: code_integer,Y3: code_integer] :
      ( ( ord_le6747313008572928689nteger @ X @ Y3 )
     => ~ ( ord_le6747313008572928689nteger @ Y3 @ X ) ) ).

% order_less_not_sym
thf(fact_1429_order__less__not__sym,axiom,
    ! [X: rat,Y3: rat] :
      ( ( ord_less_rat @ X @ Y3 )
     => ~ ( ord_less_rat @ Y3 @ X ) ) ).

% order_less_not_sym
thf(fact_1430_order__less__not__sym,axiom,
    ! [X: num,Y3: num] :
      ( ( ord_less_num @ X @ Y3 )
     => ~ ( ord_less_num @ Y3 @ X ) ) ).

% order_less_not_sym
thf(fact_1431_order__less__not__sym,axiom,
    ! [X: nat,Y3: nat] :
      ( ( ord_less_nat @ X @ Y3 )
     => ~ ( ord_less_nat @ Y3 @ X ) ) ).

% order_less_not_sym
thf(fact_1432_order__less__not__sym,axiom,
    ! [X: int,Y3: int] :
      ( ( ord_less_int @ X @ Y3 )
     => ~ ( ord_less_int @ Y3 @ X ) ) ).

% order_less_not_sym
thf(fact_1433_order__less__imp__triv,axiom,
    ! [X: code_integer,Y3: code_integer,P: $o] :
      ( ( ord_le6747313008572928689nteger @ X @ Y3 )
     => ( ( ord_le6747313008572928689nteger @ Y3 @ X )
       => P ) ) ).

% order_less_imp_triv
thf(fact_1434_order__less__imp__triv,axiom,
    ! [X: rat,Y3: rat,P: $o] :
      ( ( ord_less_rat @ X @ Y3 )
     => ( ( ord_less_rat @ Y3 @ X )
       => P ) ) ).

% order_less_imp_triv
thf(fact_1435_order__less__imp__triv,axiom,
    ! [X: num,Y3: num,P: $o] :
      ( ( ord_less_num @ X @ Y3 )
     => ( ( ord_less_num @ Y3 @ X )
       => P ) ) ).

% order_less_imp_triv
thf(fact_1436_order__less__imp__triv,axiom,
    ! [X: nat,Y3: nat,P: $o] :
      ( ( ord_less_nat @ X @ Y3 )
     => ( ( ord_less_nat @ Y3 @ X )
       => P ) ) ).

% order_less_imp_triv
thf(fact_1437_order__less__imp__triv,axiom,
    ! [X: int,Y3: int,P: $o] :
      ( ( ord_less_int @ X @ Y3 )
     => ( ( ord_less_int @ Y3 @ X )
       => P ) ) ).

% order_less_imp_triv
thf(fact_1438_linorder__less__linear,axiom,
    ! [X: code_integer,Y3: code_integer] :
      ( ( ord_le6747313008572928689nteger @ X @ Y3 )
      | ( X = Y3 )
      | ( ord_le6747313008572928689nteger @ Y3 @ X ) ) ).

% linorder_less_linear
thf(fact_1439_linorder__less__linear,axiom,
    ! [X: rat,Y3: rat] :
      ( ( ord_less_rat @ X @ Y3 )
      | ( X = Y3 )
      | ( ord_less_rat @ Y3 @ X ) ) ).

% linorder_less_linear
thf(fact_1440_linorder__less__linear,axiom,
    ! [X: num,Y3: num] :
      ( ( ord_less_num @ X @ Y3 )
      | ( X = Y3 )
      | ( ord_less_num @ Y3 @ X ) ) ).

% linorder_less_linear
thf(fact_1441_linorder__less__linear,axiom,
    ! [X: nat,Y3: nat] :
      ( ( ord_less_nat @ X @ Y3 )
      | ( X = Y3 )
      | ( ord_less_nat @ Y3 @ X ) ) ).

% linorder_less_linear
thf(fact_1442_linorder__less__linear,axiom,
    ! [X: int,Y3: int] :
      ( ( ord_less_int @ X @ Y3 )
      | ( X = Y3 )
      | ( ord_less_int @ Y3 @ X ) ) ).

% linorder_less_linear
thf(fact_1443_order__less__imp__not__eq,axiom,
    ! [X: code_integer,Y3: code_integer] :
      ( ( ord_le6747313008572928689nteger @ X @ Y3 )
     => ( X != Y3 ) ) ).

% order_less_imp_not_eq
thf(fact_1444_order__less__imp__not__eq,axiom,
    ! [X: rat,Y3: rat] :
      ( ( ord_less_rat @ X @ Y3 )
     => ( X != Y3 ) ) ).

% order_less_imp_not_eq
thf(fact_1445_order__less__imp__not__eq,axiom,
    ! [X: num,Y3: num] :
      ( ( ord_less_num @ X @ Y3 )
     => ( X != Y3 ) ) ).

% order_less_imp_not_eq
thf(fact_1446_order__less__imp__not__eq,axiom,
    ! [X: nat,Y3: nat] :
      ( ( ord_less_nat @ X @ Y3 )
     => ( X != Y3 ) ) ).

% order_less_imp_not_eq
thf(fact_1447_order__less__imp__not__eq,axiom,
    ! [X: int,Y3: int] :
      ( ( ord_less_int @ X @ Y3 )
     => ( X != Y3 ) ) ).

% order_less_imp_not_eq
thf(fact_1448_order__less__imp__not__eq2,axiom,
    ! [X: code_integer,Y3: code_integer] :
      ( ( ord_le6747313008572928689nteger @ X @ Y3 )
     => ( Y3 != X ) ) ).

% order_less_imp_not_eq2
thf(fact_1449_order__less__imp__not__eq2,axiom,
    ! [X: rat,Y3: rat] :
      ( ( ord_less_rat @ X @ Y3 )
     => ( Y3 != X ) ) ).

% order_less_imp_not_eq2
thf(fact_1450_order__less__imp__not__eq2,axiom,
    ! [X: num,Y3: num] :
      ( ( ord_less_num @ X @ Y3 )
     => ( Y3 != X ) ) ).

% order_less_imp_not_eq2
thf(fact_1451_order__less__imp__not__eq2,axiom,
    ! [X: nat,Y3: nat] :
      ( ( ord_less_nat @ X @ Y3 )
     => ( Y3 != X ) ) ).

% order_less_imp_not_eq2
thf(fact_1452_order__less__imp__not__eq2,axiom,
    ! [X: int,Y3: int] :
      ( ( ord_less_int @ X @ Y3 )
     => ( Y3 != X ) ) ).

% order_less_imp_not_eq2
thf(fact_1453_order__less__imp__not__less,axiom,
    ! [X: code_integer,Y3: code_integer] :
      ( ( ord_le6747313008572928689nteger @ X @ Y3 )
     => ~ ( ord_le6747313008572928689nteger @ Y3 @ X ) ) ).

% order_less_imp_not_less
thf(fact_1454_order__less__imp__not__less,axiom,
    ! [X: rat,Y3: rat] :
      ( ( ord_less_rat @ X @ Y3 )
     => ~ ( ord_less_rat @ Y3 @ X ) ) ).

% order_less_imp_not_less
thf(fact_1455_order__less__imp__not__less,axiom,
    ! [X: num,Y3: num] :
      ( ( ord_less_num @ X @ Y3 )
     => ~ ( ord_less_num @ Y3 @ X ) ) ).

% order_less_imp_not_less
thf(fact_1456_order__less__imp__not__less,axiom,
    ! [X: nat,Y3: nat] :
      ( ( ord_less_nat @ X @ Y3 )
     => ~ ( ord_less_nat @ Y3 @ X ) ) ).

% order_less_imp_not_less
thf(fact_1457_order__less__imp__not__less,axiom,
    ! [X: int,Y3: int] :
      ( ( ord_less_int @ X @ Y3 )
     => ~ ( ord_less_int @ Y3 @ X ) ) ).

% order_less_imp_not_less
thf(fact_1458_minf_I8_J,axiom,
    ! [T: code_integer] :
    ? [Z3: code_integer] :
    ! [X4: code_integer] :
      ( ( ord_le6747313008572928689nteger @ X4 @ Z3 )
     => ~ ( ord_le3102999989581377725nteger @ T @ X4 ) ) ).

% minf(8)
thf(fact_1459_minf_I8_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X4: rat] :
      ( ( ord_less_rat @ X4 @ Z3 )
     => ~ ( ord_less_eq_rat @ T @ X4 ) ) ).

% minf(8)
thf(fact_1460_minf_I8_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X4: num] :
      ( ( ord_less_num @ X4 @ Z3 )
     => ~ ( ord_less_eq_num @ T @ X4 ) ) ).

% minf(8)
thf(fact_1461_minf_I8_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ X4 @ Z3 )
     => ~ ( ord_less_eq_nat @ T @ X4 ) ) ).

% minf(8)
thf(fact_1462_minf_I8_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X4: int] :
      ( ( ord_less_int @ X4 @ Z3 )
     => ~ ( ord_less_eq_int @ T @ X4 ) ) ).

% minf(8)
thf(fact_1463_minf_I6_J,axiom,
    ! [T: code_integer] :
    ? [Z3: code_integer] :
    ! [X4: code_integer] :
      ( ( ord_le6747313008572928689nteger @ X4 @ Z3 )
     => ( ord_le3102999989581377725nteger @ X4 @ T ) ) ).

% minf(6)
thf(fact_1464_minf_I6_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X4: rat] :
      ( ( ord_less_rat @ X4 @ Z3 )
     => ( ord_less_eq_rat @ X4 @ T ) ) ).

% minf(6)
thf(fact_1465_minf_I6_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X4: num] :
      ( ( ord_less_num @ X4 @ Z3 )
     => ( ord_less_eq_num @ X4 @ T ) ) ).

% minf(6)
thf(fact_1466_minf_I6_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ X4 @ Z3 )
     => ( ord_less_eq_nat @ X4 @ T ) ) ).

% minf(6)
thf(fact_1467_minf_I6_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X4: int] :
      ( ( ord_less_int @ X4 @ Z3 )
     => ( ord_less_eq_int @ X4 @ T ) ) ).

% minf(6)
thf(fact_1468_pinf_I8_J,axiom,
    ! [T: code_integer] :
    ? [Z3: code_integer] :
    ! [X4: code_integer] :
      ( ( ord_le6747313008572928689nteger @ Z3 @ X4 )
     => ( ord_le3102999989581377725nteger @ T @ X4 ) ) ).

% pinf(8)
thf(fact_1469_pinf_I8_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X4: rat] :
      ( ( ord_less_rat @ Z3 @ X4 )
     => ( ord_less_eq_rat @ T @ X4 ) ) ).

% pinf(8)
thf(fact_1470_pinf_I8_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X4: num] :
      ( ( ord_less_num @ Z3 @ X4 )
     => ( ord_less_eq_num @ T @ X4 ) ) ).

% pinf(8)
thf(fact_1471_pinf_I8_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ Z3 @ X4 )
     => ( ord_less_eq_nat @ T @ X4 ) ) ).

% pinf(8)
thf(fact_1472_pinf_I8_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X4: int] :
      ( ( ord_less_int @ Z3 @ X4 )
     => ( ord_less_eq_int @ T @ X4 ) ) ).

% pinf(8)
thf(fact_1473_pinf_I6_J,axiom,
    ! [T: code_integer] :
    ? [Z3: code_integer] :
    ! [X4: code_integer] :
      ( ( ord_le6747313008572928689nteger @ Z3 @ X4 )
     => ~ ( ord_le3102999989581377725nteger @ X4 @ T ) ) ).

% pinf(6)
thf(fact_1474_pinf_I6_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X4: rat] :
      ( ( ord_less_rat @ Z3 @ X4 )
     => ~ ( ord_less_eq_rat @ X4 @ T ) ) ).

% pinf(6)
thf(fact_1475_pinf_I6_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X4: num] :
      ( ( ord_less_num @ Z3 @ X4 )
     => ~ ( ord_less_eq_num @ X4 @ T ) ) ).

% pinf(6)
thf(fact_1476_pinf_I6_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ Z3 @ X4 )
     => ~ ( ord_less_eq_nat @ X4 @ T ) ) ).

% pinf(6)
thf(fact_1477_pinf_I6_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X4: int] :
      ( ( ord_less_int @ Z3 @ X4 )
     => ~ ( ord_less_eq_int @ X4 @ T ) ) ).

% pinf(6)
thf(fact_1478_leD,axiom,
    ! [Y3: code_integer,X: code_integer] :
      ( ( ord_le3102999989581377725nteger @ Y3 @ X )
     => ~ ( ord_le6747313008572928689nteger @ X @ Y3 ) ) ).

% leD
thf(fact_1479_leD,axiom,
    ! [Y3: set_nat,X: set_nat] :
      ( ( ord_less_eq_set_nat @ Y3 @ X )
     => ~ ( ord_less_set_nat @ X @ Y3 ) ) ).

% leD
thf(fact_1480_leD,axiom,
    ! [Y3: rat,X: rat] :
      ( ( ord_less_eq_rat @ Y3 @ X )
     => ~ ( ord_less_rat @ X @ Y3 ) ) ).

% leD
thf(fact_1481_leD,axiom,
    ! [Y3: num,X: num] :
      ( ( ord_less_eq_num @ Y3 @ X )
     => ~ ( ord_less_num @ X @ Y3 ) ) ).

% leD
thf(fact_1482_leD,axiom,
    ! [Y3: nat,X: nat] :
      ( ( ord_less_eq_nat @ Y3 @ X )
     => ~ ( ord_less_nat @ X @ Y3 ) ) ).

% leD
thf(fact_1483_leD,axiom,
    ! [Y3: int,X: int] :
      ( ( ord_less_eq_int @ Y3 @ X )
     => ~ ( ord_less_int @ X @ Y3 ) ) ).

% leD
thf(fact_1484_leI,axiom,
    ! [X: code_integer,Y3: code_integer] :
      ( ~ ( ord_le6747313008572928689nteger @ X @ Y3 )
     => ( ord_le3102999989581377725nteger @ Y3 @ X ) ) ).

% leI
thf(fact_1485_leI,axiom,
    ! [X: rat,Y3: rat] :
      ( ~ ( ord_less_rat @ X @ Y3 )
     => ( ord_less_eq_rat @ Y3 @ X ) ) ).

% leI
thf(fact_1486_leI,axiom,
    ! [X: num,Y3: num] :
      ( ~ ( ord_less_num @ X @ Y3 )
     => ( ord_less_eq_num @ Y3 @ X ) ) ).

% leI
thf(fact_1487_leI,axiom,
    ! [X: nat,Y3: nat] :
      ( ~ ( ord_less_nat @ X @ Y3 )
     => ( ord_less_eq_nat @ Y3 @ X ) ) ).

% leI
thf(fact_1488_leI,axiom,
    ! [X: int,Y3: int] :
      ( ~ ( ord_less_int @ X @ Y3 )
     => ( ord_less_eq_int @ Y3 @ X ) ) ).

% leI
thf(fact_1489_nless__le,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ~ ( ord_le6747313008572928689nteger @ A @ B ) )
      = ( ~ ( ord_le3102999989581377725nteger @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_1490_nless__le,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ~ ( ord_less_set_nat @ A @ B ) )
      = ( ~ ( ord_less_eq_set_nat @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_1491_nless__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ~ ( ord_less_rat @ A @ B ) )
      = ( ~ ( ord_less_eq_rat @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_1492_nless__le,axiom,
    ! [A: num,B: num] :
      ( ( ~ ( ord_less_num @ A @ B ) )
      = ( ~ ( ord_less_eq_num @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_1493_nless__le,axiom,
    ! [A: nat,B: nat] :
      ( ( ~ ( ord_less_nat @ A @ B ) )
      = ( ~ ( ord_less_eq_nat @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_1494_nless__le,axiom,
    ! [A: int,B: int] :
      ( ( ~ ( ord_less_int @ A @ B ) )
      = ( ~ ( ord_less_eq_int @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_1495_antisym__conv1,axiom,
    ! [X: code_integer,Y3: code_integer] :
      ( ~ ( ord_le6747313008572928689nteger @ X @ Y3 )
     => ( ( ord_le3102999989581377725nteger @ X @ Y3 )
        = ( X = Y3 ) ) ) ).

% antisym_conv1
thf(fact_1496_antisym__conv1,axiom,
    ! [X: set_nat,Y3: set_nat] :
      ( ~ ( ord_less_set_nat @ X @ Y3 )
     => ( ( ord_less_eq_set_nat @ X @ Y3 )
        = ( X = Y3 ) ) ) ).

% antisym_conv1
thf(fact_1497_antisym__conv1,axiom,
    ! [X: rat,Y3: rat] :
      ( ~ ( ord_less_rat @ X @ Y3 )
     => ( ( ord_less_eq_rat @ X @ Y3 )
        = ( X = Y3 ) ) ) ).

% antisym_conv1
thf(fact_1498_antisym__conv1,axiom,
    ! [X: num,Y3: num] :
      ( ~ ( ord_less_num @ X @ Y3 )
     => ( ( ord_less_eq_num @ X @ Y3 )
        = ( X = Y3 ) ) ) ).

% antisym_conv1
thf(fact_1499_antisym__conv1,axiom,
    ! [X: nat,Y3: nat] :
      ( ~ ( ord_less_nat @ X @ Y3 )
     => ( ( ord_less_eq_nat @ X @ Y3 )
        = ( X = Y3 ) ) ) ).

% antisym_conv1
thf(fact_1500_antisym__conv1,axiom,
    ! [X: int,Y3: int] :
      ( ~ ( ord_less_int @ X @ Y3 )
     => ( ( ord_less_eq_int @ X @ Y3 )
        = ( X = Y3 ) ) ) ).

% antisym_conv1
thf(fact_1501_antisym__conv2,axiom,
    ! [X: code_integer,Y3: code_integer] :
      ( ( ord_le3102999989581377725nteger @ X @ Y3 )
     => ( ( ~ ( ord_le6747313008572928689nteger @ X @ Y3 ) )
        = ( X = Y3 ) ) ) ).

% antisym_conv2
thf(fact_1502_antisym__conv2,axiom,
    ! [X: set_nat,Y3: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ Y3 )
     => ( ( ~ ( ord_less_set_nat @ X @ Y3 ) )
        = ( X = Y3 ) ) ) ).

% antisym_conv2
thf(fact_1503_antisym__conv2,axiom,
    ! [X: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ X @ Y3 )
     => ( ( ~ ( ord_less_rat @ X @ Y3 ) )
        = ( X = Y3 ) ) ) ).

% antisym_conv2
thf(fact_1504_antisym__conv2,axiom,
    ! [X: num,Y3: num] :
      ( ( ord_less_eq_num @ X @ Y3 )
     => ( ( ~ ( ord_less_num @ X @ Y3 ) )
        = ( X = Y3 ) ) ) ).

% antisym_conv2
thf(fact_1505_antisym__conv2,axiom,
    ! [X: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ X @ Y3 )
     => ( ( ~ ( ord_less_nat @ X @ Y3 ) )
        = ( X = Y3 ) ) ) ).

% antisym_conv2
thf(fact_1506_antisym__conv2,axiom,
    ! [X: int,Y3: int] :
      ( ( ord_less_eq_int @ X @ Y3 )
     => ( ( ~ ( ord_less_int @ X @ Y3 ) )
        = ( X = Y3 ) ) ) ).

% antisym_conv2
thf(fact_1507_dense__ge,axiom,
    ! [Z2: rat,Y3: rat] :
      ( ! [X3: rat] :
          ( ( ord_less_rat @ Z2 @ X3 )
         => ( ord_less_eq_rat @ Y3 @ X3 ) )
     => ( ord_less_eq_rat @ Y3 @ Z2 ) ) ).

% dense_ge
thf(fact_1508_dense__le,axiom,
    ! [Y3: rat,Z2: rat] :
      ( ! [X3: rat] :
          ( ( ord_less_rat @ X3 @ Y3 )
         => ( ord_less_eq_rat @ X3 @ Z2 ) )
     => ( ord_less_eq_rat @ Y3 @ Z2 ) ) ).

% dense_le
thf(fact_1509_less__le__not__le,axiom,
    ( ord_le6747313008572928689nteger
    = ( ^ [X2: code_integer,Y6: code_integer] :
          ( ( ord_le3102999989581377725nteger @ X2 @ Y6 )
          & ~ ( ord_le3102999989581377725nteger @ Y6 @ X2 ) ) ) ) ).

% less_le_not_le
thf(fact_1510_less__le__not__le,axiom,
    ( ord_less_set_nat
    = ( ^ [X2: set_nat,Y6: set_nat] :
          ( ( ord_less_eq_set_nat @ X2 @ Y6 )
          & ~ ( ord_less_eq_set_nat @ Y6 @ X2 ) ) ) ) ).

% less_le_not_le
thf(fact_1511_less__le__not__le,axiom,
    ( ord_less_rat
    = ( ^ [X2: rat,Y6: rat] :
          ( ( ord_less_eq_rat @ X2 @ Y6 )
          & ~ ( ord_less_eq_rat @ Y6 @ X2 ) ) ) ) ).

% less_le_not_le
thf(fact_1512_less__le__not__le,axiom,
    ( ord_less_num
    = ( ^ [X2: num,Y6: num] :
          ( ( ord_less_eq_num @ X2 @ Y6 )
          & ~ ( ord_less_eq_num @ Y6 @ X2 ) ) ) ) ).

% less_le_not_le
thf(fact_1513_less__le__not__le,axiom,
    ( ord_less_nat
    = ( ^ [X2: nat,Y6: nat] :
          ( ( ord_less_eq_nat @ X2 @ Y6 )
          & ~ ( ord_less_eq_nat @ Y6 @ X2 ) ) ) ) ).

% less_le_not_le
thf(fact_1514_less__le__not__le,axiom,
    ( ord_less_int
    = ( ^ [X2: int,Y6: int] :
          ( ( ord_less_eq_int @ X2 @ Y6 )
          & ~ ( ord_less_eq_int @ Y6 @ X2 ) ) ) ) ).

% less_le_not_le
thf(fact_1515_not__le__imp__less,axiom,
    ! [Y3: code_integer,X: code_integer] :
      ( ~ ( ord_le3102999989581377725nteger @ Y3 @ X )
     => ( ord_le6747313008572928689nteger @ X @ Y3 ) ) ).

% not_le_imp_less
thf(fact_1516_not__le__imp__less,axiom,
    ! [Y3: rat,X: rat] :
      ( ~ ( ord_less_eq_rat @ Y3 @ X )
     => ( ord_less_rat @ X @ Y3 ) ) ).

% not_le_imp_less
thf(fact_1517_not__le__imp__less,axiom,
    ! [Y3: num,X: num] :
      ( ~ ( ord_less_eq_num @ Y3 @ X )
     => ( ord_less_num @ X @ Y3 ) ) ).

% not_le_imp_less
thf(fact_1518_not__le__imp__less,axiom,
    ! [Y3: nat,X: nat] :
      ( ~ ( ord_less_eq_nat @ Y3 @ X )
     => ( ord_less_nat @ X @ Y3 ) ) ).

% not_le_imp_less
thf(fact_1519_not__le__imp__less,axiom,
    ! [Y3: int,X: int] :
      ( ~ ( ord_less_eq_int @ Y3 @ X )
     => ( ord_less_int @ X @ Y3 ) ) ).

% not_le_imp_less
thf(fact_1520_order_Oorder__iff__strict,axiom,
    ( ord_le3102999989581377725nteger
    = ( ^ [A2: code_integer,B2: code_integer] :
          ( ( ord_le6747313008572928689nteger @ A2 @ B2 )
          | ( A2 = B2 ) ) ) ) ).

% order.order_iff_strict
thf(fact_1521_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A2: set_nat,B2: set_nat] :
          ( ( ord_less_set_nat @ A2 @ B2 )
          | ( A2 = B2 ) ) ) ) ).

% order.order_iff_strict
thf(fact_1522_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_rat
    = ( ^ [A2: rat,B2: rat] :
          ( ( ord_less_rat @ A2 @ B2 )
          | ( A2 = B2 ) ) ) ) ).

% order.order_iff_strict
thf(fact_1523_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_num
    = ( ^ [A2: num,B2: num] :
          ( ( ord_less_num @ A2 @ B2 )
          | ( A2 = B2 ) ) ) ) ).

% order.order_iff_strict
thf(fact_1524_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [A2: nat,B2: nat] :
          ( ( ord_less_nat @ A2 @ B2 )
          | ( A2 = B2 ) ) ) ) ).

% order.order_iff_strict
thf(fact_1525_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_int
    = ( ^ [A2: int,B2: int] :
          ( ( ord_less_int @ A2 @ B2 )
          | ( A2 = B2 ) ) ) ) ).

% order.order_iff_strict
thf(fact_1526_order_Ostrict__iff__order,axiom,
    ( ord_le6747313008572928689nteger
    = ( ^ [A2: code_integer,B2: code_integer] :
          ( ( ord_le3102999989581377725nteger @ A2 @ B2 )
          & ( A2 != B2 ) ) ) ) ).

% order.strict_iff_order
thf(fact_1527_order_Ostrict__iff__order,axiom,
    ( ord_less_set_nat
    = ( ^ [A2: set_nat,B2: set_nat] :
          ( ( ord_less_eq_set_nat @ A2 @ B2 )
          & ( A2 != B2 ) ) ) ) ).

% order.strict_iff_order
thf(fact_1528_order_Ostrict__iff__order,axiom,
    ( ord_less_rat
    = ( ^ [A2: rat,B2: rat] :
          ( ( ord_less_eq_rat @ A2 @ B2 )
          & ( A2 != B2 ) ) ) ) ).

% order.strict_iff_order
thf(fact_1529_order_Ostrict__iff__order,axiom,
    ( ord_less_num
    = ( ^ [A2: num,B2: num] :
          ( ( ord_less_eq_num @ A2 @ B2 )
          & ( A2 != B2 ) ) ) ) ).

% order.strict_iff_order
thf(fact_1530_order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [A2: nat,B2: nat] :
          ( ( ord_less_eq_nat @ A2 @ B2 )
          & ( A2 != B2 ) ) ) ) ).

% order.strict_iff_order
thf(fact_1531_order_Ostrict__iff__order,axiom,
    ( ord_less_int
    = ( ^ [A2: int,B2: int] :
          ( ( ord_less_eq_int @ A2 @ B2 )
          & ( A2 != B2 ) ) ) ) ).

% order.strict_iff_order
thf(fact_1532_order_Ostrict__trans1,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ B )
     => ( ( ord_le6747313008572928689nteger @ B @ C )
       => ( ord_le6747313008572928689nteger @ A @ C ) ) ) ).

% order.strict_trans1
thf(fact_1533_order_Ostrict__trans1,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( ord_less_set_nat @ B @ C )
       => ( ord_less_set_nat @ A @ C ) ) ) ).

% order.strict_trans1
thf(fact_1534_order_Ostrict__trans1,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_rat @ B @ C )
       => ( ord_less_rat @ A @ C ) ) ) ).

% order.strict_trans1
thf(fact_1535_order_Ostrict__trans1,axiom,
    ! [A: num,B: num,C: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_num @ B @ C )
       => ( ord_less_num @ A @ C ) ) ) ).

% order.strict_trans1
thf(fact_1536_order_Ostrict__trans1,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ B @ C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% order.strict_trans1
thf(fact_1537_order_Ostrict__trans1,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_int @ B @ C )
       => ( ord_less_int @ A @ C ) ) ) ).

% order.strict_trans1
thf(fact_1538_order_Ostrict__trans2,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ B )
     => ( ( ord_le3102999989581377725nteger @ B @ C )
       => ( ord_le6747313008572928689nteger @ A @ C ) ) ) ).

% order.strict_trans2
thf(fact_1539_order_Ostrict__trans2,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( ( ord_less_set_nat @ A @ B )
     => ( ( ord_less_eq_set_nat @ B @ C )
       => ( ord_less_set_nat @ A @ C ) ) ) ).

% order.strict_trans2
thf(fact_1540_order_Ostrict__trans2,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ord_less_rat @ A @ C ) ) ) ).

% order.strict_trans2
thf(fact_1541_order_Ostrict__trans2,axiom,
    ! [A: num,B: num,C: num] :
      ( ( ord_less_num @ A @ B )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ord_less_num @ A @ C ) ) ) ).

% order.strict_trans2
thf(fact_1542_order_Ostrict__trans2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% order.strict_trans2
thf(fact_1543_order_Ostrict__trans2,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_eq_int @ B @ C )
       => ( ord_less_int @ A @ C ) ) ) ).

% order.strict_trans2
thf(fact_1544_order_Ostrict__iff__not,axiom,
    ( ord_le6747313008572928689nteger
    = ( ^ [A2: code_integer,B2: code_integer] :
          ( ( ord_le3102999989581377725nteger @ A2 @ B2 )
          & ~ ( ord_le3102999989581377725nteger @ B2 @ A2 ) ) ) ) ).

% order.strict_iff_not
thf(fact_1545_order_Ostrict__iff__not,axiom,
    ( ord_less_set_nat
    = ( ^ [A2: set_nat,B2: set_nat] :
          ( ( ord_less_eq_set_nat @ A2 @ B2 )
          & ~ ( ord_less_eq_set_nat @ B2 @ A2 ) ) ) ) ).

% order.strict_iff_not
thf(fact_1546_order_Ostrict__iff__not,axiom,
    ( ord_less_rat
    = ( ^ [A2: rat,B2: rat] :
          ( ( ord_less_eq_rat @ A2 @ B2 )
          & ~ ( ord_less_eq_rat @ B2 @ A2 ) ) ) ) ).

% order.strict_iff_not
thf(fact_1547_order_Ostrict__iff__not,axiom,
    ( ord_less_num
    = ( ^ [A2: num,B2: num] :
          ( ( ord_less_eq_num @ A2 @ B2 )
          & ~ ( ord_less_eq_num @ B2 @ A2 ) ) ) ) ).

% order.strict_iff_not
thf(fact_1548_order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [A2: nat,B2: nat] :
          ( ( ord_less_eq_nat @ A2 @ B2 )
          & ~ ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).

% order.strict_iff_not
thf(fact_1549_order_Ostrict__iff__not,axiom,
    ( ord_less_int
    = ( ^ [A2: int,B2: int] :
          ( ( ord_less_eq_int @ A2 @ B2 )
          & ~ ( ord_less_eq_int @ B2 @ A2 ) ) ) ) ).

% order.strict_iff_not
thf(fact_1550_dense__ge__bounded,axiom,
    ! [Z2: rat,X: rat,Y3: rat] :
      ( ( ord_less_rat @ Z2 @ X )
     => ( ! [W: rat] :
            ( ( ord_less_rat @ Z2 @ W )
           => ( ( ord_less_rat @ W @ X )
             => ( ord_less_eq_rat @ Y3 @ W ) ) )
       => ( ord_less_eq_rat @ Y3 @ Z2 ) ) ) ).

% dense_ge_bounded
thf(fact_1551_dense__le__bounded,axiom,
    ! [X: rat,Y3: rat,Z2: rat] :
      ( ( ord_less_rat @ X @ Y3 )
     => ( ! [W: rat] :
            ( ( ord_less_rat @ X @ W )
           => ( ( ord_less_rat @ W @ Y3 )
             => ( ord_less_eq_rat @ W @ Z2 ) ) )
       => ( ord_less_eq_rat @ Y3 @ Z2 ) ) ) ).

% dense_le_bounded
thf(fact_1552_dual__order_Oorder__iff__strict,axiom,
    ( ord_le3102999989581377725nteger
    = ( ^ [B2: code_integer,A2: code_integer] :
          ( ( ord_le6747313008572928689nteger @ B2 @ A2 )
          | ( A2 = B2 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_1553_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [B2: set_nat,A2: set_nat] :
          ( ( ord_less_set_nat @ B2 @ A2 )
          | ( A2 = B2 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_1554_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_rat
    = ( ^ [B2: rat,A2: rat] :
          ( ( ord_less_rat @ B2 @ A2 )
          | ( A2 = B2 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_1555_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_num
    = ( ^ [B2: num,A2: num] :
          ( ( ord_less_num @ B2 @ A2 )
          | ( A2 = B2 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_1556_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [B2: nat,A2: nat] :
          ( ( ord_less_nat @ B2 @ A2 )
          | ( A2 = B2 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_1557_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_int
    = ( ^ [B2: int,A2: int] :
          ( ( ord_less_int @ B2 @ A2 )
          | ( A2 = B2 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_1558_dual__order_Ostrict__iff__order,axiom,
    ( ord_le6747313008572928689nteger
    = ( ^ [B2: code_integer,A2: code_integer] :
          ( ( ord_le3102999989581377725nteger @ B2 @ A2 )
          & ( A2 != B2 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_1559_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_set_nat
    = ( ^ [B2: set_nat,A2: set_nat] :
          ( ( ord_less_eq_set_nat @ B2 @ A2 )
          & ( A2 != B2 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_1560_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_rat
    = ( ^ [B2: rat,A2: rat] :
          ( ( ord_less_eq_rat @ B2 @ A2 )
          & ( A2 != B2 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_1561_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_num
    = ( ^ [B2: num,A2: num] :
          ( ( ord_less_eq_num @ B2 @ A2 )
          & ( A2 != B2 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_1562_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [B2: nat,A2: nat] :
          ( ( ord_less_eq_nat @ B2 @ A2 )
          & ( A2 != B2 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_1563_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_int
    = ( ^ [B2: int,A2: int] :
          ( ( ord_less_eq_int @ B2 @ A2 )
          & ( A2 != B2 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_1564_dual__order_Ostrict__trans1,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( ord_le3102999989581377725nteger @ B @ A )
     => ( ( ord_le6747313008572928689nteger @ C @ B )
       => ( ord_le6747313008572928689nteger @ C @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_1565_dual__order_Ostrict__trans1,axiom,
    ! [B: set_nat,A: set_nat,C: set_nat] :
      ( ( ord_less_eq_set_nat @ B @ A )
     => ( ( ord_less_set_nat @ C @ B )
       => ( ord_less_set_nat @ C @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_1566_dual__order_Ostrict__trans1,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( ord_less_rat @ C @ B )
       => ( ord_less_rat @ C @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_1567_dual__order_Ostrict__trans1,axiom,
    ! [B: num,A: num,C: num] :
      ( ( ord_less_eq_num @ B @ A )
     => ( ( ord_less_num @ C @ B )
       => ( ord_less_num @ C @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_1568_dual__order_Ostrict__trans1,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_nat @ C @ B )
       => ( ord_less_nat @ C @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_1569_dual__order_Ostrict__trans1,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_less_int @ C @ B )
       => ( ord_less_int @ C @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_1570_dual__order_Ostrict__trans2,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( ord_le6747313008572928689nteger @ B @ A )
     => ( ( ord_le3102999989581377725nteger @ C @ B )
       => ( ord_le6747313008572928689nteger @ C @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_1571_dual__order_Ostrict__trans2,axiom,
    ! [B: set_nat,A: set_nat,C: set_nat] :
      ( ( ord_less_set_nat @ B @ A )
     => ( ( ord_less_eq_set_nat @ C @ B )
       => ( ord_less_set_nat @ C @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_1572_dual__order_Ostrict__trans2,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( ord_less_eq_rat @ C @ B )
       => ( ord_less_rat @ C @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_1573_dual__order_Ostrict__trans2,axiom,
    ! [B: num,A: num,C: num] :
      ( ( ord_less_num @ B @ A )
     => ( ( ord_less_eq_num @ C @ B )
       => ( ord_less_num @ C @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_1574_dual__order_Ostrict__trans2,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( ord_less_eq_nat @ C @ B )
       => ( ord_less_nat @ C @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_1575_dual__order_Ostrict__trans2,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( ord_less_eq_int @ C @ B )
       => ( ord_less_int @ C @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_1576_dual__order_Ostrict__iff__not,axiom,
    ( ord_le6747313008572928689nteger
    = ( ^ [B2: code_integer,A2: code_integer] :
          ( ( ord_le3102999989581377725nteger @ B2 @ A2 )
          & ~ ( ord_le3102999989581377725nteger @ A2 @ B2 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_1577_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_set_nat
    = ( ^ [B2: set_nat,A2: set_nat] :
          ( ( ord_less_eq_set_nat @ B2 @ A2 )
          & ~ ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_1578_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_rat
    = ( ^ [B2: rat,A2: rat] :
          ( ( ord_less_eq_rat @ B2 @ A2 )
          & ~ ( ord_less_eq_rat @ A2 @ B2 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_1579_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_num
    = ( ^ [B2: num,A2: num] :
          ( ( ord_less_eq_num @ B2 @ A2 )
          & ~ ( ord_less_eq_num @ A2 @ B2 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_1580_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [B2: nat,A2: nat] :
          ( ( ord_less_eq_nat @ B2 @ A2 )
          & ~ ( ord_less_eq_nat @ A2 @ B2 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_1581_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_int
    = ( ^ [B2: int,A2: int] :
          ( ( ord_less_eq_int @ B2 @ A2 )
          & ~ ( ord_less_eq_int @ A2 @ B2 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_1582_order_Ostrict__implies__order,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ B )
     => ( ord_le3102999989581377725nteger @ A @ B ) ) ).

% order.strict_implies_order
thf(fact_1583_order_Ostrict__implies__order,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ord_less_set_nat @ A @ B )
     => ( ord_less_eq_set_nat @ A @ B ) ) ).

% order.strict_implies_order
thf(fact_1584_order_Ostrict__implies__order,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ord_less_eq_rat @ A @ B ) ) ).

% order.strict_implies_order
thf(fact_1585_order_Ostrict__implies__order,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_num @ A @ B )
     => ( ord_less_eq_num @ A @ B ) ) ).

% order.strict_implies_order
thf(fact_1586_order_Ostrict__implies__order,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ord_less_eq_nat @ A @ B ) ) ).

% order.strict_implies_order
thf(fact_1587_order_Ostrict__implies__order,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ( ord_less_eq_int @ A @ B ) ) ).

% order.strict_implies_order
thf(fact_1588_dual__order_Ostrict__implies__order,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ B @ A )
     => ( ord_le3102999989581377725nteger @ B @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_1589_dual__order_Ostrict__implies__order,axiom,
    ! [B: set_nat,A: set_nat] :
      ( ( ord_less_set_nat @ B @ A )
     => ( ord_less_eq_set_nat @ B @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_1590_dual__order_Ostrict__implies__order,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ord_less_eq_rat @ B @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_1591_dual__order_Ostrict__implies__order,axiom,
    ! [B: num,A: num] :
      ( ( ord_less_num @ B @ A )
     => ( ord_less_eq_num @ B @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_1592_dual__order_Ostrict__implies__order,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ord_less_eq_nat @ B @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_1593_dual__order_Ostrict__implies__order,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ A )
     => ( ord_less_eq_int @ B @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_1594_order__le__less,axiom,
    ( ord_le3102999989581377725nteger
    = ( ^ [X2: code_integer,Y6: code_integer] :
          ( ( ord_le6747313008572928689nteger @ X2 @ Y6 )
          | ( X2 = Y6 ) ) ) ) ).

% order_le_less
thf(fact_1595_order__le__less,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [X2: set_nat,Y6: set_nat] :
          ( ( ord_less_set_nat @ X2 @ Y6 )
          | ( X2 = Y6 ) ) ) ) ).

% order_le_less
thf(fact_1596_order__le__less,axiom,
    ( ord_less_eq_rat
    = ( ^ [X2: rat,Y6: rat] :
          ( ( ord_less_rat @ X2 @ Y6 )
          | ( X2 = Y6 ) ) ) ) ).

% order_le_less
thf(fact_1597_order__le__less,axiom,
    ( ord_less_eq_num
    = ( ^ [X2: num,Y6: num] :
          ( ( ord_less_num @ X2 @ Y6 )
          | ( X2 = Y6 ) ) ) ) ).

% order_le_less
thf(fact_1598_order__le__less,axiom,
    ( ord_less_eq_nat
    = ( ^ [X2: nat,Y6: nat] :
          ( ( ord_less_nat @ X2 @ Y6 )
          | ( X2 = Y6 ) ) ) ) ).

% order_le_less
thf(fact_1599_order__le__less,axiom,
    ( ord_less_eq_int
    = ( ^ [X2: int,Y6: int] :
          ( ( ord_less_int @ X2 @ Y6 )
          | ( X2 = Y6 ) ) ) ) ).

% order_le_less
thf(fact_1600_order__less__le,axiom,
    ( ord_le6747313008572928689nteger
    = ( ^ [X2: code_integer,Y6: code_integer] :
          ( ( ord_le3102999989581377725nteger @ X2 @ Y6 )
          & ( X2 != Y6 ) ) ) ) ).

% order_less_le
thf(fact_1601_order__less__le,axiom,
    ( ord_less_set_nat
    = ( ^ [X2: set_nat,Y6: set_nat] :
          ( ( ord_less_eq_set_nat @ X2 @ Y6 )
          & ( X2 != Y6 ) ) ) ) ).

% order_less_le
thf(fact_1602_order__less__le,axiom,
    ( ord_less_rat
    = ( ^ [X2: rat,Y6: rat] :
          ( ( ord_less_eq_rat @ X2 @ Y6 )
          & ( X2 != Y6 ) ) ) ) ).

% order_less_le
thf(fact_1603_order__less__le,axiom,
    ( ord_less_num
    = ( ^ [X2: num,Y6: num] :
          ( ( ord_less_eq_num @ X2 @ Y6 )
          & ( X2 != Y6 ) ) ) ) ).

% order_less_le
thf(fact_1604_order__less__le,axiom,
    ( ord_less_nat
    = ( ^ [X2: nat,Y6: nat] :
          ( ( ord_less_eq_nat @ X2 @ Y6 )
          & ( X2 != Y6 ) ) ) ) ).

% order_less_le
thf(fact_1605_order__less__le,axiom,
    ( ord_less_int
    = ( ^ [X2: int,Y6: int] :
          ( ( ord_less_eq_int @ X2 @ Y6 )
          & ( X2 != Y6 ) ) ) ) ).

% order_less_le
thf(fact_1606_linorder__not__le,axiom,
    ! [X: code_integer,Y3: code_integer] :
      ( ( ~ ( ord_le3102999989581377725nteger @ X @ Y3 ) )
      = ( ord_le6747313008572928689nteger @ Y3 @ X ) ) ).

% linorder_not_le
thf(fact_1607_linorder__not__le,axiom,
    ! [X: rat,Y3: rat] :
      ( ( ~ ( ord_less_eq_rat @ X @ Y3 ) )
      = ( ord_less_rat @ Y3 @ X ) ) ).

% linorder_not_le
thf(fact_1608_linorder__not__le,axiom,
    ! [X: num,Y3: num] :
      ( ( ~ ( ord_less_eq_num @ X @ Y3 ) )
      = ( ord_less_num @ Y3 @ X ) ) ).

% linorder_not_le
thf(fact_1609_linorder__not__le,axiom,
    ! [X: nat,Y3: nat] :
      ( ( ~ ( ord_less_eq_nat @ X @ Y3 ) )
      = ( ord_less_nat @ Y3 @ X ) ) ).

% linorder_not_le
thf(fact_1610_linorder__not__le,axiom,
    ! [X: int,Y3: int] :
      ( ( ~ ( ord_less_eq_int @ X @ Y3 ) )
      = ( ord_less_int @ Y3 @ X ) ) ).

% linorder_not_le
thf(fact_1611_linorder__not__less,axiom,
    ! [X: code_integer,Y3: code_integer] :
      ( ( ~ ( ord_le6747313008572928689nteger @ X @ Y3 ) )
      = ( ord_le3102999989581377725nteger @ Y3 @ X ) ) ).

% linorder_not_less
thf(fact_1612_linorder__not__less,axiom,
    ! [X: rat,Y3: rat] :
      ( ( ~ ( ord_less_rat @ X @ Y3 ) )
      = ( ord_less_eq_rat @ Y3 @ X ) ) ).

% linorder_not_less
thf(fact_1613_linorder__not__less,axiom,
    ! [X: num,Y3: num] :
      ( ( ~ ( ord_less_num @ X @ Y3 ) )
      = ( ord_less_eq_num @ Y3 @ X ) ) ).

% linorder_not_less
thf(fact_1614_linorder__not__less,axiom,
    ! [X: nat,Y3: nat] :
      ( ( ~ ( ord_less_nat @ X @ Y3 ) )
      = ( ord_less_eq_nat @ Y3 @ X ) ) ).

% linorder_not_less
thf(fact_1615_linorder__not__less,axiom,
    ! [X: int,Y3: int] :
      ( ( ~ ( ord_less_int @ X @ Y3 ) )
      = ( ord_less_eq_int @ Y3 @ X ) ) ).

% linorder_not_less
thf(fact_1616_order__less__imp__le,axiom,
    ! [X: code_integer,Y3: code_integer] :
      ( ( ord_le6747313008572928689nteger @ X @ Y3 )
     => ( ord_le3102999989581377725nteger @ X @ Y3 ) ) ).

% order_less_imp_le
thf(fact_1617_order__less__imp__le,axiom,
    ! [X: set_nat,Y3: set_nat] :
      ( ( ord_less_set_nat @ X @ Y3 )
     => ( ord_less_eq_set_nat @ X @ Y3 ) ) ).

% order_less_imp_le
thf(fact_1618_order__less__imp__le,axiom,
    ! [X: rat,Y3: rat] :
      ( ( ord_less_rat @ X @ Y3 )
     => ( ord_less_eq_rat @ X @ Y3 ) ) ).

% order_less_imp_le
thf(fact_1619_order__less__imp__le,axiom,
    ! [X: num,Y3: num] :
      ( ( ord_less_num @ X @ Y3 )
     => ( ord_less_eq_num @ X @ Y3 ) ) ).

% order_less_imp_le
thf(fact_1620_order__less__imp__le,axiom,
    ! [X: nat,Y3: nat] :
      ( ( ord_less_nat @ X @ Y3 )
     => ( ord_less_eq_nat @ X @ Y3 ) ) ).

% order_less_imp_le
thf(fact_1621_order__less__imp__le,axiom,
    ! [X: int,Y3: int] :
      ( ( ord_less_int @ X @ Y3 )
     => ( ord_less_eq_int @ X @ Y3 ) ) ).

% order_less_imp_le
thf(fact_1622_order__le__neq__trans,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ B )
     => ( ( A != B )
       => ( ord_le6747313008572928689nteger @ A @ B ) ) ) ).

% order_le_neq_trans
thf(fact_1623_order__le__neq__trans,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( A != B )
       => ( ord_less_set_nat @ A @ B ) ) ) ).

% order_le_neq_trans
thf(fact_1624_order__le__neq__trans,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( A != B )
       => ( ord_less_rat @ A @ B ) ) ) ).

% order_le_neq_trans
thf(fact_1625_order__le__neq__trans,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( A != B )
       => ( ord_less_num @ A @ B ) ) ) ).

% order_le_neq_trans
thf(fact_1626_order__le__neq__trans,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( A != B )
       => ( ord_less_nat @ A @ B ) ) ) ).

% order_le_neq_trans
thf(fact_1627_order__le__neq__trans,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( A != B )
       => ( ord_less_int @ A @ B ) ) ) ).

% order_le_neq_trans
thf(fact_1628_order__neq__le__trans,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A != B )
     => ( ( ord_le3102999989581377725nteger @ A @ B )
       => ( ord_le6747313008572928689nteger @ A @ B ) ) ) ).

% order_neq_le_trans
thf(fact_1629_order__neq__le__trans,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( A != B )
     => ( ( ord_less_eq_set_nat @ A @ B )
       => ( ord_less_set_nat @ A @ B ) ) ) ).

% order_neq_le_trans
thf(fact_1630_order__neq__le__trans,axiom,
    ! [A: rat,B: rat] :
      ( ( A != B )
     => ( ( ord_less_eq_rat @ A @ B )
       => ( ord_less_rat @ A @ B ) ) ) ).

% order_neq_le_trans
thf(fact_1631_order__neq__le__trans,axiom,
    ! [A: num,B: num] :
      ( ( A != B )
     => ( ( ord_less_eq_num @ A @ B )
       => ( ord_less_num @ A @ B ) ) ) ).

% order_neq_le_trans
thf(fact_1632_order__neq__le__trans,axiom,
    ! [A: nat,B: nat] :
      ( ( A != B )
     => ( ( ord_less_eq_nat @ A @ B )
       => ( ord_less_nat @ A @ B ) ) ) ).

% order_neq_le_trans
thf(fact_1633_order__neq__le__trans,axiom,
    ! [A: int,B: int] :
      ( ( A != B )
     => ( ( ord_less_eq_int @ A @ B )
       => ( ord_less_int @ A @ B ) ) ) ).

% order_neq_le_trans
thf(fact_1634_order__le__less__trans,axiom,
    ! [X: code_integer,Y3: code_integer,Z2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ X @ Y3 )
     => ( ( ord_le6747313008572928689nteger @ Y3 @ Z2 )
       => ( ord_le6747313008572928689nteger @ X @ Z2 ) ) ) ).

% order_le_less_trans
thf(fact_1635_order__le__less__trans,axiom,
    ! [X: set_nat,Y3: set_nat,Z2: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ Y3 )
     => ( ( ord_less_set_nat @ Y3 @ Z2 )
       => ( ord_less_set_nat @ X @ Z2 ) ) ) ).

% order_le_less_trans
thf(fact_1636_order__le__less__trans,axiom,
    ! [X: rat,Y3: rat,Z2: rat] :
      ( ( ord_less_eq_rat @ X @ Y3 )
     => ( ( ord_less_rat @ Y3 @ Z2 )
       => ( ord_less_rat @ X @ Z2 ) ) ) ).

% order_le_less_trans
thf(fact_1637_order__le__less__trans,axiom,
    ! [X: num,Y3: num,Z2: num] :
      ( ( ord_less_eq_num @ X @ Y3 )
     => ( ( ord_less_num @ Y3 @ Z2 )
       => ( ord_less_num @ X @ Z2 ) ) ) ).

% order_le_less_trans
thf(fact_1638_order__le__less__trans,axiom,
    ! [X: nat,Y3: nat,Z2: nat] :
      ( ( ord_less_eq_nat @ X @ Y3 )
     => ( ( ord_less_nat @ Y3 @ Z2 )
       => ( ord_less_nat @ X @ Z2 ) ) ) ).

% order_le_less_trans
thf(fact_1639_order__le__less__trans,axiom,
    ! [X: int,Y3: int,Z2: int] :
      ( ( ord_less_eq_int @ X @ Y3 )
     => ( ( ord_less_int @ Y3 @ Z2 )
       => ( ord_less_int @ X @ Z2 ) ) ) ).

% order_le_less_trans
thf(fact_1640_order__less__le__trans,axiom,
    ! [X: code_integer,Y3: code_integer,Z2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ X @ Y3 )
     => ( ( ord_le3102999989581377725nteger @ Y3 @ Z2 )
       => ( ord_le6747313008572928689nteger @ X @ Z2 ) ) ) ).

% order_less_le_trans
thf(fact_1641_order__less__le__trans,axiom,
    ! [X: set_nat,Y3: set_nat,Z2: set_nat] :
      ( ( ord_less_set_nat @ X @ Y3 )
     => ( ( ord_less_eq_set_nat @ Y3 @ Z2 )
       => ( ord_less_set_nat @ X @ Z2 ) ) ) ).

% order_less_le_trans
thf(fact_1642_order__less__le__trans,axiom,
    ! [X: rat,Y3: rat,Z2: rat] :
      ( ( ord_less_rat @ X @ Y3 )
     => ( ( ord_less_eq_rat @ Y3 @ Z2 )
       => ( ord_less_rat @ X @ Z2 ) ) ) ).

% order_less_le_trans
thf(fact_1643_order__less__le__trans,axiom,
    ! [X: num,Y3: num,Z2: num] :
      ( ( ord_less_num @ X @ Y3 )
     => ( ( ord_less_eq_num @ Y3 @ Z2 )
       => ( ord_less_num @ X @ Z2 ) ) ) ).

% order_less_le_trans
thf(fact_1644_order__less__le__trans,axiom,
    ! [X: nat,Y3: nat,Z2: nat] :
      ( ( ord_less_nat @ X @ Y3 )
     => ( ( ord_less_eq_nat @ Y3 @ Z2 )
       => ( ord_less_nat @ X @ Z2 ) ) ) ).

% order_less_le_trans
thf(fact_1645_order__less__le__trans,axiom,
    ! [X: int,Y3: int,Z2: int] :
      ( ( ord_less_int @ X @ Y3 )
     => ( ( ord_less_eq_int @ Y3 @ Z2 )
       => ( ord_less_int @ X @ Z2 ) ) ) ).

% order_less_le_trans
thf(fact_1646_order__le__less__subst1,axiom,
    ! [A: code_integer,F: code_integer > code_integer,B: code_integer,C: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ ( F @ B ) )
     => ( ( ord_le6747313008572928689nteger @ B @ C )
       => ( ! [X3: code_integer,Y4: code_integer] :
              ( ( ord_le6747313008572928689nteger @ X3 @ Y4 )
             => ( ord_le6747313008572928689nteger @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_le6747313008572928689nteger @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1647_order__le__less__subst1,axiom,
    ! [A: code_integer,F: rat > code_integer,B: rat,C: rat] :
      ( ( ord_le3102999989581377725nteger @ A @ ( F @ B ) )
     => ( ( ord_less_rat @ B @ C )
       => ( ! [X3: rat,Y4: rat] :
              ( ( ord_less_rat @ X3 @ Y4 )
             => ( ord_le6747313008572928689nteger @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_le6747313008572928689nteger @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1648_order__le__less__subst1,axiom,
    ! [A: code_integer,F: num > code_integer,B: num,C: num] :
      ( ( ord_le3102999989581377725nteger @ A @ ( F @ B ) )
     => ( ( ord_less_num @ B @ C )
       => ( ! [X3: num,Y4: num] :
              ( ( ord_less_num @ X3 @ Y4 )
             => ( ord_le6747313008572928689nteger @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_le6747313008572928689nteger @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1649_order__le__less__subst1,axiom,
    ! [A: code_integer,F: nat > code_integer,B: nat,C: nat] :
      ( ( ord_le3102999989581377725nteger @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_nat @ X3 @ Y4 )
             => ( ord_le6747313008572928689nteger @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_le6747313008572928689nteger @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1650_order__le__less__subst1,axiom,
    ! [A: code_integer,F: int > code_integer,B: int,C: int] :
      ( ( ord_le3102999989581377725nteger @ A @ ( F @ B ) )
     => ( ( ord_less_int @ B @ C )
       => ( ! [X3: int,Y4: int] :
              ( ( ord_less_int @ X3 @ Y4 )
             => ( ord_le6747313008572928689nteger @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_le6747313008572928689nteger @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1651_order__le__less__subst1,axiom,
    ! [A: rat,F: code_integer > rat,B: code_integer,C: code_integer] :
      ( ( ord_less_eq_rat @ A @ ( F @ B ) )
     => ( ( ord_le6747313008572928689nteger @ B @ C )
       => ( ! [X3: code_integer,Y4: code_integer] :
              ( ( ord_le6747313008572928689nteger @ X3 @ Y4 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1652_order__le__less__subst1,axiom,
    ! [A: rat,F: rat > rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ ( F @ B ) )
     => ( ( ord_less_rat @ B @ 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_1653_order__le__less__subst1,axiom,
    ! [A: rat,F: num > rat,B: num,C: num] :
      ( ( ord_less_eq_rat @ A @ ( F @ B ) )
     => ( ( ord_less_num @ B @ 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_1654_order__le__less__subst1,axiom,
    ! [A: rat,F: nat > rat,B: nat,C: nat] :
      ( ( ord_less_eq_rat @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ 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_1655_order__le__less__subst1,axiom,
    ! [A: rat,F: int > rat,B: int,C: int] :
      ( ( ord_less_eq_rat @ A @ ( F @ B ) )
     => ( ( ord_less_int @ B @ 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_1656_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > code_integer,C: code_integer] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_le6747313008572928689nteger @ ( F @ B ) @ C )
       => ( ! [X3: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y4 )
             => ( ord_le3102999989581377725nteger @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_le6747313008572928689nteger @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_1657_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_rat @ ( F @ B ) @ 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_1658_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > num,C: num] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_num @ ( F @ B ) @ 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_1659_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > nat,C: nat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_nat @ ( F @ B ) @ 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_1660_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > int,C: int] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_int @ ( F @ B ) @ 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_1661_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: num > code_integer,C: code_integer] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_le6747313008572928689nteger @ ( F @ B ) @ C )
       => ( ! [X3: num,Y4: num] :
              ( ( ord_less_eq_num @ X3 @ Y4 )
             => ( ord_le3102999989581377725nteger @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_le6747313008572928689nteger @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_1662_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: num > rat,C: rat] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_rat @ ( F @ B ) @ 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_1663_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: num > num,C: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_num @ ( F @ B ) @ 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_1664_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: num > nat,C: nat] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_nat @ ( F @ B ) @ 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_1665_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: num > int,C: int] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_int @ ( F @ B ) @ 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_1666_order__less__le__subst1,axiom,
    ! [A: code_integer,F: rat > code_integer,B: rat,C: rat] :
      ( ( ord_le6747313008572928689nteger @ A @ ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ! [X3: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y4 )
             => ( ord_le3102999989581377725nteger @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_le6747313008572928689nteger @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_1667_order__less__le__subst1,axiom,
    ! [A: rat,F: rat > rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ 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_1668_order__less__le__subst1,axiom,
    ! [A: num,F: rat > num,B: rat,C: rat] :
      ( ( ord_less_num @ A @ ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ 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_1669_order__less__le__subst1,axiom,
    ! [A: nat,F: rat > nat,B: rat,C: rat] :
      ( ( ord_less_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ 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_1670_order__less__le__subst1,axiom,
    ! [A: int,F: rat > int,B: rat,C: rat] :
      ( ( ord_less_int @ A @ ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ 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_1671_order__less__le__subst1,axiom,
    ! [A: code_integer,F: num > code_integer,B: num,C: num] :
      ( ( ord_le6747313008572928689nteger @ A @ ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ! [X3: num,Y4: num] :
              ( ( ord_less_eq_num @ X3 @ Y4 )
             => ( ord_le3102999989581377725nteger @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_le6747313008572928689nteger @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_1672_order__less__le__subst1,axiom,
    ! [A: rat,F: num > rat,B: num,C: num] :
      ( ( ord_less_rat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ 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_1673_order__less__le__subst1,axiom,
    ! [A: num,F: num > num,B: num,C: num] :
      ( ( ord_less_num @ A @ ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ 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_1674_order__less__le__subst1,axiom,
    ! [A: nat,F: num > nat,B: num,C: num] :
      ( ( ord_less_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ 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_1675_order__less__le__subst1,axiom,
    ! [A: int,F: num > int,B: num,C: num] :
      ( ( ord_less_int @ A @ ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ 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_1676_order__less__le__subst2,axiom,
    ! [A: code_integer,B: code_integer,F: code_integer > code_integer,C: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ B )
     => ( ( ord_le3102999989581377725nteger @ ( F @ B ) @ C )
       => ( ! [X3: code_integer,Y4: code_integer] :
              ( ( ord_le6747313008572928689nteger @ X3 @ Y4 )
             => ( ord_le6747313008572928689nteger @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_le6747313008572928689nteger @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_1677_order__less__le__subst2,axiom,
    ! [A: rat,B: rat,F: rat > code_integer,C: code_integer] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_le3102999989581377725nteger @ ( F @ B ) @ C )
       => ( ! [X3: rat,Y4: rat] :
              ( ( ord_less_rat @ X3 @ Y4 )
             => ( ord_le6747313008572928689nteger @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_le6747313008572928689nteger @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_1678_order__less__le__subst2,axiom,
    ! [A: num,B: num,F: num > code_integer,C: code_integer] :
      ( ( ord_less_num @ A @ B )
     => ( ( ord_le3102999989581377725nteger @ ( F @ B ) @ C )
       => ( ! [X3: num,Y4: num] :
              ( ( ord_less_num @ X3 @ Y4 )
             => ( ord_le6747313008572928689nteger @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_le6747313008572928689nteger @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_1679_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,F: nat > code_integer,C: code_integer] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_le3102999989581377725nteger @ ( F @ B ) @ C )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_nat @ X3 @ Y4 )
             => ( ord_le6747313008572928689nteger @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_le6747313008572928689nteger @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_1680_order__less__le__subst2,axiom,
    ! [A: int,B: int,F: int > code_integer,C: code_integer] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_le3102999989581377725nteger @ ( F @ B ) @ C )
       => ( ! [X3: int,Y4: int] :
              ( ( ord_less_int @ X3 @ Y4 )
             => ( ord_le6747313008572928689nteger @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_le6747313008572928689nteger @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_1681_order__less__le__subst2,axiom,
    ! [A: code_integer,B: code_integer,F: code_integer > rat,C: rat] :
      ( ( ord_le6747313008572928689nteger @ A @ B )
     => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
       => ( ! [X3: code_integer,Y4: code_integer] :
              ( ( ord_le6747313008572928689nteger @ X3 @ Y4 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_1682_order__less__le__subst2,axiom,
    ! [A: rat,B: rat,F: rat > rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_eq_rat @ ( F @ B ) @ 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_1683_order__less__le__subst2,axiom,
    ! [A: num,B: num,F: num > rat,C: rat] :
      ( ( ord_less_num @ A @ B )
     => ( ( ord_less_eq_rat @ ( F @ B ) @ 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_1684_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,F: nat > rat,C: rat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_rat @ ( F @ B ) @ 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_1685_order__less__le__subst2,axiom,
    ! [A: int,B: int,F: int > rat,C: rat] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_eq_rat @ ( F @ B ) @ 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_1686_linorder__le__less__linear,axiom,
    ! [X: code_integer,Y3: code_integer] :
      ( ( ord_le3102999989581377725nteger @ X @ Y3 )
      | ( ord_le6747313008572928689nteger @ Y3 @ X ) ) ).

% linorder_le_less_linear
thf(fact_1687_linorder__le__less__linear,axiom,
    ! [X: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ X @ Y3 )
      | ( ord_less_rat @ Y3 @ X ) ) ).

% linorder_le_less_linear
thf(fact_1688_linorder__le__less__linear,axiom,
    ! [X: num,Y3: num] :
      ( ( ord_less_eq_num @ X @ Y3 )
      | ( ord_less_num @ Y3 @ X ) ) ).

% linorder_le_less_linear
thf(fact_1689_linorder__le__less__linear,axiom,
    ! [X: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ X @ Y3 )
      | ( ord_less_nat @ Y3 @ X ) ) ).

% linorder_le_less_linear
thf(fact_1690_linorder__le__less__linear,axiom,
    ! [X: int,Y3: int] :
      ( ( ord_less_eq_int @ X @ Y3 )
      | ( ord_less_int @ Y3 @ X ) ) ).

% linorder_le_less_linear
thf(fact_1691_complete__interval,axiom,
    ! [A: nat,B: nat,P: nat > $o] :
      ( ( ord_less_nat @ A @ B )
     => ( ( P @ A )
       => ( ~ ( P @ B )
         => ? [C2: nat] :
              ( ( ord_less_eq_nat @ A @ C2 )
              & ( ord_less_eq_nat @ C2 @ B )
              & ! [X4: nat] :
                  ( ( ( ord_less_eq_nat @ A @ X4 )
                    & ( ord_less_nat @ X4 @ C2 ) )
                 => ( P @ X4 ) )
              & ! [D2: nat] :
                  ( ! [X3: nat] :
                      ( ( ( ord_less_eq_nat @ A @ X3 )
                        & ( ord_less_nat @ X3 @ D2 ) )
                     => ( P @ X3 ) )
                 => ( ord_less_eq_nat @ D2 @ C2 ) ) ) ) ) ) ).

% complete_interval
thf(fact_1692_complete__interval,axiom,
    ! [A: int,B: int,P: int > $o] :
      ( ( ord_less_int @ A @ B )
     => ( ( P @ A )
       => ( ~ ( P @ B )
         => ? [C2: int] :
              ( ( ord_less_eq_int @ A @ C2 )
              & ( ord_less_eq_int @ C2 @ B )
              & ! [X4: int] :
                  ( ( ( ord_less_eq_int @ A @ X4 )
                    & ( ord_less_int @ X4 @ C2 ) )
                 => ( P @ X4 ) )
              & ! [D2: int] :
                  ( ! [X3: int] :
                      ( ( ( ord_less_eq_int @ A @ X3 )
                        & ( ord_less_int @ X3 @ D2 ) )
                     => ( P @ X3 ) )
                 => ( ord_less_eq_int @ D2 @ C2 ) ) ) ) ) ) ).

% complete_interval
thf(fact_1693_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_1694_fact__reduce,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( semiri2447717529341329178atural @ N )
        = ( times_2397367101498566445atural @ ( semiri3763490453095760265atural @ N ) @ ( semiri2447717529341329178atural @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).

% fact_reduce
thf(fact_1695_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_1696_fact__reduce,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( semiri3624122377584611663nteger @ N )
        = ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ N ) @ ( semiri3624122377584611663nteger @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).

% fact_reduce
thf(fact_1697_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_1698_fact__num__eq__if,axiom,
    ( semiri1408675320244567234ct_nat
    = ( ^ [M2: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ) ).

% fact_num_eq_if
thf(fact_1699_fact__num__eq__if,axiom,
    ( semiri2447717529341329178atural
    = ( ^ [M2: nat] : ( if_Code_natural @ ( M2 = zero_zero_nat ) @ one_one_Code_natural @ ( times_2397367101498566445atural @ ( semiri3763490453095760265atural @ M2 ) @ ( semiri2447717529341329178atural @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ) ).

% fact_num_eq_if
thf(fact_1700_fact__num__eq__if,axiom,
    ( semiri773545260158071498ct_rat
    = ( ^ [M2: nat] : ( if_rat @ ( M2 = zero_zero_nat ) @ one_one_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ M2 ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ) ).

% fact_num_eq_if
thf(fact_1701_fact__num__eq__if,axiom,
    ( semiri3624122377584611663nteger
    = ( ^ [M2: nat] : ( if_Code_integer @ ( M2 = zero_zero_nat ) @ one_one_Code_integer @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M2 ) @ ( semiri3624122377584611663nteger @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ) ).

% fact_num_eq_if
thf(fact_1702_fact__num__eq__if,axiom,
    ( semiri1406184849735516958ct_int
    = ( ^ [M2: nat] : ( if_int @ ( M2 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ) ).

% fact_num_eq_if
thf(fact_1703_zero__le__ceiling,axiom,
    ! [X: rat] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( archim2889992004027027881ng_rat @ X ) )
      = ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ X ) ) ).

% zero_le_ceiling
thf(fact_1704_ceiling__less__zero,axiom,
    ! [X: rat] :
      ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ zero_zero_int )
      = ( ord_less_eq_rat @ X @ ( uminus_uminus_rat @ one_one_rat ) ) ) ).

% ceiling_less_zero
thf(fact_1705_one__less__nat__eq,axiom,
    ! [Z2: int] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( nat2 @ Z2 ) )
      = ( ord_less_int @ one_one_int @ Z2 ) ) ).

% one_less_nat_eq
thf(fact_1706_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 ) )
        | ? [Q3: nat] :
            ( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q3 ) @ M )
            & ( ord_less_nat @ M @ ( times_times_nat @ N @ ( suc @ Q3 ) ) )
            & ( P @ Q3 ) ) ) ) ).

% split_div'
thf(fact_1707_frac__eq,axiom,
    ! [X: rat] :
      ( ( ( archimedean_frac_rat @ X )
        = X )
      = ( ( ord_less_eq_rat @ zero_zero_rat @ X )
        & ( ord_less_rat @ X @ one_one_rat ) ) ) ).

% frac_eq
thf(fact_1708_euclidean__size__times__nonunit,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( B != zero_z3403309356797280102nteger )
       => ( ~ ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
         => ( ord_less_nat @ ( euclid6377331345833325938nteger @ B ) @ ( euclid6377331345833325938nteger @ ( times_3573771949741848930nteger @ A @ B ) ) ) ) ) ) ).

% euclidean_size_times_nonunit
thf(fact_1709_euclidean__size__times__nonunit,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( ( B != zero_zero_int )
       => ( ~ ( dvd_dvd_int @ A @ one_one_int )
         => ( ord_less_nat @ ( euclid4774559944035922753ze_int @ B ) @ ( euclid4774559944035922753ze_int @ ( times_times_int @ A @ B ) ) ) ) ) ) ).

% euclidean_size_times_nonunit
thf(fact_1710_euclidean__size__times__nonunit,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( ( B != zero_zero_nat )
       => ( ~ ( dvd_dvd_nat @ A @ one_one_nat )
         => ( ord_less_nat @ ( euclid4777050414544973029ze_nat @ B ) @ ( euclid4777050414544973029ze_nat @ ( times_times_nat @ A @ B ) ) ) ) ) ) ).

% euclidean_size_times_nonunit
thf(fact_1711_euclidean__size__times__nonunit,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( A != zero_z2226904508553997617atural )
     => ( ( B != zero_z2226904508553997617atural )
       => ( ~ ( dvd_dvd_Code_natural @ A @ one_one_Code_natural )
         => ( ord_less_nat @ ( euclid5200926497590043453atural @ B ) @ ( euclid5200926497590043453atural @ ( times_2397367101498566445atural @ A @ B ) ) ) ) ) ) ).

% euclidean_size_times_nonunit
thf(fact_1712_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_1713_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_1714_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_1715_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_1716_mod__mod__trivial,axiom,
    ! [A: nat,B: nat] :
      ( ( modulo_modulo_nat @ ( modulo_modulo_nat @ A @ B ) @ B )
      = ( modulo_modulo_nat @ A @ B ) ) ).

% mod_mod_trivial
thf(fact_1717_mod__mod__trivial,axiom,
    ! [A: int,B: int] :
      ( ( modulo_modulo_int @ ( modulo_modulo_int @ A @ B ) @ B )
      = ( modulo_modulo_int @ A @ B ) ) ).

% mod_mod_trivial
thf(fact_1718_mod__mod__trivial,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% mod_mod_trivial
thf(fact_1719_mod__mod__trivial,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( modulo8411746178871703098atural @ ( modulo8411746178871703098atural @ A @ B ) @ B )
      = ( modulo8411746178871703098atural @ A @ B ) ) ).

% mod_mod_trivial
thf(fact_1720_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_1721_int__dvd__int__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( dvd_dvd_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
      = ( dvd_dvd_nat @ M @ N ) ) ).

% int_dvd_int_iff
thf(fact_1722_sgn__sgn,axiom,
    ! [A: int] :
      ( ( sgn_sgn_int @ ( sgn_sgn_int @ A ) )
      = ( sgn_sgn_int @ A ) ) ).

% sgn_sgn
thf(fact_1723_sgn__sgn,axiom,
    ! [A: code_integer] :
      ( ( sgn_sgn_Code_integer @ ( sgn_sgn_Code_integer @ A ) )
      = ( sgn_sgn_Code_integer @ A ) ) ).

% sgn_sgn
thf(fact_1724_sgn__sgn,axiom,
    ! [A: rat] :
      ( ( sgn_sgn_rat @ ( sgn_sgn_rat @ A ) )
      = ( sgn_sgn_rat @ A ) ) ).

% sgn_sgn
thf(fact_1725_diff__self,axiom,
    ! [A: code_integer] :
      ( ( minus_8373710615458151222nteger @ A @ A )
      = zero_z3403309356797280102nteger ) ).

% diff_self
thf(fact_1726_diff__self,axiom,
    ! [A: rat] :
      ( ( minus_minus_rat @ A @ A )
      = zero_zero_rat ) ).

% diff_self
thf(fact_1727_diff__self,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ A @ A )
      = zero_zero_int ) ).

% diff_self
thf(fact_1728_diff__0__right,axiom,
    ! [A: code_integer] :
      ( ( minus_8373710615458151222nteger @ A @ zero_z3403309356797280102nteger )
      = A ) ).

% diff_0_right
thf(fact_1729_diff__0__right,axiom,
    ! [A: rat] :
      ( ( minus_minus_rat @ A @ zero_zero_rat )
      = A ) ).

% diff_0_right
thf(fact_1730_diff__0__right,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ A @ zero_zero_int )
      = A ) ).

% diff_0_right
thf(fact_1731_zero__diff,axiom,
    ! [A: code_natural] :
      ( ( minus_7197305767214868737atural @ zero_z2226904508553997617atural @ A )
      = zero_z2226904508553997617atural ) ).

% zero_diff
thf(fact_1732_zero__diff,axiom,
    ! [A: nat] :
      ( ( minus_minus_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% zero_diff
thf(fact_1733_diff__zero,axiom,
    ! [A: code_natural] :
      ( ( minus_7197305767214868737atural @ A @ zero_z2226904508553997617atural )
      = A ) ).

% diff_zero
thf(fact_1734_diff__zero,axiom,
    ! [A: code_integer] :
      ( ( minus_8373710615458151222nteger @ A @ zero_z3403309356797280102nteger )
      = A ) ).

% diff_zero
thf(fact_1735_diff__zero,axiom,
    ! [A: rat] :
      ( ( minus_minus_rat @ A @ zero_zero_rat )
      = A ) ).

% diff_zero
thf(fact_1736_diff__zero,axiom,
    ! [A: nat] :
      ( ( minus_minus_nat @ A @ zero_zero_nat )
      = A ) ).

% diff_zero
thf(fact_1737_diff__zero,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ A @ zero_zero_int )
      = A ) ).

% diff_zero
thf(fact_1738_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: code_natural] :
      ( ( minus_7197305767214868737atural @ A @ A )
      = zero_z2226904508553997617atural ) ).

% cancel_comm_monoid_add_class.diff_cancel
thf(fact_1739_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: code_integer] :
      ( ( minus_8373710615458151222nteger @ A @ A )
      = zero_z3403309356797280102nteger ) ).

% cancel_comm_monoid_add_class.diff_cancel
thf(fact_1740_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_1741_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_1742_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_1743_dvd__0__left__iff,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ zero_zero_nat @ A )
      = ( A = zero_zero_nat ) ) ).

% dvd_0_left_iff
thf(fact_1744_dvd__0__left__iff,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ zero_zero_int @ A )
      = ( A = zero_zero_int ) ) ).

% dvd_0_left_iff
thf(fact_1745_dvd__0__left__iff,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ A )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% dvd_0_left_iff
thf(fact_1746_dvd__0__left__iff,axiom,
    ! [A: rat] :
      ( ( dvd_dvd_rat @ zero_zero_rat @ A )
      = ( A = zero_zero_rat ) ) ).

% dvd_0_left_iff
thf(fact_1747_dvd__0__left__iff,axiom,
    ! [A: code_natural] :
      ( ( dvd_dvd_Code_natural @ zero_z2226904508553997617atural @ A )
      = ( A = zero_z2226904508553997617atural ) ) ).

% dvd_0_left_iff
thf(fact_1748_dvd__0__right,axiom,
    ! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).

% dvd_0_right
thf(fact_1749_dvd__0__right,axiom,
    ! [A: int] : ( dvd_dvd_int @ A @ zero_zero_int ) ).

% dvd_0_right
thf(fact_1750_dvd__0__right,axiom,
    ! [A: code_integer] : ( dvd_dvd_Code_integer @ A @ zero_z3403309356797280102nteger ) ).

% dvd_0_right
thf(fact_1751_dvd__0__right,axiom,
    ! [A: rat] : ( dvd_dvd_rat @ A @ zero_zero_rat ) ).

% dvd_0_right
thf(fact_1752_dvd__0__right,axiom,
    ! [A: code_natural] : ( dvd_dvd_Code_natural @ A @ zero_z2226904508553997617atural ) ).

% dvd_0_right
thf(fact_1753_division__ring__divide__zero,axiom,
    ! [A: rat] :
      ( ( divide_divide_rat @ A @ zero_zero_rat )
      = zero_zero_rat ) ).

% division_ring_divide_zero
thf(fact_1754_divide__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ( divide_divide_rat @ A @ C )
        = ( divide_divide_rat @ B @ C ) )
      = ( ( C = zero_zero_rat )
        | ( A = B ) ) ) ).

% divide_cancel_right
thf(fact_1755_divide__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ( divide_divide_rat @ C @ A )
        = ( divide_divide_rat @ C @ B ) )
      = ( ( C = zero_zero_rat )
        | ( A = B ) ) ) ).

% divide_cancel_left
thf(fact_1756_divide__eq__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ( divide_divide_rat @ A @ B )
        = zero_zero_rat )
      = ( ( A = zero_zero_rat )
        | ( B = zero_zero_rat ) ) ) ).

% divide_eq_0_iff
thf(fact_1757_div__by__0,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ A @ zero_zero_nat )
      = zero_zero_nat ) ).

% div_by_0
thf(fact_1758_div__by__0,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ A @ zero_zero_int )
      = zero_zero_int ) ).

% div_by_0
thf(fact_1759_div__by__0,axiom,
    ! [A: code_integer] :
      ( ( divide6298287555418463151nteger @ A @ zero_z3403309356797280102nteger )
      = zero_z3403309356797280102nteger ) ).

% div_by_0
thf(fact_1760_div__by__0,axiom,
    ! [A: rat] :
      ( ( divide_divide_rat @ A @ zero_zero_rat )
      = zero_zero_rat ) ).

% div_by_0
thf(fact_1761_div__by__0,axiom,
    ! [A: code_natural] :
      ( ( divide5121882707175180666atural @ A @ zero_z2226904508553997617atural )
      = zero_z2226904508553997617atural ) ).

% div_by_0
thf(fact_1762_div__0,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% div_0
thf(fact_1763_div__0,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% div_0
thf(fact_1764_div__0,axiom,
    ! [A: code_integer] :
      ( ( divide6298287555418463151nteger @ zero_z3403309356797280102nteger @ A )
      = zero_z3403309356797280102nteger ) ).

% div_0
thf(fact_1765_div__0,axiom,
    ! [A: rat] :
      ( ( divide_divide_rat @ zero_zero_rat @ A )
      = zero_zero_rat ) ).

% div_0
thf(fact_1766_div__0,axiom,
    ! [A: code_natural] :
      ( ( divide5121882707175180666atural @ zero_z2226904508553997617atural @ A )
      = zero_z2226904508553997617atural ) ).

% div_0
thf(fact_1767_times__divide__eq__right,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ A @ ( divide_divide_rat @ B @ C ) )
      = ( divide_divide_rat @ ( times_times_rat @ A @ B ) @ C ) ) ).

% times_divide_eq_right
thf(fact_1768_divide__divide__eq__right,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( divide_divide_rat @ A @ ( divide_divide_rat @ B @ C ) )
      = ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ B ) ) ).

% divide_divide_eq_right
thf(fact_1769_divide__divide__eq__left,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( divide_divide_rat @ ( divide_divide_rat @ A @ B ) @ C )
      = ( divide_divide_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).

% divide_divide_eq_left
thf(fact_1770_times__divide__eq__left,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( times_times_rat @ ( divide_divide_rat @ B @ C ) @ A )
      = ( divide_divide_rat @ ( times_times_rat @ B @ A ) @ C ) ) ).

% times_divide_eq_left
thf(fact_1771_minus__diff__eq,axiom,
    ! [A: int,B: int] :
      ( ( uminus_uminus_int @ ( minus_minus_int @ A @ B ) )
      = ( minus_minus_int @ B @ A ) ) ).

% minus_diff_eq
thf(fact_1772_minus__diff__eq,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( uminus1351360451143612070nteger @ ( minus_8373710615458151222nteger @ A @ B ) )
      = ( minus_8373710615458151222nteger @ B @ A ) ) ).

% minus_diff_eq
thf(fact_1773_minus__diff__eq,axiom,
    ! [A: rat,B: rat] :
      ( ( uminus_uminus_rat @ ( minus_minus_rat @ A @ B ) )
      = ( minus_minus_rat @ B @ A ) ) ).

% minus_diff_eq
thf(fact_1774_div__by__1,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ A @ one_one_nat )
      = A ) ).

% div_by_1
thf(fact_1775_div__by__1,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ A @ one_one_int )
      = A ) ).

% div_by_1
thf(fact_1776_div__by__1,axiom,
    ! [A: code_integer] :
      ( ( divide6298287555418463151nteger @ A @ one_one_Code_integer )
      = A ) ).

% div_by_1
thf(fact_1777_div__by__1,axiom,
    ! [A: rat] :
      ( ( divide_divide_rat @ A @ one_one_rat )
      = A ) ).

% div_by_1
thf(fact_1778_div__by__1,axiom,
    ! [A: code_natural] :
      ( ( divide5121882707175180666atural @ A @ one_one_Code_natural )
      = A ) ).

% div_by_1
thf(fact_1779_mod__self,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ A @ A )
      = zero_zero_nat ) ).

% mod_self
thf(fact_1780_mod__self,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ A @ A )
      = zero_zero_int ) ).

% mod_self
thf(fact_1781_mod__self,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger @ A @ A )
      = zero_z3403309356797280102nteger ) ).

% mod_self
thf(fact_1782_mod__self,axiom,
    ! [A: code_natural] :
      ( ( modulo8411746178871703098atural @ A @ A )
      = zero_z2226904508553997617atural ) ).

% mod_self
thf(fact_1783_mod__by__0,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ A @ zero_zero_nat )
      = A ) ).

% mod_by_0
thf(fact_1784_mod__by__0,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ A @ zero_zero_int )
      = A ) ).

% mod_by_0
thf(fact_1785_mod__by__0,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger @ A @ zero_z3403309356797280102nteger )
      = A ) ).

% mod_by_0
thf(fact_1786_mod__by__0,axiom,
    ! [A: code_natural] :
      ( ( modulo8411746178871703098atural @ A @ zero_z2226904508553997617atural )
      = A ) ).

% mod_by_0
thf(fact_1787_mod__0,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% mod_0
thf(fact_1788_mod__0,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% mod_0
thf(fact_1789_mod__0,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger @ zero_z3403309356797280102nteger @ A )
      = zero_z3403309356797280102nteger ) ).

% mod_0
thf(fact_1790_mod__0,axiom,
    ! [A: code_natural] :
      ( ( modulo8411746178871703098atural @ zero_z2226904508553997617atural @ A )
      = zero_z2226904508553997617atural ) ).

% mod_0
thf(fact_1791_minus__dvd__iff,axiom,
    ! [X: int,Y3: int] :
      ( ( dvd_dvd_int @ ( uminus_uminus_int @ X ) @ Y3 )
      = ( dvd_dvd_int @ X @ Y3 ) ) ).

% minus_dvd_iff
thf(fact_1792_minus__dvd__iff,axiom,
    ! [X: code_integer,Y3: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( uminus1351360451143612070nteger @ X ) @ Y3 )
      = ( dvd_dvd_Code_integer @ X @ Y3 ) ) ).

% minus_dvd_iff
thf(fact_1793_minus__dvd__iff,axiom,
    ! [X: rat,Y3: rat] :
      ( ( dvd_dvd_rat @ ( uminus_uminus_rat @ X ) @ Y3 )
      = ( dvd_dvd_rat @ X @ Y3 ) ) ).

% minus_dvd_iff
thf(fact_1794_dvd__minus__iff,axiom,
    ! [X: int,Y3: int] :
      ( ( dvd_dvd_int @ X @ ( uminus_uminus_int @ Y3 ) )
      = ( dvd_dvd_int @ X @ Y3 ) ) ).

% dvd_minus_iff
thf(fact_1795_dvd__minus__iff,axiom,
    ! [X: code_integer,Y3: code_integer] :
      ( ( dvd_dvd_Code_integer @ X @ ( uminus1351360451143612070nteger @ Y3 ) )
      = ( dvd_dvd_Code_integer @ X @ Y3 ) ) ).

% dvd_minus_iff
thf(fact_1796_dvd__minus__iff,axiom,
    ! [X: rat,Y3: rat] :
      ( ( dvd_dvd_rat @ X @ ( uminus_uminus_rat @ Y3 ) )
      = ( dvd_dvd_rat @ X @ Y3 ) ) ).

% dvd_minus_iff
thf(fact_1797_div__minus__minus,axiom,
    ! [A: int,B: int] :
      ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
      = ( divide_divide_int @ A @ B ) ) ).

% div_minus_minus
thf(fact_1798_div__minus__minus,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) )
      = ( divide6298287555418463151nteger @ A @ B ) ) ).

% div_minus_minus
thf(fact_1799_dvd__1__left,axiom,
    ! [K: nat] : ( dvd_dvd_nat @ ( suc @ zero_zero_nat ) @ K ) ).

% dvd_1_left
thf(fact_1800_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_1801_div__dvd__div,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( ( dvd_dvd_nat @ A @ C )
       => ( ( dvd_dvd_nat @ ( divide_divide_nat @ B @ A ) @ ( divide_divide_nat @ C @ A ) )
          = ( dvd_dvd_nat @ B @ C ) ) ) ) ).

% div_dvd_div
thf(fact_1802_div__dvd__div,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ A @ B )
     => ( ( dvd_dvd_int @ A @ C )
       => ( ( dvd_dvd_int @ ( divide_divide_int @ B @ A ) @ ( divide_divide_int @ C @ A ) )
          = ( dvd_dvd_int @ B @ C ) ) ) ) ).

% div_dvd_div
thf(fact_1803_div__dvd__div,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ B )
     => ( ( dvd_dvd_Code_integer @ A @ C )
       => ( ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ B @ A ) @ ( divide6298287555418463151nteger @ C @ A ) )
          = ( dvd_dvd_Code_integer @ B @ C ) ) ) ) ).

% div_dvd_div
thf(fact_1804_div__dvd__div,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( ( dvd_dvd_Code_natural @ A @ B )
     => ( ( dvd_dvd_Code_natural @ A @ C )
       => ( ( dvd_dvd_Code_natural @ ( divide5121882707175180666atural @ B @ A ) @ ( divide5121882707175180666atural @ C @ A ) )
          = ( dvd_dvd_Code_natural @ B @ C ) ) ) ) ).

% div_dvd_div
thf(fact_1805_minus__mod__self2,axiom,
    ! [A: int,B: int] :
      ( ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ B )
      = ( modulo_modulo_int @ A @ B ) ) ).

% minus_mod_self2
thf(fact_1806_minus__mod__self2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ B )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% minus_mod_self2
thf(fact_1807_mod__minus__minus,axiom,
    ! [A: int,B: int] :
      ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
      = ( uminus_uminus_int @ ( modulo_modulo_int @ A @ B ) ) ) ).

% mod_minus_minus
thf(fact_1808_mod__minus__minus,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) )
      = ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).

% mod_minus_minus
thf(fact_1809_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_1810_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_1811_diff__0__eq__0,axiom,
    ! [N: nat] :
      ( ( minus_minus_nat @ zero_zero_nat @ N )
      = zero_zero_nat ) ).

% diff_0_eq_0
thf(fact_1812_diff__self__eq__0,axiom,
    ! [M: nat] :
      ( ( minus_minus_nat @ M @ M )
      = zero_zero_nat ) ).

% diff_self_eq_0
thf(fact_1813_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_1814_diff__diff__cancel,axiom,
    ! [I2: nat,N: nat] :
      ( ( ord_less_eq_nat @ I2 @ N )
     => ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I2 ) )
        = I2 ) ) ).

% diff_diff_cancel
thf(fact_1815_sgn__0,axiom,
    ( ( sgn_sgn_int @ zero_zero_int )
    = zero_zero_int ) ).

% sgn_0
thf(fact_1816_sgn__0,axiom,
    ( ( sgn_sgn_Code_integer @ zero_z3403309356797280102nteger )
    = zero_z3403309356797280102nteger ) ).

% sgn_0
thf(fact_1817_sgn__0,axiom,
    ( ( sgn_sgn_rat @ zero_zero_rat )
    = zero_zero_rat ) ).

% sgn_0
thf(fact_1818_sgn__mult__dvd__iff,axiom,
    ! [R: int,L: int,K: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ ( sgn_sgn_int @ R ) @ L ) @ K )
      = ( ( dvd_dvd_int @ L @ K )
        & ( ( R = zero_zero_int )
         => ( K = zero_zero_int ) ) ) ) ).

% sgn_mult_dvd_iff
thf(fact_1819_mult__sgn__dvd__iff,axiom,
    ! [L: int,R: int,K: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ L @ ( sgn_sgn_int @ R ) ) @ K )
      = ( ( dvd_dvd_int @ L @ K )
        & ( ( R = zero_zero_int )
         => ( K = zero_zero_int ) ) ) ) ).

% mult_sgn_dvd_iff
thf(fact_1820_dvd__sgn__mult__iff,axiom,
    ! [L: int,R: int,K: int] :
      ( ( dvd_dvd_int @ L @ ( times_times_int @ ( sgn_sgn_int @ R ) @ K ) )
      = ( ( dvd_dvd_int @ L @ K )
        | ( R = zero_zero_int ) ) ) ).

% dvd_sgn_mult_iff
thf(fact_1821_dvd__mult__sgn__iff,axiom,
    ! [L: int,K: int,R: int] :
      ( ( dvd_dvd_int @ L @ ( times_times_int @ K @ ( sgn_sgn_int @ R ) ) )
      = ( ( dvd_dvd_int @ L @ K )
        | ( R = zero_zero_int ) ) ) ).

% dvd_mult_sgn_iff
thf(fact_1822_sgn__1,axiom,
    ( ( sgn_sgn_int @ one_one_int )
    = one_one_int ) ).

% sgn_1
thf(fact_1823_sgn__1,axiom,
    ( ( sgn_sgn_Code_integer @ one_one_Code_integer )
    = one_one_Code_integer ) ).

% sgn_1
thf(fact_1824_sgn__1,axiom,
    ( ( sgn_sgn_rat @ one_one_rat )
    = one_one_rat ) ).

% sgn_1
thf(fact_1825_sgn__minus,axiom,
    ! [A: int] :
      ( ( sgn_sgn_int @ ( uminus_uminus_int @ A ) )
      = ( uminus_uminus_int @ ( sgn_sgn_int @ A ) ) ) ).

% sgn_minus
thf(fact_1826_sgn__minus,axiom,
    ! [A: code_integer] :
      ( ( sgn_sgn_Code_integer @ ( uminus1351360451143612070nteger @ A ) )
      = ( uminus1351360451143612070nteger @ ( sgn_sgn_Code_integer @ A ) ) ) ).

% sgn_minus
thf(fact_1827_sgn__minus,axiom,
    ! [A: rat] :
      ( ( sgn_sgn_rat @ ( uminus_uminus_rat @ A ) )
      = ( uminus_uminus_rat @ ( sgn_sgn_rat @ A ) ) ) ).

% sgn_minus
thf(fact_1828_sgn__divide,axiom,
    ! [A: rat,B: rat] :
      ( ( sgn_sgn_rat @ ( divide_divide_rat @ A @ B ) )
      = ( divide_divide_rat @ ( sgn_sgn_rat @ A ) @ ( sgn_sgn_rat @ B ) ) ) ).

% sgn_divide
thf(fact_1829_mod__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ( modulo_modulo_nat @ M @ N )
        = M ) ) ).

% mod_less
thf(fact_1830_nat__int,axiom,
    ! [N: nat] :
      ( ( nat2 @ ( semiri1314217659103216013at_int @ N ) )
      = N ) ).

% nat_int
thf(fact_1831_diff__ge__0__iff__ge,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( minus_8373710615458151222nteger @ A @ B ) )
      = ( ord_le3102999989581377725nteger @ B @ A ) ) ).

% diff_ge_0_iff_ge
thf(fact_1832_diff__ge__0__iff__ge,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( minus_minus_rat @ A @ B ) )
      = ( ord_less_eq_rat @ B @ A ) ) ).

% diff_ge_0_iff_ge
thf(fact_1833_diff__ge__0__iff__ge,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
      = ( ord_less_eq_int @ B @ A ) ) ).

% diff_ge_0_iff_ge
thf(fact_1834_diff__gt__0__iff__gt,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( minus_8373710615458151222nteger @ A @ B ) )
      = ( ord_le6747313008572928689nteger @ B @ A ) ) ).

% diff_gt_0_iff_gt
thf(fact_1835_diff__gt__0__iff__gt,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( minus_minus_rat @ A @ B ) )
      = ( ord_less_rat @ B @ A ) ) ).

% diff_gt_0_iff_gt
thf(fact_1836_diff__gt__0__iff__gt,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
      = ( ord_less_int @ B @ A ) ) ).

% diff_gt_0_iff_gt
thf(fact_1837_diff__numeral__special_I9_J,axiom,
    ( ( minus_8373710615458151222nteger @ one_one_Code_integer @ one_one_Code_integer )
    = zero_z3403309356797280102nteger ) ).

% diff_numeral_special(9)
thf(fact_1838_diff__numeral__special_I9_J,axiom,
    ( ( minus_minus_rat @ one_one_rat @ one_one_rat )
    = zero_zero_rat ) ).

% diff_numeral_special(9)
thf(fact_1839_diff__numeral__special_I9_J,axiom,
    ( ( minus_minus_int @ one_one_int @ one_one_int )
    = zero_zero_int ) ).

% diff_numeral_special(9)
thf(fact_1840_dvd__mult__cancel__left,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) )
      = ( ( C = zero_z3403309356797280102nteger )
        | ( dvd_dvd_Code_integer @ A @ B ) ) ) ).

% dvd_mult_cancel_left
thf(fact_1841_dvd__mult__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( dvd_dvd_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
      = ( ( C = zero_zero_rat )
        | ( dvd_dvd_rat @ A @ B ) ) ) ).

% dvd_mult_cancel_left
thf(fact_1842_dvd__mult__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
      = ( ( C = zero_zero_int )
        | ( dvd_dvd_int @ A @ B ) ) ) ).

% dvd_mult_cancel_left
thf(fact_1843_dvd__mult__cancel__right,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) )
      = ( ( C = zero_z3403309356797280102nteger )
        | ( dvd_dvd_Code_integer @ A @ B ) ) ) ).

% dvd_mult_cancel_right
thf(fact_1844_dvd__mult__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( dvd_dvd_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
      = ( ( C = zero_zero_rat )
        | ( dvd_dvd_rat @ A @ B ) ) ) ).

% dvd_mult_cancel_right
thf(fact_1845_dvd__mult__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
      = ( ( C = zero_zero_int )
        | ( dvd_dvd_int @ A @ B ) ) ) ).

% dvd_mult_cancel_right
thf(fact_1846_dvd__times__left__cancel__iff,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( ( A != zero_z2226904508553997617atural )
     => ( ( dvd_dvd_Code_natural @ ( times_2397367101498566445atural @ A @ B ) @ ( times_2397367101498566445atural @ A @ C ) )
        = ( dvd_dvd_Code_natural @ B @ C ) ) ) ).

% dvd_times_left_cancel_iff
thf(fact_1847_dvd__times__left__cancel__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ ( times_3573771949741848930nteger @ A @ C ) )
        = ( dvd_dvd_Code_integer @ B @ C ) ) ) ).

% dvd_times_left_cancel_iff
thf(fact_1848_dvd__times__left__cancel__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A != zero_zero_nat )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) )
        = ( dvd_dvd_nat @ B @ C ) ) ) ).

% dvd_times_left_cancel_iff
thf(fact_1849_dvd__times__left__cancel__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A != zero_zero_int )
     => ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) )
        = ( dvd_dvd_int @ B @ C ) ) ) ).

% dvd_times_left_cancel_iff
thf(fact_1850_dvd__times__right__cancel__iff,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( ( A != zero_z2226904508553997617atural )
     => ( ( dvd_dvd_Code_natural @ ( times_2397367101498566445atural @ B @ A ) @ ( times_2397367101498566445atural @ C @ A ) )
        = ( dvd_dvd_Code_natural @ B @ C ) ) ) ).

% dvd_times_right_cancel_iff
thf(fact_1851_dvd__times__right__cancel__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ B @ A ) @ ( times_3573771949741848930nteger @ C @ A ) )
        = ( dvd_dvd_Code_integer @ B @ C ) ) ) ).

% dvd_times_right_cancel_iff
thf(fact_1852_dvd__times__right__cancel__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A != zero_zero_nat )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C @ A ) )
        = ( dvd_dvd_nat @ B @ C ) ) ) ).

% dvd_times_right_cancel_iff
thf(fact_1853_dvd__times__right__cancel__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A != zero_zero_int )
     => ( ( dvd_dvd_int @ ( times_times_int @ B @ A ) @ ( times_times_int @ C @ A ) )
        = ( dvd_dvd_int @ B @ C ) ) ) ).

% dvd_times_right_cancel_iff
thf(fact_1854_mult__divide__mult__cancel__left__if,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ( C = zero_zero_rat )
       => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
          = zero_zero_rat ) )
      & ( ( C != zero_zero_rat )
       => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
          = ( divide_divide_rat @ A @ B ) ) ) ) ).

% mult_divide_mult_cancel_left_if
thf(fact_1855_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
        = ( divide_divide_rat @ A @ B ) ) ) ).

% nonzero_mult_divide_mult_cancel_left
thf(fact_1856_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ B @ C ) )
        = ( divide_divide_rat @ A @ B ) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
thf(fact_1857_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
        = ( divide_divide_rat @ A @ B ) ) ) ).

% nonzero_mult_divide_mult_cancel_right
thf(fact_1858_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ C @ B ) )
        = ( divide_divide_rat @ A @ B ) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
thf(fact_1859_div__mult__mult1,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( C != zero_zero_nat )
     => ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
        = ( divide_divide_nat @ A @ B ) ) ) ).

% div_mult_mult1
thf(fact_1860_div__mult__mult1,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
        = ( divide_divide_int @ A @ B ) ) ) ).

% div_mult_mult1
thf(fact_1861_div__mult__mult1,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( C != zero_z3403309356797280102nteger )
     => ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) )
        = ( divide6298287555418463151nteger @ A @ B ) ) ) ).

% div_mult_mult1
thf(fact_1862_div__mult__mult1,axiom,
    ! [C: code_natural,A: code_natural,B: code_natural] :
      ( ( C != zero_z2226904508553997617atural )
     => ( ( divide5121882707175180666atural @ ( times_2397367101498566445atural @ C @ A ) @ ( times_2397367101498566445atural @ C @ B ) )
        = ( divide5121882707175180666atural @ A @ B ) ) ) ).

% div_mult_mult1
thf(fact_1863_div__mult__mult2,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( C != zero_zero_nat )
     => ( ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
        = ( divide_divide_nat @ A @ B ) ) ) ).

% div_mult_mult2
thf(fact_1864_div__mult__mult2,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( ( divide_divide_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
        = ( divide_divide_int @ A @ B ) ) ) ).

% div_mult_mult2
thf(fact_1865_div__mult__mult2,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( C != zero_z3403309356797280102nteger )
     => ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) )
        = ( divide6298287555418463151nteger @ A @ B ) ) ) ).

% div_mult_mult2
thf(fact_1866_div__mult__mult2,axiom,
    ! [C: code_natural,A: code_natural,B: code_natural] :
      ( ( C != zero_z2226904508553997617atural )
     => ( ( divide5121882707175180666atural @ ( times_2397367101498566445atural @ A @ C ) @ ( times_2397367101498566445atural @ B @ C ) )
        = ( divide5121882707175180666atural @ A @ B ) ) ) ).

% div_mult_mult2
thf(fact_1867_div__mult__mult1__if,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ( C = zero_zero_nat )
       => ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
          = zero_zero_nat ) )
      & ( ( C != zero_zero_nat )
       => ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
          = ( divide_divide_nat @ A @ B ) ) ) ) ).

% div_mult_mult1_if
thf(fact_1868_div__mult__mult1__if,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ( C = zero_zero_int )
       => ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
          = zero_zero_int ) )
      & ( ( C != zero_zero_int )
       => ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
          = ( divide_divide_int @ A @ B ) ) ) ) ).

% div_mult_mult1_if
thf(fact_1869_div__mult__mult1__if,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( ( C = zero_z3403309356797280102nteger )
       => ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) )
          = zero_z3403309356797280102nteger ) )
      & ( ( C != zero_z3403309356797280102nteger )
       => ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) )
          = ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).

% div_mult_mult1_if
thf(fact_1870_div__mult__mult1__if,axiom,
    ! [C: code_natural,A: code_natural,B: code_natural] :
      ( ( ( C = zero_z2226904508553997617atural )
       => ( ( divide5121882707175180666atural @ ( times_2397367101498566445atural @ C @ A ) @ ( times_2397367101498566445atural @ C @ B ) )
          = zero_z2226904508553997617atural ) )
      & ( ( C != zero_z2226904508553997617atural )
       => ( ( divide5121882707175180666atural @ ( times_2397367101498566445atural @ C @ A ) @ ( times_2397367101498566445atural @ C @ B ) )
          = ( divide5121882707175180666atural @ A @ B ) ) ) ) ).

% div_mult_mult1_if
thf(fact_1871_nonzero__mult__div__cancel__left,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ A )
        = B ) ) ).

% nonzero_mult_div_cancel_left
thf(fact_1872_nonzero__mult__div__cancel__left,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ A )
        = B ) ) ).

% nonzero_mult_div_cancel_left
thf(fact_1873_nonzero__mult__div__cancel__left,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B ) @ A )
        = B ) ) ).

% nonzero_mult_div_cancel_left
thf(fact_1874_nonzero__mult__div__cancel__left,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( divide_divide_rat @ ( times_times_rat @ A @ B ) @ A )
        = B ) ) ).

% nonzero_mult_div_cancel_left
thf(fact_1875_nonzero__mult__div__cancel__left,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( A != zero_z2226904508553997617atural )
     => ( ( divide5121882707175180666atural @ ( times_2397367101498566445atural @ A @ B ) @ A )
        = B ) ) ).

% nonzero_mult_div_cancel_left
thf(fact_1876_nonzero__mult__div__cancel__right,axiom,
    ! [B: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ B )
        = A ) ) ).

% nonzero_mult_div_cancel_right
thf(fact_1877_nonzero__mult__div__cancel__right,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ B )
        = A ) ) ).

% nonzero_mult_div_cancel_right
thf(fact_1878_nonzero__mult__div__cancel__right,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B ) @ B )
        = A ) ) ).

% nonzero_mult_div_cancel_right
thf(fact_1879_nonzero__mult__div__cancel__right,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( ( divide_divide_rat @ ( times_times_rat @ A @ B ) @ B )
        = A ) ) ).

% nonzero_mult_div_cancel_right
thf(fact_1880_nonzero__mult__div__cancel__right,axiom,
    ! [B: code_natural,A: code_natural] :
      ( ( B != zero_z2226904508553997617atural )
     => ( ( divide5121882707175180666atural @ ( times_2397367101498566445atural @ A @ B ) @ B )
        = A ) ) ).

% nonzero_mult_div_cancel_right
thf(fact_1881_diff__0,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ zero_zero_int @ A )
      = ( uminus_uminus_int @ A ) ) ).

% diff_0
thf(fact_1882_diff__0,axiom,
    ! [A: code_integer] :
      ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ A )
      = ( uminus1351360451143612070nteger @ A ) ) ).

% diff_0
thf(fact_1883_diff__0,axiom,
    ! [A: rat] :
      ( ( minus_minus_rat @ zero_zero_rat @ A )
      = ( uminus_uminus_rat @ A ) ) ).

% diff_0
thf(fact_1884_verit__minus__simplify_I3_J,axiom,
    ! [B: int] :
      ( ( minus_minus_int @ zero_zero_int @ B )
      = ( uminus_uminus_int @ B ) ) ).

% verit_minus_simplify(3)
thf(fact_1885_verit__minus__simplify_I3_J,axiom,
    ! [B: code_integer] :
      ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ B )
      = ( uminus1351360451143612070nteger @ B ) ) ).

% verit_minus_simplify(3)
thf(fact_1886_verit__minus__simplify_I3_J,axiom,
    ! [B: rat] :
      ( ( minus_minus_rat @ zero_zero_rat @ B )
      = ( uminus_uminus_rat @ B ) ) ).

% verit_minus_simplify(3)
thf(fact_1887_divide__eq__1__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ( divide_divide_rat @ A @ B )
        = one_one_rat )
      = ( ( B != zero_zero_rat )
        & ( A = B ) ) ) ).

% divide_eq_1_iff
thf(fact_1888_one__eq__divide__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( one_one_rat
        = ( divide_divide_rat @ A @ B ) )
      = ( ( B != zero_zero_rat )
        & ( A = B ) ) ) ).

% one_eq_divide_iff
thf(fact_1889_divide__self,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( ( divide_divide_rat @ A @ A )
        = one_one_rat ) ) ).

% divide_self
thf(fact_1890_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_1891_divide__eq__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( ( ( divide_divide_rat @ B @ A )
        = one_one_rat )
      = ( ( A != zero_zero_rat )
        & ( A = B ) ) ) ).

% divide_eq_eq_1
thf(fact_1892_eq__divide__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( ( one_one_rat
        = ( divide_divide_rat @ B @ A ) )
      = ( ( A != zero_zero_rat )
        & ( A = B ) ) ) ).

% eq_divide_eq_1
thf(fact_1893_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_1894_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_1895_div__self,axiom,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
     => ( ( divide_divide_nat @ A @ A )
        = one_one_nat ) ) ).

% div_self
thf(fact_1896_div__self,axiom,
    ! [A: int] :
      ( ( A != zero_zero_int )
     => ( ( divide_divide_int @ A @ A )
        = one_one_int ) ) ).

% div_self
thf(fact_1897_div__self,axiom,
    ! [A: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( divide6298287555418463151nteger @ A @ A )
        = one_one_Code_integer ) ) ).

% div_self
thf(fact_1898_div__self,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( ( divide_divide_rat @ A @ A )
        = one_one_rat ) ) ).

% div_self
thf(fact_1899_div__self,axiom,
    ! [A: code_natural] :
      ( ( A != zero_z2226904508553997617atural )
     => ( ( divide5121882707175180666atural @ A @ A )
        = one_one_Code_natural ) ) ).

% div_self
thf(fact_1900_unit__prod,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( dvd_dvd_Code_natural @ A @ one_one_Code_natural )
     => ( ( dvd_dvd_Code_natural @ B @ one_one_Code_natural )
       => ( dvd_dvd_Code_natural @ ( times_2397367101498566445atural @ A @ B ) @ one_one_Code_natural ) ) ) ).

% unit_prod
thf(fact_1901_unit__prod,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
       => ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ one_one_Code_integer ) ) ) ).

% unit_prod
thf(fact_1902_unit__prod,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( dvd_dvd_nat @ B @ one_one_nat )
       => ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ).

% unit_prod
thf(fact_1903_unit__prod,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( dvd_dvd_int @ B @ one_one_int )
       => ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ one_one_int ) ) ) ).

% unit_prod
thf(fact_1904_mod__mult__self1__is__0,axiom,
    ! [B: nat,A: nat] :
      ( ( modulo_modulo_nat @ ( times_times_nat @ B @ A ) @ B )
      = zero_zero_nat ) ).

% mod_mult_self1_is_0
thf(fact_1905_mod__mult__self1__is__0,axiom,
    ! [B: int,A: int] :
      ( ( modulo_modulo_int @ ( times_times_int @ B @ A ) @ B )
      = zero_zero_int ) ).

% mod_mult_self1_is_0
thf(fact_1906_mod__mult__self1__is__0,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ B @ A ) @ B )
      = zero_z3403309356797280102nteger ) ).

% mod_mult_self1_is_0
thf(fact_1907_mod__mult__self1__is__0,axiom,
    ! [B: code_natural,A: code_natural] :
      ( ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ B @ A ) @ B )
      = zero_z2226904508553997617atural ) ).

% mod_mult_self1_is_0
thf(fact_1908_mod__mult__self2__is__0,axiom,
    ! [A: nat,B: nat] :
      ( ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ B )
      = zero_zero_nat ) ).

% mod_mult_self2_is_0
thf(fact_1909_mod__mult__self2__is__0,axiom,
    ! [A: int,B: int] :
      ( ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ B )
      = zero_zero_int ) ).

% mod_mult_self2_is_0
thf(fact_1910_mod__mult__self2__is__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ B )
      = zero_z3403309356797280102nteger ) ).

% mod_mult_self2_is_0
thf(fact_1911_mod__mult__self2__is__0,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ A @ B ) @ B )
      = zero_z2226904508553997617atural ) ).

% mod_mult_self2_is_0
thf(fact_1912_mod__by__1,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ A @ one_one_nat )
      = zero_zero_nat ) ).

% mod_by_1
thf(fact_1913_mod__by__1,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ A @ one_one_int )
      = zero_zero_int ) ).

% mod_by_1
thf(fact_1914_mod__by__1,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger @ A @ one_one_Code_integer )
      = zero_z3403309356797280102nteger ) ).

% mod_by_1
thf(fact_1915_mod__by__1,axiom,
    ! [A: code_natural] :
      ( ( modulo8411746178871703098atural @ A @ one_one_Code_natural )
      = zero_z2226904508553997617atural ) ).

% mod_by_1
thf(fact_1916_divide__minus1,axiom,
    ! [X: rat] :
      ( ( divide_divide_rat @ X @ ( uminus_uminus_rat @ one_one_rat ) )
      = ( uminus_uminus_rat @ X ) ) ).

% divide_minus1
thf(fact_1917_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_1918_div__minus1__right,axiom,
    ! [A: code_integer] :
      ( ( divide6298287555418463151nteger @ A @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( uminus1351360451143612070nteger @ A ) ) ).

% div_minus1_right
thf(fact_1919_dvd__mult__div__cancel,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( ( times_times_nat @ A @ ( divide_divide_nat @ B @ A ) )
        = B ) ) ).

% dvd_mult_div_cancel
thf(fact_1920_dvd__mult__div__cancel,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ B )
     => ( ( times_times_int @ A @ ( divide_divide_int @ B @ A ) )
        = B ) ) ).

% dvd_mult_div_cancel
thf(fact_1921_dvd__mult__div__cancel,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ B )
     => ( ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B @ A ) )
        = B ) ) ).

% dvd_mult_div_cancel
thf(fact_1922_dvd__mult__div__cancel,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( dvd_dvd_Code_natural @ A @ B )
     => ( ( times_2397367101498566445atural @ A @ ( divide5121882707175180666atural @ B @ A ) )
        = B ) ) ).

% dvd_mult_div_cancel
thf(fact_1923_dvd__div__mult__self,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( ( times_times_nat @ ( divide_divide_nat @ B @ A ) @ A )
        = B ) ) ).

% dvd_div_mult_self
thf(fact_1924_dvd__div__mult__self,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ B )
     => ( ( times_times_int @ ( divide_divide_int @ B @ A ) @ A )
        = B ) ) ).

% dvd_div_mult_self
thf(fact_1925_dvd__div__mult__self,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ B )
     => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ B @ A ) @ A )
        = B ) ) ).

% dvd_div_mult_self
thf(fact_1926_dvd__div__mult__self,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( dvd_dvd_Code_natural @ A @ B )
     => ( ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ B @ A ) @ A )
        = B ) ) ).

% dvd_div_mult_self
thf(fact_1927_unit__div,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( dvd_dvd_nat @ B @ one_one_nat )
       => ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).

% unit_div
thf(fact_1928_unit__div,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( dvd_dvd_int @ B @ one_one_int )
       => ( dvd_dvd_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).

% unit_div
thf(fact_1929_unit__div,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
       => ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A @ B ) @ one_one_Code_integer ) ) ) ).

% unit_div
thf(fact_1930_unit__div,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( dvd_dvd_Code_natural @ A @ one_one_Code_natural )
     => ( ( dvd_dvd_Code_natural @ B @ one_one_Code_natural )
       => ( dvd_dvd_Code_natural @ ( divide5121882707175180666atural @ A @ B ) @ one_one_Code_natural ) ) ) ).

% unit_div
thf(fact_1931_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_1932_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_1933_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_1934_unit__div__1__unit,axiom,
    ! [A: code_natural] :
      ( ( dvd_dvd_Code_natural @ A @ one_one_Code_natural )
     => ( dvd_dvd_Code_natural @ ( divide5121882707175180666atural @ one_one_Code_natural @ A ) @ one_one_Code_natural ) ) ).

% unit_div_1_unit
thf(fact_1935_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_1936_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_1937_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_1938_unit__div__1__div__1,axiom,
    ! [A: code_natural] :
      ( ( dvd_dvd_Code_natural @ A @ one_one_Code_natural )
     => ( ( divide5121882707175180666atural @ one_one_Code_natural @ ( divide5121882707175180666atural @ one_one_Code_natural @ A ) )
        = A ) ) ).

% unit_div_1_div_1
thf(fact_1939_div__diff,axiom,
    ! [C: int,A: int,B: int] :
      ( ( dvd_dvd_int @ C @ A )
     => ( ( dvd_dvd_int @ C @ B )
       => ( ( divide_divide_int @ ( minus_minus_int @ A @ B ) @ C )
          = ( minus_minus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ) ).

% div_diff
thf(fact_1940_div__diff,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ A )
     => ( ( dvd_dvd_Code_integer @ C @ B )
       => ( ( divide6298287555418463151nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C )
          = ( minus_8373710615458151222nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) ) ) ) ).

% div_diff
thf(fact_1941_dvd__imp__mod__0,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( ( modulo_modulo_nat @ B @ A )
        = zero_zero_nat ) ) ).

% dvd_imp_mod_0
thf(fact_1942_dvd__imp__mod__0,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ B )
     => ( ( modulo_modulo_int @ B @ A )
        = zero_zero_int ) ) ).

% dvd_imp_mod_0
thf(fact_1943_dvd__imp__mod__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ B )
     => ( ( modulo364778990260209775nteger @ B @ A )
        = zero_z3403309356797280102nteger ) ) ).

% dvd_imp_mod_0
thf(fact_1944_dvd__imp__mod__0,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( dvd_dvd_Code_natural @ A @ B )
     => ( ( modulo8411746178871703098atural @ B @ A )
        = zero_z2226904508553997617atural ) ) ).

% dvd_imp_mod_0
thf(fact_1945_mod__div__trivial,axiom,
    ! [A: nat,B: nat] :
      ( ( divide_divide_nat @ ( modulo_modulo_nat @ A @ B ) @ B )
      = zero_zero_nat ) ).

% mod_div_trivial
thf(fact_1946_mod__div__trivial,axiom,
    ! [A: int,B: int] :
      ( ( divide_divide_int @ ( modulo_modulo_int @ A @ B ) @ B )
      = zero_zero_int ) ).

% mod_div_trivial
thf(fact_1947_mod__div__trivial,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B )
      = zero_z3403309356797280102nteger ) ).

% mod_div_trivial
thf(fact_1948_mod__div__trivial,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( divide5121882707175180666atural @ ( modulo8411746178871703098atural @ A @ B ) @ B )
      = zero_z2226904508553997617atural ) ).

% mod_div_trivial
thf(fact_1949_minus__mod__self1,axiom,
    ! [B: int,A: int] :
      ( ( modulo_modulo_int @ ( minus_minus_int @ B @ A ) @ B )
      = ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B ) ) ).

% minus_mod_self1
thf(fact_1950_minus__mod__self1,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ B @ A ) @ B )
      = ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).

% minus_mod_self1
thf(fact_1951_sgn__less,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( sgn_sgn_Code_integer @ A ) @ zero_z3403309356797280102nteger )
      = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).

% sgn_less
thf(fact_1952_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_1953_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_1954_sgn__greater,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( sgn_sgn_Code_integer @ A ) )
      = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).

% sgn_greater
thf(fact_1955_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_1956_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_1957_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_1958_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_1959_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_1960_divide__sgn,axiom,
    ! [A: rat,B: rat] :
      ( ( divide_divide_rat @ A @ ( sgn_sgn_rat @ B ) )
      = ( times_times_rat @ A @ ( sgn_sgn_rat @ B ) ) ) ).

% divide_sgn
thf(fact_1961_diff__Suc__1,axiom,
    ! [N: nat] :
      ( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
      = N ) ).

% diff_Suc_1
thf(fact_1962_mod__by__Suc__0,axiom,
    ! [M: nat] :
      ( ( modulo_modulo_nat @ M @ ( suc @ zero_zero_nat ) )
      = zero_zero_nat ) ).

% mod_by_Suc_0
thf(fact_1963_div__by__Suc__0,axiom,
    ! [M: nat] :
      ( ( divide_divide_nat @ M @ ( suc @ zero_zero_nat ) )
      = M ) ).

% div_by_Suc_0
thf(fact_1964_div__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ( divide_divide_nat @ M @ N )
        = zero_zero_nat ) ) ).

% div_less
thf(fact_1965_ceiling__zero,axiom,
    ( ( archim2889992004027027881ng_rat @ zero_zero_rat )
    = zero_zero_int ) ).

% ceiling_zero
thf(fact_1966_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_1967_ceiling__one,axiom,
    ( ( archim2889992004027027881ng_rat @ one_one_rat )
    = one_one_int ) ).

% ceiling_one
thf(fact_1968_ceiling__of__nat,axiom,
    ! [N: nat] :
      ( ( archim2889992004027027881ng_rat @ ( semiri681578069525770553at_rat @ N ) )
      = ( semiri1314217659103216013at_int @ N ) ) ).

% ceiling_of_nat
thf(fact_1969_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_1970_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_1971_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_1972_divide__less__eq__1__neg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( ord_less_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
        = ( ord_less_rat @ A @ B ) ) ) ).

% divide_less_eq_1_neg
thf(fact_1973_divide__less__eq__1__pos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
        = ( ord_less_rat @ B @ A ) ) ) ).

% divide_less_eq_1_pos
thf(fact_1974_less__divide__eq__1__neg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
        = ( ord_less_rat @ B @ A ) ) ) ).

% less_divide_eq_1_neg
thf(fact_1975_less__divide__eq__1__pos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
        = ( ord_less_rat @ A @ B ) ) ) ).

% less_divide_eq_1_pos
thf(fact_1976_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_1977_nonzero__divide__mult__cancel__left,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( divide_divide_rat @ A @ ( times_times_rat @ A @ B ) )
        = ( divide_divide_rat @ one_one_rat @ B ) ) ) ).

% nonzero_divide_mult_cancel_left
thf(fact_1978_nonzero__divide__mult__cancel__right,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( ( divide_divide_rat @ B @ ( times_times_rat @ A @ B ) )
        = ( divide_divide_rat @ one_one_rat @ A ) ) ) ).

% nonzero_divide_mult_cancel_right
thf(fact_1979_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_1980_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_1981_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_1982_unit__mult__div__div,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( times_times_nat @ B @ ( divide_divide_nat @ one_one_nat @ A ) )
        = ( divide_divide_nat @ B @ A ) ) ) ).

% unit_mult_div_div
thf(fact_1983_unit__mult__div__div,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( times_times_int @ B @ ( divide_divide_int @ one_one_int @ A ) )
        = ( divide_divide_int @ B @ A ) ) ) ).

% unit_mult_div_div
thf(fact_1984_unit__mult__div__div,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ( ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ one_one_Code_integer @ A ) )
        = ( divide6298287555418463151nteger @ B @ A ) ) ) ).

% unit_mult_div_div
thf(fact_1985_unit__mult__div__div,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( dvd_dvd_Code_natural @ A @ one_one_Code_natural )
     => ( ( times_2397367101498566445atural @ B @ ( divide5121882707175180666atural @ one_one_Code_natural @ A ) )
        = ( divide5121882707175180666atural @ B @ A ) ) ) ).

% unit_mult_div_div
thf(fact_1986_unit__div__mult__self,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( times_times_nat @ ( divide_divide_nat @ B @ A ) @ A )
        = B ) ) ).

% unit_div_mult_self
thf(fact_1987_unit__div__mult__self,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( times_times_int @ ( divide_divide_int @ B @ A ) @ A )
        = B ) ) ).

% unit_div_mult_self
thf(fact_1988_unit__div__mult__self,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ B @ A ) @ A )
        = B ) ) ).

% unit_div_mult_self
thf(fact_1989_unit__div__mult__self,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( dvd_dvd_Code_natural @ A @ one_one_Code_natural )
     => ( ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ B @ A ) @ A )
        = B ) ) ).

% unit_div_mult_self
thf(fact_1990_mod__minus1__right,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ one_one_int ) )
      = zero_zero_int ) ).

% mod_minus1_right
thf(fact_1991_mod__minus1__right,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger @ A @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = zero_z3403309356797280102nteger ) ).

% mod_minus1_right
thf(fact_1992_sgn__pos,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( sgn_sgn_Code_integer @ A )
        = one_one_Code_integer ) ) ).

% sgn_pos
thf(fact_1993_sgn__pos,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( sgn_sgn_rat @ A )
        = one_one_rat ) ) ).

% sgn_pos
thf(fact_1994_sgn__pos,axiom,
    ! [A: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( sgn_sgn_int @ A )
        = one_one_int ) ) ).

% sgn_pos
thf(fact_1995_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_1996_nat__1,axiom,
    ( ( nat2 @ one_one_int )
    = ( suc @ zero_zero_nat ) ) ).

% nat_1
thf(fact_1997_nat__le__0,axiom,
    ! [Z2: int] :
      ( ( ord_less_eq_int @ Z2 @ zero_zero_int )
     => ( ( nat2 @ Z2 )
        = zero_zero_nat ) ) ).

% nat_le_0
thf(fact_1998_nat__0__iff,axiom,
    ! [I2: int] :
      ( ( ( nat2 @ I2 )
        = zero_zero_nat )
      = ( ord_less_eq_int @ I2 @ zero_zero_int ) ) ).

% nat_0_iff
thf(fact_1999_zless__nat__conj,axiom,
    ! [W2: int,Z2: int] :
      ( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z2 ) )
      = ( ( ord_less_int @ zero_zero_int @ Z2 )
        & ( ord_less_int @ W2 @ Z2 ) ) ) ).

% zless_nat_conj
thf(fact_2000_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_2001_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_2002_ceiling__diff__one,axiom,
    ! [X: rat] :
      ( ( archim2889992004027027881ng_rat @ ( minus_minus_rat @ X @ one_one_rat ) )
      = ( minus_minus_int @ ( archim2889992004027027881ng_rat @ X ) @ one_one_int ) ) ).

% ceiling_diff_one
thf(fact_2003_int__nat__eq,axiom,
    ! [Z2: int] :
      ( ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
       => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z2 ) )
          = Z2 ) )
      & ( ~ ( ord_less_eq_int @ zero_zero_int @ Z2 )
       => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z2 ) )
          = zero_zero_int ) ) ) ).

% int_nat_eq
thf(fact_2004_nat__zminus__int,axiom,
    ! [N: nat] :
      ( ( nat2 @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) )
      = zero_zero_nat ) ).

% nat_zminus_int
thf(fact_2005_divide__le__eq__1__neg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
        = ( ord_less_eq_rat @ A @ B ) ) ) ).

% divide_le_eq_1_neg
thf(fact_2006_divide__le__eq__1__pos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
        = ( ord_less_eq_rat @ B @ A ) ) ) ).

% divide_le_eq_1_pos
thf(fact_2007_le__divide__eq__1__neg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
        = ( ord_less_eq_rat @ B @ A ) ) ) ).

% le_divide_eq_1_neg
thf(fact_2008_le__divide__eq__1__pos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
        = ( ord_less_eq_rat @ A @ B ) ) ) ).

% le_divide_eq_1_pos
thf(fact_2009_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_2010_ceiling__le__zero,axiom,
    ! [X: rat] :
      ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ zero_zero_int )
      = ( ord_less_eq_rat @ X @ zero_zero_rat ) ) ).

% ceiling_le_zero
thf(fact_2011_zero__less__ceiling,axiom,
    ! [X: rat] :
      ( ( ord_less_int @ zero_zero_int @ ( archim2889992004027027881ng_rat @ X ) )
      = ( ord_less_rat @ zero_zero_rat @ X ) ) ).

% zero_less_ceiling
thf(fact_2012_zero__less__nat__eq,axiom,
    ! [Z2: int] :
      ( ( ord_less_nat @ zero_zero_nat @ ( nat2 @ Z2 ) )
      = ( ord_less_int @ zero_zero_int @ Z2 ) ) ).

% zero_less_nat_eq
thf(fact_2013_ceiling__less__one,axiom,
    ! [X: rat] :
      ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ one_one_int )
      = ( ord_less_eq_rat @ X @ zero_zero_rat ) ) ).

% ceiling_less_one
thf(fact_2014_one__le__ceiling,axiom,
    ! [X: rat] :
      ( ( ord_less_eq_int @ one_one_int @ ( archim2889992004027027881ng_rat @ X ) )
      = ( ord_less_rat @ zero_zero_rat @ X ) ) ).

% one_le_ceiling
thf(fact_2015_ceiling__le__one,axiom,
    ! [X: rat] :
      ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ one_one_int )
      = ( ord_less_eq_rat @ X @ one_one_rat ) ) ).

% ceiling_le_one
thf(fact_2016_one__less__ceiling,axiom,
    ! [X: rat] :
      ( ( ord_less_int @ one_one_int @ ( archim2889992004027027881ng_rat @ X ) )
      = ( ord_less_rat @ one_one_rat @ X ) ) ).

% one_less_ceiling
thf(fact_2017_mod__eq__0__iff__dvd,axiom,
    ! [A: nat,B: nat] :
      ( ( ( modulo_modulo_nat @ A @ B )
        = zero_zero_nat )
      = ( dvd_dvd_nat @ B @ A ) ) ).

% mod_eq_0_iff_dvd
thf(fact_2018_mod__eq__0__iff__dvd,axiom,
    ! [A: int,B: int] :
      ( ( ( modulo_modulo_int @ A @ B )
        = zero_zero_int )
      = ( dvd_dvd_int @ B @ A ) ) ).

% mod_eq_0_iff_dvd
thf(fact_2019_mod__eq__0__iff__dvd,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ B )
        = zero_z3403309356797280102nteger )
      = ( dvd_dvd_Code_integer @ B @ A ) ) ).

% mod_eq_0_iff_dvd
thf(fact_2020_mod__eq__0__iff__dvd,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( ( modulo8411746178871703098atural @ A @ B )
        = zero_z2226904508553997617atural )
      = ( dvd_dvd_Code_natural @ B @ A ) ) ).

% mod_eq_0_iff_dvd
thf(fact_2021_dvd__eq__mod__eq__0,axiom,
    ( dvd_dvd_nat
    = ( ^ [A2: nat,B2: nat] :
          ( ( modulo_modulo_nat @ B2 @ A2 )
          = zero_zero_nat ) ) ) ).

% dvd_eq_mod_eq_0
thf(fact_2022_dvd__eq__mod__eq__0,axiom,
    ( dvd_dvd_int
    = ( ^ [A2: int,B2: int] :
          ( ( modulo_modulo_int @ B2 @ A2 )
          = zero_zero_int ) ) ) ).

% dvd_eq_mod_eq_0
thf(fact_2023_dvd__eq__mod__eq__0,axiom,
    ( dvd_dvd_Code_integer
    = ( ^ [A2: code_integer,B2: code_integer] :
          ( ( modulo364778990260209775nteger @ B2 @ A2 )
          = zero_z3403309356797280102nteger ) ) ) ).

% dvd_eq_mod_eq_0
thf(fact_2024_dvd__eq__mod__eq__0,axiom,
    ( dvd_dvd_Code_natural
    = ( ^ [A2: code_natural,B2: code_natural] :
          ( ( modulo8411746178871703098atural @ B2 @ A2 )
          = zero_z2226904508553997617atural ) ) ) ).

% dvd_eq_mod_eq_0
thf(fact_2025_mod__0__imp__dvd,axiom,
    ! [A: nat,B: nat] :
      ( ( ( modulo_modulo_nat @ A @ B )
        = zero_zero_nat )
     => ( dvd_dvd_nat @ B @ A ) ) ).

% mod_0_imp_dvd
thf(fact_2026_mod__0__imp__dvd,axiom,
    ! [A: int,B: int] :
      ( ( ( modulo_modulo_int @ A @ B )
        = zero_zero_int )
     => ( dvd_dvd_int @ B @ A ) ) ).

% mod_0_imp_dvd
thf(fact_2027_mod__0__imp__dvd,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ B )
        = zero_z3403309356797280102nteger )
     => ( dvd_dvd_Code_integer @ B @ A ) ) ).

% mod_0_imp_dvd
thf(fact_2028_mod__0__imp__dvd,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( ( modulo8411746178871703098atural @ A @ B )
        = zero_z2226904508553997617atural )
     => ( dvd_dvd_Code_natural @ B @ A ) ) ).

% mod_0_imp_dvd
thf(fact_2029_mod__eq__self__iff__div__eq__0,axiom,
    ! [A: nat,B: nat] :
      ( ( ( modulo_modulo_nat @ A @ B )
        = A )
      = ( ( divide_divide_nat @ A @ B )
        = zero_zero_nat ) ) ).

% mod_eq_self_iff_div_eq_0
thf(fact_2030_mod__eq__self__iff__div__eq__0,axiom,
    ! [A: int,B: int] :
      ( ( ( modulo_modulo_int @ A @ B )
        = A )
      = ( ( divide_divide_int @ A @ B )
        = zero_zero_int ) ) ).

% mod_eq_self_iff_div_eq_0
thf(fact_2031_mod__eq__self__iff__div__eq__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ B )
        = A )
      = ( ( divide6298287555418463151nteger @ A @ B )
        = zero_z3403309356797280102nteger ) ) ).

% mod_eq_self_iff_div_eq_0
thf(fact_2032_mod__eq__self__iff__div__eq__0,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( ( modulo8411746178871703098atural @ A @ B )
        = A )
      = ( ( divide5121882707175180666atural @ A @ B )
        = zero_z2226904508553997617atural ) ) ).

% mod_eq_self_iff_div_eq_0
thf(fact_2033_dvd__div__eq__0__iff,axiom,
    ! [B: nat,A: nat] :
      ( ( dvd_dvd_nat @ B @ A )
     => ( ( ( divide_divide_nat @ A @ B )
          = zero_zero_nat )
        = ( A = zero_zero_nat ) ) ) ).

% dvd_div_eq_0_iff
thf(fact_2034_dvd__div__eq__0__iff,axiom,
    ! [B: int,A: int] :
      ( ( dvd_dvd_int @ B @ A )
     => ( ( ( divide_divide_int @ A @ B )
          = zero_zero_int )
        = ( A = zero_zero_int ) ) ) ).

% dvd_div_eq_0_iff
thf(fact_2035_dvd__div__eq__0__iff,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ A )
     => ( ( ( divide6298287555418463151nteger @ A @ B )
          = zero_z3403309356797280102nteger )
        = ( A = zero_z3403309356797280102nteger ) ) ) ).

% dvd_div_eq_0_iff
thf(fact_2036_dvd__div__eq__0__iff,axiom,
    ! [B: rat,A: rat] :
      ( ( dvd_dvd_rat @ B @ A )
     => ( ( ( divide_divide_rat @ A @ B )
          = zero_zero_rat )
        = ( A = zero_zero_rat ) ) ) ).

% dvd_div_eq_0_iff
thf(fact_2037_dvd__div__eq__0__iff,axiom,
    ! [B: code_natural,A: code_natural] :
      ( ( dvd_dvd_Code_natural @ B @ A )
     => ( ( ( divide5121882707175180666atural @ A @ B )
          = zero_z2226904508553997617atural )
        = ( A = zero_z2226904508553997617atural ) ) ) ).

% dvd_div_eq_0_iff
thf(fact_2038_dvd__diff,axiom,
    ! [X: code_integer,Y3: code_integer,Z2: code_integer] :
      ( ( dvd_dvd_Code_integer @ X @ Y3 )
     => ( ( dvd_dvd_Code_integer @ X @ Z2 )
       => ( dvd_dvd_Code_integer @ X @ ( minus_8373710615458151222nteger @ Y3 @ Z2 ) ) ) ) ).

% dvd_diff
thf(fact_2039_dvd__diff,axiom,
    ! [X: rat,Y3: rat,Z2: rat] :
      ( ( dvd_dvd_rat @ X @ Y3 )
     => ( ( dvd_dvd_rat @ X @ Z2 )
       => ( dvd_dvd_rat @ X @ ( minus_minus_rat @ Y3 @ Z2 ) ) ) ) ).

% dvd_diff
thf(fact_2040_dvd__diff,axiom,
    ! [X: int,Y3: int,Z2: int] :
      ( ( dvd_dvd_int @ X @ Y3 )
     => ( ( dvd_dvd_int @ X @ Z2 )
       => ( dvd_dvd_int @ X @ ( minus_minus_int @ Y3 @ Z2 ) ) ) ) ).

% dvd_diff
thf(fact_2041_dvd__refl,axiom,
    ! [A: nat] : ( dvd_dvd_nat @ A @ A ) ).

% dvd_refl
thf(fact_2042_dvd__refl,axiom,
    ! [A: int] : ( dvd_dvd_int @ A @ A ) ).

% dvd_refl
thf(fact_2043_dvd__refl,axiom,
    ! [A: code_integer] : ( dvd_dvd_Code_integer @ A @ A ) ).

% dvd_refl
thf(fact_2044_dvd__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( ( dvd_dvd_nat @ B @ C )
       => ( dvd_dvd_nat @ A @ C ) ) ) ).

% dvd_trans
thf(fact_2045_dvd__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ A @ B )
     => ( ( dvd_dvd_int @ B @ C )
       => ( dvd_dvd_int @ A @ C ) ) ) ).

% dvd_trans
thf(fact_2046_dvd__trans,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ B )
     => ( ( dvd_dvd_Code_integer @ B @ C )
       => ( dvd_dvd_Code_integer @ A @ C ) ) ) ).

% dvd_trans
thf(fact_2047_dvd__mod__iff,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( dvd_dvd_nat @ C @ B )
     => ( ( dvd_dvd_nat @ C @ ( modulo_modulo_nat @ A @ B ) )
        = ( dvd_dvd_nat @ C @ A ) ) ) ).

% dvd_mod_iff
thf(fact_2048_dvd__mod__iff,axiom,
    ! [C: int,B: int,A: int] :
      ( ( dvd_dvd_int @ C @ B )
     => ( ( dvd_dvd_int @ C @ ( modulo_modulo_int @ A @ B ) )
        = ( dvd_dvd_int @ C @ A ) ) ) ).

% dvd_mod_iff
thf(fact_2049_dvd__mod__iff,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ B )
     => ( ( dvd_dvd_Code_integer @ C @ ( modulo364778990260209775nteger @ A @ B ) )
        = ( dvd_dvd_Code_integer @ C @ A ) ) ) ).

% dvd_mod_iff
thf(fact_2050_dvd__mod__iff,axiom,
    ! [C: code_natural,B: code_natural,A: code_natural] :
      ( ( dvd_dvd_Code_natural @ C @ B )
     => ( ( dvd_dvd_Code_natural @ C @ ( modulo8411746178871703098atural @ A @ B ) )
        = ( dvd_dvd_Code_natural @ C @ A ) ) ) ).

% dvd_mod_iff
thf(fact_2051_dvd__minus__mod,axiom,
    ! [B: nat,A: nat] : ( dvd_dvd_nat @ B @ ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B ) ) ) ).

% dvd_minus_mod
thf(fact_2052_dvd__minus__mod,axiom,
    ! [B: int,A: int] : ( dvd_dvd_int @ B @ ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B ) ) ) ).

% dvd_minus_mod
thf(fact_2053_dvd__minus__mod,axiom,
    ! [B: code_integer,A: code_integer] : ( dvd_dvd_Code_integer @ B @ ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).

% dvd_minus_mod
thf(fact_2054_dvd__minus__mod,axiom,
    ! [B: code_natural,A: code_natural] : ( dvd_dvd_Code_natural @ B @ ( minus_7197305767214868737atural @ A @ ( modulo8411746178871703098atural @ A @ B ) ) ) ).

% dvd_minus_mod
thf(fact_2055_dvd__div__eq__iff,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( dvd_dvd_nat @ C @ A )
     => ( ( dvd_dvd_nat @ C @ B )
       => ( ( ( divide_divide_nat @ A @ C )
            = ( divide_divide_nat @ B @ C ) )
          = ( A = B ) ) ) ) ).

% dvd_div_eq_iff
thf(fact_2056_dvd__div__eq__iff,axiom,
    ! [C: int,A: int,B: int] :
      ( ( dvd_dvd_int @ C @ A )
     => ( ( dvd_dvd_int @ C @ B )
       => ( ( ( divide_divide_int @ A @ C )
            = ( divide_divide_int @ B @ C ) )
          = ( A = B ) ) ) ) ).

% dvd_div_eq_iff
thf(fact_2057_dvd__div__eq__iff,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ A )
     => ( ( dvd_dvd_Code_integer @ C @ B )
       => ( ( ( divide6298287555418463151nteger @ A @ C )
            = ( divide6298287555418463151nteger @ B @ C ) )
          = ( A = B ) ) ) ) ).

% dvd_div_eq_iff
thf(fact_2058_dvd__div__eq__iff,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( dvd_dvd_rat @ C @ A )
     => ( ( dvd_dvd_rat @ C @ B )
       => ( ( ( divide_divide_rat @ A @ C )
            = ( divide_divide_rat @ B @ C ) )
          = ( A = B ) ) ) ) ).

% dvd_div_eq_iff
thf(fact_2059_dvd__div__eq__iff,axiom,
    ! [C: code_natural,A: code_natural,B: code_natural] :
      ( ( dvd_dvd_Code_natural @ C @ A )
     => ( ( dvd_dvd_Code_natural @ C @ B )
       => ( ( ( divide5121882707175180666atural @ A @ C )
            = ( divide5121882707175180666atural @ B @ C ) )
          = ( A = B ) ) ) ) ).

% dvd_div_eq_iff
thf(fact_2060_dvd__mod__imp__dvd,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( dvd_dvd_nat @ C @ ( modulo_modulo_nat @ A @ B ) )
     => ( ( dvd_dvd_nat @ C @ B )
       => ( dvd_dvd_nat @ C @ A ) ) ) ).

% dvd_mod_imp_dvd
thf(fact_2061_dvd__mod__imp__dvd,axiom,
    ! [C: int,A: int,B: int] :
      ( ( dvd_dvd_int @ C @ ( modulo_modulo_int @ A @ B ) )
     => ( ( dvd_dvd_int @ C @ B )
       => ( dvd_dvd_int @ C @ A ) ) ) ).

% dvd_mod_imp_dvd
thf(fact_2062_dvd__mod__imp__dvd,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ ( modulo364778990260209775nteger @ A @ B ) )
     => ( ( dvd_dvd_Code_integer @ C @ B )
       => ( dvd_dvd_Code_integer @ C @ A ) ) ) ).

% dvd_mod_imp_dvd
thf(fact_2063_dvd__mod__imp__dvd,axiom,
    ! [C: code_natural,A: code_natural,B: code_natural] :
      ( ( dvd_dvd_Code_natural @ C @ ( modulo8411746178871703098atural @ A @ B ) )
     => ( ( dvd_dvd_Code_natural @ C @ B )
       => ( dvd_dvd_Code_natural @ C @ A ) ) ) ).

% dvd_mod_imp_dvd
thf(fact_2064_dvd__div__eq__cancel,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ( divide_divide_nat @ A @ C )
        = ( divide_divide_nat @ B @ C ) )
     => ( ( dvd_dvd_nat @ C @ A )
       => ( ( dvd_dvd_nat @ C @ B )
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
thf(fact_2065_dvd__div__eq__cancel,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ( divide_divide_int @ A @ C )
        = ( divide_divide_int @ B @ C ) )
     => ( ( dvd_dvd_int @ C @ A )
       => ( ( dvd_dvd_int @ C @ B )
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
thf(fact_2066_dvd__div__eq__cancel,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( ( divide6298287555418463151nteger @ A @ C )
        = ( divide6298287555418463151nteger @ B @ C ) )
     => ( ( dvd_dvd_Code_integer @ C @ A )
       => ( ( dvd_dvd_Code_integer @ C @ B )
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
thf(fact_2067_dvd__div__eq__cancel,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ( divide_divide_rat @ A @ C )
        = ( divide_divide_rat @ B @ C ) )
     => ( ( dvd_dvd_rat @ C @ A )
       => ( ( dvd_dvd_rat @ C @ B )
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
thf(fact_2068_dvd__div__eq__cancel,axiom,
    ! [A: code_natural,C: code_natural,B: code_natural] :
      ( ( ( divide5121882707175180666atural @ A @ C )
        = ( divide5121882707175180666atural @ B @ C ) )
     => ( ( dvd_dvd_Code_natural @ C @ A )
       => ( ( dvd_dvd_Code_natural @ C @ B )
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
thf(fact_2069_div__div__div__same,axiom,
    ! [D: nat,B: nat,A: nat] :
      ( ( dvd_dvd_nat @ D @ B )
     => ( ( dvd_dvd_nat @ B @ A )
       => ( ( divide_divide_nat @ ( divide_divide_nat @ A @ D ) @ ( divide_divide_nat @ B @ D ) )
          = ( divide_divide_nat @ A @ B ) ) ) ) ).

% div_div_div_same
thf(fact_2070_div__div__div__same,axiom,
    ! [D: int,B: int,A: int] :
      ( ( dvd_dvd_int @ D @ B )
     => ( ( dvd_dvd_int @ B @ A )
       => ( ( divide_divide_int @ ( divide_divide_int @ A @ D ) @ ( divide_divide_int @ B @ D ) )
          = ( divide_divide_int @ A @ B ) ) ) ) ).

% div_div_div_same
thf(fact_2071_div__div__div__same,axiom,
    ! [D: code_integer,B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ D @ B )
     => ( ( dvd_dvd_Code_integer @ B @ A )
       => ( ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ D ) @ ( divide6298287555418463151nteger @ B @ D ) )
          = ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).

% div_div_div_same
thf(fact_2072_div__div__div__same,axiom,
    ! [D: code_natural,B: code_natural,A: code_natural] :
      ( ( dvd_dvd_Code_natural @ D @ B )
     => ( ( dvd_dvd_Code_natural @ B @ A )
       => ( ( divide5121882707175180666atural @ ( divide5121882707175180666atural @ A @ D ) @ ( divide5121882707175180666atural @ B @ D ) )
          = ( divide5121882707175180666atural @ A @ B ) ) ) ) ).

% div_div_div_same
thf(fact_2073_diff__eq__diff__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
      ( ( ( minus_8373710615458151222nteger @ A @ B )
        = ( minus_8373710615458151222nteger @ C @ D ) )
     => ( ( A = B )
        = ( C = D ) ) ) ).

% diff_eq_diff_eq
thf(fact_2074_diff__eq__diff__eq,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ( minus_minus_rat @ A @ B )
        = ( minus_minus_rat @ C @ D ) )
     => ( ( A = B )
        = ( C = D ) ) ) ).

% diff_eq_diff_eq
thf(fact_2075_diff__eq__diff__eq,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ( minus_minus_int @ A @ B )
        = ( minus_minus_int @ C @ D ) )
     => ( ( A = B )
        = ( C = D ) ) ) ).

% diff_eq_diff_eq
thf(fact_2076_diff__right__commute,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( minus_8373710615458151222nteger @ ( minus_8373710615458151222nteger @ A @ C ) @ B )
      = ( minus_8373710615458151222nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C ) ) ).

% diff_right_commute
thf(fact_2077_diff__right__commute,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( minus_minus_rat @ ( minus_minus_rat @ A @ C ) @ B )
      = ( minus_minus_rat @ ( minus_minus_rat @ A @ B ) @ C ) ) ).

% diff_right_commute
thf(fact_2078_diff__right__commute,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
      = ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).

% diff_right_commute
thf(fact_2079_diff__right__commute,axiom,
    ! [A: int,C: int,B: int] :
      ( ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B )
      = ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).

% diff_right_commute
thf(fact_2080_dvd__diff__commute,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ ( minus_8373710615458151222nteger @ C @ B ) )
      = ( dvd_dvd_Code_integer @ A @ ( minus_8373710615458151222nteger @ B @ C ) ) ) ).

% dvd_diff_commute
thf(fact_2081_dvd__diff__commute,axiom,
    ! [A: int,C: int,B: int] :
      ( ( dvd_dvd_int @ A @ ( minus_minus_int @ C @ B ) )
      = ( dvd_dvd_int @ A @ ( minus_minus_int @ B @ C ) ) ) ).

% dvd_diff_commute
thf(fact_2082_mod__diff__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( ( modulo_modulo_int @ ( minus_minus_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B @ C ) ) @ C )
      = ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).

% mod_diff_eq
thf(fact_2083_mod__diff__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
      = ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C ) ) ).

% mod_diff_eq
thf(fact_2084_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_2085_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_2086_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_2087_dvd__mod,axiom,
    ! [K: code_natural,M: code_natural,N: code_natural] :
      ( ( dvd_dvd_Code_natural @ K @ M )
     => ( ( dvd_dvd_Code_natural @ K @ N )
       => ( dvd_dvd_Code_natural @ K @ ( modulo8411746178871703098atural @ M @ N ) ) ) ) ).

% dvd_mod
thf(fact_2088_mod__diff__cong,axiom,
    ! [A: int,C: int,A4: int,B: int,B3: int] :
      ( ( ( modulo_modulo_int @ A @ C )
        = ( modulo_modulo_int @ A4 @ C ) )
     => ( ( ( modulo_modulo_int @ B @ C )
          = ( modulo_modulo_int @ B3 @ C ) )
       => ( ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ C )
          = ( modulo_modulo_int @ ( minus_minus_int @ A4 @ B3 ) @ C ) ) ) ) ).

% mod_diff_cong
thf(fact_2089_mod__diff__cong,axiom,
    ! [A: code_integer,C: code_integer,A4: code_integer,B: code_integer,B3: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ C )
        = ( modulo364778990260209775nteger @ A4 @ C ) )
     => ( ( ( modulo364778990260209775nteger @ B @ C )
          = ( modulo364778990260209775nteger @ B3 @ C ) )
       => ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C )
          = ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A4 @ B3 ) @ C ) ) ) ) ).

% mod_diff_cong
thf(fact_2090_mod__eq__dvd__iff,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ( modulo_modulo_int @ A @ C )
        = ( modulo_modulo_int @ B @ C ) )
      = ( dvd_dvd_int @ C @ ( minus_minus_int @ A @ B ) ) ) ).

% mod_eq_dvd_iff
thf(fact_2091_mod__eq__dvd__iff,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ C )
        = ( modulo364778990260209775nteger @ B @ C ) )
      = ( dvd_dvd_Code_integer @ C @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ).

% mod_eq_dvd_iff
thf(fact_2092_mod__diff__left__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( ( modulo_modulo_int @ ( minus_minus_int @ ( modulo_modulo_int @ A @ C ) @ B ) @ C )
      = ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).

% mod_diff_left_eq
thf(fact_2093_mod__diff__left__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ ( modulo364778990260209775nteger @ A @ C ) @ B ) @ C )
      = ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C ) ) ).

% mod_diff_left_eq
thf(fact_2094_mod__diff__right__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( modulo_modulo_int @ ( minus_minus_int @ A @ ( modulo_modulo_int @ B @ C ) ) @ C )
      = ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).

% mod_diff_right_eq
thf(fact_2095_mod__diff__right__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
      = ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C ) ) ).

% mod_diff_right_eq
thf(fact_2096_mod__mod__cancel,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( dvd_dvd_nat @ C @ B )
     => ( ( modulo_modulo_nat @ ( modulo_modulo_nat @ A @ B ) @ C )
        = ( modulo_modulo_nat @ A @ C ) ) ) ).

% mod_mod_cancel
thf(fact_2097_mod__mod__cancel,axiom,
    ! [C: int,B: int,A: int] :
      ( ( dvd_dvd_int @ C @ B )
     => ( ( modulo_modulo_int @ ( modulo_modulo_int @ A @ B ) @ C )
        = ( modulo_modulo_int @ A @ C ) ) ) ).

% mod_mod_cancel
thf(fact_2098_mod__mod__cancel,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ B )
     => ( ( modulo364778990260209775nteger @ ( modulo364778990260209775nteger @ A @ B ) @ C )
        = ( modulo364778990260209775nteger @ A @ C ) ) ) ).

% mod_mod_cancel
thf(fact_2099_mod__mod__cancel,axiom,
    ! [C: code_natural,B: code_natural,A: code_natural] :
      ( ( dvd_dvd_Code_natural @ C @ B )
     => ( ( modulo8411746178871703098atural @ ( modulo8411746178871703098atural @ A @ B ) @ C )
        = ( modulo8411746178871703098atural @ A @ C ) ) ) ).

% mod_mod_cancel
thf(fact_2100_diff__commute,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J ) @ K )
      = ( minus_minus_nat @ ( minus_minus_nat @ I2 @ K ) @ J ) ) ).

% diff_commute
thf(fact_2101_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_2102_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_2103_diff__divide__distrib,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( divide_divide_rat @ ( minus_minus_rat @ A @ B ) @ C )
      = ( minus_minus_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ).

% diff_divide_distrib
thf(fact_2104_modulo__nat__def,axiom,
    ( modulo_modulo_nat
    = ( ^ [M2: nat,N3: nat] : ( minus_minus_nat @ M2 @ ( times_times_nat @ ( divide_divide_nat @ M2 @ N3 ) @ N3 ) ) ) ) ).

% modulo_nat_def
thf(fact_2105_minus__mult__div__eq__mod,axiom,
    ! [A: nat,B: nat] :
      ( ( minus_minus_nat @ A @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) )
      = ( modulo_modulo_nat @ A @ B ) ) ).

% minus_mult_div_eq_mod
thf(fact_2106_minus__mult__div__eq__mod,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ A @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) )
      = ( modulo_modulo_int @ A @ B ) ) ).

% minus_mult_div_eq_mod
thf(fact_2107_minus__mult__div__eq__mod,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( minus_8373710615458151222nteger @ A @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% minus_mult_div_eq_mod
thf(fact_2108_minus__mult__div__eq__mod,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( minus_7197305767214868737atural @ A @ ( times_2397367101498566445atural @ B @ ( divide5121882707175180666atural @ A @ B ) ) )
      = ( modulo8411746178871703098atural @ A @ B ) ) ).

% minus_mult_div_eq_mod
thf(fact_2109_minus__mod__eq__mult__div,axiom,
    ! [A: nat,B: nat] :
      ( ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B ) )
      = ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) ) ).

% minus_mod_eq_mult_div
thf(fact_2110_minus__mod__eq__mult__div,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B ) )
      = ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) ) ).

% minus_mod_eq_mult_div
thf(fact_2111_minus__mod__eq__mult__div,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ A @ B ) )
      = ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) ) ).

% minus_mod_eq_mult_div
thf(fact_2112_minus__mod__eq__mult__div,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( minus_7197305767214868737atural @ A @ ( modulo8411746178871703098atural @ A @ B ) )
      = ( times_2397367101498566445atural @ B @ ( divide5121882707175180666atural @ A @ B ) ) ) ).

% minus_mod_eq_mult_div
thf(fact_2113_minus__mod__eq__div__mult,axiom,
    ! [A: nat,B: nat] :
      ( ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B ) )
      = ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) ) ).

% minus_mod_eq_div_mult
thf(fact_2114_minus__mod__eq__div__mult,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B ) )
      = ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) ) ).

% minus_mod_eq_div_mult
thf(fact_2115_minus__mod__eq__div__mult,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ A @ B ) )
      = ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) ) ).

% minus_mod_eq_div_mult
thf(fact_2116_minus__mod__eq__div__mult,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( minus_7197305767214868737atural @ A @ ( modulo8411746178871703098atural @ A @ B ) )
      = ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ A @ B ) @ B ) ) ).

% minus_mod_eq_div_mult
thf(fact_2117_minus__div__mult__eq__mod,axiom,
    ! [A: nat,B: nat] :
      ( ( minus_minus_nat @ A @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) )
      = ( modulo_modulo_nat @ A @ B ) ) ).

% minus_div_mult_eq_mod
thf(fact_2118_minus__div__mult__eq__mod,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ A @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) )
      = ( modulo_modulo_int @ A @ B ) ) ).

% minus_div_mult_eq_mod
thf(fact_2119_minus__div__mult__eq__mod,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( minus_8373710615458151222nteger @ A @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% minus_div_mult_eq_mod
thf(fact_2120_minus__div__mult__eq__mod,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( minus_7197305767214868737atural @ A @ ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ A @ B ) @ B ) )
      = ( modulo8411746178871703098atural @ A @ B ) ) ).

% minus_div_mult_eq_mod
thf(fact_2121_div__mult__div__if__dvd,axiom,
    ! [B: nat,A: nat,D: nat,C: nat] :
      ( ( dvd_dvd_nat @ B @ A )
     => ( ( dvd_dvd_nat @ D @ C )
       => ( ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ ( divide_divide_nat @ C @ D ) )
          = ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ).

% div_mult_div_if_dvd
thf(fact_2122_div__mult__div__if__dvd,axiom,
    ! [B: int,A: int,D: int,C: int] :
      ( ( dvd_dvd_int @ B @ A )
     => ( ( dvd_dvd_int @ D @ C )
       => ( ( times_times_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ C @ D ) )
          = ( divide_divide_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ).

% div_mult_div_if_dvd
thf(fact_2123_div__mult__div__if__dvd,axiom,
    ! [B: code_integer,A: code_integer,D: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ A )
     => ( ( dvd_dvd_Code_integer @ D @ C )
       => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ ( divide6298287555418463151nteger @ C @ D ) )
          = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ D ) ) ) ) ) ).

% div_mult_div_if_dvd
thf(fact_2124_div__mult__div__if__dvd,axiom,
    ! [B: code_natural,A: code_natural,D: code_natural,C: code_natural] :
      ( ( dvd_dvd_Code_natural @ B @ A )
     => ( ( dvd_dvd_Code_natural @ D @ C )
       => ( ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ A @ B ) @ ( divide5121882707175180666atural @ C @ D ) )
          = ( divide5121882707175180666atural @ ( times_2397367101498566445atural @ A @ C ) @ ( times_2397367101498566445atural @ B @ D ) ) ) ) ) ).

% div_mult_div_if_dvd
thf(fact_2125_dvd__mult__imp__div,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ B )
     => ( dvd_dvd_nat @ A @ ( divide_divide_nat @ B @ C ) ) ) ).

% dvd_mult_imp_div
thf(fact_2126_dvd__mult__imp__div,axiom,
    ! [A: int,C: int,B: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ B )
     => ( dvd_dvd_int @ A @ ( divide_divide_int @ B @ C ) ) ) ).

% dvd_mult_imp_div
thf(fact_2127_dvd__mult__imp__div,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ B )
     => ( dvd_dvd_Code_integer @ A @ ( divide6298287555418463151nteger @ B @ C ) ) ) ).

% dvd_mult_imp_div
thf(fact_2128_dvd__mult__imp__div,axiom,
    ! [A: code_natural,C: code_natural,B: code_natural] :
      ( ( dvd_dvd_Code_natural @ ( times_2397367101498566445atural @ A @ C ) @ B )
     => ( dvd_dvd_Code_natural @ A @ ( divide5121882707175180666atural @ B @ C ) ) ) ).

% dvd_mult_imp_div
thf(fact_2129_dvd__div__mult2__eq,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ B @ C ) @ A )
     => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
        = ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ).

% dvd_div_mult2_eq
thf(fact_2130_dvd__div__mult2__eq,axiom,
    ! [B: int,C: int,A: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ B @ C ) @ A )
     => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
        = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ).

% dvd_div_mult2_eq
thf(fact_2131_dvd__div__mult2__eq,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ B @ C ) @ A )
     => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
        = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ).

% dvd_div_mult2_eq
thf(fact_2132_dvd__div__mult2__eq,axiom,
    ! [B: code_natural,C: code_natural,A: code_natural] :
      ( ( dvd_dvd_Code_natural @ ( times_2397367101498566445atural @ B @ C ) @ A )
     => ( ( divide5121882707175180666atural @ A @ ( times_2397367101498566445atural @ B @ C ) )
        = ( divide5121882707175180666atural @ ( divide5121882707175180666atural @ A @ B ) @ C ) ) ) ).

% dvd_div_mult2_eq
thf(fact_2133_div__div__eq__right,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( dvd_dvd_nat @ C @ B )
     => ( ( dvd_dvd_nat @ B @ A )
       => ( ( divide_divide_nat @ A @ ( divide_divide_nat @ B @ C ) )
          = ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ) ).

% div_div_eq_right
thf(fact_2134_div__div__eq__right,axiom,
    ! [C: int,B: int,A: int] :
      ( ( dvd_dvd_int @ C @ B )
     => ( ( dvd_dvd_int @ B @ A )
       => ( ( divide_divide_int @ A @ ( divide_divide_int @ B @ C ) )
          = ( times_times_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ) ).

% div_div_eq_right
thf(fact_2135_div__div__eq__right,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ B )
     => ( ( dvd_dvd_Code_integer @ B @ A )
       => ( ( divide6298287555418463151nteger @ A @ ( divide6298287555418463151nteger @ B @ C ) )
          = ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ) ).

% div_div_eq_right
thf(fact_2136_div__div__eq__right,axiom,
    ! [C: code_natural,B: code_natural,A: code_natural] :
      ( ( dvd_dvd_Code_natural @ C @ B )
     => ( ( dvd_dvd_Code_natural @ B @ A )
       => ( ( divide5121882707175180666atural @ A @ ( divide5121882707175180666atural @ B @ C ) )
          = ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ A @ B ) @ C ) ) ) ) ).

% div_div_eq_right
thf(fact_2137_div__mult__swap,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( dvd_dvd_nat @ C @ B )
     => ( ( times_times_nat @ A @ ( divide_divide_nat @ B @ C ) )
        = ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ C ) ) ) ).

% div_mult_swap
thf(fact_2138_div__mult__swap,axiom,
    ! [C: int,B: int,A: int] :
      ( ( dvd_dvd_int @ C @ B )
     => ( ( times_times_int @ A @ ( divide_divide_int @ B @ C ) )
        = ( divide_divide_int @ ( times_times_int @ A @ B ) @ C ) ) ) ).

% div_mult_swap
thf(fact_2139_div__mult__swap,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ B )
     => ( ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B @ C ) )
        = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ) ).

% div_mult_swap
thf(fact_2140_div__mult__swap,axiom,
    ! [C: code_natural,B: code_natural,A: code_natural] :
      ( ( dvd_dvd_Code_natural @ C @ B )
     => ( ( times_2397367101498566445atural @ A @ ( divide5121882707175180666atural @ B @ C ) )
        = ( divide5121882707175180666atural @ ( times_2397367101498566445atural @ A @ B ) @ C ) ) ) ).

% div_mult_swap
thf(fact_2141_dvd__div__mult,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( dvd_dvd_nat @ C @ B )
     => ( ( times_times_nat @ ( divide_divide_nat @ B @ C ) @ A )
        = ( divide_divide_nat @ ( times_times_nat @ B @ A ) @ C ) ) ) ).

% dvd_div_mult
thf(fact_2142_dvd__div__mult,axiom,
    ! [C: int,B: int,A: int] :
      ( ( dvd_dvd_int @ C @ B )
     => ( ( times_times_int @ ( divide_divide_int @ B @ C ) @ A )
        = ( divide_divide_int @ ( times_times_int @ B @ A ) @ C ) ) ) ).

% dvd_div_mult
thf(fact_2143_dvd__div__mult,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ B )
     => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ B @ C ) @ A )
        = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ B @ A ) @ C ) ) ) ).

% dvd_div_mult
thf(fact_2144_dvd__div__mult,axiom,
    ! [C: code_natural,B: code_natural,A: code_natural] :
      ( ( dvd_dvd_Code_natural @ C @ B )
     => ( ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ B @ C ) @ A )
        = ( divide5121882707175180666atural @ ( times_2397367101498566445atural @ B @ A ) @ C ) ) ) ).

% dvd_div_mult
thf(fact_2145_unit__div__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( ( divide_divide_nat @ B @ A )
          = ( divide_divide_nat @ C @ A ) )
        = ( B = C ) ) ) ).

% unit_div_cancel
thf(fact_2146_unit__div__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( ( divide_divide_int @ B @ A )
          = ( divide_divide_int @ C @ A ) )
        = ( B = C ) ) ) ).

% unit_div_cancel
thf(fact_2147_unit__div__cancel,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ( ( ( divide6298287555418463151nteger @ B @ A )
          = ( divide6298287555418463151nteger @ C @ A ) )
        = ( B = C ) ) ) ).

% unit_div_cancel
thf(fact_2148_unit__div__cancel,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( ( dvd_dvd_Code_natural @ A @ one_one_Code_natural )
     => ( ( ( divide5121882707175180666atural @ B @ A )
          = ( divide5121882707175180666atural @ C @ A ) )
        = ( B = C ) ) ) ).

% unit_div_cancel
thf(fact_2149_div__unit__dvd__iff,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B ) @ C )
        = ( dvd_dvd_nat @ A @ C ) ) ) ).

% div_unit_dvd_iff
thf(fact_2150_div__unit__dvd__iff,axiom,
    ! [B: int,A: int,C: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( dvd_dvd_int @ ( divide_divide_int @ A @ B ) @ C )
        = ( dvd_dvd_int @ A @ C ) ) ) ).

% div_unit_dvd_iff
thf(fact_2151_div__unit__dvd__iff,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A @ B ) @ C )
        = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).

% div_unit_dvd_iff
thf(fact_2152_div__unit__dvd__iff,axiom,
    ! [B: code_natural,A: code_natural,C: code_natural] :
      ( ( dvd_dvd_Code_natural @ B @ one_one_Code_natural )
     => ( ( dvd_dvd_Code_natural @ ( divide5121882707175180666atural @ A @ B ) @ C )
        = ( dvd_dvd_Code_natural @ A @ C ) ) ) ).

% div_unit_dvd_iff
thf(fact_2153_dvd__div__unit__iff,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( dvd_dvd_nat @ A @ ( divide_divide_nat @ C @ B ) )
        = ( dvd_dvd_nat @ A @ C ) ) ) ).

% dvd_div_unit_iff
thf(fact_2154_dvd__div__unit__iff,axiom,
    ! [B: int,A: int,C: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( dvd_dvd_int @ A @ ( divide_divide_int @ C @ B ) )
        = ( dvd_dvd_int @ A @ C ) ) ) ).

% dvd_div_unit_iff
thf(fact_2155_dvd__div__unit__iff,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ A @ ( divide6298287555418463151nteger @ C @ B ) )
        = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).

% dvd_div_unit_iff
thf(fact_2156_dvd__div__unit__iff,axiom,
    ! [B: code_natural,A: code_natural,C: code_natural] :
      ( ( dvd_dvd_Code_natural @ B @ one_one_Code_natural )
     => ( ( dvd_dvd_Code_natural @ A @ ( divide5121882707175180666atural @ C @ B ) )
        = ( dvd_dvd_Code_natural @ A @ C ) ) ) ).

% dvd_div_unit_iff
thf(fact_2157_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_2158_of__nat__mod,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri4939895301339042750nteger @ ( modulo_modulo_nat @ M @ N ) )
      = ( modulo364778990260209775nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) ) ) ).

% of_nat_mod
thf(fact_2159_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_2160_of__nat__mod,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri3763490453095760265atural @ ( modulo_modulo_nat @ M @ N ) )
      = ( modulo8411746178871703098atural @ ( semiri3763490453095760265atural @ M ) @ ( semiri3763490453095760265atural @ N ) ) ) ).

% of_nat_mod
thf(fact_2161_dvd__neg__div,axiom,
    ! [B: int,A: int] :
      ( ( dvd_dvd_int @ B @ A )
     => ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B )
        = ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) ) ) ).

% dvd_neg_div
thf(fact_2162_dvd__neg__div,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ A )
     => ( ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
        = ( uminus1351360451143612070nteger @ ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).

% dvd_neg_div
thf(fact_2163_dvd__neg__div,axiom,
    ! [B: rat,A: rat] :
      ( ( dvd_dvd_rat @ B @ A )
     => ( ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ B )
        = ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) ) ) ) ).

% dvd_neg_div
thf(fact_2164_dvd__div__neg,axiom,
    ! [B: int,A: int] :
      ( ( dvd_dvd_int @ B @ A )
     => ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
        = ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) ) ) ).

% dvd_div_neg
thf(fact_2165_dvd__div__neg,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ A )
     => ( ( divide6298287555418463151nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
        = ( uminus1351360451143612070nteger @ ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).

% dvd_div_neg
thf(fact_2166_dvd__div__neg,axiom,
    ! [B: rat,A: rat] :
      ( ( dvd_dvd_rat @ B @ A )
     => ( ( divide_divide_rat @ A @ ( uminus_uminus_rat @ B ) )
        = ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) ) ) ) ).

% dvd_div_neg
thf(fact_2167_mod__if,axiom,
    ( modulo_modulo_nat
    = ( ^ [M2: nat,N3: nat] : ( if_nat @ ( ord_less_nat @ M2 @ N3 ) @ M2 @ ( modulo_modulo_nat @ ( minus_minus_nat @ M2 @ N3 ) @ N3 ) ) ) ) ).

% mod_if
thf(fact_2168_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_2169_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_2170_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_2171_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri4939895301339042750nteger @ ( divide_divide_nat @ M @ N ) )
      = ( divide6298287555418463151nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) ) ) ).

% unique_euclidean_semiring_with_nat_class.of_nat_div
thf(fact_2172_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_2173_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri3763490453095760265atural @ ( divide_divide_nat @ M @ N ) )
      = ( divide5121882707175180666atural @ ( semiri3763490453095760265atural @ M ) @ ( semiri3763490453095760265atural @ N ) ) ) ).

% unique_euclidean_semiring_with_nat_class.of_nat_div
thf(fact_2174_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_2175_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_2176_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_2177_of__nat__dvd__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( dvd_dvd_Code_natural @ ( semiri3763490453095760265atural @ M ) @ ( semiri3763490453095760265atural @ N ) )
      = ( dvd_dvd_nat @ M @ N ) ) ).

% of_nat_dvd_iff
thf(fact_2178_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_2179_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_2180_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_2181_mod__nat__eqI,axiom,
    ! [R: nat,N: nat,M: nat] :
      ( ( ord_less_nat @ R @ N )
     => ( ( ord_less_eq_nat @ R @ M )
       => ( ( dvd_dvd_nat @ N @ ( minus_minus_nat @ M @ R ) )
         => ( ( modulo_modulo_nat @ M @ N )
            = R ) ) ) ) ).

% mod_nat_eqI
thf(fact_2182_unit__imp__mod__eq__0,axiom,
    ! [B: nat,A: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( modulo_modulo_nat @ A @ B )
        = zero_zero_nat ) ) ).

% unit_imp_mod_eq_0
thf(fact_2183_unit__imp__mod__eq__0,axiom,
    ! [B: int,A: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( modulo_modulo_int @ A @ B )
        = zero_zero_int ) ) ).

% unit_imp_mod_eq_0
thf(fact_2184_unit__imp__mod__eq__0,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
     => ( ( modulo364778990260209775nteger @ A @ B )
        = zero_z3403309356797280102nteger ) ) ).

% unit_imp_mod_eq_0
thf(fact_2185_unit__imp__mod__eq__0,axiom,
    ! [B: code_natural,A: code_natural] :
      ( ( dvd_dvd_Code_natural @ B @ one_one_Code_natural )
     => ( ( modulo8411746178871703098atural @ A @ B )
        = zero_z2226904508553997617atural ) ) ).

% unit_imp_mod_eq_0
thf(fact_2186_of__nat__ceiling,axiom,
    ! [R: rat] : ( ord_less_eq_rat @ R @ ( semiri681578069525770553at_rat @ ( nat2 @ ( archim2889992004027027881ng_rat @ R ) ) ) ) ).

% of_nat_ceiling
thf(fact_2187_add__divide__eq__if__simps_I4_J,axiom,
    ! [Z2: rat,A: rat,B: rat] :
      ( ( ( Z2 = zero_zero_rat )
       => ( ( minus_minus_rat @ A @ ( divide_divide_rat @ B @ Z2 ) )
          = A ) )
      & ( ( Z2 != zero_zero_rat )
       => ( ( minus_minus_rat @ A @ ( divide_divide_rat @ B @ Z2 ) )
          = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ A @ Z2 ) @ B ) @ Z2 ) ) ) ) ).

% add_divide_eq_if_simps(4)
thf(fact_2188_diff__frac__eq,axiom,
    ! [Y3: rat,Z2: rat,X: rat,W2: rat] :
      ( ( Y3 != zero_zero_rat )
     => ( ( Z2 != zero_zero_rat )
       => ( ( minus_minus_rat @ ( divide_divide_rat @ X @ Y3 ) @ ( divide_divide_rat @ W2 @ Z2 ) )
          = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z2 ) @ ( times_times_rat @ W2 @ Y3 ) ) @ ( times_times_rat @ Y3 @ Z2 ) ) ) ) ) ).

% diff_frac_eq
thf(fact_2189_diff__divide__eq__iff,axiom,
    ! [Z2: rat,X: rat,Y3: rat] :
      ( ( Z2 != zero_zero_rat )
     => ( ( minus_minus_rat @ X @ ( divide_divide_rat @ Y3 @ Z2 ) )
        = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z2 ) @ Y3 ) @ Z2 ) ) ) ).

% diff_divide_eq_iff
thf(fact_2190_divide__diff__eq__iff,axiom,
    ! [Z2: rat,X: rat,Y3: rat] :
      ( ( Z2 != zero_zero_rat )
     => ( ( minus_minus_rat @ ( divide_divide_rat @ X @ Z2 ) @ Y3 )
        = ( divide_divide_rat @ ( minus_minus_rat @ X @ ( times_times_rat @ Y3 @ Z2 ) ) @ Z2 ) ) ) ).

% divide_diff_eq_iff
thf(fact_2191_dvd__div__eq__mult,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A != zero_zero_nat )
     => ( ( dvd_dvd_nat @ A @ B )
       => ( ( ( divide_divide_nat @ B @ A )
            = C )
          = ( B
            = ( times_times_nat @ C @ A ) ) ) ) ) ).

% dvd_div_eq_mult
thf(fact_2192_dvd__div__eq__mult,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A != zero_zero_int )
     => ( ( dvd_dvd_int @ A @ B )
       => ( ( ( divide_divide_int @ B @ A )
            = C )
          = ( B
            = ( times_times_int @ C @ A ) ) ) ) ) ).

% dvd_div_eq_mult
thf(fact_2193_dvd__div__eq__mult,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ A @ B )
       => ( ( ( divide6298287555418463151nteger @ B @ A )
            = C )
          = ( B
            = ( times_3573771949741848930nteger @ C @ A ) ) ) ) ) ).

% dvd_div_eq_mult
thf(fact_2194_dvd__div__eq__mult,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( ( A != zero_z2226904508553997617atural )
     => ( ( dvd_dvd_Code_natural @ A @ B )
       => ( ( ( divide5121882707175180666atural @ B @ A )
            = C )
          = ( B
            = ( times_2397367101498566445atural @ C @ A ) ) ) ) ) ).

% dvd_div_eq_mult
thf(fact_2195_div__dvd__iff__mult,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( B != zero_zero_nat )
     => ( ( dvd_dvd_nat @ B @ A )
       => ( ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B ) @ C )
          = ( dvd_dvd_nat @ A @ ( times_times_nat @ C @ B ) ) ) ) ) ).

% div_dvd_iff_mult
thf(fact_2196_div__dvd__iff__mult,axiom,
    ! [B: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( dvd_dvd_int @ B @ A )
       => ( ( dvd_dvd_int @ ( divide_divide_int @ A @ B ) @ C )
          = ( dvd_dvd_int @ A @ ( times_times_int @ C @ B ) ) ) ) ) ).

% div_dvd_iff_mult
thf(fact_2197_div__dvd__iff__mult,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ B @ A )
       => ( ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A @ B ) @ C )
          = ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ C @ B ) ) ) ) ) ).

% div_dvd_iff_mult
thf(fact_2198_div__dvd__iff__mult,axiom,
    ! [B: code_natural,A: code_natural,C: code_natural] :
      ( ( B != zero_z2226904508553997617atural )
     => ( ( dvd_dvd_Code_natural @ B @ A )
       => ( ( dvd_dvd_Code_natural @ ( divide5121882707175180666atural @ A @ B ) @ C )
          = ( dvd_dvd_Code_natural @ A @ ( times_2397367101498566445atural @ C @ B ) ) ) ) ) ).

% div_dvd_iff_mult
thf(fact_2199_dvd__div__iff__mult,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( C != zero_zero_nat )
     => ( ( dvd_dvd_nat @ C @ B )
       => ( ( dvd_dvd_nat @ A @ ( divide_divide_nat @ B @ C ) )
          = ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ B ) ) ) ) ).

% dvd_div_iff_mult
thf(fact_2200_dvd__div__iff__mult,axiom,
    ! [C: int,B: int,A: int] :
      ( ( C != zero_zero_int )
     => ( ( dvd_dvd_int @ C @ B )
       => ( ( dvd_dvd_int @ A @ ( divide_divide_int @ B @ C ) )
          = ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ B ) ) ) ) ).

% dvd_div_iff_mult
thf(fact_2201_dvd__div__iff__mult,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( ( C != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ C @ B )
       => ( ( dvd_dvd_Code_integer @ A @ ( divide6298287555418463151nteger @ B @ C ) )
          = ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ B ) ) ) ) ).

% dvd_div_iff_mult
thf(fact_2202_dvd__div__iff__mult,axiom,
    ! [C: code_natural,B: code_natural,A: code_natural] :
      ( ( C != zero_z2226904508553997617atural )
     => ( ( dvd_dvd_Code_natural @ C @ B )
       => ( ( dvd_dvd_Code_natural @ A @ ( divide5121882707175180666atural @ B @ C ) )
          = ( dvd_dvd_Code_natural @ ( times_2397367101498566445atural @ A @ C ) @ B ) ) ) ) ).

% dvd_div_iff_mult
thf(fact_2203_dvd__div__div__eq__mult,axiom,
    ! [A: nat,C: nat,B: nat,D: nat] :
      ( ( A != zero_zero_nat )
     => ( ( C != zero_zero_nat )
       => ( ( dvd_dvd_nat @ A @ B )
         => ( ( dvd_dvd_nat @ C @ D )
           => ( ( ( divide_divide_nat @ B @ A )
                = ( divide_divide_nat @ D @ C ) )
              = ( ( times_times_nat @ B @ C )
                = ( times_times_nat @ A @ D ) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
thf(fact_2204_dvd__div__div__eq__mult,axiom,
    ! [A: int,C: int,B: int,D: int] :
      ( ( A != zero_zero_int )
     => ( ( C != zero_zero_int )
       => ( ( dvd_dvd_int @ A @ B )
         => ( ( dvd_dvd_int @ C @ D )
           => ( ( ( divide_divide_int @ B @ A )
                = ( divide_divide_int @ D @ C ) )
              = ( ( times_times_int @ B @ C )
                = ( times_times_int @ A @ D ) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
thf(fact_2205_dvd__div__div__eq__mult,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer,D: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( C != zero_z3403309356797280102nteger )
       => ( ( dvd_dvd_Code_integer @ A @ B )
         => ( ( dvd_dvd_Code_integer @ C @ D )
           => ( ( ( divide6298287555418463151nteger @ B @ A )
                = ( divide6298287555418463151nteger @ D @ C ) )
              = ( ( times_3573771949741848930nteger @ B @ C )
                = ( times_3573771949741848930nteger @ A @ D ) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
thf(fact_2206_dvd__div__div__eq__mult,axiom,
    ! [A: code_natural,C: code_natural,B: code_natural,D: code_natural] :
      ( ( A != zero_z2226904508553997617atural )
     => ( ( C != zero_z2226904508553997617atural )
       => ( ( dvd_dvd_Code_natural @ A @ B )
         => ( ( dvd_dvd_Code_natural @ C @ D )
           => ( ( ( divide5121882707175180666atural @ B @ A )
                = ( divide5121882707175180666atural @ D @ C ) )
              = ( ( times_2397367101498566445atural @ B @ C )
                = ( times_2397367101498566445atural @ A @ D ) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
thf(fact_2207_unit__div__eq__0__iff,axiom,
    ! [B: nat,A: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( ( divide_divide_nat @ A @ B )
          = zero_zero_nat )
        = ( A = zero_zero_nat ) ) ) ).

% unit_div_eq_0_iff
thf(fact_2208_unit__div__eq__0__iff,axiom,
    ! [B: int,A: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( ( divide_divide_int @ A @ B )
          = zero_zero_int )
        = ( A = zero_zero_int ) ) ) ).

% unit_div_eq_0_iff
thf(fact_2209_unit__div__eq__0__iff,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
     => ( ( ( divide6298287555418463151nteger @ A @ B )
          = zero_z3403309356797280102nteger )
        = ( A = zero_z3403309356797280102nteger ) ) ) ).

% unit_div_eq_0_iff
thf(fact_2210_unit__div__eq__0__iff,axiom,
    ! [B: code_natural,A: code_natural] :
      ( ( dvd_dvd_Code_natural @ B @ one_one_Code_natural )
     => ( ( ( divide5121882707175180666atural @ A @ B )
          = zero_z2226904508553997617atural )
        = ( A = zero_z2226904508553997617atural ) ) ) ).

% unit_div_eq_0_iff
thf(fact_2211_unit__eq__div1,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( ( divide_divide_nat @ A @ B )
          = C )
        = ( A
          = ( times_times_nat @ C @ B ) ) ) ) ).

% unit_eq_div1
thf(fact_2212_unit__eq__div1,axiom,
    ! [B: int,A: int,C: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( ( divide_divide_int @ A @ B )
          = C )
        = ( A
          = ( times_times_int @ C @ B ) ) ) ) ).

% unit_eq_div1
thf(fact_2213_unit__eq__div1,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
     => ( ( ( divide6298287555418463151nteger @ A @ B )
          = C )
        = ( A
          = ( times_3573771949741848930nteger @ C @ B ) ) ) ) ).

% unit_eq_div1
thf(fact_2214_unit__eq__div1,axiom,
    ! [B: code_natural,A: code_natural,C: code_natural] :
      ( ( dvd_dvd_Code_natural @ B @ one_one_Code_natural )
     => ( ( ( divide5121882707175180666atural @ A @ B )
          = C )
        = ( A
          = ( times_2397367101498566445atural @ C @ B ) ) ) ) ).

% unit_eq_div1
thf(fact_2215_unit__eq__div2,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( A
          = ( divide_divide_nat @ C @ B ) )
        = ( ( times_times_nat @ A @ B )
          = C ) ) ) ).

% unit_eq_div2
thf(fact_2216_unit__eq__div2,axiom,
    ! [B: int,A: int,C: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( A
          = ( divide_divide_int @ C @ B ) )
        = ( ( times_times_int @ A @ B )
          = C ) ) ) ).

% unit_eq_div2
thf(fact_2217_unit__eq__div2,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
     => ( ( A
          = ( divide6298287555418463151nteger @ C @ B ) )
        = ( ( times_3573771949741848930nteger @ A @ B )
          = C ) ) ) ).

% unit_eq_div2
thf(fact_2218_unit__eq__div2,axiom,
    ! [B: code_natural,A: code_natural,C: code_natural] :
      ( ( dvd_dvd_Code_natural @ B @ one_one_Code_natural )
     => ( ( A
          = ( divide5121882707175180666atural @ C @ B ) )
        = ( ( times_2397367101498566445atural @ A @ B )
          = C ) ) ) ).

% unit_eq_div2
thf(fact_2219_div__mult__unit2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( dvd_dvd_nat @ C @ one_one_nat )
     => ( ( dvd_dvd_nat @ B @ A )
       => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
          = ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ) ).

% div_mult_unit2
thf(fact_2220_div__mult__unit2,axiom,
    ! [C: int,B: int,A: int] :
      ( ( dvd_dvd_int @ C @ one_one_int )
     => ( ( dvd_dvd_int @ B @ A )
       => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
          = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ) ).

% div_mult_unit2
thf(fact_2221_div__mult__unit2,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ B @ A )
       => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
          = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ) ).

% div_mult_unit2
thf(fact_2222_div__mult__unit2,axiom,
    ! [C: code_natural,B: code_natural,A: code_natural] :
      ( ( dvd_dvd_Code_natural @ C @ one_one_Code_natural )
     => ( ( dvd_dvd_Code_natural @ B @ A )
       => ( ( divide5121882707175180666atural @ A @ ( times_2397367101498566445atural @ B @ C ) )
          = ( divide5121882707175180666atural @ ( divide5121882707175180666atural @ A @ B ) @ C ) ) ) ) ).

% div_mult_unit2
thf(fact_2223_unit__div__commute,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ C )
        = ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ B ) ) ) ).

% unit_div_commute
thf(fact_2224_unit__div__commute,axiom,
    ! [B: int,A: int,C: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( times_times_int @ ( divide_divide_int @ A @ B ) @ C )
        = ( divide_divide_int @ ( times_times_int @ A @ C ) @ B ) ) ) ).

% unit_div_commute
thf(fact_2225_unit__div__commute,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
     => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C )
        = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ C ) @ B ) ) ) ).

% unit_div_commute
thf(fact_2226_unit__div__commute,axiom,
    ! [B: code_natural,A: code_natural,C: code_natural] :
      ( ( dvd_dvd_Code_natural @ B @ one_one_Code_natural )
     => ( ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ A @ B ) @ C )
        = ( divide5121882707175180666atural @ ( times_2397367101498566445atural @ A @ C ) @ B ) ) ) ).

% unit_div_commute
thf(fact_2227_unit__div__mult__swap,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( dvd_dvd_nat @ C @ one_one_nat )
     => ( ( times_times_nat @ A @ ( divide_divide_nat @ B @ C ) )
        = ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ C ) ) ) ).

% unit_div_mult_swap
thf(fact_2228_unit__div__mult__swap,axiom,
    ! [C: int,A: int,B: int] :
      ( ( dvd_dvd_int @ C @ one_one_int )
     => ( ( times_times_int @ A @ ( divide_divide_int @ B @ C ) )
        = ( divide_divide_int @ ( times_times_int @ A @ B ) @ C ) ) ) ).

% unit_div_mult_swap
thf(fact_2229_unit__div__mult__swap,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ one_one_Code_integer )
     => ( ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B @ C ) )
        = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ) ).

% unit_div_mult_swap
thf(fact_2230_unit__div__mult__swap,axiom,
    ! [C: code_natural,A: code_natural,B: code_natural] :
      ( ( dvd_dvd_Code_natural @ C @ one_one_Code_natural )
     => ( ( times_2397367101498566445atural @ A @ ( divide5121882707175180666atural @ B @ C ) )
        = ( divide5121882707175180666atural @ ( times_2397367101498566445atural @ A @ B ) @ C ) ) ) ).

% unit_div_mult_swap
thf(fact_2231_is__unit__div__mult2__eq,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( dvd_dvd_nat @ C @ one_one_nat )
       => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
          = ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ) ).

% is_unit_div_mult2_eq
thf(fact_2232_is__unit__div__mult2__eq,axiom,
    ! [B: int,C: int,A: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( dvd_dvd_int @ C @ one_one_int )
       => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
          = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ) ).

% is_unit_div_mult2_eq
thf(fact_2233_is__unit__div__mult2__eq,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ C @ one_one_Code_integer )
       => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
          = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ) ).

% is_unit_div_mult2_eq
thf(fact_2234_is__unit__div__mult2__eq,axiom,
    ! [B: code_natural,C: code_natural,A: code_natural] :
      ( ( dvd_dvd_Code_natural @ B @ one_one_Code_natural )
     => ( ( dvd_dvd_Code_natural @ C @ one_one_Code_natural )
       => ( ( divide5121882707175180666atural @ A @ ( times_2397367101498566445atural @ B @ C ) )
          = ( divide5121882707175180666atural @ ( divide5121882707175180666atural @ A @ B ) @ C ) ) ) ) ).

% is_unit_div_mult2_eq
thf(fact_2235_div__less__mono,axiom,
    ! [A3: nat,B5: nat,N: nat] :
      ( ( ord_less_nat @ A3 @ B5 )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( ( modulo_modulo_nat @ A3 @ N )
            = zero_zero_nat )
         => ( ( ( modulo_modulo_nat @ B5 @ N )
              = zero_zero_nat )
           => ( ord_less_nat @ ( divide_divide_nat @ A3 @ N ) @ ( divide_divide_nat @ B5 @ N ) ) ) ) ) ) ).

% div_less_mono
thf(fact_2236_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_2237_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_2238_of__nat__diff,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( semiri4939895301339042750nteger @ ( minus_minus_nat @ M @ N ) )
        = ( minus_8373710615458151222nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) ) ) ) ).

% of_nat_diff
thf(fact_2239_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_2240_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_2241_of__nat__diff,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( semiri3763490453095760265atural @ ( minus_minus_nat @ M @ N ) )
        = ( minus_7197305767214868737atural @ ( semiri3763490453095760265atural @ M ) @ ( semiri3763490453095760265atural @ N ) ) ) ) ).

% of_nat_diff
thf(fact_2242_division__segment__mod,axiom,
    ! [B: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( ~ ( dvd_dvd_nat @ B @ A )
       => ( ( euclid3398187327856392827nt_nat @ ( modulo_modulo_nat @ A @ B ) )
          = ( euclid3398187327856392827nt_nat @ B ) ) ) ) ).

% division_segment_mod
thf(fact_2243_division__segment__mod,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( ~ ( dvd_dvd_int @ B @ A )
       => ( ( euclid3395696857347342551nt_int @ ( modulo_modulo_int @ A @ B ) )
          = ( euclid3395696857347342551nt_int @ B ) ) ) ) ).

% division_segment_mod
thf(fact_2244_division__segment__mod,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( ~ ( dvd_dvd_Code_integer @ B @ A )
       => ( ( euclid6289375185220004616nteger @ ( modulo364778990260209775nteger @ A @ B ) )
          = ( euclid6289375185220004616nteger @ B ) ) ) ) ).

% division_segment_mod
thf(fact_2245_division__segment__mod,axiom,
    ! [B: code_natural,A: code_natural] :
      ( ( B != zero_z2226904508553997617atural )
     => ( ~ ( dvd_dvd_Code_natural @ B @ A )
       => ( ( euclid5112970336976722131atural @ ( modulo8411746178871703098atural @ A @ B ) )
          = ( euclid5112970336976722131atural @ B ) ) ) ) ).

% division_segment_mod
thf(fact_2246_mod__mult__eq,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( modulo_modulo_nat @ ( times_times_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B @ C ) ) @ C )
      = ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C ) ) ).

% mod_mult_eq
thf(fact_2247_mod__mult__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( ( modulo_modulo_int @ ( times_times_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B @ C ) ) @ C )
      = ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C ) ) ).

% mod_mult_eq
thf(fact_2248_mod__mult__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
      = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ).

% mod_mult_eq
thf(fact_2249_mod__mult__eq,axiom,
    ! [A: code_natural,C: code_natural,B: code_natural] :
      ( ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ ( modulo8411746178871703098atural @ A @ C ) @ ( modulo8411746178871703098atural @ B @ C ) ) @ C )
      = ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ A @ B ) @ C ) ) ).

% mod_mult_eq
thf(fact_2250_mod__mult__cong,axiom,
    ! [A: nat,C: nat,A4: nat,B: nat,B3: nat] :
      ( ( ( modulo_modulo_nat @ A @ C )
        = ( modulo_modulo_nat @ A4 @ C ) )
     => ( ( ( modulo_modulo_nat @ B @ C )
          = ( modulo_modulo_nat @ B3 @ C ) )
       => ( ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C )
          = ( modulo_modulo_nat @ ( times_times_nat @ A4 @ B3 ) @ C ) ) ) ) ).

% mod_mult_cong
thf(fact_2251_mod__mult__cong,axiom,
    ! [A: int,C: int,A4: int,B: int,B3: int] :
      ( ( ( modulo_modulo_int @ A @ C )
        = ( modulo_modulo_int @ A4 @ C ) )
     => ( ( ( modulo_modulo_int @ B @ C )
          = ( modulo_modulo_int @ B3 @ C ) )
       => ( ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C )
          = ( modulo_modulo_int @ ( times_times_int @ A4 @ B3 ) @ C ) ) ) ) ).

% mod_mult_cong
thf(fact_2252_mod__mult__cong,axiom,
    ! [A: code_integer,C: code_integer,A4: code_integer,B: code_integer,B3: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ C )
        = ( modulo364778990260209775nteger @ A4 @ C ) )
     => ( ( ( modulo364778990260209775nteger @ B @ C )
          = ( modulo364778990260209775nteger @ B3 @ C ) )
       => ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C )
          = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A4 @ B3 ) @ C ) ) ) ) ).

% mod_mult_cong
thf(fact_2253_mod__mult__cong,axiom,
    ! [A: code_natural,C: code_natural,A4: code_natural,B: code_natural,B3: code_natural] :
      ( ( ( modulo8411746178871703098atural @ A @ C )
        = ( modulo8411746178871703098atural @ A4 @ C ) )
     => ( ( ( modulo8411746178871703098atural @ B @ C )
          = ( modulo8411746178871703098atural @ B3 @ C ) )
       => ( ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ A @ B ) @ C )
          = ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ A4 @ B3 ) @ C ) ) ) ) ).

% mod_mult_cong
thf(fact_2254_mod__mult__mult2,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( modulo_modulo_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
      = ( times_times_nat @ ( modulo_modulo_nat @ A @ B ) @ C ) ) ).

% mod_mult_mult2
thf(fact_2255_mod__mult__mult2,axiom,
    ! [A: int,C: int,B: int] :
      ( ( modulo_modulo_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
      = ( times_times_int @ ( modulo_modulo_int @ A @ B ) @ C ) ) ).

% mod_mult_mult2
thf(fact_2256_mod__mult__mult2,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) )
      = ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ B ) @ C ) ) ).

% mod_mult_mult2
thf(fact_2257_mod__mult__mult2,axiom,
    ! [A: code_natural,C: code_natural,B: code_natural] :
      ( ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ A @ C ) @ ( times_2397367101498566445atural @ B @ C ) )
      = ( times_2397367101498566445atural @ ( modulo8411746178871703098atural @ A @ B ) @ C ) ) ).

% mod_mult_mult2
thf(fact_2258_mult__mod__right,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( times_times_nat @ C @ ( modulo_modulo_nat @ A @ B ) )
      = ( modulo_modulo_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ).

% mult_mod_right
thf(fact_2259_mult__mod__right,axiom,
    ! [C: int,A: int,B: int] :
      ( ( times_times_int @ C @ ( modulo_modulo_int @ A @ B ) )
      = ( modulo_modulo_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ).

% mult_mod_right
thf(fact_2260_mult__mod__right,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( times_3573771949741848930nteger @ C @ ( modulo364778990260209775nteger @ A @ B ) )
      = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) ) ) ).

% mult_mod_right
thf(fact_2261_mult__mod__right,axiom,
    ! [C: code_natural,A: code_natural,B: code_natural] :
      ( ( times_2397367101498566445atural @ C @ ( modulo8411746178871703098atural @ A @ B ) )
      = ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ C @ A ) @ ( times_2397367101498566445atural @ C @ B ) ) ) ).

% mult_mod_right
thf(fact_2262_mod__mult__left__eq,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( modulo_modulo_nat @ ( times_times_nat @ ( modulo_modulo_nat @ A @ C ) @ B ) @ C )
      = ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C ) ) ).

% mod_mult_left_eq
thf(fact_2263_mod__mult__left__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( ( modulo_modulo_int @ ( times_times_int @ ( modulo_modulo_int @ A @ C ) @ B ) @ C )
      = ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C ) ) ).

% mod_mult_left_eq
thf(fact_2264_mod__mult__left__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ C ) @ B ) @ C )
      = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ).

% mod_mult_left_eq
thf(fact_2265_mod__mult__left__eq,axiom,
    ! [A: code_natural,C: code_natural,B: code_natural] :
      ( ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ ( modulo8411746178871703098atural @ A @ C ) @ B ) @ C )
      = ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ A @ B ) @ C ) ) ).

% mod_mult_left_eq
thf(fact_2266_mod__mult__right__eq,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( modulo_modulo_nat @ ( times_times_nat @ A @ ( modulo_modulo_nat @ B @ C ) ) @ C )
      = ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C ) ) ).

% mod_mult_right_eq
thf(fact_2267_mod__mult__right__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( modulo_modulo_int @ ( times_times_int @ A @ ( modulo_modulo_int @ B @ C ) ) @ C )
      = ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C ) ) ).

% mod_mult_right_eq
thf(fact_2268_mod__mult__right__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
      = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ).

% mod_mult_right_eq
thf(fact_2269_mod__mult__right__eq,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ A @ ( modulo8411746178871703098atural @ B @ C ) ) @ C )
      = ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ A @ B ) @ C ) ) ).

% mod_mult_right_eq
thf(fact_2270_diff__mono,axiom,
    ! [A: code_integer,B: code_integer,D: code_integer,C: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ B )
     => ( ( ord_le3102999989581377725nteger @ D @ C )
       => ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ A @ C ) @ ( minus_8373710615458151222nteger @ B @ D ) ) ) ) ).

% diff_mono
thf(fact_2271_diff__mono,axiom,
    ! [A: rat,B: rat,D: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ D @ C )
       => ( ord_less_eq_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ D ) ) ) ) ).

% diff_mono
thf(fact_2272_diff__mono,axiom,
    ! [A: int,B: int,D: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ D @ C )
       => ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).

% diff_mono
thf(fact_2273_diff__left__mono,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( ord_le3102999989581377725nteger @ B @ A )
     => ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ C @ A ) @ ( minus_8373710615458151222nteger @ C @ B ) ) ) ).

% diff_left_mono
thf(fact_2274_diff__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ord_less_eq_rat @ ( minus_minus_rat @ C @ A ) @ ( minus_minus_rat @ C @ B ) ) ) ).

% diff_left_mono
thf(fact_2275_diff__left__mono,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ord_less_eq_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).

% diff_left_mono
thf(fact_2276_diff__right__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ B )
     => ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ A @ C ) @ ( minus_8373710615458151222nteger @ B @ C ) ) ) ).

% diff_right_mono
thf(fact_2277_diff__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ord_less_eq_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ C ) ) ) ).

% diff_right_mono
thf(fact_2278_diff__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).

% diff_right_mono
thf(fact_2279_diff__eq__diff__less__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
      ( ( ( minus_8373710615458151222nteger @ A @ B )
        = ( minus_8373710615458151222nteger @ C @ D ) )
     => ( ( ord_le3102999989581377725nteger @ A @ B )
        = ( ord_le3102999989581377725nteger @ C @ D ) ) ) ).

% diff_eq_diff_less_eq
thf(fact_2280_diff__eq__diff__less__eq,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ( minus_minus_rat @ A @ B )
        = ( minus_minus_rat @ C @ D ) )
     => ( ( ord_less_eq_rat @ A @ B )
        = ( ord_less_eq_rat @ C @ D ) ) ) ).

% diff_eq_diff_less_eq
thf(fact_2281_diff__eq__diff__less__eq,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ( minus_minus_int @ A @ B )
        = ( minus_minus_int @ C @ D ) )
     => ( ( ord_less_eq_int @ A @ B )
        = ( ord_less_eq_int @ C @ D ) ) ) ).

% diff_eq_diff_less_eq
thf(fact_2282_mod__minus__eq,axiom,
    ! [A: int,B: int] :
      ( ( modulo_modulo_int @ ( uminus_uminus_int @ ( modulo_modulo_int @ A @ B ) ) @ B )
      = ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B ) ) ).

% mod_minus_eq
thf(fact_2283_mod__minus__eq,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A @ B ) ) @ B )
      = ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).

% mod_minus_eq
thf(fact_2284_mod__minus__cong,axiom,
    ! [A: int,B: int,A4: int] :
      ( ( ( modulo_modulo_int @ A @ B )
        = ( modulo_modulo_int @ A4 @ B ) )
     => ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B )
        = ( modulo_modulo_int @ ( uminus_uminus_int @ A4 ) @ B ) ) ) ).

% mod_minus_cong
thf(fact_2285_mod__minus__cong,axiom,
    ! [A: code_integer,B: code_integer,A4: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ B )
        = ( modulo364778990260209775nteger @ A4 @ B ) )
     => ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
        = ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A4 ) @ B ) ) ) ).

% mod_minus_cong
thf(fact_2286_mod__minus__right,axiom,
    ! [A: int,B: int] :
      ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ B ) )
      = ( uminus_uminus_int @ ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).

% mod_minus_right
thf(fact_2287_mod__minus__right,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
      = ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ) ).

% mod_minus_right
thf(fact_2288_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y: code_integer,Z: code_integer] : ( Y = Z ) )
    = ( ^ [A2: code_integer,B2: code_integer] :
          ( ( minus_8373710615458151222nteger @ A2 @ B2 )
          = zero_z3403309356797280102nteger ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_2289_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y: rat,Z: rat] : ( Y = Z ) )
    = ( ^ [A2: rat,B2: rat] :
          ( ( minus_minus_rat @ A2 @ B2 )
          = zero_zero_rat ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_2290_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y: int,Z: int] : ( Y = Z ) )
    = ( ^ [A2: int,B2: int] :
          ( ( minus_minus_int @ A2 @ B2 )
          = zero_zero_int ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_2291_diff__strict__mono,axiom,
    ! [A: code_integer,B: code_integer,D: code_integer,C: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ B )
     => ( ( ord_le6747313008572928689nteger @ D @ C )
       => ( ord_le6747313008572928689nteger @ ( minus_8373710615458151222nteger @ A @ C ) @ ( minus_8373710615458151222nteger @ B @ D ) ) ) ) ).

% diff_strict_mono
thf(fact_2292_diff__strict__mono,axiom,
    ! [A: rat,B: rat,D: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ D @ C )
       => ( ord_less_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ D ) ) ) ) ).

% diff_strict_mono
thf(fact_2293_diff__strict__mono,axiom,
    ! [A: int,B: int,D: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ D @ C )
       => ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).

% diff_strict_mono
thf(fact_2294_diff__eq__diff__less,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
      ( ( ( minus_8373710615458151222nteger @ A @ B )
        = ( minus_8373710615458151222nteger @ C @ D ) )
     => ( ( ord_le6747313008572928689nteger @ A @ B )
        = ( ord_le6747313008572928689nteger @ C @ D ) ) ) ).

% diff_eq_diff_less
thf(fact_2295_diff__eq__diff__less,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ( minus_minus_rat @ A @ B )
        = ( minus_minus_rat @ C @ D ) )
     => ( ( ord_less_rat @ A @ B )
        = ( ord_less_rat @ C @ D ) ) ) ).

% diff_eq_diff_less
thf(fact_2296_diff__eq__diff__less,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ( minus_minus_int @ A @ B )
        = ( minus_minus_int @ C @ D ) )
     => ( ( ord_less_int @ A @ B )
        = ( ord_less_int @ C @ D ) ) ) ).

% diff_eq_diff_less
thf(fact_2297_diff__strict__left__mono,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( ord_le6747313008572928689nteger @ B @ A )
     => ( ord_le6747313008572928689nteger @ ( minus_8373710615458151222nteger @ C @ A ) @ ( minus_8373710615458151222nteger @ C @ B ) ) ) ).

% diff_strict_left_mono
thf(fact_2298_diff__strict__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ord_less_rat @ ( minus_minus_rat @ C @ A ) @ ( minus_minus_rat @ C @ B ) ) ) ).

% diff_strict_left_mono
thf(fact_2299_diff__strict__left__mono,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_int @ B @ A )
     => ( ord_less_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).

% diff_strict_left_mono
thf(fact_2300_diff__strict__right__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ B )
     => ( ord_le6747313008572928689nteger @ ( minus_8373710615458151222nteger @ A @ C ) @ ( minus_8373710615458151222nteger @ B @ C ) ) ) ).

% diff_strict_right_mono
thf(fact_2301_diff__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ord_less_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ C ) ) ) ).

% diff_strict_right_mono
thf(fact_2302_diff__strict__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).

% diff_strict_right_mono
thf(fact_2303_right__diff__distrib_H,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( times_3573771949741848930nteger @ A @ ( minus_8373710615458151222nteger @ B @ C ) )
      = ( minus_8373710615458151222nteger @ ( times_3573771949741848930nteger @ A @ B ) @ ( times_3573771949741848930nteger @ A @ C ) ) ) ).

% right_diff_distrib'
thf(fact_2304_right__diff__distrib_H,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ A @ ( minus_minus_rat @ B @ C ) )
      = ( minus_minus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).

% right_diff_distrib'
thf(fact_2305_right__diff__distrib_H,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( times_times_nat @ A @ ( minus_minus_nat @ B @ C ) )
      = ( minus_minus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).

% right_diff_distrib'
thf(fact_2306_right__diff__distrib_H,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
      = ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).

% right_diff_distrib'
thf(fact_2307_left__diff__distrib_H,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] :
      ( ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ B @ C ) @ A )
      = ( minus_8373710615458151222nteger @ ( times_3573771949741848930nteger @ B @ A ) @ ( times_3573771949741848930nteger @ C @ A ) ) ) ).

% left_diff_distrib'
thf(fact_2308_left__diff__distrib_H,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( times_times_rat @ ( minus_minus_rat @ B @ C ) @ A )
      = ( minus_minus_rat @ ( times_times_rat @ B @ A ) @ ( times_times_rat @ C @ A ) ) ) ).

% left_diff_distrib'
thf(fact_2309_left__diff__distrib_H,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( times_times_nat @ ( minus_minus_nat @ B @ C ) @ A )
      = ( minus_minus_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C @ A ) ) ) ).

% left_diff_distrib'
thf(fact_2310_left__diff__distrib_H,axiom,
    ! [B: int,C: int,A: int] :
      ( ( times_times_int @ ( minus_minus_int @ B @ C ) @ A )
      = ( minus_minus_int @ ( times_times_int @ B @ A ) @ ( times_times_int @ C @ A ) ) ) ).

% left_diff_distrib'
thf(fact_2311_right__diff__distrib,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( times_3573771949741848930nteger @ A @ ( minus_8373710615458151222nteger @ B @ C ) )
      = ( minus_8373710615458151222nteger @ ( times_3573771949741848930nteger @ A @ B ) @ ( times_3573771949741848930nteger @ A @ C ) ) ) ).

% right_diff_distrib
thf(fact_2312_right__diff__distrib,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ A @ ( minus_minus_rat @ B @ C ) )
      = ( minus_minus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).

% right_diff_distrib
thf(fact_2313_right__diff__distrib,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
      = ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).

% right_diff_distrib
thf(fact_2314_left__diff__distrib,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C )
      = ( minus_8373710615458151222nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) ) ) ).

% left_diff_distrib
thf(fact_2315_left__diff__distrib,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ C )
      = ( minus_minus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).

% left_diff_distrib
thf(fact_2316_left__diff__distrib,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ ( minus_minus_int @ A @ B ) @ C )
      = ( minus_minus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).

% left_diff_distrib
thf(fact_2317_inf__period_I1_J,axiom,
    ! [P: code_integer > $o,D3: code_integer,Q: code_integer > $o] :
      ( ! [X3: code_integer,K2: code_integer] :
          ( ( P @ X3 )
          = ( P @ ( minus_8373710615458151222nteger @ X3 @ ( times_3573771949741848930nteger @ K2 @ D3 ) ) ) )
     => ( ! [X3: code_integer,K2: code_integer] :
            ( ( Q @ X3 )
            = ( Q @ ( minus_8373710615458151222nteger @ X3 @ ( times_3573771949741848930nteger @ K2 @ D3 ) ) ) )
       => ! [X4: code_integer,K4: code_integer] :
            ( ( ( P @ X4 )
              & ( Q @ X4 ) )
            = ( ( P @ ( minus_8373710615458151222nteger @ X4 @ ( times_3573771949741848930nteger @ K4 @ D3 ) ) )
              & ( Q @ ( minus_8373710615458151222nteger @ X4 @ ( times_3573771949741848930nteger @ K4 @ D3 ) ) ) ) ) ) ) ).

% inf_period(1)
thf(fact_2318_inf__period_I1_J,axiom,
    ! [P: rat > $o,D3: rat,Q: rat > $o] :
      ( ! [X3: rat,K2: rat] :
          ( ( P @ X3 )
          = ( P @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K2 @ D3 ) ) ) )
     => ( ! [X3: rat,K2: rat] :
            ( ( Q @ X3 )
            = ( Q @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K2 @ D3 ) ) ) )
       => ! [X4: rat,K4: rat] :
            ( ( ( P @ X4 )
              & ( Q @ X4 ) )
            = ( ( P @ ( minus_minus_rat @ X4 @ ( times_times_rat @ K4 @ D3 ) ) )
              & ( Q @ ( minus_minus_rat @ X4 @ ( times_times_rat @ K4 @ D3 ) ) ) ) ) ) ) ).

% inf_period(1)
thf(fact_2319_inf__period_I1_J,axiom,
    ! [P: int > $o,D3: int,Q: int > $o] :
      ( ! [X3: int,K2: int] :
          ( ( P @ X3 )
          = ( P @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D3 ) ) ) )
     => ( ! [X3: int,K2: int] :
            ( ( Q @ X3 )
            = ( Q @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D3 ) ) ) )
       => ! [X4: int,K4: int] :
            ( ( ( P @ X4 )
              & ( Q @ X4 ) )
            = ( ( P @ ( minus_minus_int @ X4 @ ( times_times_int @ K4 @ D3 ) ) )
              & ( Q @ ( minus_minus_int @ X4 @ ( times_times_int @ K4 @ D3 ) ) ) ) ) ) ) ).

% inf_period(1)
thf(fact_2320_inf__period_I2_J,axiom,
    ! [P: code_integer > $o,D3: code_integer,Q: code_integer > $o] :
      ( ! [X3: code_integer,K2: code_integer] :
          ( ( P @ X3 )
          = ( P @ ( minus_8373710615458151222nteger @ X3 @ ( times_3573771949741848930nteger @ K2 @ D3 ) ) ) )
     => ( ! [X3: code_integer,K2: code_integer] :
            ( ( Q @ X3 )
            = ( Q @ ( minus_8373710615458151222nteger @ X3 @ ( times_3573771949741848930nteger @ K2 @ D3 ) ) ) )
       => ! [X4: code_integer,K4: code_integer] :
            ( ( ( P @ X4 )
              | ( Q @ X4 ) )
            = ( ( P @ ( minus_8373710615458151222nteger @ X4 @ ( times_3573771949741848930nteger @ K4 @ D3 ) ) )
              | ( Q @ ( minus_8373710615458151222nteger @ X4 @ ( times_3573771949741848930nteger @ K4 @ D3 ) ) ) ) ) ) ) ).

% inf_period(2)
thf(fact_2321_inf__period_I2_J,axiom,
    ! [P: rat > $o,D3: rat,Q: rat > $o] :
      ( ! [X3: rat,K2: rat] :
          ( ( P @ X3 )
          = ( P @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K2 @ D3 ) ) ) )
     => ( ! [X3: rat,K2: rat] :
            ( ( Q @ X3 )
            = ( Q @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K2 @ D3 ) ) ) )
       => ! [X4: rat,K4: rat] :
            ( ( ( P @ X4 )
              | ( Q @ X4 ) )
            = ( ( P @ ( minus_minus_rat @ X4 @ ( times_times_rat @ K4 @ D3 ) ) )
              | ( Q @ ( minus_minus_rat @ X4 @ ( times_times_rat @ K4 @ D3 ) ) ) ) ) ) ) ).

% inf_period(2)
thf(fact_2322_inf__period_I2_J,axiom,
    ! [P: int > $o,D3: int,Q: int > $o] :
      ( ! [X3: int,K2: int] :
          ( ( P @ X3 )
          = ( P @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D3 ) ) ) )
     => ( ! [X3: int,K2: int] :
            ( ( Q @ X3 )
            = ( Q @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D3 ) ) ) )
       => ! [X4: int,K4: int] :
            ( ( ( P @ X4 )
              | ( Q @ X4 ) )
            = ( ( P @ ( minus_minus_int @ X4 @ ( times_times_int @ K4 @ D3 ) ) )
              | ( Q @ ( minus_minus_int @ X4 @ ( times_times_int @ K4 @ D3 ) ) ) ) ) ) ) ).

% inf_period(2)
thf(fact_2323_dvd__field__iff,axiom,
    ( dvd_dvd_rat
    = ( ^ [A2: rat,B2: rat] :
          ( ( A2 = zero_zero_rat )
         => ( B2 = zero_zero_rat ) ) ) ) ).

% dvd_field_iff
thf(fact_2324_dvd__0__left,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ zero_zero_nat @ A )
     => ( A = zero_zero_nat ) ) ).

% dvd_0_left
thf(fact_2325_dvd__0__left,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ zero_zero_int @ A )
     => ( A = zero_zero_int ) ) ).

% dvd_0_left
thf(fact_2326_dvd__0__left,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ A )
     => ( A = zero_z3403309356797280102nteger ) ) ).

% dvd_0_left
thf(fact_2327_dvd__0__left,axiom,
    ! [A: rat] :
      ( ( dvd_dvd_rat @ zero_zero_rat @ A )
     => ( A = zero_zero_rat ) ) ).

% dvd_0_left
thf(fact_2328_dvd__0__left,axiom,
    ! [A: code_natural] :
      ( ( dvd_dvd_Code_natural @ zero_z2226904508553997617atural @ A )
     => ( A = zero_z2226904508553997617atural ) ) ).

% dvd_0_left
thf(fact_2329_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_2330_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_2331_minus__diff__commute,axiom,
    ! [B: int,A: int] :
      ( ( minus_minus_int @ ( uminus_uminus_int @ B ) @ A )
      = ( minus_minus_int @ ( uminus_uminus_int @ A ) @ B ) ) ).

% minus_diff_commute
thf(fact_2332_minus__diff__commute,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ B ) @ A )
      = ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).

% minus_diff_commute
thf(fact_2333_minus__diff__commute,axiom,
    ! [B: rat,A: rat] :
      ( ( minus_minus_rat @ ( uminus_uminus_rat @ B ) @ A )
      = ( minus_minus_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ).

% minus_diff_commute
thf(fact_2334_dvd__triv__right,axiom,
    ! [A: code_integer,B: code_integer] : ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B @ A ) ) ).

% dvd_triv_right
thf(fact_2335_dvd__triv__right,axiom,
    ! [A: rat,B: rat] : ( dvd_dvd_rat @ A @ ( times_times_rat @ B @ A ) ) ).

% dvd_triv_right
thf(fact_2336_dvd__triv__right,axiom,
    ! [A: nat,B: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ A ) ) ).

% dvd_triv_right
thf(fact_2337_dvd__triv__right,axiom,
    ! [A: int,B: int] : ( dvd_dvd_int @ A @ ( times_times_int @ B @ A ) ) ).

% dvd_triv_right
thf(fact_2338_dvd__mult__right,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ C )
     => ( dvd_dvd_Code_integer @ B @ C ) ) ).

% dvd_mult_right
thf(fact_2339_dvd__mult__right,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( dvd_dvd_rat @ ( times_times_rat @ A @ B ) @ C )
     => ( dvd_dvd_rat @ B @ C ) ) ).

% dvd_mult_right
thf(fact_2340_dvd__mult__right,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
     => ( dvd_dvd_nat @ B @ C ) ) ).

% dvd_mult_right
thf(fact_2341_dvd__mult__right,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
     => ( dvd_dvd_int @ B @ C ) ) ).

% dvd_mult_right
thf(fact_2342_mult__dvd__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ B )
     => ( ( dvd_dvd_Code_integer @ C @ D )
       => ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ D ) ) ) ) ).

% mult_dvd_mono
thf(fact_2343_mult__dvd__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( dvd_dvd_rat @ A @ B )
     => ( ( dvd_dvd_rat @ C @ D )
       => ( dvd_dvd_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ).

% mult_dvd_mono
thf(fact_2344_mult__dvd__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( ( dvd_dvd_nat @ C @ D )
       => ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ).

% mult_dvd_mono
thf(fact_2345_mult__dvd__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( dvd_dvd_int @ A @ B )
     => ( ( dvd_dvd_int @ C @ D )
       => ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ).

% mult_dvd_mono
thf(fact_2346_dvd__triv__left,axiom,
    ! [A: code_integer,B: code_integer] : ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ A @ B ) ) ).

% dvd_triv_left
thf(fact_2347_dvd__triv__left,axiom,
    ! [A: rat,B: rat] : ( dvd_dvd_rat @ A @ ( times_times_rat @ A @ B ) ) ).

% dvd_triv_left
thf(fact_2348_dvd__triv__left,axiom,
    ! [A: nat,B: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ A @ B ) ) ).

% dvd_triv_left
thf(fact_2349_dvd__triv__left,axiom,
    ! [A: int,B: int] : ( dvd_dvd_int @ A @ ( times_times_int @ A @ B ) ) ).

% dvd_triv_left
thf(fact_2350_dvd__mult__left,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ C )
     => ( dvd_dvd_Code_integer @ A @ C ) ) ).

% dvd_mult_left
thf(fact_2351_dvd__mult__left,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( dvd_dvd_rat @ ( times_times_rat @ A @ B ) @ C )
     => ( dvd_dvd_rat @ A @ C ) ) ).

% dvd_mult_left
thf(fact_2352_dvd__mult__left,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
     => ( dvd_dvd_nat @ A @ C ) ) ).

% dvd_mult_left
thf(fact_2353_dvd__mult__left,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
     => ( dvd_dvd_int @ A @ C ) ) ).

% dvd_mult_left
thf(fact_2354_dvd__mult2,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ B )
     => ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B @ C ) ) ) ).

% dvd_mult2
thf(fact_2355_dvd__mult2,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( dvd_dvd_rat @ A @ B )
     => ( dvd_dvd_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).

% dvd_mult2
thf(fact_2356_dvd__mult2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).

% dvd_mult2
thf(fact_2357_dvd__mult2,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ A @ B )
     => ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) ) ) ).

% dvd_mult2
thf(fact_2358_dvd__mult,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ C )
     => ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B @ C ) ) ) ).

% dvd_mult
thf(fact_2359_dvd__mult,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( dvd_dvd_rat @ A @ C )
     => ( dvd_dvd_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).

% dvd_mult
thf(fact_2360_dvd__mult,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ C )
     => ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).

% dvd_mult
thf(fact_2361_dvd__mult,axiom,
    ! [A: int,C: int,B: int] :
      ( ( dvd_dvd_int @ A @ C )
     => ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) ) ) ).

% dvd_mult
thf(fact_2362_dvd__def,axiom,
    ( dvd_dvd_Code_integer
    = ( ^ [B2: code_integer,A2: code_integer] :
        ? [K3: code_integer] :
          ( A2
          = ( times_3573771949741848930nteger @ B2 @ K3 ) ) ) ) ).

% dvd_def
thf(fact_2363_dvd__def,axiom,
    ( dvd_dvd_rat
    = ( ^ [B2: rat,A2: rat] :
        ? [K3: rat] :
          ( A2
          = ( times_times_rat @ B2 @ K3 ) ) ) ) ).

% dvd_def
thf(fact_2364_dvd__def,axiom,
    ( dvd_dvd_nat
    = ( ^ [B2: nat,A2: nat] :
        ? [K3: nat] :
          ( A2
          = ( times_times_nat @ B2 @ K3 ) ) ) ) ).

% dvd_def
thf(fact_2365_dvd__def,axiom,
    ( dvd_dvd_int
    = ( ^ [B2: int,A2: int] :
        ? [K3: int] :
          ( A2
          = ( times_times_int @ B2 @ K3 ) ) ) ) ).

% dvd_def
thf(fact_2366_dvdI,axiom,
    ! [A: code_integer,B: code_integer,K: code_integer] :
      ( ( A
        = ( times_3573771949741848930nteger @ B @ K ) )
     => ( dvd_dvd_Code_integer @ B @ A ) ) ).

% dvdI
thf(fact_2367_dvdI,axiom,
    ! [A: rat,B: rat,K: rat] :
      ( ( A
        = ( times_times_rat @ B @ K ) )
     => ( dvd_dvd_rat @ B @ A ) ) ).

% dvdI
thf(fact_2368_dvdI,axiom,
    ! [A: nat,B: nat,K: nat] :
      ( ( A
        = ( times_times_nat @ B @ K ) )
     => ( dvd_dvd_nat @ B @ A ) ) ).

% dvdI
thf(fact_2369_dvdI,axiom,
    ! [A: int,B: int,K: int] :
      ( ( A
        = ( times_times_int @ B @ K ) )
     => ( dvd_dvd_int @ B @ A ) ) ).

% dvdI
thf(fact_2370_dvdE,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ A )
     => ~ ! [K2: code_integer] :
            ( A
           != ( times_3573771949741848930nteger @ B @ K2 ) ) ) ).

% dvdE
thf(fact_2371_dvdE,axiom,
    ! [B: rat,A: rat] :
      ( ( dvd_dvd_rat @ B @ A )
     => ~ ! [K2: rat] :
            ( A
           != ( times_times_rat @ B @ K2 ) ) ) ).

% dvdE
thf(fact_2372_dvdE,axiom,
    ! [B: nat,A: nat] :
      ( ( dvd_dvd_nat @ B @ A )
     => ~ ! [K2: nat] :
            ( A
           != ( times_times_nat @ B @ K2 ) ) ) ).

% dvdE
thf(fact_2373_dvdE,axiom,
    ! [B: int,A: int] :
      ( ( dvd_dvd_int @ B @ A )
     => ~ ! [K2: int] :
            ( A
           != ( times_times_int @ B @ K2 ) ) ) ).

% dvdE
thf(fact_2374_one__dvd,axiom,
    ! [A: nat] : ( dvd_dvd_nat @ one_one_nat @ A ) ).

% one_dvd
thf(fact_2375_one__dvd,axiom,
    ! [A: int] : ( dvd_dvd_int @ one_one_int @ A ) ).

% one_dvd
thf(fact_2376_one__dvd,axiom,
    ! [A: code_integer] : ( dvd_dvd_Code_integer @ one_one_Code_integer @ A ) ).

% one_dvd
thf(fact_2377_one__dvd,axiom,
    ! [A: rat] : ( dvd_dvd_rat @ one_one_rat @ A ) ).

% one_dvd
thf(fact_2378_one__dvd,axiom,
    ! [A: code_natural] : ( dvd_dvd_Code_natural @ one_one_Code_natural @ A ) ).

% one_dvd
thf(fact_2379_unit__imp__dvd,axiom,
    ! [B: nat,A: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( dvd_dvd_nat @ B @ A ) ) ).

% unit_imp_dvd
thf(fact_2380_unit__imp__dvd,axiom,
    ! [B: int,A: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( dvd_dvd_int @ B @ A ) ) ).

% unit_imp_dvd
thf(fact_2381_unit__imp__dvd,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
     => ( dvd_dvd_Code_integer @ B @ A ) ) ).

% unit_imp_dvd
thf(fact_2382_unit__imp__dvd,axiom,
    ! [B: code_natural,A: code_natural] :
      ( ( dvd_dvd_Code_natural @ B @ one_one_Code_natural )
     => ( dvd_dvd_Code_natural @ B @ A ) ) ).

% unit_imp_dvd
thf(fact_2383_dvd__unit__imp__unit,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( ( dvd_dvd_nat @ B @ one_one_nat )
       => ( dvd_dvd_nat @ A @ one_one_nat ) ) ) ).

% dvd_unit_imp_unit
thf(fact_2384_dvd__unit__imp__unit,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ B )
     => ( ( dvd_dvd_int @ B @ one_one_int )
       => ( dvd_dvd_int @ A @ one_one_int ) ) ) ).

% dvd_unit_imp_unit
thf(fact_2385_dvd__unit__imp__unit,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ B )
     => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
       => ( dvd_dvd_Code_integer @ A @ one_one_Code_integer ) ) ) ).

% dvd_unit_imp_unit
thf(fact_2386_dvd__unit__imp__unit,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( dvd_dvd_Code_natural @ A @ B )
     => ( ( dvd_dvd_Code_natural @ B @ one_one_Code_natural )
       => ( dvd_dvd_Code_natural @ A @ one_one_Code_natural ) ) ) ).

% dvd_unit_imp_unit
thf(fact_2387_divide__divide__eq__left_H,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( divide_divide_rat @ ( divide_divide_rat @ A @ B ) @ C )
      = ( divide_divide_rat @ A @ ( times_times_rat @ C @ B ) ) ) ).

% divide_divide_eq_left'
thf(fact_2388_divide__divide__times__eq,axiom,
    ! [X: rat,Y3: rat,Z2: rat,W2: rat] :
      ( ( divide_divide_rat @ ( divide_divide_rat @ X @ Y3 ) @ ( divide_divide_rat @ Z2 @ W2 ) )
      = ( divide_divide_rat @ ( times_times_rat @ X @ W2 ) @ ( times_times_rat @ Y3 @ Z2 ) ) ) ).

% divide_divide_times_eq
thf(fact_2389_times__divide__times__eq,axiom,
    ! [X: rat,Y3: rat,Z2: rat,W2: rat] :
      ( ( times_times_rat @ ( divide_divide_rat @ X @ Y3 ) @ ( divide_divide_rat @ Z2 @ W2 ) )
      = ( divide_divide_rat @ ( times_times_rat @ X @ Z2 ) @ ( times_times_rat @ Y3 @ W2 ) ) ) ).

% times_divide_times_eq
thf(fact_2390_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_2391_minus__divide__left,axiom,
    ! [A: rat,B: rat] :
      ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) )
      = ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ).

% minus_divide_left
thf(fact_2392_minus__divide__divide,axiom,
    ! [A: rat,B: rat] :
      ( ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) )
      = ( divide_divide_rat @ A @ B ) ) ).

% minus_divide_divide
thf(fact_2393_minus__divide__right,axiom,
    ! [A: rat,B: rat] :
      ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) )
      = ( divide_divide_rat @ A @ ( uminus_uminus_rat @ B ) ) ) ).

% minus_divide_right
thf(fact_2394_div__minus__right,axiom,
    ! [A: int,B: int] :
      ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
      = ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B ) ) ).

% div_minus_right
thf(fact_2395_div__minus__right,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( divide6298287555418463151nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
      = ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).

% div_minus_right
thf(fact_2396_zero__induct__lemma,axiom,
    ! [P: nat > $o,K: nat,I2: nat] :
      ( ( P @ K )
     => ( ! [N4: nat] :
            ( ( P @ ( suc @ N4 ) )
           => ( P @ N4 ) )
       => ( P @ ( minus_minus_nat @ K @ I2 ) ) ) ) ).

% zero_induct_lemma
thf(fact_2397_minus__nat_Odiff__0,axiom,
    ! [M: nat] :
      ( ( minus_minus_nat @ M @ zero_zero_nat )
      = M ) ).

% minus_nat.diff_0
thf(fact_2398_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_2399_sgn__eq__0__iff,axiom,
    ! [A: int] :
      ( ( ( sgn_sgn_int @ A )
        = zero_zero_int )
      = ( A = zero_zero_int ) ) ).

% sgn_eq_0_iff
thf(fact_2400_sgn__eq__0__iff,axiom,
    ! [A: code_integer] :
      ( ( ( sgn_sgn_Code_integer @ A )
        = zero_z3403309356797280102nteger )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% sgn_eq_0_iff
thf(fact_2401_sgn__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( ( sgn_sgn_rat @ A )
        = zero_zero_rat )
      = ( A = zero_zero_rat ) ) ).

% sgn_eq_0_iff
thf(fact_2402_sgn__0__0,axiom,
    ! [A: int] :
      ( ( ( sgn_sgn_int @ A )
        = zero_zero_int )
      = ( A = zero_zero_int ) ) ).

% sgn_0_0
thf(fact_2403_sgn__0__0,axiom,
    ! [A: code_integer] :
      ( ( ( sgn_sgn_Code_integer @ A )
        = zero_z3403309356797280102nteger )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% sgn_0_0
thf(fact_2404_sgn__0__0,axiom,
    ! [A: rat] :
      ( ( ( sgn_sgn_rat @ A )
        = zero_zero_rat )
      = ( A = zero_zero_rat ) ) ).

% sgn_0_0
thf(fact_2405_diff__less__mono2,axiom,
    ! [M: nat,N: nat,L: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ( ord_less_nat @ M @ L )
       => ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).

% diff_less_mono2
thf(fact_2406_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_2407_div__le__dividend,axiom,
    ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ M ) ).

% div_le_dividend
thf(fact_2408_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_2409_diff__le__mono2,axiom,
    ! [M: nat,N: nat,L: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).

% diff_le_mono2
thf(fact_2410_le__diff__iff_H,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ C )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
          = ( ord_less_eq_nat @ B @ A ) ) ) ) ).

% le_diff_iff'
thf(fact_2411_diff__le__self,axiom,
    ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).

% diff_le_self
thf(fact_2412_diff__le__mono,axiom,
    ! [M: nat,N: nat,L: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).

% diff_le_mono
thf(fact_2413_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_2414_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_2415_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_2416_sgn__mult,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( sgn_sgn_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) )
      = ( times_3573771949741848930nteger @ ( sgn_sgn_Code_integer @ A ) @ ( sgn_sgn_Code_integer @ B ) ) ) ).

% sgn_mult
thf(fact_2417_sgn__mult,axiom,
    ! [A: rat,B: rat] :
      ( ( sgn_sgn_rat @ ( times_times_rat @ A @ B ) )
      = ( times_times_rat @ ( sgn_sgn_rat @ A ) @ ( sgn_sgn_rat @ B ) ) ) ).

% sgn_mult
thf(fact_2418_sgn__mult,axiom,
    ! [A: int,B: int] :
      ( ( sgn_sgn_int @ ( times_times_int @ A @ B ) )
      = ( times_times_int @ ( sgn_sgn_int @ A ) @ ( sgn_sgn_int @ B ) ) ) ).

% sgn_mult
thf(fact_2419_div__mult2__eq,axiom,
    ! [M: nat,N: nat,Q4: nat] :
      ( ( divide_divide_nat @ M @ ( times_times_nat @ N @ Q4 ) )
      = ( divide_divide_nat @ ( divide_divide_nat @ M @ N ) @ Q4 ) ) ).

% div_mult2_eq
thf(fact_2420_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_2421_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_2422_uminus__dvd__conv_I2_J,axiom,
    ( dvd_dvd_int
    = ( ^ [D4: int,T2: int] : ( dvd_dvd_int @ D4 @ ( uminus_uminus_int @ T2 ) ) ) ) ).

% uminus_dvd_conv(2)
thf(fact_2423_uminus__dvd__conv_I1_J,axiom,
    ( dvd_dvd_int
    = ( ^ [D4: int] : ( dvd_dvd_int @ ( uminus_uminus_int @ D4 ) ) ) ) ).

% uminus_dvd_conv(1)
thf(fact_2424_nat__dvd__iff,axiom,
    ! [Z2: int,M: nat] :
      ( ( dvd_dvd_nat @ ( nat2 @ Z2 ) @ M )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
         => ( dvd_dvd_int @ Z2 @ ( semiri1314217659103216013at_int @ M ) ) )
        & ( ~ ( ord_less_eq_int @ zero_zero_int @ Z2 )
         => ( M = zero_zero_nat ) ) ) ) ).

% nat_dvd_iff
thf(fact_2425_frac__le__eq,axiom,
    ! [Y3: rat,Z2: rat,X: rat,W2: rat] :
      ( ( Y3 != zero_zero_rat )
     => ( ( Z2 != zero_zero_rat )
       => ( ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y3 ) @ ( divide_divide_rat @ W2 @ Z2 ) )
          = ( ord_less_eq_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z2 ) @ ( times_times_rat @ W2 @ Y3 ) ) @ ( times_times_rat @ Y3 @ Z2 ) ) @ zero_zero_rat ) ) ) ) ).

% frac_le_eq
thf(fact_2426_frac__less__eq,axiom,
    ! [Y3: rat,Z2: rat,X: rat,W2: rat] :
      ( ( Y3 != zero_zero_rat )
     => ( ( Z2 != zero_zero_rat )
       => ( ( ord_less_rat @ ( divide_divide_rat @ X @ Y3 ) @ ( divide_divide_rat @ W2 @ Z2 ) )
          = ( ord_less_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z2 ) @ ( times_times_rat @ W2 @ Y3 ) ) @ ( times_times_rat @ Y3 @ Z2 ) ) @ zero_zero_rat ) ) ) ) ).

% frac_less_eq
thf(fact_2427_add__divide__eq__if__simps_I6_J,axiom,
    ! [Z2: rat,A: rat,B: rat] :
      ( ( ( Z2 = zero_zero_rat )
       => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z2 ) ) @ B )
          = ( uminus_uminus_rat @ B ) ) )
      & ( ( Z2 != zero_zero_rat )
       => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z2 ) ) @ B )
          = ( divide_divide_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_rat @ B @ Z2 ) ) @ Z2 ) ) ) ) ).

% add_divide_eq_if_simps(6)
thf(fact_2428_add__divide__eq__if__simps_I5_J,axiom,
    ! [Z2: rat,A: rat,B: rat] :
      ( ( ( Z2 = zero_zero_rat )
       => ( ( minus_minus_rat @ ( divide_divide_rat @ A @ Z2 ) @ B )
          = ( uminus_uminus_rat @ B ) ) )
      & ( ( Z2 != zero_zero_rat )
       => ( ( minus_minus_rat @ ( divide_divide_rat @ A @ Z2 ) @ B )
          = ( divide_divide_rat @ ( minus_minus_rat @ A @ ( times_times_rat @ B @ Z2 ) ) @ Z2 ) ) ) ) ).

% add_divide_eq_if_simps(5)
thf(fact_2429_minus__divide__diff__eq__iff,axiom,
    ! [Z2: rat,X: rat,Y3: rat] :
      ( ( Z2 != zero_zero_rat )
     => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ X @ Z2 ) ) @ Y3 )
        = ( divide_divide_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ X ) @ ( times_times_rat @ Y3 @ Z2 ) ) @ Z2 ) ) ) ).

% minus_divide_diff_eq_iff
thf(fact_2430_is__unitE,axiom,
    ! [A: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ~ ( ( A != zero_zero_nat )
         => ! [B4: nat] :
              ( ( B4 != zero_zero_nat )
             => ( ( dvd_dvd_nat @ B4 @ one_one_nat )
               => ( ( ( divide_divide_nat @ one_one_nat @ A )
                    = B4 )
                 => ( ( ( divide_divide_nat @ one_one_nat @ B4 )
                      = A )
                   => ( ( ( times_times_nat @ A @ B4 )
                        = one_one_nat )
                     => ( ( divide_divide_nat @ C @ A )
                       != ( times_times_nat @ C @ B4 ) ) ) ) ) ) ) ) ) ).

% is_unitE
thf(fact_2431_is__unitE,axiom,
    ! [A: int,C: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ~ ( ( A != zero_zero_int )
         => ! [B4: int] :
              ( ( B4 != zero_zero_int )
             => ( ( dvd_dvd_int @ B4 @ one_one_int )
               => ( ( ( divide_divide_int @ one_one_int @ A )
                    = B4 )
                 => ( ( ( divide_divide_int @ one_one_int @ B4 )
                      = A )
                   => ( ( ( times_times_int @ A @ B4 )
                        = one_one_int )
                     => ( ( divide_divide_int @ C @ A )
                       != ( times_times_int @ C @ B4 ) ) ) ) ) ) ) ) ) ).

% is_unitE
thf(fact_2432_is__unitE,axiom,
    ! [A: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ~ ( ( A != zero_z3403309356797280102nteger )
         => ! [B4: code_integer] :
              ( ( B4 != zero_z3403309356797280102nteger )
             => ( ( dvd_dvd_Code_integer @ B4 @ one_one_Code_integer )
               => ( ( ( divide6298287555418463151nteger @ one_one_Code_integer @ A )
                    = B4 )
                 => ( ( ( divide6298287555418463151nteger @ one_one_Code_integer @ B4 )
                      = A )
                   => ( ( ( times_3573771949741848930nteger @ A @ B4 )
                        = one_one_Code_integer )
                     => ( ( divide6298287555418463151nteger @ C @ A )
                       != ( times_3573771949741848930nteger @ C @ B4 ) ) ) ) ) ) ) ) ) ).

% is_unitE
thf(fact_2433_is__unitE,axiom,
    ! [A: code_natural,C: code_natural] :
      ( ( dvd_dvd_Code_natural @ A @ one_one_Code_natural )
     => ~ ( ( A != zero_z2226904508553997617atural )
         => ! [B4: code_natural] :
              ( ( B4 != zero_z2226904508553997617atural )
             => ( ( dvd_dvd_Code_natural @ B4 @ one_one_Code_natural )
               => ( ( ( divide5121882707175180666atural @ one_one_Code_natural @ A )
                    = B4 )
                 => ( ( ( divide5121882707175180666atural @ one_one_Code_natural @ B4 )
                      = A )
                   => ( ( ( times_2397367101498566445atural @ A @ B4 )
                        = one_one_Code_natural )
                     => ( ( divide5121882707175180666atural @ C @ A )
                       != ( times_2397367101498566445atural @ C @ B4 ) ) ) ) ) ) ) ) ) ).

% is_unitE
thf(fact_2434_is__unit__div__mult__cancel__left,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( ( dvd_dvd_nat @ B @ one_one_nat )
       => ( ( divide_divide_nat @ A @ ( times_times_nat @ A @ B ) )
          = ( divide_divide_nat @ one_one_nat @ B ) ) ) ) ).

% is_unit_div_mult_cancel_left
thf(fact_2435_is__unit__div__mult__cancel__left,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( ( dvd_dvd_int @ B @ one_one_int )
       => ( ( divide_divide_int @ A @ ( times_times_int @ A @ B ) )
          = ( divide_divide_int @ one_one_int @ B ) ) ) ) ).

% is_unit_div_mult_cancel_left
thf(fact_2436_is__unit__div__mult__cancel__left,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
       => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ A @ B ) )
          = ( divide6298287555418463151nteger @ one_one_Code_integer @ B ) ) ) ) ).

% is_unit_div_mult_cancel_left
thf(fact_2437_is__unit__div__mult__cancel__left,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( A != zero_z2226904508553997617atural )
     => ( ( dvd_dvd_Code_natural @ B @ one_one_Code_natural )
       => ( ( divide5121882707175180666atural @ A @ ( times_2397367101498566445atural @ A @ B ) )
          = ( divide5121882707175180666atural @ one_one_Code_natural @ B ) ) ) ) ).

% is_unit_div_mult_cancel_left
thf(fact_2438_is__unit__div__mult__cancel__right,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( ( dvd_dvd_nat @ B @ one_one_nat )
       => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ A ) )
          = ( divide_divide_nat @ one_one_nat @ B ) ) ) ) ).

% is_unit_div_mult_cancel_right
thf(fact_2439_is__unit__div__mult__cancel__right,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( ( dvd_dvd_int @ B @ one_one_int )
       => ( ( divide_divide_int @ A @ ( times_times_int @ B @ A ) )
          = ( divide_divide_int @ one_one_int @ B ) ) ) ) ).

% is_unit_div_mult_cancel_right
thf(fact_2440_is__unit__div__mult__cancel__right,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
       => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ A ) )
          = ( divide6298287555418463151nteger @ one_one_Code_integer @ B ) ) ) ) ).

% is_unit_div_mult_cancel_right
thf(fact_2441_is__unit__div__mult__cancel__right,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( A != zero_z2226904508553997617atural )
     => ( ( dvd_dvd_Code_natural @ B @ one_one_Code_natural )
       => ( ( divide5121882707175180666atural @ A @ ( times_2397367101498566445atural @ B @ A ) )
          = ( divide5121882707175180666atural @ one_one_Code_natural @ B ) ) ) ) ).

% is_unit_div_mult_cancel_right
thf(fact_2442_div__if,axiom,
    ( divide_divide_nat
    = ( ^ [M2: nat,N3: nat] :
          ( if_nat
          @ ( ( ord_less_nat @ M2 @ N3 )
            | ( N3 = zero_zero_nat ) )
          @ zero_zero_nat
          @ ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M2 @ N3 ) @ N3 ) ) ) ) ) ).

% div_if
thf(fact_2443_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_2444_nat__zero__as__int,axiom,
    ( zero_zero_nat
    = ( nat2 @ zero_zero_int ) ) ).

% nat_zero_as_int
thf(fact_2445_le__iff__diff__le__0,axiom,
    ( ord_le3102999989581377725nteger
    = ( ^ [A2: code_integer,B2: code_integer] : ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ A2 @ B2 ) @ zero_z3403309356797280102nteger ) ) ) ).

% le_iff_diff_le_0
thf(fact_2446_le__iff__diff__le__0,axiom,
    ( ord_less_eq_rat
    = ( ^ [A2: rat,B2: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ A2 @ B2 ) @ zero_zero_rat ) ) ) ).

% le_iff_diff_le_0
thf(fact_2447_le__iff__diff__le__0,axiom,
    ( ord_less_eq_int
    = ( ^ [A2: int,B2: int] : ( ord_less_eq_int @ ( minus_minus_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).

% le_iff_diff_le_0
thf(fact_2448_less__iff__diff__less__0,axiom,
    ( ord_le6747313008572928689nteger
    = ( ^ [A2: code_integer,B2: code_integer] : ( ord_le6747313008572928689nteger @ ( minus_8373710615458151222nteger @ A2 @ B2 ) @ zero_z3403309356797280102nteger ) ) ) ).

% less_iff_diff_less_0
thf(fact_2449_less__iff__diff__less__0,axiom,
    ( ord_less_rat
    = ( ^ [A2: rat,B2: rat] : ( ord_less_rat @ ( minus_minus_rat @ A2 @ B2 ) @ zero_zero_rat ) ) ) ).

% less_iff_diff_less_0
thf(fact_2450_less__iff__diff__less__0,axiom,
    ( ord_less_int
    = ( ^ [A2: int,B2: int] : ( ord_less_int @ ( minus_minus_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).

% less_iff_diff_less_0
thf(fact_2451_int__sgnE,axiom,
    ! [K: int] :
      ~ ! [N4: nat,L2: int] :
          ( K
         != ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ N4 ) ) ) ).

% int_sgnE
thf(fact_2452_ex__nat,axiom,
    ( ( ^ [P3: nat > $o] :
        ? [X6: nat] : ( P3 @ X6 ) )
    = ( ^ [P4: nat > $o] :
        ? [X2: int] :
          ( ( ord_less_eq_int @ zero_zero_int @ X2 )
          & ( P4 @ ( nat2 @ X2 ) ) ) ) ) ).

% ex_nat
thf(fact_2453_all__nat,axiom,
    ( ( ^ [P3: nat > $o] :
        ! [X6: nat] : ( P3 @ X6 ) )
    = ( ^ [P4: nat > $o] :
        ! [X2: int] :
          ( ( ord_less_eq_int @ zero_zero_int @ X2 )
         => ( P4 @ ( nat2 @ X2 ) ) ) ) ) ).

% all_nat
thf(fact_2454_eq__nat__nat__iff,axiom,
    ! [Z2: int,Z5: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ Z5 )
       => ( ( ( nat2 @ Z2 )
            = ( nat2 @ Z5 ) )
          = ( Z2 = Z5 ) ) ) ) ).

% eq_nat_nat_iff
thf(fact_2455_divide__le__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ B ) @ zero_zero_rat )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
          & ( ord_less_eq_rat @ B @ zero_zero_rat ) )
        | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
          & ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ) ) ).

% divide_le_0_iff
thf(fact_2456_divide__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ) ).

% divide_right_mono
thf(fact_2457_zero__le__divide__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ B ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
          & ( ord_less_eq_rat @ zero_zero_rat @ B ) )
        | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
          & ( ord_less_eq_rat @ B @ zero_zero_rat ) ) ) ) ).

% zero_le_divide_iff
thf(fact_2458_divide__nonneg__nonneg,axiom,
    ! [X: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y3 ) ) ) ) ).

% divide_nonneg_nonneg
thf(fact_2459_divide__nonneg__nonpos,axiom,
    ! [X: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X )
     => ( ( ord_less_eq_rat @ Y3 @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y3 ) @ zero_zero_rat ) ) ) ).

% divide_nonneg_nonpos
thf(fact_2460_divide__nonpos__nonneg,axiom,
    ! [X: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ X @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y3 ) @ zero_zero_rat ) ) ) ).

% divide_nonpos_nonneg
thf(fact_2461_divide__nonpos__nonpos,axiom,
    ! [X: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ X @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ Y3 @ zero_zero_rat )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y3 ) ) ) ) ).

% divide_nonpos_nonpos
thf(fact_2462_divide__right__mono__neg,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ ( divide_divide_rat @ A @ C ) ) ) ) ).

% divide_right_mono_neg
thf(fact_2463_nat__mono,axiom,
    ! [X: int,Y3: int] :
      ( ( ord_less_eq_int @ X @ Y3 )
     => ( ord_less_eq_nat @ ( nat2 @ X ) @ ( nat2 @ Y3 ) ) ) ).

% nat_mono
thf(fact_2464_divide__neg__neg,axiom,
    ! [X: rat,Y3: rat] :
      ( ( ord_less_rat @ X @ zero_zero_rat )
     => ( ( ord_less_rat @ Y3 @ zero_zero_rat )
       => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y3 ) ) ) ) ).

% divide_neg_neg
thf(fact_2465_divide__neg__pos,axiom,
    ! [X: rat,Y3: rat] :
      ( ( ord_less_rat @ X @ zero_zero_rat )
     => ( ( ord_less_rat @ zero_zero_rat @ Y3 )
       => ( ord_less_rat @ ( divide_divide_rat @ X @ Y3 ) @ zero_zero_rat ) ) ) ).

% divide_neg_pos
thf(fact_2466_divide__pos__neg,axiom,
    ! [X: rat,Y3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ X )
     => ( ( ord_less_rat @ Y3 @ zero_zero_rat )
       => ( ord_less_rat @ ( divide_divide_rat @ X @ Y3 ) @ zero_zero_rat ) ) ) ).

% divide_pos_neg
thf(fact_2467_divide__pos__pos,axiom,
    ! [X: rat,Y3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ X )
     => ( ( ord_less_rat @ zero_zero_rat @ Y3 )
       => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y3 ) ) ) ) ).

% divide_pos_pos
thf(fact_2468_divide__less__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ A @ B ) @ zero_zero_rat )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A )
          & ( ord_less_rat @ B @ zero_zero_rat ) )
        | ( ( ord_less_rat @ A @ zero_zero_rat )
          & ( ord_less_rat @ zero_zero_rat @ B ) ) ) ) ).

% divide_less_0_iff
thf(fact_2469_divide__less__cancel,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ A @ B ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ B @ A ) )
        & ( C != zero_zero_rat ) ) ) ).

% divide_less_cancel
thf(fact_2470_zero__less__divide__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ B ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A )
          & ( ord_less_rat @ zero_zero_rat @ B ) )
        | ( ( ord_less_rat @ A @ zero_zero_rat )
          & ( ord_less_rat @ B @ zero_zero_rat ) ) ) ) ).

% zero_less_divide_iff
thf(fact_2471_divide__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ord_less_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ) ).

% divide_strict_right_mono
thf(fact_2472_divide__strict__right__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( ord_less_rat @ C @ zero_zero_rat )
       => ( ord_less_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ) ).

% divide_strict_right_mono_neg
thf(fact_2473_not__is__unit__0,axiom,
    ~ ( dvd_dvd_nat @ zero_zero_nat @ one_one_nat ) ).

% not_is_unit_0
thf(fact_2474_not__is__unit__0,axiom,
    ~ ( dvd_dvd_int @ zero_zero_int @ one_one_int ) ).

% not_is_unit_0
thf(fact_2475_not__is__unit__0,axiom,
    ~ ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ one_one_Code_integer ) ).

% not_is_unit_0
thf(fact_2476_not__is__unit__0,axiom,
    ~ ( dvd_dvd_Code_natural @ zero_z2226904508553997617atural @ one_one_Code_natural ) ).

% not_is_unit_0
thf(fact_2477_frac__eq__eq,axiom,
    ! [Y3: rat,Z2: rat,X: rat,W2: rat] :
      ( ( Y3 != zero_zero_rat )
     => ( ( Z2 != zero_zero_rat )
       => ( ( ( divide_divide_rat @ X @ Y3 )
            = ( divide_divide_rat @ W2 @ Z2 ) )
          = ( ( times_times_rat @ X @ Z2 )
            = ( times_times_rat @ W2 @ Y3 ) ) ) ) ) ).

% frac_eq_eq
thf(fact_2478_divide__eq__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( ( divide_divide_rat @ B @ C )
        = A )
      = ( ( ( C != zero_zero_rat )
         => ( B
            = ( times_times_rat @ A @ C ) ) )
        & ( ( C = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% divide_eq_eq
thf(fact_2479_eq__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( A
        = ( divide_divide_rat @ B @ C ) )
      = ( ( ( C != zero_zero_rat )
         => ( ( times_times_rat @ A @ C )
            = B ) )
        & ( ( C = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% eq_divide_eq
thf(fact_2480_divide__eq__imp,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( C != zero_zero_rat )
     => ( ( B
          = ( times_times_rat @ A @ C ) )
       => ( ( divide_divide_rat @ B @ C )
          = A ) ) ) ).

% divide_eq_imp
thf(fact_2481_eq__divide__imp,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( ( times_times_rat @ A @ C )
          = B )
       => ( A
          = ( divide_divide_rat @ B @ C ) ) ) ) ).

% eq_divide_imp
thf(fact_2482_nonzero__divide__eq__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( C != zero_zero_rat )
     => ( ( ( divide_divide_rat @ B @ C )
          = A )
        = ( B
          = ( times_times_rat @ A @ C ) ) ) ) ).

% nonzero_divide_eq_eq
thf(fact_2483_nonzero__eq__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( A
          = ( divide_divide_rat @ B @ C ) )
        = ( ( times_times_rat @ A @ C )
          = B ) ) ) ).

% nonzero_eq_divide_eq
thf(fact_2484_right__inverse__eq,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( ( ( divide_divide_rat @ A @ B )
          = one_one_rat )
        = ( A = B ) ) ) ).

% right_inverse_eq
thf(fact_2485_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_2486_is__unit__mult__iff,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( dvd_dvd_Code_natural @ ( times_2397367101498566445atural @ A @ B ) @ one_one_Code_natural )
      = ( ( dvd_dvd_Code_natural @ A @ one_one_Code_natural )
        & ( dvd_dvd_Code_natural @ B @ one_one_Code_natural ) ) ) ).

% is_unit_mult_iff
thf(fact_2487_is__unit__mult__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ one_one_Code_integer )
      = ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
        & ( dvd_dvd_Code_integer @ B @ one_one_Code_integer ) ) ) ).

% is_unit_mult_iff
thf(fact_2488_is__unit__mult__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ one_one_nat )
      = ( ( dvd_dvd_nat @ A @ one_one_nat )
        & ( dvd_dvd_nat @ B @ one_one_nat ) ) ) ).

% is_unit_mult_iff
thf(fact_2489_is__unit__mult__iff,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ one_one_int )
      = ( ( dvd_dvd_int @ A @ one_one_int )
        & ( dvd_dvd_int @ B @ one_one_int ) ) ) ).

% is_unit_mult_iff
thf(fact_2490_dvd__mult__unit__iff,axiom,
    ! [B: code_natural,A: code_natural,C: code_natural] :
      ( ( dvd_dvd_Code_natural @ B @ one_one_Code_natural )
     => ( ( dvd_dvd_Code_natural @ A @ ( times_2397367101498566445atural @ C @ B ) )
        = ( dvd_dvd_Code_natural @ A @ C ) ) ) ).

% dvd_mult_unit_iff
thf(fact_2491_dvd__mult__unit__iff,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ C @ B ) )
        = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).

% dvd_mult_unit_iff
thf(fact_2492_dvd__mult__unit__iff,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( dvd_dvd_nat @ A @ ( times_times_nat @ C @ B ) )
        = ( dvd_dvd_nat @ A @ C ) ) ) ).

% dvd_mult_unit_iff
thf(fact_2493_dvd__mult__unit__iff,axiom,
    ! [B: int,A: int,C: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( dvd_dvd_int @ A @ ( times_times_int @ C @ B ) )
        = ( dvd_dvd_int @ A @ C ) ) ) ).

% dvd_mult_unit_iff
thf(fact_2494_mult__unit__dvd__iff,axiom,
    ! [B: code_natural,A: code_natural,C: code_natural] :
      ( ( dvd_dvd_Code_natural @ B @ one_one_Code_natural )
     => ( ( dvd_dvd_Code_natural @ ( times_2397367101498566445atural @ A @ B ) @ C )
        = ( dvd_dvd_Code_natural @ A @ C ) ) ) ).

% mult_unit_dvd_iff
thf(fact_2495_mult__unit__dvd__iff,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ C )
        = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).

% mult_unit_dvd_iff
thf(fact_2496_mult__unit__dvd__iff,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
        = ( dvd_dvd_nat @ A @ C ) ) ) ).

% mult_unit_dvd_iff
thf(fact_2497_mult__unit__dvd__iff,axiom,
    ! [B: int,A: int,C: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
        = ( dvd_dvd_int @ A @ C ) ) ) ).

% mult_unit_dvd_iff
thf(fact_2498_dvd__mult__unit__iff_H,axiom,
    ! [B: code_natural,A: code_natural,C: code_natural] :
      ( ( dvd_dvd_Code_natural @ B @ one_one_Code_natural )
     => ( ( dvd_dvd_Code_natural @ A @ ( times_2397367101498566445atural @ B @ C ) )
        = ( dvd_dvd_Code_natural @ A @ C ) ) ) ).

% dvd_mult_unit_iff'
thf(fact_2499_dvd__mult__unit__iff_H,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B @ C ) )
        = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).

% dvd_mult_unit_iff'
thf(fact_2500_dvd__mult__unit__iff_H,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) )
        = ( dvd_dvd_nat @ A @ C ) ) ) ).

% dvd_mult_unit_iff'
thf(fact_2501_dvd__mult__unit__iff_H,axiom,
    ! [B: int,A: int,C: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) )
        = ( dvd_dvd_int @ A @ C ) ) ) ).

% dvd_mult_unit_iff'
thf(fact_2502_mult__unit__dvd__iff_H,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( ( dvd_dvd_Code_natural @ A @ one_one_Code_natural )
     => ( ( dvd_dvd_Code_natural @ ( times_2397367101498566445atural @ A @ B ) @ C )
        = ( dvd_dvd_Code_natural @ B @ C ) ) ) ).

% mult_unit_dvd_iff'
thf(fact_2503_mult__unit__dvd__iff_H,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ C )
        = ( dvd_dvd_Code_integer @ B @ C ) ) ) ).

% mult_unit_dvd_iff'
thf(fact_2504_mult__unit__dvd__iff_H,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
        = ( dvd_dvd_nat @ B @ C ) ) ) ).

% mult_unit_dvd_iff'
thf(fact_2505_mult__unit__dvd__iff_H,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
        = ( dvd_dvd_int @ B @ C ) ) ) ).

% mult_unit_dvd_iff'
thf(fact_2506_unit__mult__left__cancel,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( ( dvd_dvd_Code_natural @ A @ one_one_Code_natural )
     => ( ( ( times_2397367101498566445atural @ A @ B )
          = ( times_2397367101498566445atural @ A @ C ) )
        = ( B = C ) ) ) ).

% unit_mult_left_cancel
thf(fact_2507_unit__mult__left__cancel,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ( ( ( times_3573771949741848930nteger @ A @ B )
          = ( times_3573771949741848930nteger @ A @ C ) )
        = ( B = C ) ) ) ).

% unit_mult_left_cancel
thf(fact_2508_unit__mult__left__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( ( times_times_nat @ A @ B )
          = ( times_times_nat @ A @ C ) )
        = ( B = C ) ) ) ).

% unit_mult_left_cancel
thf(fact_2509_unit__mult__left__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( ( times_times_int @ A @ B )
          = ( times_times_int @ A @ C ) )
        = ( B = C ) ) ) ).

% unit_mult_left_cancel
thf(fact_2510_unit__mult__right__cancel,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( ( dvd_dvd_Code_natural @ A @ one_one_Code_natural )
     => ( ( ( times_2397367101498566445atural @ B @ A )
          = ( times_2397367101498566445atural @ C @ A ) )
        = ( B = C ) ) ) ).

% unit_mult_right_cancel
thf(fact_2511_unit__mult__right__cancel,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ( ( ( times_3573771949741848930nteger @ B @ A )
          = ( times_3573771949741848930nteger @ C @ A ) )
        = ( B = C ) ) ) ).

% unit_mult_right_cancel
thf(fact_2512_unit__mult__right__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( ( times_times_nat @ B @ A )
          = ( times_times_nat @ C @ A ) )
        = ( B = C ) ) ) ).

% unit_mult_right_cancel
thf(fact_2513_unit__mult__right__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( ( times_times_int @ B @ A )
          = ( times_times_int @ C @ A ) )
        = ( B = C ) ) ) ).

% unit_mult_right_cancel
thf(fact_2514_nonzero__minus__divide__right,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) )
        = ( divide_divide_rat @ A @ ( uminus_uminus_rat @ B ) ) ) ) ).

% nonzero_minus_divide_right
thf(fact_2515_nonzero__minus__divide__divide,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) )
        = ( divide_divide_rat @ A @ B ) ) ) ).

% nonzero_minus_divide_divide
thf(fact_2516_mod__induct,axiom,
    ! [P: nat > $o,N: nat,P5: nat,M: nat] :
      ( ( P @ N )
     => ( ( ord_less_nat @ N @ P5 )
       => ( ( ord_less_nat @ M @ P5 )
         => ( ! [N4: nat] :
                ( ( ord_less_nat @ N4 @ P5 )
               => ( ( P @ N4 )
                 => ( P @ ( modulo_modulo_nat @ ( suc @ N4 ) @ P5 ) ) ) )
           => ( P @ M ) ) ) ) ) ).

% mod_induct
thf(fact_2517_nat__one__as__int,axiom,
    ( one_one_nat
    = ( nat2 @ one_one_int ) ) ).

% nat_one_as_int
thf(fact_2518_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_2519_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_2520_ceiling__mono,axiom,
    ! [Y3: rat,X: rat] :
      ( ( ord_less_eq_rat @ Y3 @ X )
     => ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ Y3 ) @ ( archim2889992004027027881ng_rat @ X ) ) ) ).

% ceiling_mono
thf(fact_2521_ceiling__less__cancel,axiom,
    ! [X: rat,Y3: rat] :
      ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ ( archim2889992004027027881ng_rat @ Y3 ) )
     => ( ord_less_rat @ X @ Y3 ) ) ).

% ceiling_less_cancel
thf(fact_2522_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_2523_div__mult2__eq_H,axiom,
    ! [A: code_integer,M: nat,N: nat] :
      ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) ) )
      = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ ( semiri4939895301339042750nteger @ M ) ) @ ( semiri4939895301339042750nteger @ N ) ) ) ).

% div_mult2_eq'
thf(fact_2524_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_2525_div__mult2__eq_H,axiom,
    ! [A: code_natural,M: nat,N: nat] :
      ( ( divide5121882707175180666atural @ A @ ( times_2397367101498566445atural @ ( semiri3763490453095760265atural @ M ) @ ( semiri3763490453095760265atural @ N ) ) )
      = ( divide5121882707175180666atural @ ( divide5121882707175180666atural @ A @ ( semiri3763490453095760265atural @ M ) ) @ ( semiri3763490453095760265atural @ N ) ) ) ).

% div_mult2_eq'
thf(fact_2526_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_2527_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_2528_diff__less__Suc,axiom,
    ! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).

% diff_less_Suc
thf(fact_2529_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_2530_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_2531_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_2532_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_2533_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_2534_diff__less__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ A )
       => ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).

% diff_less_mono
thf(fact_2535_zdvd__antisym__nonneg,axiom,
    ! [M: int,N: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ M )
     => ( ( ord_less_eq_int @ zero_zero_int @ N )
       => ( ( dvd_dvd_int @ M @ N )
         => ( ( dvd_dvd_int @ N @ M )
           => ( M = N ) ) ) ) ) ).

% zdvd_antisym_nonneg
thf(fact_2536_sgn__not__eq__imp,axiom,
    ! [B: int,A: int] :
      ( ( ( sgn_sgn_int @ B )
       != ( sgn_sgn_int @ A ) )
     => ( ( ( sgn_sgn_int @ A )
         != zero_zero_int )
       => ( ( ( sgn_sgn_int @ B )
           != zero_zero_int )
         => ( ( sgn_sgn_int @ A )
            = ( uminus_uminus_int @ ( sgn_sgn_int @ B ) ) ) ) ) ) ).

% sgn_not_eq_imp
thf(fact_2537_sgn__not__eq__imp,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ( sgn_sgn_Code_integer @ B )
       != ( sgn_sgn_Code_integer @ A ) )
     => ( ( ( sgn_sgn_Code_integer @ A )
         != zero_z3403309356797280102nteger )
       => ( ( ( sgn_sgn_Code_integer @ B )
           != zero_z3403309356797280102nteger )
         => ( ( sgn_sgn_Code_integer @ A )
            = ( uminus1351360451143612070nteger @ ( sgn_sgn_Code_integer @ B ) ) ) ) ) ) ).

% sgn_not_eq_imp
thf(fact_2538_sgn__not__eq__imp,axiom,
    ! [B: rat,A: rat] :
      ( ( ( sgn_sgn_rat @ B )
       != ( sgn_sgn_rat @ A ) )
     => ( ( ( sgn_sgn_rat @ A )
         != zero_zero_rat )
       => ( ( ( sgn_sgn_rat @ B )
           != zero_zero_rat )
         => ( ( sgn_sgn_rat @ A )
            = ( uminus_uminus_rat @ ( sgn_sgn_rat @ B ) ) ) ) ) ) ).

% sgn_not_eq_imp
thf(fact_2539_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_2540_less__mult__imp__div__less,axiom,
    ! [M: nat,I2: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( times_times_nat @ I2 @ N ) )
     => ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ I2 ) ) ).

% less_mult_imp_div_less
thf(fact_2541_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_2542_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_2543_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_2544_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_2545_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_2546_sgn__minus__1,axiom,
    ( ( sgn_sgn_int @ ( uminus_uminus_int @ one_one_int ) )
    = ( uminus_uminus_int @ one_one_int ) ) ).

% sgn_minus_1
thf(fact_2547_sgn__minus__1,axiom,
    ( ( sgn_sgn_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% sgn_minus_1
thf(fact_2548_sgn__minus__1,axiom,
    ( ( sgn_sgn_rat @ ( uminus_uminus_rat @ one_one_rat ) )
    = ( uminus_uminus_rat @ one_one_rat ) ) ).

% sgn_minus_1
thf(fact_2549_dvd__euclidean__size__eq__imp__dvd,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( ( euclid6377331345833325938nteger @ A )
          = ( euclid6377331345833325938nteger @ B ) )
       => ( ( dvd_dvd_Code_integer @ B @ A )
         => ( dvd_dvd_Code_integer @ A @ B ) ) ) ) ).

% dvd_euclidean_size_eq_imp_dvd
thf(fact_2550_dvd__euclidean__size__eq__imp__dvd,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( ( ( euclid4774559944035922753ze_int @ A )
          = ( euclid4774559944035922753ze_int @ B ) )
       => ( ( dvd_dvd_int @ B @ A )
         => ( dvd_dvd_int @ A @ B ) ) ) ) ).

% dvd_euclidean_size_eq_imp_dvd
thf(fact_2551_dvd__euclidean__size__eq__imp__dvd,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( ( ( euclid4777050414544973029ze_nat @ A )
          = ( euclid4777050414544973029ze_nat @ B ) )
       => ( ( dvd_dvd_nat @ B @ A )
         => ( dvd_dvd_nat @ A @ B ) ) ) ) ).

% dvd_euclidean_size_eq_imp_dvd
thf(fact_2552_dvd__euclidean__size__eq__imp__dvd,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( A != zero_z2226904508553997617atural )
     => ( ( ( euclid5200926497590043453atural @ A )
          = ( euclid5200926497590043453atural @ B ) )
       => ( ( dvd_dvd_Code_natural @ B @ A )
         => ( dvd_dvd_Code_natural @ A @ B ) ) ) ) ).

% dvd_euclidean_size_eq_imp_dvd
thf(fact_2553_euclidean__size__unit,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ( ( euclid6377331345833325938nteger @ A )
        = ( euclid6377331345833325938nteger @ one_one_Code_integer ) ) ) ).

% euclidean_size_unit
thf(fact_2554_euclidean__size__unit,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( euclid4774559944035922753ze_int @ A )
        = ( euclid4774559944035922753ze_int @ one_one_int ) ) ) ).

% euclidean_size_unit
thf(fact_2555_euclidean__size__unit,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( euclid4777050414544973029ze_nat @ A )
        = ( euclid4777050414544973029ze_nat @ one_one_nat ) ) ) ).

% euclidean_size_unit
thf(fact_2556_euclidean__size__unit,axiom,
    ! [A: code_natural] :
      ( ( dvd_dvd_Code_natural @ A @ one_one_Code_natural )
     => ( ( euclid5200926497590043453atural @ A )
        = ( euclid5200926497590043453atural @ one_one_Code_natural ) ) ) ).

% euclidean_size_unit
thf(fact_2557_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_2558_fact__dvd,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( dvd_dvd_Code_natural @ ( semiri2447717529341329178atural @ N ) @ ( semiri2447717529341329178atural @ M ) ) ) ).

% fact_dvd
thf(fact_2559_fact__dvd,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( dvd_dvd_rat @ ( semiri773545260158071498ct_rat @ N ) @ ( semiri773545260158071498ct_rat @ M ) ) ) ).

% fact_dvd
thf(fact_2560_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_2561_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_2562_is__unit__division__segment,axiom,
    ! [A: nat] : ( dvd_dvd_nat @ ( euclid3398187327856392827nt_nat @ A ) @ one_one_nat ) ).

% is_unit_division_segment
thf(fact_2563_is__unit__division__segment,axiom,
    ! [A: int] : ( dvd_dvd_int @ ( euclid3395696857347342551nt_int @ A ) @ one_one_int ) ).

% is_unit_division_segment
thf(fact_2564_is__unit__division__segment,axiom,
    ! [A: code_integer] : ( dvd_dvd_Code_integer @ ( euclid6289375185220004616nteger @ A ) @ one_one_Code_integer ) ).

% is_unit_division_segment
thf(fact_2565_is__unit__division__segment,axiom,
    ! [A: code_natural] : ( dvd_dvd_Code_natural @ ( euclid5112970336976722131atural @ A ) @ one_one_Code_natural ) ).

% is_unit_division_segment
thf(fact_2566_division__segment__eq__sgn,axiom,
    ! [K: int] :
      ( ( K != zero_zero_int )
     => ( ( euclid3395696857347342551nt_int @ K )
        = ( sgn_sgn_int @ K ) ) ) ).

% division_segment_eq_sgn
thf(fact_2567_frac__ge__0,axiom,
    ! [X: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( archimedean_frac_rat @ X ) ) ).

% frac_ge_0
thf(fact_2568_frac__lt__1,axiom,
    ! [X: rat] : ( ord_less_rat @ ( archimedean_frac_rat @ X ) @ one_one_rat ) ).

% frac_lt_1
thf(fact_2569_nat__mono__iff,axiom,
    ! [Z2: int,W2: int] :
      ( ( ord_less_int @ zero_zero_int @ Z2 )
     => ( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z2 ) )
        = ( ord_less_int @ W2 @ Z2 ) ) ) ).

% nat_mono_iff
thf(fact_2570_frac__le,axiom,
    ! [Y3: rat,X: rat,W2: rat,Z2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
     => ( ( ord_less_eq_rat @ X @ Y3 )
       => ( ( ord_less_rat @ zero_zero_rat @ W2 )
         => ( ( ord_less_eq_rat @ W2 @ Z2 )
           => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Z2 ) @ ( divide_divide_rat @ Y3 @ W2 ) ) ) ) ) ) ).

% frac_le
thf(fact_2571_frac__less,axiom,
    ! [X: rat,Y3: rat,W2: rat,Z2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X )
     => ( ( ord_less_rat @ X @ Y3 )
       => ( ( ord_less_rat @ zero_zero_rat @ W2 )
         => ( ( ord_less_eq_rat @ W2 @ Z2 )
           => ( ord_less_rat @ ( divide_divide_rat @ X @ Z2 ) @ ( divide_divide_rat @ Y3 @ W2 ) ) ) ) ) ) ).

% frac_less
thf(fact_2572_frac__less2,axiom,
    ! [X: rat,Y3: rat,W2: rat,Z2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ X )
     => ( ( ord_less_eq_rat @ X @ Y3 )
       => ( ( ord_less_rat @ zero_zero_rat @ W2 )
         => ( ( ord_less_rat @ W2 @ Z2 )
           => ( ord_less_rat @ ( divide_divide_rat @ X @ Z2 ) @ ( divide_divide_rat @ Y3 @ W2 ) ) ) ) ) ) ).

% frac_less2
thf(fact_2573_divide__le__cancel,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ A @ B ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ B @ A ) ) ) ) ).

% divide_le_cancel
thf(fact_2574_divide__nonneg__neg,axiom,
    ! [X: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X )
     => ( ( ord_less_rat @ Y3 @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y3 ) @ zero_zero_rat ) ) ) ).

% divide_nonneg_neg
thf(fact_2575_divide__nonneg__pos,axiom,
    ! [X: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X )
     => ( ( ord_less_rat @ zero_zero_rat @ Y3 )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y3 ) ) ) ) ).

% divide_nonneg_pos
thf(fact_2576_divide__nonpos__neg,axiom,
    ! [X: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ X @ zero_zero_rat )
     => ( ( ord_less_rat @ Y3 @ zero_zero_rat )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y3 ) ) ) ) ).

% divide_nonpos_neg
thf(fact_2577_divide__nonpos__pos,axiom,
    ! [X: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ X @ zero_zero_rat )
     => ( ( ord_less_rat @ zero_zero_rat @ Y3 )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y3 ) @ zero_zero_rat ) ) ) ).

% divide_nonpos_pos
thf(fact_2578_divide__less__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ A )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ B @ ( 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 ) @ B ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).

% divide_less_eq
thf(fact_2579_less__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).

% less_divide_eq
thf(fact_2580_neg__divide__less__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ A )
        = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).

% neg_divide_less_eq
thf(fact_2581_neg__less__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ C ) )
        = ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).

% neg_less_divide_eq
thf(fact_2582_pos__divide__less__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ A )
        = ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).

% pos_divide_less_eq
thf(fact_2583_pos__less__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ C ) )
        = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).

% pos_less_divide_eq
thf(fact_2584_mult__imp__div__pos__less,axiom,
    ! [Y3: rat,X: rat,Z2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Y3 )
     => ( ( ord_less_rat @ X @ ( times_times_rat @ Z2 @ Y3 ) )
       => ( ord_less_rat @ ( divide_divide_rat @ X @ Y3 ) @ Z2 ) ) ) ).

% mult_imp_div_pos_less
thf(fact_2585_mult__imp__less__div__pos,axiom,
    ! [Y3: rat,Z2: rat,X: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Y3 )
     => ( ( ord_less_rat @ ( times_times_rat @ Z2 @ Y3 ) @ X )
       => ( ord_less_rat @ Z2 @ ( divide_divide_rat @ X @ Y3 ) ) ) ) ).

% mult_imp_less_div_pos
thf(fact_2586_divide__strict__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
         => ( ord_less_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).

% divide_strict_left_mono
thf(fact_2587_divide__strict__left__mono__neg,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ C @ zero_zero_rat )
       => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
         => ( ord_less_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).

% divide_strict_left_mono_neg
thf(fact_2588_unit__dvdE,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( dvd_dvd_Code_natural @ A @ one_one_Code_natural )
     => ~ ( ( A != zero_z2226904508553997617atural )
         => ! [C2: code_natural] :
              ( B
             != ( times_2397367101498566445atural @ A @ C2 ) ) ) ) ).

% unit_dvdE
thf(fact_2589_unit__dvdE,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ~ ( ( A != zero_z3403309356797280102nteger )
         => ! [C2: code_integer] :
              ( B
             != ( times_3573771949741848930nteger @ A @ C2 ) ) ) ) ).

% unit_dvdE
thf(fact_2590_unit__dvdE,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ~ ( ( A != zero_zero_nat )
         => ! [C2: nat] :
              ( B
             != ( times_times_nat @ A @ C2 ) ) ) ) ).

% unit_dvdE
thf(fact_2591_unit__dvdE,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ~ ( ( A != zero_zero_int )
         => ! [C2: int] :
              ( B
             != ( times_times_int @ A @ C2 ) ) ) ) ).

% unit_dvdE
thf(fact_2592_divide__less__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A )
          & ( ord_less_rat @ B @ A ) )
        | ( ( ord_less_rat @ A @ zero_zero_rat )
          & ( ord_less_rat @ A @ B ) )
        | ( A = zero_zero_rat ) ) ) ).

% divide_less_eq_1
thf(fact_2593_less__divide__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A )
          & ( ord_less_rat @ A @ B ) )
        | ( ( ord_less_rat @ A @ zero_zero_rat )
          & ( ord_less_rat @ B @ A ) ) ) ) ).

% less_divide_eq_1
thf(fact_2594_zless__nat__eq__int__zless,axiom,
    ! [M: nat,Z2: int] :
      ( ( ord_less_nat @ M @ ( nat2 @ Z2 ) )
      = ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ Z2 ) ) ).

% zless_nat_eq_int_zless
thf(fact_2595_int__eq__iff,axiom,
    ! [M: nat,Z2: int] :
      ( ( ( semiri1314217659103216013at_int @ M )
        = Z2 )
      = ( ( M
          = ( nat2 @ Z2 ) )
        & ( ord_less_eq_int @ zero_zero_int @ Z2 ) ) ) ).

% int_eq_iff
thf(fact_2596_nat__0__le,axiom,
    ! [Z2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
     => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z2 ) )
        = Z2 ) ) ).

% nat_0_le
thf(fact_2597_nat__le__iff,axiom,
    ! [X: int,N: nat] :
      ( ( ord_less_eq_nat @ ( nat2 @ X ) @ N )
      = ( ord_less_eq_int @ X @ ( semiri1314217659103216013at_int @ N ) ) ) ).

% nat_le_iff
thf(fact_2598_eq__minus__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( A
        = ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
      = ( ( ( C != zero_zero_rat )
         => ( ( times_times_rat @ A @ C )
            = ( uminus_uminus_rat @ B ) ) )
        & ( ( C = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% eq_minus_divide_eq
thf(fact_2599_minus__divide__eq__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) )
        = A )
      = ( ( ( C != zero_zero_rat )
         => ( ( uminus_uminus_rat @ B )
            = ( times_times_rat @ A @ C ) ) )
        & ( ( C = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% minus_divide_eq_eq
thf(fact_2600_nonzero__neg__divide__eq__eq,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( B != zero_zero_rat )
     => ( ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) )
          = C )
        = ( ( uminus_uminus_rat @ A )
          = ( times_times_rat @ C @ B ) ) ) ) ).

% nonzero_neg_divide_eq_eq
thf(fact_2601_nonzero__neg__divide__eq__eq2,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( ( C
          = ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) ) )
        = ( ( times_times_rat @ C @ B )
          = ( uminus_uminus_rat @ A ) ) ) ) ).

% nonzero_neg_divide_eq_eq2
thf(fact_2602_divide__eq__minus__1__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ( divide_divide_rat @ A @ B )
        = ( uminus_uminus_rat @ one_one_rat ) )
      = ( ( B != zero_zero_rat )
        & ( A
          = ( uminus_uminus_rat @ B ) ) ) ) ).

% divide_eq_minus_1_iff
thf(fact_2603_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_2604_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_2605_fact__mod,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( modulo8411746178871703098atural @ ( semiri2447717529341329178atural @ N ) @ ( semiri2447717529341329178atural @ M ) )
        = zero_z2226904508553997617atural ) ) ).

% fact_mod
thf(fact_2606_fact__mod,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( modulo364778990260209775nteger @ ( semiri3624122377584611663nteger @ N ) @ ( semiri3624122377584611663nteger @ M ) )
        = zero_z3403309356797280102nteger ) ) ).

% fact_mod
thf(fact_2607_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_2608_mod__size__less,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( ord_less_nat @ ( euclid6377331345833325938nteger @ ( modulo364778990260209775nteger @ A @ B ) ) @ ( euclid6377331345833325938nteger @ B ) ) ) ).

% mod_size_less
thf(fact_2609_mod__size__less,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( ord_less_nat @ ( euclid4774559944035922753ze_int @ ( modulo_modulo_int @ A @ B ) ) @ ( euclid4774559944035922753ze_int @ B ) ) ) ).

% mod_size_less
thf(fact_2610_mod__size__less,axiom,
    ! [B: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( ord_less_nat @ ( euclid4777050414544973029ze_nat @ ( modulo_modulo_nat @ A @ B ) ) @ ( euclid4777050414544973029ze_nat @ B ) ) ) ).

% mod_size_less
thf(fact_2611_mod__size__less,axiom,
    ! [B: code_natural,A: code_natural] :
      ( ( B != zero_z2226904508553997617atural )
     => ( ord_less_nat @ ( euclid5200926497590043453atural @ ( modulo8411746178871703098atural @ A @ B ) ) @ ( euclid5200926497590043453atural @ B ) ) ) ).

% mod_size_less
thf(fact_2612_diff__Suc__less,axiom,
    ! [N: nat,I2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I2 ) ) @ N ) ) ).

% diff_Suc_less
thf(fact_2613_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_2614_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_2615_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_2616_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_2617_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_2618_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_2619_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_2620_div__less__iff__less__mult,axiom,
    ! [Q4: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ Q4 )
     => ( ( ord_less_nat @ ( divide_divide_nat @ M @ Q4 ) @ N )
        = ( ord_less_nat @ M @ ( times_times_nat @ N @ Q4 ) ) ) ) ).

% div_less_iff_less_mult
thf(fact_2621_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_2622_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_2623_unit__iff__euclidean__size,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
      = ( ( ( euclid6377331345833325938nteger @ A )
          = ( euclid6377331345833325938nteger @ one_one_Code_integer ) )
        & ( A != zero_z3403309356797280102nteger ) ) ) ).

% unit_iff_euclidean_size
thf(fact_2624_unit__iff__euclidean__size,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
      = ( ( ( euclid4774559944035922753ze_int @ A )
          = ( euclid4774559944035922753ze_int @ one_one_int ) )
        & ( A != zero_zero_int ) ) ) ).

% unit_iff_euclidean_size
thf(fact_2625_unit__iff__euclidean__size,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
      = ( ( ( euclid4777050414544973029ze_nat @ A )
          = ( euclid4777050414544973029ze_nat @ one_one_nat ) )
        & ( A != zero_zero_nat ) ) ) ).

% unit_iff_euclidean_size
thf(fact_2626_unit__iff__euclidean__size,axiom,
    ! [A: code_natural] :
      ( ( dvd_dvd_Code_natural @ A @ one_one_Code_natural )
      = ( ( ( euclid5200926497590043453atural @ A )
          = ( euclid5200926497590043453atural @ one_one_Code_natural ) )
        & ( A != zero_z2226904508553997617atural ) ) ) ).

% unit_iff_euclidean_size
thf(fact_2627_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_2628_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_2629_euclidean__size__times__unit,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ( ( euclid6377331345833325938nteger @ ( times_3573771949741848930nteger @ A @ B ) )
        = ( euclid6377331345833325938nteger @ B ) ) ) ).

% euclidean_size_times_unit
thf(fact_2630_euclidean__size__times__unit,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( euclid4774559944035922753ze_int @ ( times_times_int @ A @ B ) )
        = ( euclid4774559944035922753ze_int @ B ) ) ) ).

% euclidean_size_times_unit
thf(fact_2631_euclidean__size__times__unit,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( euclid4777050414544973029ze_nat @ ( times_times_nat @ A @ B ) )
        = ( euclid4777050414544973029ze_nat @ B ) ) ) ).

% euclidean_size_times_unit
thf(fact_2632_euclidean__size__times__unit,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( dvd_dvd_Code_natural @ A @ one_one_Code_natural )
     => ( ( euclid5200926497590043453atural @ ( times_2397367101498566445atural @ A @ B ) )
        = ( euclid5200926497590043453atural @ B ) ) ) ).

% euclidean_size_times_unit
thf(fact_2633_zdvd__imp__le,axiom,
    ! [Z2: int,N: int] :
      ( ( dvd_dvd_int @ Z2 @ N )
     => ( ( ord_less_int @ zero_zero_int @ N )
       => ( ord_less_eq_int @ Z2 @ N ) ) ) ).

% zdvd_imp_le
thf(fact_2634_dvd__proper__imp__size__less,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ B )
     => ( ~ ( dvd_dvd_Code_integer @ B @ A )
       => ( ( B != zero_z3403309356797280102nteger )
         => ( ord_less_nat @ ( euclid6377331345833325938nteger @ A ) @ ( euclid6377331345833325938nteger @ B ) ) ) ) ) ).

% dvd_proper_imp_size_less
thf(fact_2635_dvd__proper__imp__size__less,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ B )
     => ( ~ ( dvd_dvd_int @ B @ A )
       => ( ( B != zero_zero_int )
         => ( ord_less_nat @ ( euclid4774559944035922753ze_int @ A ) @ ( euclid4774559944035922753ze_int @ B ) ) ) ) ) ).

% dvd_proper_imp_size_less
thf(fact_2636_dvd__proper__imp__size__less,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( ~ ( dvd_dvd_nat @ B @ A )
       => ( ( B != zero_zero_nat )
         => ( ord_less_nat @ ( euclid4777050414544973029ze_nat @ A ) @ ( euclid4777050414544973029ze_nat @ B ) ) ) ) ) ).

% dvd_proper_imp_size_less
thf(fact_2637_dvd__proper__imp__size__less,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( dvd_dvd_Code_natural @ A @ B )
     => ( ~ ( dvd_dvd_Code_natural @ B @ A )
       => ( ( B != zero_z2226904508553997617atural )
         => ( ord_less_nat @ ( euclid5200926497590043453atural @ A ) @ ( euclid5200926497590043453atural @ B ) ) ) ) ) ).

% dvd_proper_imp_size_less
thf(fact_2638_dvd__imp__size__le,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ B )
     => ( ( B != zero_z3403309356797280102nteger )
       => ( ord_less_eq_nat @ ( euclid6377331345833325938nteger @ A ) @ ( euclid6377331345833325938nteger @ B ) ) ) ) ).

% dvd_imp_size_le
thf(fact_2639_dvd__imp__size__le,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ B )
     => ( ( B != zero_zero_int )
       => ( ord_less_eq_nat @ ( euclid4774559944035922753ze_int @ A ) @ ( euclid4774559944035922753ze_int @ B ) ) ) ) ).

% dvd_imp_size_le
thf(fact_2640_dvd__imp__size__le,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( ( B != zero_zero_nat )
       => ( ord_less_eq_nat @ ( euclid4777050414544973029ze_nat @ A ) @ ( euclid4777050414544973029ze_nat @ B ) ) ) ) ).

% dvd_imp_size_le
thf(fact_2641_dvd__imp__size__le,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( dvd_dvd_Code_natural @ A @ B )
     => ( ( B != zero_z2226904508553997617atural )
       => ( ord_less_eq_nat @ ( euclid5200926497590043453atural @ A ) @ ( euclid5200926497590043453atural @ B ) ) ) ) ).

% dvd_imp_size_le
thf(fact_2642_of__nat__euclidean__size,axiom,
    ! [A: nat] :
      ( ( semiri1316708129612266289at_nat @ ( euclid4777050414544973029ze_nat @ A ) )
      = ( divide_divide_nat @ A @ ( euclid3398187327856392827nt_nat @ A ) ) ) ).

% of_nat_euclidean_size
thf(fact_2643_of__nat__euclidean__size,axiom,
    ! [A: int] :
      ( ( semiri1314217659103216013at_int @ ( euclid4774559944035922753ze_int @ A ) )
      = ( divide_divide_int @ A @ ( euclid3395696857347342551nt_int @ A ) ) ) ).

% of_nat_euclidean_size
thf(fact_2644_of__nat__euclidean__size,axiom,
    ! [A: code_integer] :
      ( ( semiri4939895301339042750nteger @ ( euclid6377331345833325938nteger @ A ) )
      = ( divide6298287555418463151nteger @ A @ ( euclid6289375185220004616nteger @ A ) ) ) ).

% of_nat_euclidean_size
thf(fact_2645_of__nat__euclidean__size,axiom,
    ! [A: code_natural] :
      ( ( semiri3763490453095760265atural @ ( euclid5200926497590043453atural @ A ) )
      = ( divide5121882707175180666atural @ A @ ( euclid5112970336976722131atural @ A ) ) ) ).

% of_nat_euclidean_size
thf(fact_2646_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_2647_zsgn__def,axiom,
    ( sgn_sgn_int
    = ( ^ [I4: int] : ( if_int @ ( I4 = zero_zero_int ) @ zero_zero_int @ ( if_int @ ( ord_less_int @ zero_zero_int @ I4 ) @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).

% zsgn_def
thf(fact_2648_nat__less__eq__zless,axiom,
    ! [W2: int,Z2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ W2 )
     => ( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z2 ) )
        = ( ord_less_int @ W2 @ Z2 ) ) ) ).

% nat_less_eq_zless
thf(fact_2649_nat__le__eq__zle,axiom,
    ! [W2: int,Z2: int] :
      ( ( ( ord_less_int @ zero_zero_int @ W2 )
        | ( ord_less_eq_int @ zero_zero_int @ Z2 ) )
     => ( ( ord_less_eq_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z2 ) )
        = ( ord_less_eq_int @ W2 @ Z2 ) ) ) ).

% nat_le_eq_zle
thf(fact_2650_nat__eq__iff,axiom,
    ! [W2: int,M: nat] :
      ( ( ( nat2 @ W2 )
        = M )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ W2 )
         => ( W2
            = ( semiri1314217659103216013at_int @ M ) ) )
        & ( ~ ( ord_less_eq_int @ zero_zero_int @ W2 )
         => ( M = zero_zero_nat ) ) ) ) ).

% nat_eq_iff
thf(fact_2651_nat__eq__iff2,axiom,
    ! [M: nat,W2: int] :
      ( ( M
        = ( nat2 @ W2 ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ W2 )
         => ( W2
            = ( semiri1314217659103216013at_int @ M ) ) )
        & ( ~ ( ord_less_eq_int @ zero_zero_int @ W2 )
         => ( M = zero_zero_nat ) ) ) ) ).

% nat_eq_iff2
thf(fact_2652_divide__le__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ A )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ B @ ( 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 ) @ B ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).

% divide_le_eq
thf(fact_2653_le__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ B @ ( 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_2654_divide__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
         => ( ord_less_eq_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).

% divide_left_mono
thf(fact_2655_neg__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ A )
        = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).

% neg_divide_le_eq
thf(fact_2656_neg__le__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ C ) )
        = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).

% neg_le_divide_eq
thf(fact_2657_pos__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ A )
        = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).

% pos_divide_le_eq
thf(fact_2658_pos__le__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ C ) )
        = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).

% pos_le_divide_eq
thf(fact_2659_mult__imp__div__pos__le,axiom,
    ! [Y3: rat,X: rat,Z2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Y3 )
     => ( ( ord_less_eq_rat @ X @ ( times_times_rat @ Z2 @ Y3 ) )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y3 ) @ Z2 ) ) ) ).

% mult_imp_div_pos_le
thf(fact_2660_mult__imp__le__div__pos,axiom,
    ! [Y3: rat,Z2: rat,X: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Y3 )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ Z2 @ Y3 ) @ X )
       => ( ord_less_eq_rat @ Z2 @ ( divide_divide_rat @ X @ Y3 ) ) ) ) ).

% mult_imp_le_div_pos
thf(fact_2661_divide__left__mono__neg,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
       => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
         => ( ord_less_eq_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).

% divide_left_mono_neg
thf(fact_2662_split__nat,axiom,
    ! [P: nat > $o,I2: int] :
      ( ( P @ ( nat2 @ I2 ) )
      = ( ! [N3: nat] :
            ( ( I2
              = ( semiri1314217659103216013at_int @ N3 ) )
           => ( P @ N3 ) )
        & ( ( ord_less_int @ I2 @ zero_zero_int )
         => ( P @ zero_zero_nat ) ) ) ) ).

% split_nat
thf(fact_2663_divide__le__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A )
          & ( ord_less_eq_rat @ B @ A ) )
        | ( ( ord_less_rat @ A @ zero_zero_rat )
          & ( ord_less_eq_rat @ A @ B ) )
        | ( A = zero_zero_rat ) ) ) ).

% divide_le_eq_1
thf(fact_2664_le__divide__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A )
          & ( ord_less_eq_rat @ A @ B ) )
        | ( ( ord_less_rat @ A @ zero_zero_rat )
          & ( ord_less_eq_rat @ B @ A ) ) ) ) ).

% le_divide_eq_1
thf(fact_2665_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_2666_nat__mult__distrib,axiom,
    ! [Z2: int,Z5: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
     => ( ( nat2 @ ( times_times_int @ Z2 @ Z5 ) )
        = ( times_times_nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z5 ) ) ) ) ).

% nat_mult_distrib
thf(fact_2667_less__minus__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( 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_2668_minus__divide__less__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( 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 @ B ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).

% minus_divide_less_eq
thf(fact_2669_neg__less__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
        = ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).

% neg_less_minus_divide_eq
thf(fact_2670_neg__minus__divide__less__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
        = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).

% neg_minus_divide_less_eq
thf(fact_2671_pos__less__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
        = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).

% pos_less_minus_divide_eq
thf(fact_2672_pos__minus__divide__less__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
        = ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).

% pos_minus_divide_less_eq
thf(fact_2673_div__nat__eqI,axiom,
    ! [N: nat,Q4: nat,M: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q4 ) @ M )
     => ( ( ord_less_nat @ M @ ( times_times_nat @ N @ ( suc @ Q4 ) ) )
       => ( ( divide_divide_nat @ M @ N )
          = Q4 ) ) ) ).

% div_nat_eqI
thf(fact_2674_less__eq__div__iff__mult__less__eq,axiom,
    ! [Q4: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ Q4 )
     => ( ( ord_less_eq_nat @ M @ ( divide_divide_nat @ N @ Q4 ) )
        = ( ord_less_eq_nat @ ( times_times_nat @ M @ Q4 ) @ N ) ) ) ).

% less_eq_div_iff_mult_less_eq
thf(fact_2675_sgn__if,axiom,
    ( sgn_sgn_int
    = ( ^ [X2: int] : ( if_int @ ( X2 = zero_zero_int ) @ zero_zero_int @ ( if_int @ ( ord_less_int @ zero_zero_int @ X2 ) @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).

% sgn_if
thf(fact_2676_sgn__if,axiom,
    ( sgn_sgn_Code_integer
    = ( ^ [X2: code_integer] : ( if_Code_integer @ ( X2 = zero_z3403309356797280102nteger ) @ zero_z3403309356797280102nteger @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ X2 ) @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ) ).

% sgn_if
thf(fact_2677_sgn__if,axiom,
    ( sgn_sgn_rat
    = ( ^ [X2: rat] : ( if_rat @ ( X2 = zero_zero_rat ) @ zero_zero_rat @ ( if_rat @ ( ord_less_rat @ zero_zero_rat @ X2 ) @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ) ) ) ).

% sgn_if
thf(fact_2678_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_2679_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_2680_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_2681_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_2682_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_2683_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_2684_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_2685_unique__euclidean__semiring__class_Odiv__eq__0__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( ( euclid3398187327856392827nt_nat @ A )
        = ( euclid3398187327856392827nt_nat @ B ) )
     => ( ( ( divide_divide_nat @ A @ B )
          = zero_zero_nat )
        = ( ( ord_less_nat @ ( euclid4777050414544973029ze_nat @ A ) @ ( euclid4777050414544973029ze_nat @ B ) )
          | ( B = zero_zero_nat ) ) ) ) ).

% unique_euclidean_semiring_class.div_eq_0_iff
thf(fact_2686_unique__euclidean__semiring__class_Odiv__eq__0__iff,axiom,
    ! [A: int,B: int] :
      ( ( ( euclid3395696857347342551nt_int @ A )
        = ( euclid3395696857347342551nt_int @ B ) )
     => ( ( ( divide_divide_int @ A @ B )
          = zero_zero_int )
        = ( ( ord_less_nat @ ( euclid4774559944035922753ze_int @ A ) @ ( euclid4774559944035922753ze_int @ B ) )
          | ( B = zero_zero_int ) ) ) ) ).

% unique_euclidean_semiring_class.div_eq_0_iff
thf(fact_2687_unique__euclidean__semiring__class_Odiv__eq__0__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( euclid6289375185220004616nteger @ A )
        = ( euclid6289375185220004616nteger @ B ) )
     => ( ( ( divide6298287555418463151nteger @ A @ B )
          = zero_z3403309356797280102nteger )
        = ( ( ord_less_nat @ ( euclid6377331345833325938nteger @ A ) @ ( euclid6377331345833325938nteger @ B ) )
          | ( B = zero_z3403309356797280102nteger ) ) ) ) ).

% unique_euclidean_semiring_class.div_eq_0_iff
thf(fact_2688_unique__euclidean__semiring__class_Odiv__eq__0__iff,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( ( euclid5112970336976722131atural @ A )
        = ( euclid5112970336976722131atural @ B ) )
     => ( ( ( divide5121882707175180666atural @ A @ B )
          = zero_z2226904508553997617atural )
        = ( ( ord_less_nat @ ( euclid5200926497590043453atural @ A ) @ ( euclid5200926497590043453atural @ B ) )
          | ( B = zero_z2226904508553997617atural ) ) ) ) ).

% unique_euclidean_semiring_class.div_eq_0_iff
thf(fact_2689_nat__less__iff,axiom,
    ! [W2: int,M: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ W2 )
     => ( ( ord_less_nat @ ( nat2 @ W2 ) @ M )
        = ( ord_less_int @ W2 @ ( semiri1314217659103216013at_int @ M ) ) ) ) ).

% nat_less_iff
thf(fact_2690_le__minus__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( 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_2691_minus__divide__le__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( 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 @ B ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).

% minus_divide_le_eq
thf(fact_2692_neg__le__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
        = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).

% neg_le_minus_divide_eq
thf(fact_2693_neg__minus__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
        = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).

% neg_minus_divide_le_eq
thf(fact_2694_pos__le__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
        = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).

% pos_le_minus_divide_eq
thf(fact_2695_pos__minus__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
        = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).

% pos_minus_divide_le_eq
thf(fact_2696_nat__mult__distrib__neg,axiom,
    ! [Z2: int,Z5: int] :
      ( ( ord_less_eq_int @ Z2 @ zero_zero_int )
     => ( ( nat2 @ ( times_times_int @ Z2 @ Z5 ) )
        = ( times_times_nat @ ( nat2 @ ( uminus_uminus_int @ Z2 ) ) @ ( nat2 @ ( uminus_uminus_int @ Z5 ) ) ) ) ) ).

% nat_mult_distrib_neg
thf(fact_2697_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_2698_mult__ceiling__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
       => ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ ( times_times_rat @ A @ B ) ) @ ( times_times_int @ ( archim2889992004027027881ng_rat @ A ) @ ( archim2889992004027027881ng_rat @ B ) ) ) ) ) ).

% mult_ceiling_le
thf(fact_2699_bits__mod__div__trivial,axiom,
    ! [A: nat,B: nat] :
      ( ( divide_divide_nat @ ( modulo_modulo_nat @ A @ B ) @ B )
      = zero_zero_nat ) ).

% bits_mod_div_trivial
thf(fact_2700_bits__mod__div__trivial,axiom,
    ! [A: int,B: int] :
      ( ( divide_divide_int @ ( modulo_modulo_int @ A @ B ) @ B )
      = zero_zero_int ) ).

% bits_mod_div_trivial
thf(fact_2701_bits__mod__div__trivial,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B )
      = zero_z3403309356797280102nteger ) ).

% bits_mod_div_trivial
thf(fact_2702_bits__mod__div__trivial,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( divide5121882707175180666atural @ ( modulo8411746178871703098atural @ A @ B ) @ B )
      = zero_z2226904508553997617atural ) ).

% bits_mod_div_trivial
thf(fact_2703_bits__mod__by__1,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ A @ one_one_nat )
      = zero_zero_nat ) ).

% bits_mod_by_1
thf(fact_2704_bits__mod__by__1,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ A @ one_one_int )
      = zero_zero_int ) ).

% bits_mod_by_1
thf(fact_2705_bits__mod__by__1,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger @ A @ one_one_Code_integer )
      = zero_z3403309356797280102nteger ) ).

% bits_mod_by_1
thf(fact_2706_bits__mod__by__1,axiom,
    ! [A: code_natural] :
      ( ( modulo8411746178871703098atural @ A @ one_one_Code_natural )
      = zero_z2226904508553997617atural ) ).

% bits_mod_by_1
thf(fact_2707_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_2708_choose__dvd,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( dvd_dvd_Code_natural @ ( times_2397367101498566445atural @ ( semiri2447717529341329178atural @ K ) @ ( semiri2447717529341329178atural @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri2447717529341329178atural @ N ) ) ) ).

% choose_dvd
thf(fact_2709_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_2710_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_2711_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_2712_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_2713_bits__mod__0,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% bits_mod_0
thf(fact_2714_bits__mod__0,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% bits_mod_0
thf(fact_2715_bits__mod__0,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger @ zero_z3403309356797280102nteger @ A )
      = zero_z3403309356797280102nteger ) ).

% bits_mod_0
thf(fact_2716_bits__mod__0,axiom,
    ! [A: code_natural] :
      ( ( modulo8411746178871703098atural @ zero_z2226904508553997617atural @ A )
      = zero_z2226904508553997617atural ) ).

% bits_mod_0
thf(fact_2717_bits__div__by__1,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ A @ one_one_nat )
      = A ) ).

% bits_div_by_1
thf(fact_2718_bits__div__by__1,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ A @ one_one_int )
      = A ) ).

% bits_div_by_1
thf(fact_2719_bits__div__by__1,axiom,
    ! [A: code_integer] :
      ( ( divide6298287555418463151nteger @ A @ one_one_Code_integer )
      = A ) ).

% bits_div_by_1
thf(fact_2720_bits__div__by__1,axiom,
    ! [A: code_natural] :
      ( ( divide5121882707175180666atural @ A @ one_one_Code_natural )
      = A ) ).

% bits_div_by_1
thf(fact_2721_bits__div__by__0,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ A @ zero_zero_nat )
      = zero_zero_nat ) ).

% bits_div_by_0
thf(fact_2722_bits__div__by__0,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ A @ zero_zero_int )
      = zero_zero_int ) ).

% bits_div_by_0
thf(fact_2723_bits__div__by__0,axiom,
    ! [A: code_integer] :
      ( ( divide6298287555418463151nteger @ A @ zero_z3403309356797280102nteger )
      = zero_z3403309356797280102nteger ) ).

% bits_div_by_0
thf(fact_2724_bits__div__by__0,axiom,
    ! [A: code_natural] :
      ( ( divide5121882707175180666atural @ A @ zero_z2226904508553997617atural )
      = zero_z2226904508553997617atural ) ).

% bits_div_by_0
thf(fact_2725_bits__div__0,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% bits_div_0
thf(fact_2726_bits__div__0,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% bits_div_0
thf(fact_2727_bits__div__0,axiom,
    ! [A: code_integer] :
      ( ( divide6298287555418463151nteger @ zero_z3403309356797280102nteger @ A )
      = zero_z3403309356797280102nteger ) ).

% bits_div_0
thf(fact_2728_bits__div__0,axiom,
    ! [A: code_natural] :
      ( ( divide5121882707175180666atural @ zero_z2226904508553997617atural @ A )
      = zero_z2226904508553997617atural ) ).

% bits_div_0
thf(fact_2729_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le6747313008572928689nteger @ A @ B )
       => ( ( modulo364778990260209775nteger @ A @ B )
          = A ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
thf(fact_2730_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ A @ B )
       => ( ( modulo_modulo_nat @ A @ B )
          = A ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
thf(fact_2731_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ A @ B )
       => ( ( modulo_modulo_int @ A @ B )
          = A ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
thf(fact_2732_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
     => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
thf(fact_2733_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( modulo_modulo_nat @ A @ B ) ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
thf(fact_2734_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ A @ B ) ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
thf(fact_2735_mod__neg__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_eq_int @ K @ zero_zero_int )
     => ( ( ord_less_int @ L @ K )
       => ( ( modulo_modulo_int @ K @ L )
          = K ) ) ) ).

% mod_neg_neg_trivial
thf(fact_2736_mod__pos__pos__trivial,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ( ( ord_less_int @ K @ L )
       => ( ( modulo_modulo_int @ K @ L )
          = K ) ) ) ).

% mod_pos_pos_trivial
thf(fact_2737_div__neg__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_eq_int @ K @ zero_zero_int )
     => ( ( ord_less_int @ L @ K )
       => ( ( divide_divide_int @ K @ L )
          = zero_zero_int ) ) ) ).

% div_neg_neg_trivial
thf(fact_2738_div__pos__pos__trivial,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ( ( ord_less_int @ K @ L )
       => ( ( divide_divide_int @ K @ L )
          = zero_zero_int ) ) ) ).

% div_pos_pos_trivial
thf(fact_2739_zle__diff1__eq,axiom,
    ! [W2: int,Z2: int] :
      ( ( ord_less_eq_int @ W2 @ ( minus_minus_int @ Z2 @ one_one_int ) )
      = ( ord_less_int @ W2 @ Z2 ) ) ).

% zle_diff1_eq
thf(fact_2740_dvd__antisym,axiom,
    ! [M: nat,N: nat] :
      ( ( dvd_dvd_nat @ M @ N )
     => ( ( dvd_dvd_nat @ N @ M )
       => ( M = N ) ) ) ).

% dvd_antisym
thf(fact_2741_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_2742_zmod__zminus1__eq__if,axiom,
    ! [A: int,B: int] :
      ( ( ( ( modulo_modulo_int @ A @ B )
          = zero_zero_int )
       => ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B )
          = zero_zero_int ) )
      & ( ( ( modulo_modulo_int @ A @ B )
         != zero_zero_int )
       => ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B )
          = ( minus_minus_int @ B @ ( modulo_modulo_int @ A @ B ) ) ) ) ) ).

% zmod_zminus1_eq_if
thf(fact_2743_zmod__zminus2__eq__if,axiom,
    ! [A: int,B: int] :
      ( ( ( ( modulo_modulo_int @ A @ B )
          = zero_zero_int )
       => ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ B ) )
          = zero_zero_int ) )
      & ( ( ( modulo_modulo_int @ A @ B )
         != zero_zero_int )
       => ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ B ) )
          = ( minus_minus_int @ ( modulo_modulo_int @ A @ B ) @ B ) ) ) ) ).

% zmod_zminus2_eq_if
thf(fact_2744_zdiv__zminus1__eq__if,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( ( ( ( modulo_modulo_int @ A @ B )
            = zero_zero_int )
         => ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B )
            = ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) ) )
        & ( ( ( modulo_modulo_int @ A @ B )
           != zero_zero_int )
         => ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B )
            = ( minus_minus_int @ ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) @ one_one_int ) ) ) ) ) ).

% zdiv_zminus1_eq_if
thf(fact_2745_zdiv__zminus2__eq__if,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( ( ( ( modulo_modulo_int @ A @ B )
            = zero_zero_int )
         => ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
            = ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) ) )
        & ( ( ( modulo_modulo_int @ A @ B )
           != zero_zero_int )
         => ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
            = ( minus_minus_int @ ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) @ one_one_int ) ) ) ) ) ).

% zdiv_zminus2_eq_if
thf(fact_2746_zdiv__mono__strict,axiom,
    ! [A3: int,B5: int,N: int] :
      ( ( ord_less_int @ A3 @ B5 )
     => ( ( ord_less_int @ zero_zero_int @ N )
       => ( ( ( modulo_modulo_int @ A3 @ N )
            = zero_zero_int )
         => ( ( ( modulo_modulo_int @ B5 @ N )
              = zero_zero_int )
           => ( ord_less_int @ ( divide_divide_int @ A3 @ N ) @ ( divide_divide_int @ B5 @ N ) ) ) ) ) ) ).

% zdiv_mono_strict
thf(fact_2747_mod__pos__geq,axiom,
    ! [L: int,K: int] :
      ( ( ord_less_int @ zero_zero_int @ L )
     => ( ( ord_less_eq_int @ L @ K )
       => ( ( modulo_modulo_int @ K @ L )
          = ( modulo_modulo_int @ ( minus_minus_int @ K @ L ) @ L ) ) ) ) ).

% mod_pos_geq
thf(fact_2748_sgn__mod,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero_int )
     => ( ~ ( dvd_dvd_int @ L @ K )
       => ( ( sgn_sgn_int @ ( modulo_modulo_int @ K @ L ) )
          = ( sgn_sgn_int @ L ) ) ) ) ).

% sgn_mod
thf(fact_2749_minus__int__code_I1_J,axiom,
    ! [K: int] :
      ( ( minus_minus_int @ K @ zero_zero_int )
      = K ) ).

% minus_int_code(1)
thf(fact_2750_int__distrib_I3_J,axiom,
    ! [Z1: int,Z22: int,W2: int] :
      ( ( times_times_int @ ( minus_minus_int @ Z1 @ Z22 ) @ W2 )
      = ( minus_minus_int @ ( times_times_int @ Z1 @ W2 ) @ ( times_times_int @ Z22 @ W2 ) ) ) ).

% int_distrib(3)
thf(fact_2751_int__distrib_I4_J,axiom,
    ! [W2: int,Z1: int,Z22: int] :
      ( ( times_times_int @ W2 @ ( minus_minus_int @ Z1 @ Z22 ) )
      = ( minus_minus_int @ ( times_times_int @ W2 @ Z1 ) @ ( times_times_int @ W2 @ Z22 ) ) ) ).

% int_distrib(4)
thf(fact_2752_int__diff__cases,axiom,
    ! [Z2: int] :
      ~ ! [M6: nat,N4: nat] :
          ( Z2
         != ( minus_minus_int @ ( semiri1314217659103216013at_int @ M6 ) @ ( semiri1314217659103216013at_int @ N4 ) ) ) ).

% int_diff_cases
thf(fact_2753_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_2754_zmod__eq__0__iff,axiom,
    ! [M: int,D: int] :
      ( ( ( modulo_modulo_int @ M @ D )
        = zero_zero_int )
      = ( ? [Q3: int] :
            ( M
            = ( times_times_int @ D @ Q3 ) ) ) ) ).

% zmod_eq_0_iff
thf(fact_2755_zmod__eq__0D,axiom,
    ! [M: int,D: int] :
      ( ( ( modulo_modulo_int @ M @ D )
        = zero_zero_int )
     => ? [Q5: int] :
          ( M
          = ( times_times_int @ D @ Q5 ) ) ) ).

% zmod_eq_0D
thf(fact_2756_zmod__zminus2__not__zero,axiom,
    ! [K: int,L: int] :
      ( ( ( modulo_modulo_int @ K @ ( uminus_uminus_int @ L ) )
       != zero_zero_int )
     => ( ( modulo_modulo_int @ K @ L )
       != zero_zero_int ) ) ).

% zmod_zminus2_not_zero
thf(fact_2757_zmod__zminus1__not__zero,axiom,
    ! [K: int,L: int] :
      ( ( ( modulo_modulo_int @ ( uminus_uminus_int @ K ) @ L )
       != zero_zero_int )
     => ( ( modulo_modulo_int @ K @ L )
       != zero_zero_int ) ) ).

% zmod_zminus1_not_zero
thf(fact_2758_div__neg__pos__less0,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).

% div_neg_pos_less0
thf(fact_2759_neg__imp__zdiv__neg__iff,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ zero_zero_int )
     => ( ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int )
        = ( ord_less_int @ zero_zero_int @ A ) ) ) ).

% neg_imp_zdiv_neg_iff
thf(fact_2760_pos__imp__zdiv__neg__iff,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int )
        = ( ord_less_int @ A @ zero_zero_int ) ) ) ).

% pos_imp_zdiv_neg_iff
thf(fact_2761_zmod__int,axiom,
    ! [A: nat,B: nat] :
      ( ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ A @ B ) )
      = ( modulo_modulo_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).

% zmod_int
thf(fact_2762_zdiv__int,axiom,
    ! [A: nat,B: nat] :
      ( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ A @ B ) )
      = ( divide_divide_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).

% zdiv_int
thf(fact_2763_minus__mod__int__eq,axiom,
    ! [L: int,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ L )
     => ( ( modulo_modulo_int @ ( uminus_uminus_int @ K ) @ L )
        = ( minus_minus_int @ ( minus_minus_int @ L @ one_one_int ) @ ( modulo_modulo_int @ ( minus_minus_int @ K @ one_one_int ) @ L ) ) ) ) ).

% minus_mod_int_eq
thf(fact_2764_zmod__minus1,axiom,
    ! [B: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ B )
        = ( minus_minus_int @ B @ one_one_int ) ) ) ).

% zmod_minus1
thf(fact_2765_neg__mod__conj,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ zero_zero_int )
     => ( ( ord_less_eq_int @ ( modulo_modulo_int @ A @ B ) @ zero_zero_int )
        & ( ord_less_int @ B @ ( modulo_modulo_int @ A @ B ) ) ) ) ).

% neg_mod_conj
thf(fact_2766_pos__mod__conj,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ A @ B ) )
        & ( ord_less_int @ ( modulo_modulo_int @ A @ B ) @ B ) ) ) ).

% pos_mod_conj
thf(fact_2767_zmod__trivial__iff,axiom,
    ! [I2: int,K: int] :
      ( ( ( modulo_modulo_int @ I2 @ K )
        = I2 )
      = ( ( K = zero_zero_int )
        | ( ( ord_less_eq_int @ zero_zero_int @ I2 )
          & ( ord_less_int @ I2 @ K ) )
        | ( ( ord_less_eq_int @ I2 @ zero_zero_int )
          & ( ord_less_int @ K @ I2 ) ) ) ) ).

% zmod_trivial_iff
thf(fact_2768_mod__int__pos__iff,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ K @ L ) )
      = ( ( dvd_dvd_int @ L @ K )
        | ( ( L = zero_zero_int )
          & ( ord_less_eq_int @ zero_zero_int @ K ) )
        | ( ord_less_int @ zero_zero_int @ L ) ) ) ).

% mod_int_pos_iff
thf(fact_2769_neg__mod__bound,axiom,
    ! [L: int,K: int] :
      ( ( ord_less_int @ L @ zero_zero_int )
     => ( ord_less_int @ L @ ( modulo_modulo_int @ K @ L ) ) ) ).

% neg_mod_bound
thf(fact_2770_Euclidean__Division_Opos__mod__bound,axiom,
    ! [L: int,K: int] :
      ( ( ord_less_int @ zero_zero_int @ L )
     => ( ord_less_int @ ( modulo_modulo_int @ K @ L ) @ L ) ) ).

% Euclidean_Division.pos_mod_bound
thf(fact_2771_zdiv__mono1,axiom,
    ! [A: int,A4: int,B: int] :
      ( ( ord_less_eq_int @ A @ A4 )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A4 @ B ) ) ) ) ).

% zdiv_mono1
thf(fact_2772_zdiv__mono2,axiom,
    ! [A: int,B3: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ zero_zero_int @ B3 )
       => ( ( ord_less_eq_int @ B3 @ B )
         => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A @ B3 ) ) ) ) ) ).

% zdiv_mono2
thf(fact_2773_zdiv__eq__0__iff,axiom,
    ! [I2: int,K: int] :
      ( ( ( divide_divide_int @ I2 @ K )
        = zero_zero_int )
      = ( ( K = zero_zero_int )
        | ( ( ord_less_eq_int @ zero_zero_int @ I2 )
          & ( ord_less_int @ I2 @ K ) )
        | ( ( ord_less_eq_int @ I2 @ zero_zero_int )
          & ( ord_less_int @ K @ I2 ) ) ) ) ).

% zdiv_eq_0_iff
thf(fact_2774_zdiv__mono1__neg,axiom,
    ! [A: int,A4: int,B: int] :
      ( ( ord_less_eq_int @ A @ A4 )
     => ( ( ord_less_int @ B @ zero_zero_int )
       => ( ord_less_eq_int @ ( divide_divide_int @ A4 @ B ) @ ( divide_divide_int @ A @ B ) ) ) ) ).

% zdiv_mono1_neg
thf(fact_2775_zdiv__mono2__neg,axiom,
    ! [A: int,B3: int,B: int] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ zero_zero_int @ B3 )
       => ( ( ord_less_eq_int @ B3 @ B )
         => ( ord_less_eq_int @ ( divide_divide_int @ A @ B3 ) @ ( divide_divide_int @ A @ B ) ) ) ) ) ).

% zdiv_mono2_neg
thf(fact_2776_div__int__pos__iff,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ L ) )
      = ( ( K = zero_zero_int )
        | ( L = zero_zero_int )
        | ( ( ord_less_eq_int @ zero_zero_int @ K )
          & ( ord_less_eq_int @ zero_zero_int @ L ) )
        | ( ( ord_less_int @ K @ zero_zero_int )
          & ( ord_less_int @ L @ zero_zero_int ) ) ) ) ).

% div_int_pos_iff
thf(fact_2777_div__positive__int,axiom,
    ! [L: int,K: int] :
      ( ( ord_less_eq_int @ L @ K )
     => ( ( ord_less_int @ zero_zero_int @ L )
       => ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ K @ L ) ) ) ) ).

% div_positive_int
thf(fact_2778_div__nonneg__neg__le0,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ B @ zero_zero_int )
       => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).

% div_nonneg_neg_le0
thf(fact_2779_div__nonpos__pos__le0,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).

% div_nonpos_pos_le0
thf(fact_2780_pos__imp__zdiv__pos__iff,axiom,
    ! [K: int,I2: int] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ I2 @ K ) )
        = ( ord_less_eq_int @ K @ I2 ) ) ) ).

% pos_imp_zdiv_pos_iff
thf(fact_2781_neg__imp__zdiv__nonneg__iff,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ zero_zero_int )
     => ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
        = ( ord_less_eq_int @ A @ zero_zero_int ) ) ) ).

% neg_imp_zdiv_nonneg_iff
thf(fact_2782_pos__imp__zdiv__nonneg__iff,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
        = ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).

% pos_imp_zdiv_nonneg_iff
thf(fact_2783_nonneg1__imp__zdiv__pos__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
        = ( ( ord_less_eq_int @ B @ A )
          & ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).

% nonneg1_imp_zdiv_pos_iff
thf(fact_2784_int__div__less__self,axiom,
    ! [X: int,K: int] :
      ( ( ord_less_int @ zero_zero_int @ X )
     => ( ( ord_less_int @ one_one_int @ K )
       => ( ord_less_int @ ( divide_divide_int @ X @ K ) @ X ) ) ) ).

% int_div_less_self
thf(fact_2785_minus__int__code_I2_J,axiom,
    ! [L: int] :
      ( ( minus_minus_int @ zero_zero_int @ L )
      = ( uminus_uminus_int @ L ) ) ).

% minus_int_code(2)
thf(fact_2786_int__le__induct,axiom,
    ! [I2: int,K: int,P: int > $o] :
      ( ( ord_less_eq_int @ I2 @ K )
     => ( ( P @ K )
       => ( ! [I3: int] :
              ( ( ord_less_eq_int @ I3 @ K )
             => ( ( P @ I3 )
               => ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
         => ( P @ I2 ) ) ) ) ).

% int_le_induct
thf(fact_2787_int__less__induct,axiom,
    ! [I2: int,K: int,P: int > $o] :
      ( ( ord_less_int @ I2 @ 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 @ I2 ) ) ) ) ).

% int_less_induct
thf(fact_2788_div__eq__minus1,axiom,
    ! [B: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ B )
        = ( uminus_uminus_int @ one_one_int ) ) ) ).

% div_eq_minus1
thf(fact_2789_neg__mod__sign,axiom,
    ! [L: int,K: int] :
      ( ( ord_less_int @ L @ zero_zero_int )
     => ( ord_less_eq_int @ ( modulo_modulo_int @ K @ L ) @ zero_zero_int ) ) ).

% neg_mod_sign
thf(fact_2790_Euclidean__Division_Opos__mod__sign,axiom,
    ! [L: int,K: int] :
      ( ( ord_less_int @ zero_zero_int @ L )
     => ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ K @ L ) ) ) ).

% Euclidean_Division.pos_mod_sign
thf(fact_2791_plusinfinity,axiom,
    ! [D: int,P2: int > $o,P: int > $o] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ! [X3: int,K2: int] :
            ( ( P2 @ X3 )
            = ( P2 @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D ) ) ) )
       => ( ? [Z4: int] :
            ! [X3: int] :
              ( ( ord_less_int @ Z4 @ X3 )
             => ( ( P @ X3 )
                = ( P2 @ X3 ) ) )
         => ( ? [X_1: int] : ( P2 @ X_1 )
           => ? [X_12: int] : ( P @ X_12 ) ) ) ) ) ).

% plusinfinity
thf(fact_2792_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 ) ) ) )
       => ( ? [Z4: int] :
            ! [X3: int] :
              ( ( ord_less_int @ X3 @ Z4 )
             => ( ( P @ X3 )
                = ( P1 @ X3 ) ) )
         => ( ? [X_1: int] : ( P1 @ X_1 )
           => ? [X_12: int] : ( P @ X_12 ) ) ) ) ) ).

% minusinfinity
thf(fact_2793_zdiv__zmult2__eq,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ C )
     => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
        = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ).

% zdiv_zmult2_eq
thf(fact_2794_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_2795_nat__mod__distrib,axiom,
    ! [X: int,Y3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
       => ( ( nat2 @ ( modulo_modulo_int @ X @ Y3 ) )
          = ( modulo_modulo_nat @ ( nat2 @ X ) @ ( nat2 @ Y3 ) ) ) ) ) ).

% nat_mod_distrib
thf(fact_2796_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 )
         => ! [X4: int] :
              ( ( P @ X4 )
             => ( P @ ( minus_minus_int @ X4 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).

% decr_mult_lemma
thf(fact_2797_int__ops_I6_J,axiom,
    ! [A: nat,B: nat] :
      ( ( ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
       => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
          = zero_zero_int ) )
      & ( ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
       => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
          = ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ) ) ).

% int_ops(6)
thf(fact_2798_nat__diff__distrib,axiom,
    ! [Z5: int,Z2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z5 )
     => ( ( ord_less_eq_int @ Z5 @ Z2 )
       => ( ( nat2 @ ( minus_minus_int @ Z2 @ Z5 ) )
          = ( minus_minus_nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z5 ) ) ) ) ) ).

% nat_diff_distrib
thf(fact_2799_nat__diff__distrib_H,axiom,
    ! [X: int,Y3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
       => ( ( nat2 @ ( minus_minus_int @ X @ Y3 ) )
          = ( minus_minus_nat @ ( nat2 @ X ) @ ( nat2 @ Y3 ) ) ) ) ) ).

% nat_diff_distrib'
thf(fact_2800_verit__less__mono__div__int2,axiom,
    ! [A3: int,B5: int,N: int] :
      ( ( ord_less_eq_int @ A3 @ B5 )
     => ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ N ) )
       => ( ord_less_eq_int @ ( divide_divide_int @ B5 @ N ) @ ( divide_divide_int @ A3 @ N ) ) ) ) ).

% verit_less_mono_div_int2
thf(fact_2801_nat__div__distrib_H,axiom,
    ! [Y3: int,X: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
     => ( ( nat2 @ ( divide_divide_int @ X @ Y3 ) )
        = ( divide_divide_nat @ ( nat2 @ X ) @ ( nat2 @ Y3 ) ) ) ) ).

% nat_div_distrib'
thf(fact_2802_nat__div__distrib,axiom,
    ! [X: int,Y3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( nat2 @ ( divide_divide_int @ X @ Y3 ) )
        = ( divide_divide_nat @ ( nat2 @ X ) @ ( nat2 @ Y3 ) ) ) ) ).

% nat_div_distrib
thf(fact_2803_zdiff__int__split,axiom,
    ! [P: int > $o,X: nat,Y3: nat] :
      ( ( P @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X @ Y3 ) ) )
      = ( ( ( ord_less_eq_nat @ Y3 @ X )
         => ( P @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X ) @ ( semiri1314217659103216013at_int @ Y3 ) ) ) )
        & ( ( ord_less_nat @ X @ Y3 )
         => ( P @ zero_zero_int ) ) ) ) ).

% zdiff_int_split
thf(fact_2804_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ord_le3102999989581377725nteger @ ( modulo364778990260209775nteger @ A @ B ) @ A ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
thf(fact_2805_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ord_less_eq_nat @ ( modulo_modulo_nat @ A @ B ) @ A ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
thf(fact_2806_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ord_less_eq_int @ ( modulo_modulo_int @ A @ B ) @ A ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
thf(fact_2807_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B )
     => ( ord_less_nat @ ( modulo_modulo_nat @ A @ B ) @ B ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
thf(fact_2808_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ord_less_int @ ( modulo_modulo_int @ A @ B ) @ B ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
thf(fact_2809_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
     => ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
thf(fact_2810_mod__eq__0D,axiom,
    ! [M: nat,D: nat] :
      ( ( ( modulo_modulo_nat @ M @ D )
        = zero_zero_nat )
     => ? [Q5: nat] :
          ( M
          = ( times_times_nat @ D @ Q5 ) ) ) ).

% mod_eq_0D
thf(fact_2811_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_2812_mod__eq__dvd__iff__nat,axiom,
    ! [N: nat,M: nat,Q4: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( ( modulo_modulo_nat @ M @ Q4 )
          = ( modulo_modulo_nat @ N @ Q4 ) )
        = ( dvd_dvd_nat @ Q4 @ ( minus_minus_nat @ M @ N ) ) ) ) ).

% mod_eq_dvd_iff_nat
thf(fact_2813_div__positive,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
     => ( ( ord_le3102999989581377725nteger @ B @ A )
       => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).

% div_positive
thf(fact_2814_div__positive,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B )
     => ( ( ord_less_eq_nat @ B @ A )
       => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ A @ B ) ) ) ) ).

% div_positive
thf(fact_2815_div__positive,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_eq_int @ B @ A )
       => ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) ) ) ) ).

% div_positive
thf(fact_2816_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le6747313008572928689nteger @ A @ B )
       => ( ( divide6298287555418463151nteger @ A @ B )
          = zero_z3403309356797280102nteger ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
thf(fact_2817_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ A @ B )
       => ( ( divide_divide_nat @ A @ B )
          = zero_zero_nat ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
thf(fact_2818_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ A @ B )
       => ( ( divide_divide_int @ A @ B )
          = zero_zero_int ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
thf(fact_2819_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
     => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
        = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
thf(fact_2820_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ C )
     => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
        = ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
thf(fact_2821_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ C )
     => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
        = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
thf(fact_2822_bezout1__nat,axiom,
    ! [A: nat,B: nat] :
    ? [D5: nat,X3: nat,Y4: nat] :
      ( ( dvd_dvd_nat @ D5 @ A )
      & ( dvd_dvd_nat @ D5 @ B )
      & ( ( ( minus_minus_nat @ ( times_times_nat @ A @ X3 ) @ ( times_times_nat @ B @ Y4 ) )
          = D5 )
        | ( ( minus_minus_nat @ ( times_times_nat @ B @ X3 ) @ ( times_times_nat @ A @ Y4 ) )
          = D5 ) ) ) ).

% bezout1_nat
thf(fact_2823_gcd__nat__induct,axiom,
    ! [P: nat > nat > $o,M: nat,N: nat] :
      ( ! [M6: nat] : ( P @ M6 @ zero_zero_nat )
     => ( ! [M6: nat,N4: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N4 )
           => ( ( P @ N4 @ ( modulo_modulo_nat @ M6 @ N4 ) )
             => ( P @ M6 @ N4 ) ) )
       => ( P @ M @ N ) ) ) ).

% gcd_nat_induct
thf(fact_2824_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_2825_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_2826_modulo__int__unfold,axiom,
    ! [L: int,K: int,N: nat,M: nat] :
      ( ( ( ( ( sgn_sgn_int @ L )
            = 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 @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
          = ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) ) )
      & ( ~ ( ( ( sgn_sgn_int @ L )
              = zero_zero_int )
            | ( ( sgn_sgn_int @ K )
              = zero_zero_int )
            | ( N = zero_zero_nat ) )
       => ( ( ( ( sgn_sgn_int @ K )
              = ( sgn_sgn_int @ L ) )
           => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
              = ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N ) ) ) ) )
          & ( ( ( sgn_sgn_int @ K )
             != ( sgn_sgn_int @ L ) )
           => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
              = ( times_times_int @ ( sgn_sgn_int @ L )
                @ ( 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_2827_divmod__cases,axiom,
    ! [B: nat,A: nat] :
      ( ( ( B != zero_zero_nat )
       => ( ( ( modulo_modulo_nat @ A @ B )
            = zero_zero_nat )
         => ( A
           != ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) ) ) )
     => ( ( ( B != zero_zero_nat )
         => ! [Q5: nat,R3: nat] :
              ( ( ( euclid3398187327856392827nt_nat @ R3 )
                = ( euclid3398187327856392827nt_nat @ B ) )
             => ( ( ord_less_nat @ ( euclid4777050414544973029ze_nat @ R3 ) @ ( euclid4777050414544973029ze_nat @ B ) )
               => ( ( R3 != zero_zero_nat )
                 => ( ( ( divide_divide_nat @ A @ B )
                      = Q5 )
                   => ( ( ( modulo_modulo_nat @ A @ B )
                        = R3 )
                     => ( A
                       != ( plus_plus_nat @ ( times_times_nat @ Q5 @ B ) @ R3 ) ) ) ) ) ) ) )
       => ( B = zero_zero_nat ) ) ) ).

% divmod_cases
thf(fact_2828_divmod__cases,axiom,
    ! [B: int,A: int] :
      ( ( ( B != zero_zero_int )
       => ( ( ( modulo_modulo_int @ A @ B )
            = zero_zero_int )
         => ( A
           != ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) ) ) )
     => ( ( ( B != zero_zero_int )
         => ! [Q5: int,R3: int] :
              ( ( ( euclid3395696857347342551nt_int @ R3 )
                = ( euclid3395696857347342551nt_int @ B ) )
             => ( ( ord_less_nat @ ( euclid4774559944035922753ze_int @ R3 ) @ ( euclid4774559944035922753ze_int @ B ) )
               => ( ( R3 != zero_zero_int )
                 => ( ( ( divide_divide_int @ A @ B )
                      = Q5 )
                   => ( ( ( modulo_modulo_int @ A @ B )
                        = R3 )
                     => ( A
                       != ( plus_plus_int @ ( times_times_int @ Q5 @ B ) @ R3 ) ) ) ) ) ) ) )
       => ( B = zero_zero_int ) ) ) ).

% divmod_cases
thf(fact_2829_divmod__cases,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ( B != zero_z3403309356797280102nteger )
       => ( ( ( modulo364778990260209775nteger @ A @ B )
            = zero_z3403309356797280102nteger )
         => ( A
           != ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) ) ) )
     => ( ( ( B != zero_z3403309356797280102nteger )
         => ! [Q5: code_integer,R3: code_integer] :
              ( ( ( euclid6289375185220004616nteger @ R3 )
                = ( euclid6289375185220004616nteger @ B ) )
             => ( ( ord_less_nat @ ( euclid6377331345833325938nteger @ R3 ) @ ( euclid6377331345833325938nteger @ B ) )
               => ( ( R3 != zero_z3403309356797280102nteger )
                 => ( ( ( divide6298287555418463151nteger @ A @ B )
                      = Q5 )
                   => ( ( ( modulo364778990260209775nteger @ A @ B )
                        = R3 )
                     => ( A
                       != ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ Q5 @ B ) @ R3 ) ) ) ) ) ) ) )
       => ( B = zero_z3403309356797280102nteger ) ) ) ).

% divmod_cases
thf(fact_2830_divmod__cases,axiom,
    ! [B: code_natural,A: code_natural] :
      ( ( ( B != zero_z2226904508553997617atural )
       => ( ( ( modulo8411746178871703098atural @ A @ B )
            = zero_z2226904508553997617atural )
         => ( A
           != ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ A @ B ) @ B ) ) ) )
     => ( ( ( B != zero_z2226904508553997617atural )
         => ! [Q5: code_natural,R3: code_natural] :
              ( ( ( euclid5112970336976722131atural @ R3 )
                = ( euclid5112970336976722131atural @ B ) )
             => ( ( ord_less_nat @ ( euclid5200926497590043453atural @ R3 ) @ ( euclid5200926497590043453atural @ B ) )
               => ( ( R3 != zero_z2226904508553997617atural )
                 => ( ( ( divide5121882707175180666atural @ A @ B )
                      = Q5 )
                   => ( ( ( modulo8411746178871703098atural @ A @ B )
                        = R3 )
                     => ( A
                       != ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ Q5 @ B ) @ R3 ) ) ) ) ) ) ) )
       => ( B = zero_z2226904508553997617atural ) ) ) ).

% divmod_cases
thf(fact_2831_gcd__nat_Oextremum,axiom,
    ! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).

% gcd_nat.extremum
thf(fact_2832_add__left__cancel,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( ( plus_p5714425477246183910nteger @ A @ B )
        = ( plus_p5714425477246183910nteger @ A @ C ) )
      = ( B = C ) ) ).

% add_left_cancel
thf(fact_2833_add__left__cancel,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ( plus_plus_rat @ A @ B )
        = ( plus_plus_rat @ A @ C ) )
      = ( B = C ) ) ).

% add_left_cancel
thf(fact_2834_add__left__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( plus_plus_nat @ A @ B )
        = ( plus_plus_nat @ A @ C ) )
      = ( B = C ) ) ).

% add_left_cancel
thf(fact_2835_add__left__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ( plus_plus_int @ A @ B )
        = ( plus_plus_int @ A @ C ) )
      = ( B = C ) ) ).

% add_left_cancel
thf(fact_2836_add__right__cancel,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( ( plus_p5714425477246183910nteger @ B @ A )
        = ( plus_p5714425477246183910nteger @ C @ A ) )
      = ( B = C ) ) ).

% add_right_cancel
thf(fact_2837_add__right__cancel,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ( plus_plus_rat @ B @ A )
        = ( plus_plus_rat @ C @ A ) )
      = ( B = C ) ) ).

% add_right_cancel
thf(fact_2838_add__right__cancel,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ( plus_plus_nat @ B @ A )
        = ( plus_plus_nat @ C @ A ) )
      = ( B = C ) ) ).

% add_right_cancel
thf(fact_2839_add__right__cancel,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ( plus_plus_int @ B @ A )
        = ( plus_plus_int @ C @ A ) )
      = ( B = C ) ) ).

% add_right_cancel
thf(fact_2840_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_2841_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_2842_Nat_Oadd__0__right,axiom,
    ! [M: nat] :
      ( ( plus_plus_nat @ M @ zero_zero_nat )
      = M ) ).

% Nat.add_0_right
thf(fact_2843_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_2844_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_2845_diff__diff__left,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J ) @ K )
      = ( minus_minus_nat @ I2 @ ( plus_plus_nat @ J @ K ) ) ) ).

% diff_diff_left
thf(fact_2846_add__le__cancel__left,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ C @ A ) @ ( plus_p5714425477246183910nteger @ C @ B ) )
      = ( ord_le3102999989581377725nteger @ A @ B ) ) ).

% add_le_cancel_left
thf(fact_2847_add__le__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
      = ( ord_less_eq_rat @ A @ B ) ) ).

% add_le_cancel_left
thf(fact_2848_add__le__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
      = ( ord_less_eq_nat @ A @ B ) ) ).

% add_le_cancel_left
thf(fact_2849_add__le__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
      = ( ord_less_eq_int @ A @ B ) ) ).

% add_le_cancel_left
thf(fact_2850_add__le__cancel__right,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ A @ C ) @ ( plus_p5714425477246183910nteger @ B @ C ) )
      = ( ord_le3102999989581377725nteger @ A @ B ) ) ).

% add_le_cancel_right
thf(fact_2851_add__le__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
      = ( ord_less_eq_rat @ A @ B ) ) ).

% add_le_cancel_right
thf(fact_2852_add__le__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
      = ( ord_less_eq_nat @ A @ B ) ) ).

% add_le_cancel_right
thf(fact_2853_add__le__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
      = ( ord_less_eq_int @ A @ B ) ) ).

% add_le_cancel_right
thf(fact_2854_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_2855_double__eq__0__iff,axiom,
    ! [A: code_integer] :
      ( ( ( plus_p5714425477246183910nteger @ A @ A )
        = zero_z3403309356797280102nteger )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% double_eq_0_iff
thf(fact_2856_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_2857_add_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% add.right_neutral
thf(fact_2858_add_Oright__neutral,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ zero_zero_int )
      = A ) ).

% add.right_neutral
thf(fact_2859_add_Oright__neutral,axiom,
    ! [A: code_integer] :
      ( ( plus_p5714425477246183910nteger @ A @ zero_z3403309356797280102nteger )
      = A ) ).

% add.right_neutral
thf(fact_2860_add_Oright__neutral,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ A @ zero_zero_rat )
      = A ) ).

% add.right_neutral
thf(fact_2861_add_Oright__neutral,axiom,
    ! [A: code_natural] :
      ( ( plus_p4538020629002901425atural @ A @ zero_z2226904508553997617atural )
      = A ) ).

% add.right_neutral
thf(fact_2862_double__zero__sym,axiom,
    ! [A: int] :
      ( ( zero_zero_int
        = ( plus_plus_int @ A @ A ) )
      = ( A = zero_zero_int ) ) ).

% double_zero_sym
thf(fact_2863_double__zero__sym,axiom,
    ! [A: code_integer] :
      ( ( zero_z3403309356797280102nteger
        = ( plus_p5714425477246183910nteger @ A @ A ) )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% double_zero_sym
thf(fact_2864_double__zero__sym,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat
        = ( plus_plus_rat @ A @ A ) )
      = ( A = zero_zero_rat ) ) ).

% double_zero_sym
thf(fact_2865_add__cancel__left__left,axiom,
    ! [B: nat,A: nat] :
      ( ( ( plus_plus_nat @ B @ A )
        = A )
      = ( B = zero_zero_nat ) ) ).

% add_cancel_left_left
thf(fact_2866_add__cancel__left__left,axiom,
    ! [B: int,A: int] :
      ( ( ( plus_plus_int @ B @ A )
        = A )
      = ( B = zero_zero_int ) ) ).

% add_cancel_left_left
thf(fact_2867_add__cancel__left__left,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ( plus_p5714425477246183910nteger @ B @ A )
        = A )
      = ( B = zero_z3403309356797280102nteger ) ) ).

% add_cancel_left_left
thf(fact_2868_add__cancel__left__left,axiom,
    ! [B: rat,A: rat] :
      ( ( ( plus_plus_rat @ B @ A )
        = A )
      = ( B = zero_zero_rat ) ) ).

% add_cancel_left_left
thf(fact_2869_add__cancel__left__left,axiom,
    ! [B: code_natural,A: code_natural] :
      ( ( ( plus_p4538020629002901425atural @ B @ A )
        = A )
      = ( B = zero_z2226904508553997617atural ) ) ).

% add_cancel_left_left
thf(fact_2870_add__cancel__left__right,axiom,
    ! [A: nat,B: nat] :
      ( ( ( plus_plus_nat @ A @ B )
        = A )
      = ( B = zero_zero_nat ) ) ).

% add_cancel_left_right
thf(fact_2871_add__cancel__left__right,axiom,
    ! [A: int,B: int] :
      ( ( ( plus_plus_int @ A @ B )
        = A )
      = ( B = zero_zero_int ) ) ).

% add_cancel_left_right
thf(fact_2872_add__cancel__left__right,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( plus_p5714425477246183910nteger @ A @ B )
        = A )
      = ( B = zero_z3403309356797280102nteger ) ) ).

% add_cancel_left_right
thf(fact_2873_add__cancel__left__right,axiom,
    ! [A: rat,B: rat] :
      ( ( ( plus_plus_rat @ A @ B )
        = A )
      = ( B = zero_zero_rat ) ) ).

% add_cancel_left_right
thf(fact_2874_add__cancel__left__right,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( ( plus_p4538020629002901425atural @ A @ B )
        = A )
      = ( B = zero_z2226904508553997617atural ) ) ).

% add_cancel_left_right
thf(fact_2875_add__cancel__right__left,axiom,
    ! [A: nat,B: nat] :
      ( ( A
        = ( plus_plus_nat @ B @ A ) )
      = ( B = zero_zero_nat ) ) ).

% add_cancel_right_left
thf(fact_2876_add__cancel__right__left,axiom,
    ! [A: int,B: int] :
      ( ( A
        = ( plus_plus_int @ B @ A ) )
      = ( B = zero_zero_int ) ) ).

% add_cancel_right_left
thf(fact_2877_add__cancel__right__left,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A
        = ( plus_p5714425477246183910nteger @ B @ A ) )
      = ( B = zero_z3403309356797280102nteger ) ) ).

% add_cancel_right_left
thf(fact_2878_add__cancel__right__left,axiom,
    ! [A: rat,B: rat] :
      ( ( A
        = ( plus_plus_rat @ B @ A ) )
      = ( B = zero_zero_rat ) ) ).

% add_cancel_right_left
thf(fact_2879_add__cancel__right__left,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( A
        = ( plus_p4538020629002901425atural @ B @ A ) )
      = ( B = zero_z2226904508553997617atural ) ) ).

% add_cancel_right_left
thf(fact_2880_add__cancel__right__right,axiom,
    ! [A: nat,B: nat] :
      ( ( A
        = ( plus_plus_nat @ A @ B ) )
      = ( B = zero_zero_nat ) ) ).

% add_cancel_right_right
thf(fact_2881_add__cancel__right__right,axiom,
    ! [A: int,B: int] :
      ( ( A
        = ( plus_plus_int @ A @ B ) )
      = ( B = zero_zero_int ) ) ).

% add_cancel_right_right
thf(fact_2882_add__cancel__right__right,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A
        = ( plus_p5714425477246183910nteger @ A @ B ) )
      = ( B = zero_z3403309356797280102nteger ) ) ).

% add_cancel_right_right
thf(fact_2883_add__cancel__right__right,axiom,
    ! [A: rat,B: rat] :
      ( ( A
        = ( plus_plus_rat @ A @ B ) )
      = ( B = zero_zero_rat ) ) ).

% add_cancel_right_right
thf(fact_2884_add__cancel__right__right,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( A
        = ( plus_p4538020629002901425atural @ A @ B ) )
      = ( B = zero_z2226904508553997617atural ) ) ).

% add_cancel_right_right
thf(fact_2885_add__eq__0__iff__both__eq__0,axiom,
    ! [X: nat,Y3: nat] :
      ( ( ( plus_plus_nat @ X @ Y3 )
        = zero_zero_nat )
      = ( ( X = zero_zero_nat )
        & ( Y3 = zero_zero_nat ) ) ) ).

% add_eq_0_iff_both_eq_0
thf(fact_2886_zero__eq__add__iff__both__eq__0,axiom,
    ! [X: nat,Y3: nat] :
      ( ( zero_zero_nat
        = ( plus_plus_nat @ X @ Y3 ) )
      = ( ( X = zero_zero_nat )
        & ( Y3 = zero_zero_nat ) ) ) ).

% zero_eq_add_iff_both_eq_0
thf(fact_2887_add__0,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ A )
      = A ) ).

% add_0
thf(fact_2888_add__0,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ zero_zero_int @ A )
      = A ) ).

% add_0
thf(fact_2889_add__0,axiom,
    ! [A: code_integer] :
      ( ( plus_p5714425477246183910nteger @ zero_z3403309356797280102nteger @ A )
      = A ) ).

% add_0
thf(fact_2890_add__0,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ zero_zero_rat @ A )
      = A ) ).

% add_0
thf(fact_2891_add__0,axiom,
    ! [A: code_natural] :
      ( ( plus_p4538020629002901425atural @ zero_z2226904508553997617atural @ A )
      = A ) ).

% add_0
thf(fact_2892_add__less__cancel__left,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ C @ A ) @ ( plus_p5714425477246183910nteger @ C @ B ) )
      = ( ord_le6747313008572928689nteger @ A @ B ) ) ).

% add_less_cancel_left
thf(fact_2893_add__less__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
      = ( ord_less_rat @ A @ B ) ) ).

% add_less_cancel_left
thf(fact_2894_add__less__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
      = ( ord_less_nat @ A @ B ) ) ).

% add_less_cancel_left
thf(fact_2895_add__less__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
      = ( ord_less_int @ A @ B ) ) ).

% add_less_cancel_left
thf(fact_2896_add__less__cancel__right,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A @ C ) @ ( plus_p5714425477246183910nteger @ B @ C ) )
      = ( ord_le6747313008572928689nteger @ A @ B ) ) ).

% add_less_cancel_right
thf(fact_2897_add__less__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
      = ( ord_less_rat @ A @ B ) ) ).

% add_less_cancel_right
thf(fact_2898_add__less__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
      = ( ord_less_nat @ A @ B ) ) ).

% add_less_cancel_right
thf(fact_2899_add__less__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
      = ( ord_less_int @ A @ B ) ) ).

% add_less_cancel_right
thf(fact_2900_add__diff__cancel,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel
thf(fact_2901_add__diff__cancel,axiom,
    ! [A: rat,B: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel
thf(fact_2902_add__diff__cancel,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel
thf(fact_2903_diff__add__cancel,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ B )
      = A ) ).

% diff_add_cancel
thf(fact_2904_diff__add__cancel,axiom,
    ! [A: rat,B: rat] :
      ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ B )
      = A ) ).

% diff_add_cancel
thf(fact_2905_diff__add__cancel,axiom,
    ! [A: int,B: int] :
      ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
      = A ) ).

% diff_add_cancel
thf(fact_2906_add__diff__cancel__left,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ C @ A ) @ ( plus_p5714425477246183910nteger @ C @ B ) )
      = ( minus_8373710615458151222nteger @ A @ B ) ) ).

% add_diff_cancel_left
thf(fact_2907_add__diff__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
      = ( minus_minus_rat @ A @ B ) ) ).

% add_diff_cancel_left
thf(fact_2908_add__diff__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
      = ( minus_minus_nat @ A @ B ) ) ).

% add_diff_cancel_left
thf(fact_2909_add__diff__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
      = ( minus_minus_int @ A @ B ) ) ).

% add_diff_cancel_left
thf(fact_2910_add__diff__cancel__left_H,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ A )
      = B ) ).

% add_diff_cancel_left'
thf(fact_2911_add__diff__cancel__left_H,axiom,
    ! [A: rat,B: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ A )
      = B ) ).

% add_diff_cancel_left'
thf(fact_2912_add__diff__cancel__left_H,axiom,
    ! [A: nat,B: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
      = B ) ).

% add_diff_cancel_left'
thf(fact_2913_add__diff__cancel__left_H,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ A )
      = B ) ).

% add_diff_cancel_left'
thf(fact_2914_add__diff__cancel__right,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A @ C ) @ ( plus_p5714425477246183910nteger @ B @ C ) )
      = ( minus_8373710615458151222nteger @ A @ B ) ) ).

% add_diff_cancel_right
thf(fact_2915_add__diff__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
      = ( minus_minus_rat @ A @ B ) ) ).

% add_diff_cancel_right
thf(fact_2916_add__diff__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
      = ( minus_minus_nat @ A @ B ) ) ).

% add_diff_cancel_right
thf(fact_2917_add__diff__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
      = ( minus_minus_int @ A @ B ) ) ).

% add_diff_cancel_right
thf(fact_2918_add__diff__cancel__right_H,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel_right'
thf(fact_2919_add__diff__cancel__right_H,axiom,
    ! [A: rat,B: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel_right'
thf(fact_2920_add__diff__cancel__right_H,axiom,
    ! [A: nat,B: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel_right'
thf(fact_2921_add__diff__cancel__right_H,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel_right'
thf(fact_2922_add__minus__cancel,axiom,
    ! [A: int,B: int] :
      ( ( plus_plus_int @ A @ ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B ) )
      = B ) ).

% add_minus_cancel
thf(fact_2923_add__minus__cancel,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger @ A @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) )
      = B ) ).

% add_minus_cancel
thf(fact_2924_add__minus__cancel,axiom,
    ! [A: rat,B: rat] :
      ( ( plus_plus_rat @ A @ ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ B ) )
      = B ) ).

% add_minus_cancel
thf(fact_2925_minus__add__cancel,axiom,
    ! [A: int,B: int] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( plus_plus_int @ A @ B ) )
      = B ) ).

% minus_add_cancel
thf(fact_2926_minus__add__cancel,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ ( plus_p5714425477246183910nteger @ A @ B ) )
      = B ) ).

% minus_add_cancel
thf(fact_2927_minus__add__cancel,axiom,
    ! [A: rat,B: rat] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( plus_plus_rat @ A @ B ) )
      = B ) ).

% minus_add_cancel
thf(fact_2928_minus__add__distrib,axiom,
    ! [A: int,B: int] :
      ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
      = ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) ) ) ).

% minus_add_distrib
thf(fact_2929_minus__add__distrib,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A @ B ) )
      = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) ) ) ).

% minus_add_distrib
thf(fact_2930_minus__add__distrib,axiom,
    ! [A: rat,B: rat] :
      ( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B ) )
      = ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) ) ) ).

% minus_add_distrib
thf(fact_2931_dvd__add__triv__left__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ A @ B ) )
      = ( dvd_dvd_Code_integer @ A @ B ) ) ).

% dvd_add_triv_left_iff
thf(fact_2932_dvd__add__triv__left__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ A @ B ) )
      = ( dvd_dvd_rat @ A @ B ) ) ).

% dvd_add_triv_left_iff
thf(fact_2933_dvd__add__triv__left__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ A @ B ) )
      = ( dvd_dvd_nat @ A @ B ) ) ).

% dvd_add_triv_left_iff
thf(fact_2934_dvd__add__triv__left__iff,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ ( plus_plus_int @ A @ B ) )
      = ( dvd_dvd_int @ A @ B ) ) ).

% dvd_add_triv_left_iff
thf(fact_2935_dvd__add__triv__right__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ A ) )
      = ( dvd_dvd_Code_integer @ A @ B ) ) ).

% dvd_add_triv_right_iff
thf(fact_2936_dvd__add__triv__right__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ A ) )
      = ( dvd_dvd_rat @ A @ B ) ) ).

% dvd_add_triv_right_iff
thf(fact_2937_dvd__add__triv__right__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ A ) )
      = ( dvd_dvd_nat @ A @ B ) ) ).

% dvd_add_triv_right_iff
thf(fact_2938_dvd__add__triv__right__iff,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ A ) )
      = ( dvd_dvd_int @ A @ B ) ) ).

% dvd_add_triv_right_iff
thf(fact_2939_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_2940_of__nat__add,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri4939895301339042750nteger @ ( plus_plus_nat @ M @ N ) )
      = ( plus_p5714425477246183910nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) ) ) ).

% of_nat_add
thf(fact_2941_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_2942_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_2943_of__nat__add,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri3763490453095760265atural @ ( plus_plus_nat @ M @ N ) )
      = ( plus_p4538020629002901425atural @ ( semiri3763490453095760265atural @ M ) @ ( semiri3763490453095760265atural @ N ) ) ) ).

% of_nat_add
thf(fact_2944_mod__add__self1,axiom,
    ! [B: nat,A: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ B @ A ) @ B )
      = ( modulo_modulo_nat @ A @ B ) ) ).

% mod_add_self1
thf(fact_2945_mod__add__self1,axiom,
    ! [B: int,A: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ B @ A ) @ B )
      = ( modulo_modulo_int @ A @ B ) ) ).

% mod_add_self1
thf(fact_2946_mod__add__self1,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ B @ A ) @ B )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% mod_add_self1
thf(fact_2947_mod__add__self1,axiom,
    ! [B: code_natural,A: code_natural] :
      ( ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ B @ A ) @ B )
      = ( modulo8411746178871703098atural @ A @ B ) ) ).

% mod_add_self1
thf(fact_2948_mod__add__self2,axiom,
    ! [A: nat,B: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ B )
      = ( modulo_modulo_nat @ A @ B ) ) ).

% mod_add_self2
thf(fact_2949_mod__add__self2,axiom,
    ! [A: int,B: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ B )
      = ( modulo_modulo_int @ A @ B ) ) ).

% mod_add_self2
thf(fact_2950_mod__add__self2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ B )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% mod_add_self2
thf(fact_2951_mod__add__self2,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ A @ B ) @ B )
      = ( modulo8411746178871703098atural @ A @ B ) ) ).

% mod_add_self2
thf(fact_2952_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_2953_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_2954_Nat_Odiff__diff__right,axiom,
    ! [K: nat,J: nat,I2: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( minus_minus_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ I2 @ K ) @ J ) ) ) ).

% Nat.diff_diff_right
thf(fact_2955_Nat_Oadd__diff__assoc2,axiom,
    ! [K: nat,J: nat,I2: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I2 )
        = ( minus_minus_nat @ ( plus_plus_nat @ J @ I2 ) @ K ) ) ) ).

% Nat.add_diff_assoc2
thf(fact_2956_Nat_Oadd__diff__assoc,axiom,
    ! [K: nat,J: nat,I2: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( plus_plus_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ I2 @ J ) @ K ) ) ) ).

% Nat.add_diff_assoc
thf(fact_2957_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_2958_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_2959_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_2960_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n356916108424825756nteger @ P )
        = zero_z3403309356797280102nteger )
      = ~ P ) ).

% of_bool_eq_0_iff
thf(fact_2961_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n2052037380579107095ol_rat @ P )
        = zero_zero_rat )
      = ~ P ) ).

% of_bool_eq_0_iff
thf(fact_2962_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n8403883297036319079atural @ P )
        = zero_z2226904508553997617atural )
      = ~ P ) ).

% of_bool_eq_0_iff
thf(fact_2963_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n2687167440665602831ol_nat @ P )
        = zero_zero_nat )
      = ~ P ) ).

% of_bool_eq_0_iff
thf(fact_2964_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n2684676970156552555ol_int @ P )
        = zero_zero_int )
      = ~ P ) ).

% of_bool_eq_0_iff
thf(fact_2965_of__bool__eq_I1_J,axiom,
    ( ( zero_n356916108424825756nteger @ $false )
    = zero_z3403309356797280102nteger ) ).

% of_bool_eq(1)
thf(fact_2966_of__bool__eq_I1_J,axiom,
    ( ( zero_n2052037380579107095ol_rat @ $false )
    = zero_zero_rat ) ).

% of_bool_eq(1)
thf(fact_2967_of__bool__eq_I1_J,axiom,
    ( ( zero_n8403883297036319079atural @ $false )
    = zero_z2226904508553997617atural ) ).

% of_bool_eq(1)
thf(fact_2968_of__bool__eq_I1_J,axiom,
    ( ( zero_n2687167440665602831ol_nat @ $false )
    = zero_zero_nat ) ).

% of_bool_eq(1)
thf(fact_2969_of__bool__eq_I1_J,axiom,
    ( ( zero_n2684676970156552555ol_int @ $false )
    = zero_zero_int ) ).

% of_bool_eq(1)
thf(fact_2970_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_2971_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_2972_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_2973_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_2974_of__bool__eq_I2_J,axiom,
    ( ( zero_n356916108424825756nteger @ $true )
    = one_one_Code_integer ) ).

% of_bool_eq(2)
thf(fact_2975_of__bool__eq_I2_J,axiom,
    ( ( zero_n2052037380579107095ol_rat @ $true )
    = one_one_rat ) ).

% of_bool_eq(2)
thf(fact_2976_of__bool__eq_I2_J,axiom,
    ( ( zero_n8403883297036319079atural @ $true )
    = one_one_Code_natural ) ).

% of_bool_eq(2)
thf(fact_2977_of__bool__eq_I2_J,axiom,
    ( ( zero_n2687167440665602831ol_nat @ $true )
    = one_one_nat ) ).

% of_bool_eq(2)
thf(fact_2978_of__bool__eq_I2_J,axiom,
    ( ( zero_n2684676970156552555ol_int @ $true )
    = one_one_int ) ).

% of_bool_eq(2)
thf(fact_2979_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n356916108424825756nteger @ P )
        = one_one_Code_integer )
      = P ) ).

% of_bool_eq_1_iff
thf(fact_2980_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n2052037380579107095ol_rat @ P )
        = one_one_rat )
      = P ) ).

% of_bool_eq_1_iff
thf(fact_2981_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n8403883297036319079atural @ P )
        = one_one_Code_natural )
      = P ) ).

% of_bool_eq_1_iff
thf(fact_2982_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n2687167440665602831ol_nat @ P )
        = one_one_nat )
      = P ) ).

% of_bool_eq_1_iff
thf(fact_2983_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n2684676970156552555ol_int @ P )
        = one_one_int )
      = P ) ).

% of_bool_eq_1_iff
thf(fact_2984_add__le__same__cancel1,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ B @ A ) @ B )
      = ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) ) ).

% add_le_same_cancel1
thf(fact_2985_add__le__same__cancel1,axiom,
    ! [B: code_natural,A: code_natural] :
      ( ( ord_le1926595141338095240atural @ ( plus_p4538020629002901425atural @ B @ A ) @ B )
      = ( ord_le1926595141338095240atural @ A @ zero_z2226904508553997617atural ) ) ).

% add_le_same_cancel1
thf(fact_2986_add__le__same__cancel1,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ B @ A ) @ B )
      = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).

% add_le_same_cancel1
thf(fact_2987_add__le__same__cancel1,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
      = ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).

% add_le_same_cancel1
thf(fact_2988_add__le__same__cancel1,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ B @ A ) @ B )
      = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).

% add_le_same_cancel1
thf(fact_2989_add__le__same__cancel2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ B )
      = ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) ) ).

% add_le_same_cancel2
thf(fact_2990_add__le__same__cancel2,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( ord_le1926595141338095240atural @ ( plus_p4538020629002901425atural @ A @ B ) @ B )
      = ( ord_le1926595141338095240atural @ A @ zero_z2226904508553997617atural ) ) ).

% add_le_same_cancel2
thf(fact_2991_add__le__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B ) @ B )
      = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).

% add_le_same_cancel2
thf(fact_2992_add__le__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
      = ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).

% add_le_same_cancel2
thf(fact_2993_add__le__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ B )
      = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).

% add_le_same_cancel2
thf(fact_2994_le__add__same__cancel1,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ ( plus_p5714425477246183910nteger @ A @ B ) )
      = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B ) ) ).

% le_add_same_cancel1
thf(fact_2995_le__add__same__cancel1,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( ord_le1926595141338095240atural @ A @ ( plus_p4538020629002901425atural @ A @ B ) )
      = ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ B ) ) ).

% le_add_same_cancel1
thf(fact_2996_le__add__same__cancel1,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ ( plus_plus_rat @ A @ B ) )
      = ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ).

% le_add_same_cancel1
thf(fact_2997_le__add__same__cancel1,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
      = ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).

% le_add_same_cancel1
thf(fact_2998_le__add__same__cancel1,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ ( plus_plus_int @ A @ B ) )
      = ( ord_less_eq_int @ zero_zero_int @ B ) ) ).

% le_add_same_cancel1
thf(fact_2999_le__add__same__cancel2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ ( plus_p5714425477246183910nteger @ B @ A ) )
      = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B ) ) ).

% le_add_same_cancel2
thf(fact_3000_le__add__same__cancel2,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( ord_le1926595141338095240atural @ A @ ( plus_p4538020629002901425atural @ B @ A ) )
      = ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ B ) ) ).

% le_add_same_cancel2
thf(fact_3001_le__add__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ ( plus_plus_rat @ B @ A ) )
      = ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ).

% le_add_same_cancel2
thf(fact_3002_le__add__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
      = ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).

% le_add_same_cancel2
thf(fact_3003_le__add__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ ( plus_plus_int @ B @ A ) )
      = ( ord_less_eq_int @ zero_zero_int @ B ) ) ).

% le_add_same_cancel2
thf(fact_3004_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ A @ A ) @ zero_z3403309356797280102nteger )
      = ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) ) ).

% double_add_le_zero_iff_single_add_le_zero
thf(fact_3005_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_3006_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_3007_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ A @ A ) )
      = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).

% zero_le_double_add_iff_zero_le_single_add
thf(fact_3008_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_3009_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_3010_add__less__same__cancel1,axiom,
    ! [B: code_natural,A: code_natural] :
      ( ( ord_le5570908160329646204atural @ ( plus_p4538020629002901425atural @ B @ A ) @ B )
      = ( ord_le5570908160329646204atural @ A @ zero_z2226904508553997617atural ) ) ).

% add_less_same_cancel1
thf(fact_3011_add__less__same__cancel1,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ B @ A ) @ B )
      = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).

% add_less_same_cancel1
thf(fact_3012_add__less__same__cancel1,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ B @ A ) @ B )
      = ( ord_less_rat @ A @ zero_zero_rat ) ) ).

% add_less_same_cancel1
thf(fact_3013_add__less__same__cancel1,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
      = ( ord_less_nat @ A @ zero_zero_nat ) ) ).

% add_less_same_cancel1
thf(fact_3014_add__less__same__cancel1,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ ( plus_plus_int @ B @ A ) @ B )
      = ( ord_less_int @ A @ zero_zero_int ) ) ).

% add_less_same_cancel1
thf(fact_3015_add__less__same__cancel2,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( ord_le5570908160329646204atural @ ( plus_p4538020629002901425atural @ A @ B ) @ B )
      = ( ord_le5570908160329646204atural @ A @ zero_z2226904508553997617atural ) ) ).

% add_less_same_cancel2
thf(fact_3016_add__less__same__cancel2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ B )
      = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).

% add_less_same_cancel2
thf(fact_3017_add__less__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ B )
      = ( ord_less_rat @ A @ zero_zero_rat ) ) ).

% add_less_same_cancel2
thf(fact_3018_add__less__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
      = ( ord_less_nat @ A @ zero_zero_nat ) ) ).

% add_less_same_cancel2
thf(fact_3019_add__less__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ ( plus_plus_int @ A @ B ) @ B )
      = ( ord_less_int @ A @ zero_zero_int ) ) ).

% add_less_same_cancel2
thf(fact_3020_less__add__same__cancel1,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( ord_le5570908160329646204atural @ A @ ( plus_p4538020629002901425atural @ A @ B ) )
      = ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ B ) ) ).

% less_add_same_cancel1
thf(fact_3021_less__add__same__cancel1,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ ( plus_p5714425477246183910nteger @ A @ B ) )
      = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B ) ) ).

% less_add_same_cancel1
thf(fact_3022_less__add__same__cancel1,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ ( plus_plus_rat @ A @ B ) )
      = ( ord_less_rat @ zero_zero_rat @ B ) ) ).

% less_add_same_cancel1
thf(fact_3023_less__add__same__cancel1,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
      = ( ord_less_nat @ zero_zero_nat @ B ) ) ).

% less_add_same_cancel1
thf(fact_3024_less__add__same__cancel1,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ ( plus_plus_int @ A @ B ) )
      = ( ord_less_int @ zero_zero_int @ B ) ) ).

% less_add_same_cancel1
thf(fact_3025_less__add__same__cancel2,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( ord_le5570908160329646204atural @ A @ ( plus_p4538020629002901425atural @ B @ A ) )
      = ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ B ) ) ).

% less_add_same_cancel2
thf(fact_3026_less__add__same__cancel2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ ( plus_p5714425477246183910nteger @ B @ A ) )
      = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B ) ) ).

% less_add_same_cancel2
thf(fact_3027_less__add__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ ( plus_plus_rat @ B @ A ) )
      = ( ord_less_rat @ zero_zero_rat @ B ) ) ).

% less_add_same_cancel2
thf(fact_3028_less__add__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
      = ( ord_less_nat @ zero_zero_nat @ B ) ) ).

% less_add_same_cancel2
thf(fact_3029_less__add__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ ( plus_plus_int @ B @ A ) )
      = ( ord_less_int @ zero_zero_int @ B ) ) ).

% less_add_same_cancel2
thf(fact_3030_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A @ A ) @ zero_z3403309356797280102nteger )
      = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).

% double_add_less_zero_iff_single_add_less_zero
thf(fact_3031_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_3032_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_3033_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ A @ A ) )
      = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).

% zero_less_double_add_iff_zero_less_single_add
thf(fact_3034_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_3035_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_3036_le__add__diff__inverse2,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ B @ A )
     => ( ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ B )
        = A ) ) ).

% le_add_diff_inverse2
thf(fact_3037_le__add__diff__inverse2,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ B )
        = A ) ) ).

% le_add_diff_inverse2
thf(fact_3038_le__add__diff__inverse2,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B ) @ B )
        = A ) ) ).

% le_add_diff_inverse2
thf(fact_3039_le__add__diff__inverse2,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
        = A ) ) ).

% le_add_diff_inverse2
thf(fact_3040_le__add__diff__inverse,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ B @ A )
     => ( ( plus_p5714425477246183910nteger @ B @ ( minus_8373710615458151222nteger @ A @ B ) )
        = A ) ) ).

% le_add_diff_inverse
thf(fact_3041_le__add__diff__inverse,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( plus_plus_rat @ B @ ( minus_minus_rat @ A @ B ) )
        = A ) ) ).

% le_add_diff_inverse
thf(fact_3042_le__add__diff__inverse,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
        = A ) ) ).

% le_add_diff_inverse
thf(fact_3043_le__add__diff__inverse,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
        = A ) ) ).

% le_add_diff_inverse
thf(fact_3044_diff__add__zero,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( minus_7197305767214868737atural @ A @ ( plus_p4538020629002901425atural @ A @ B ) )
      = zero_z2226904508553997617atural ) ).

% diff_add_zero
thf(fact_3045_diff__add__zero,axiom,
    ! [A: nat,B: nat] :
      ( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
      = zero_zero_nat ) ).

% diff_add_zero
thf(fact_3046_add_Oright__inverse,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ ( uminus_uminus_int @ A ) )
      = zero_zero_int ) ).

% add.right_inverse
thf(fact_3047_add_Oright__inverse,axiom,
    ! [A: code_integer] :
      ( ( plus_p5714425477246183910nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
      = zero_z3403309356797280102nteger ) ).

% add.right_inverse
thf(fact_3048_add_Oright__inverse,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ A @ ( uminus_uminus_rat @ A ) )
      = zero_zero_rat ) ).

% add.right_inverse
thf(fact_3049_ab__left__minus,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
      = zero_zero_int ) ).

% ab_left_minus
thf(fact_3050_ab__left__minus,axiom,
    ! [A: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
      = zero_z3403309356797280102nteger ) ).

% ab_left_minus
thf(fact_3051_ab__left__minus,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ A )
      = zero_zero_rat ) ).

% ab_left_minus
thf(fact_3052_dvd__add__times__triv__right__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ ( times_3573771949741848930nteger @ C @ A ) ) )
      = ( dvd_dvd_Code_integer @ A @ B ) ) ).

% dvd_add_times_triv_right_iff
thf(fact_3053_dvd__add__times__triv__right__iff,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ ( times_times_rat @ C @ A ) ) )
      = ( dvd_dvd_rat @ A @ B ) ) ).

% dvd_add_times_triv_right_iff
thf(fact_3054_dvd__add__times__triv__right__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ ( times_times_nat @ C @ A ) ) )
      = ( dvd_dvd_nat @ A @ B ) ) ).

% dvd_add_times_triv_right_iff
thf(fact_3055_dvd__add__times__triv__right__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ ( times_times_int @ C @ A ) ) )
      = ( dvd_dvd_int @ A @ B ) ) ).

% dvd_add_times_triv_right_iff
thf(fact_3056_dvd__add__times__triv__left__iff,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ C @ A ) @ B ) )
      = ( dvd_dvd_Code_integer @ A @ B ) ) ).

% dvd_add_times_triv_left_iff
thf(fact_3057_dvd__add__times__triv__left__iff,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ ( times_times_rat @ C @ A ) @ B ) )
      = ( dvd_dvd_rat @ A @ B ) ) ).

% dvd_add_times_triv_left_iff
thf(fact_3058_dvd__add__times__triv__left__iff,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ ( times_times_nat @ C @ A ) @ B ) )
      = ( dvd_dvd_nat @ A @ B ) ) ).

% dvd_add_times_triv_left_iff
thf(fact_3059_dvd__add__times__triv__left__iff,axiom,
    ! [A: int,C: int,B: int] :
      ( ( dvd_dvd_int @ A @ ( plus_plus_int @ ( times_times_int @ C @ A ) @ B ) )
      = ( dvd_dvd_int @ A @ B ) ) ).

% dvd_add_times_triv_left_iff
thf(fact_3060_diff__minus__eq__add,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ A @ ( uminus_uminus_int @ B ) )
      = ( plus_plus_int @ A @ B ) ) ).

% diff_minus_eq_add
thf(fact_3061_diff__minus__eq__add,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( minus_8373710615458151222nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
      = ( plus_p5714425477246183910nteger @ A @ B ) ) ).

% diff_minus_eq_add
thf(fact_3062_diff__minus__eq__add,axiom,
    ! [A: rat,B: rat] :
      ( ( minus_minus_rat @ A @ ( uminus_uminus_rat @ B ) )
      = ( plus_plus_rat @ A @ B ) ) ).

% diff_minus_eq_add
thf(fact_3063_uminus__add__conv__diff,axiom,
    ! [A: int,B: int] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B )
      = ( minus_minus_int @ B @ A ) ) ).

% uminus_add_conv_diff
thf(fact_3064_uminus__add__conv__diff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
      = ( minus_8373710615458151222nteger @ B @ A ) ) ).

% uminus_add_conv_diff
thf(fact_3065_uminus__add__conv__diff,axiom,
    ! [A: rat,B: rat] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ B )
      = ( minus_minus_rat @ B @ A ) ) ).

% uminus_add_conv_diff
thf(fact_3066_div__add,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( dvd_dvd_nat @ C @ A )
     => ( ( dvd_dvd_nat @ C @ B )
       => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
          = ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) ) ) ) ).

% div_add
thf(fact_3067_div__add,axiom,
    ! [C: int,A: int,B: int] :
      ( ( dvd_dvd_int @ C @ A )
     => ( ( dvd_dvd_int @ C @ B )
       => ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
          = ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ) ).

% div_add
thf(fact_3068_div__add,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ A )
     => ( ( dvd_dvd_Code_integer @ C @ B )
       => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
          = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) ) ) ) ).

% div_add
thf(fact_3069_div__add,axiom,
    ! [C: code_natural,A: code_natural,B: code_natural] :
      ( ( dvd_dvd_Code_natural @ C @ A )
     => ( ( dvd_dvd_Code_natural @ C @ B )
       => ( ( divide5121882707175180666atural @ ( plus_p4538020629002901425atural @ A @ B ) @ C )
          = ( plus_p4538020629002901425atural @ ( divide5121882707175180666atural @ A @ C ) @ ( divide5121882707175180666atural @ B @ C ) ) ) ) ) ).

% div_add
thf(fact_3070_mod__mult__self1,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C @ B ) ) @ B )
      = ( modulo_modulo_nat @ A @ B ) ) ).

% mod_mult_self1
thf(fact_3071_mod__mult__self1,axiom,
    ! [A: int,C: int,B: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( times_times_int @ C @ B ) ) @ B )
      = ( modulo_modulo_int @ A @ B ) ) ).

% mod_mult_self1
thf(fact_3072_mod__mult__self1,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ C @ B ) ) @ B )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% mod_mult_self1
thf(fact_3073_mod__mult__self1,axiom,
    ! [A: code_natural,C: code_natural,B: code_natural] :
      ( ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ A @ ( times_2397367101498566445atural @ C @ B ) ) @ B )
      = ( modulo8411746178871703098atural @ A @ B ) ) ).

% mod_mult_self1
thf(fact_3074_mod__mult__self2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B @ C ) ) @ B )
      = ( modulo_modulo_nat @ A @ B ) ) ).

% mod_mult_self2
thf(fact_3075_mod__mult__self2,axiom,
    ! [A: int,B: int,C: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( times_times_int @ B @ C ) ) @ B )
      = ( modulo_modulo_int @ A @ B ) ) ).

% mod_mult_self2
thf(fact_3076_mod__mult__self2,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) ) @ B )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% mod_mult_self2
thf(fact_3077_mod__mult__self2,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ A @ ( times_2397367101498566445atural @ B @ C ) ) @ B )
      = ( modulo8411746178871703098atural @ A @ B ) ) ).

% mod_mult_self2
thf(fact_3078_mod__mult__self3,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B ) @ A ) @ B )
      = ( modulo_modulo_nat @ A @ B ) ) ).

% mod_mult_self3
thf(fact_3079_mod__mult__self3,axiom,
    ! [C: int,B: int,A: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ C @ B ) @ A ) @ B )
      = ( modulo_modulo_int @ A @ B ) ) ).

% mod_mult_self3
thf(fact_3080_mod__mult__self3,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ C @ B ) @ A ) @ B )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% mod_mult_self3
thf(fact_3081_mod__mult__self3,axiom,
    ! [C: code_natural,B: code_natural,A: code_natural] :
      ( ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ C @ B ) @ A ) @ B )
      = ( modulo8411746178871703098atural @ A @ B ) ) ).

% mod_mult_self3
thf(fact_3082_mod__mult__self4,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ A ) @ B )
      = ( modulo_modulo_nat @ A @ B ) ) ).

% mod_mult_self4
thf(fact_3083_mod__mult__self4,axiom,
    ! [B: int,C: int,A: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ B @ C ) @ A ) @ B )
      = ( modulo_modulo_int @ A @ B ) ) ).

% mod_mult_self4
thf(fact_3084_mod__mult__self4,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ C ) @ A ) @ B )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% mod_mult_self4
thf(fact_3085_mod__mult__self4,axiom,
    ! [B: code_natural,C: code_natural,A: code_natural] :
      ( ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ B @ C ) @ A ) @ B )
      = ( modulo8411746178871703098atural @ A @ B ) ) ).

% mod_mult_self4
thf(fact_3086_zero__less__of__bool__iff,axiom,
    ! [P: $o] :
      ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ ( zero_n8403883297036319079atural @ P ) )
      = P ) ).

% zero_less_of_bool_iff
thf(fact_3087_zero__less__of__bool__iff,axiom,
    ! [P: $o] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( zero_n356916108424825756nteger @ P ) )
      = P ) ).

% zero_less_of_bool_iff
thf(fact_3088_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_3089_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_3090_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_3091_diff__Suc__diff__eq1,axiom,
    ! [K: nat,J: nat,I2: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( minus_minus_nat @ I2 @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ I2 @ K ) @ ( suc @ J ) ) ) ) ).

% diff_Suc_diff_eq1
thf(fact_3092_diff__Suc__diff__eq2,axiom,
    ! [K: nat,J: nat,I2: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I2 )
        = ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I2 ) ) ) ) ).

% diff_Suc_diff_eq2
thf(fact_3093_of__bool__less__one__iff,axiom,
    ! [P: $o] :
      ( ( ord_le5570908160329646204atural @ ( zero_n8403883297036319079atural @ P ) @ one_one_Code_natural )
      = ~ P ) ).

% of_bool_less_one_iff
thf(fact_3094_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_3095_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_3096_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_3097_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_3098_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_3099_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_3100_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_3101_gbinomial__0_I2_J,axiom,
    ! [K: nat] :
      ( ( gbinomial_nat @ zero_zero_nat @ ( suc @ K ) )
      = zero_zero_nat ) ).

% gbinomial_0(2)
thf(fact_3102_gbinomial__0_I2_J,axiom,
    ! [K: nat] :
      ( ( gbinomial_int @ zero_zero_int @ ( suc @ K ) )
      = zero_zero_int ) ).

% gbinomial_0(2)
thf(fact_3103_gbinomial__0_I2_J,axiom,
    ! [K: nat] :
      ( ( gbinom8545251970709558553nteger @ zero_z3403309356797280102nteger @ ( suc @ K ) )
      = zero_z3403309356797280102nteger ) ).

% gbinomial_0(2)
thf(fact_3104_gbinomial__0_I2_J,axiom,
    ! [K: nat] :
      ( ( gbinomial_rat @ zero_zero_rat @ ( suc @ K ) )
      = zero_zero_rat ) ).

% gbinomial_0(2)
thf(fact_3105_gbinomial__0_I2_J,axiom,
    ! [K: nat] :
      ( ( gbinom7368847122466276068atural @ zero_z2226904508553997617atural @ ( suc @ K ) )
      = zero_z2226904508553997617atural ) ).

% gbinomial_0(2)
thf(fact_3106_zle__add1__eq__le,axiom,
    ! [W2: int,Z2: int] :
      ( ( ord_less_int @ W2 @ ( plus_plus_int @ Z2 @ one_one_int ) )
      = ( ord_less_eq_int @ W2 @ Z2 ) ) ).

% zle_add1_eq_le
thf(fact_3107_gbinomial__0_I1_J,axiom,
    ! [A: nat] :
      ( ( gbinomial_nat @ A @ zero_zero_nat )
      = one_one_nat ) ).

% gbinomial_0(1)
thf(fact_3108_gbinomial__0_I1_J,axiom,
    ! [A: int] :
      ( ( gbinomial_int @ A @ zero_zero_nat )
      = one_one_int ) ).

% gbinomial_0(1)
thf(fact_3109_gbinomial__0_I1_J,axiom,
    ! [A: code_integer] :
      ( ( gbinom8545251970709558553nteger @ A @ zero_zero_nat )
      = one_one_Code_integer ) ).

% gbinomial_0(1)
thf(fact_3110_gbinomial__0_I1_J,axiom,
    ! [A: rat] :
      ( ( gbinomial_rat @ A @ zero_zero_nat )
      = one_one_rat ) ).

% gbinomial_0(1)
thf(fact_3111_gbinomial__0_I1_J,axiom,
    ! [A: code_natural] :
      ( ( gbinom7368847122466276068atural @ A @ zero_zero_nat )
      = one_one_Code_natural ) ).

% gbinomial_0(1)
thf(fact_3112_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_3113_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_3114_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_3115_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_3116_of__nat__of__bool,axiom,
    ! [P: $o] :
      ( ( semiri4939895301339042750nteger @ ( zero_n2687167440665602831ol_nat @ P ) )
      = ( zero_n356916108424825756nteger @ P ) ) ).

% of_nat_of_bool
thf(fact_3117_of__nat__of__bool,axiom,
    ! [P: $o] :
      ( ( semiri681578069525770553at_rat @ ( zero_n2687167440665602831ol_nat @ P ) )
      = ( zero_n2052037380579107095ol_rat @ P ) ) ).

% of_nat_of_bool
thf(fact_3118_of__nat__of__bool,axiom,
    ! [P: $o] :
      ( ( semiri3763490453095760265atural @ ( zero_n2687167440665602831ol_nat @ P ) )
      = ( zero_n8403883297036319079atural @ P ) ) ).

% of_nat_of_bool
thf(fact_3119_of__nat__of__bool,axiom,
    ! [P: $o] :
      ( ( semiri1316708129612266289at_nat @ ( zero_n2687167440665602831ol_nat @ P ) )
      = ( zero_n2687167440665602831ol_nat @ P ) ) ).

% of_nat_of_bool
thf(fact_3120_of__nat__of__bool,axiom,
    ! [P: $o] :
      ( ( semiri1314217659103216013at_int @ ( zero_n2687167440665602831ol_nat @ P ) )
      = ( zero_n2684676970156552555ol_int @ P ) ) ).

% of_nat_of_bool
thf(fact_3121_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_3122_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_3123_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_3124_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_3125_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_3126_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_3127_div__mult__self4,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ A ) @ B )
        = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).

% div_mult_self4
thf(fact_3128_div__mult__self4,axiom,
    ! [B: int,C: int,A: int] :
      ( ( B != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ B @ C ) @ A ) @ B )
        = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).

% div_mult_self4
thf(fact_3129_div__mult__self4,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ C ) @ A ) @ B )
        = ( plus_p5714425477246183910nteger @ C @ ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).

% div_mult_self4
thf(fact_3130_div__mult__self4,axiom,
    ! [B: code_natural,C: code_natural,A: code_natural] :
      ( ( B != zero_z2226904508553997617atural )
     => ( ( divide5121882707175180666atural @ ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ B @ C ) @ A ) @ B )
        = ( plus_p4538020629002901425atural @ C @ ( divide5121882707175180666atural @ A @ B ) ) ) ) ).

% div_mult_self4
thf(fact_3131_div__mult__self3,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B ) @ A ) @ B )
        = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).

% div_mult_self3
thf(fact_3132_div__mult__self3,axiom,
    ! [B: int,C: int,A: int] :
      ( ( B != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ C @ B ) @ A ) @ B )
        = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).

% div_mult_self3
thf(fact_3133_div__mult__self3,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ C @ B ) @ A ) @ B )
        = ( plus_p5714425477246183910nteger @ C @ ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).

% div_mult_self3
thf(fact_3134_div__mult__self3,axiom,
    ! [B: code_natural,C: code_natural,A: code_natural] :
      ( ( B != zero_z2226904508553997617atural )
     => ( ( divide5121882707175180666atural @ ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ C @ B ) @ A ) @ B )
        = ( plus_p4538020629002901425atural @ C @ ( divide5121882707175180666atural @ A @ B ) ) ) ) ).

% div_mult_self3
thf(fact_3135_div__mult__self2,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( B != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B @ C ) ) @ B )
        = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).

% div_mult_self2
thf(fact_3136_div__mult__self2,axiom,
    ! [B: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ B @ C ) ) @ B )
        = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).

% div_mult_self2
thf(fact_3137_div__mult__self2,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) ) @ B )
        = ( plus_p5714425477246183910nteger @ C @ ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).

% div_mult_self2
thf(fact_3138_div__mult__self2,axiom,
    ! [B: code_natural,A: code_natural,C: code_natural] :
      ( ( B != zero_z2226904508553997617atural )
     => ( ( divide5121882707175180666atural @ ( plus_p4538020629002901425atural @ A @ ( times_2397367101498566445atural @ B @ C ) ) @ B )
        = ( plus_p4538020629002901425atural @ C @ ( divide5121882707175180666atural @ A @ B ) ) ) ) ).

% div_mult_self2
thf(fact_3139_div__mult__self1,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( B != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C @ B ) ) @ B )
        = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).

% div_mult_self1
thf(fact_3140_div__mult__self1,axiom,
    ! [B: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ C @ B ) ) @ B )
        = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).

% div_mult_self1
thf(fact_3141_div__mult__self1,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ C @ B ) ) @ B )
        = ( plus_p5714425477246183910nteger @ C @ ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).

% div_mult_self1
thf(fact_3142_div__mult__self1,axiom,
    ! [B: code_natural,A: code_natural,C: code_natural] :
      ( ( B != zero_z2226904508553997617atural )
     => ( ( divide5121882707175180666atural @ ( plus_p4538020629002901425atural @ A @ ( times_2397367101498566445atural @ C @ B ) ) @ B )
        = ( plus_p4538020629002901425atural @ C @ ( divide5121882707175180666atural @ A @ B ) ) ) ) ).

% div_mult_self1
thf(fact_3143_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri1314217659103216013at_int @ ( suc @ M ) )
      = ( plus_plus_int @ one_one_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).

% of_nat_Suc
thf(fact_3144_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri4939895301339042750nteger @ ( suc @ M ) )
      = ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( semiri4939895301339042750nteger @ M ) ) ) ).

% of_nat_Suc
thf(fact_3145_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri1316708129612266289at_nat @ ( suc @ M ) )
      = ( plus_plus_nat @ one_one_nat @ ( semiri1316708129612266289at_nat @ M ) ) ) ).

% of_nat_Suc
thf(fact_3146_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri681578069525770553at_rat @ ( suc @ M ) )
      = ( plus_plus_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ M ) ) ) ).

% of_nat_Suc
thf(fact_3147_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri3763490453095760265atural @ ( suc @ M ) )
      = ( plus_p4538020629002901425atural @ one_one_Code_natural @ ( semiri3763490453095760265atural @ M ) ) ) ).

% of_nat_Suc
thf(fact_3148_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_3149_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_3150_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_3151_ceiling__add__one,axiom,
    ! [X: rat] :
      ( ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X @ one_one_rat ) )
      = ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X ) @ one_one_int ) ) ).

% ceiling_add_one
thf(fact_3152_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_3153_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_3154_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_3155_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_3156_of__nat__gbinomial,axiom,
    ! [N: nat,K: nat] :
      ( ( semiri681578069525770553at_rat @ ( gbinomial_nat @ N @ K ) )
      = ( gbinomial_rat @ ( semiri681578069525770553at_rat @ N ) @ K ) ) ).

% of_nat_gbinomial
thf(fact_3157_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_3158_int__ops_I5_J,axiom,
    ! [A: nat,B: nat] :
      ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ A @ B ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).

% int_ops(5)
thf(fact_3159_zadd__int__left,axiom,
    ! [M: nat,N: nat,Z2: int] :
      ( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ Z2 ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) ) @ Z2 ) ) ).

% zadd_int_left
thf(fact_3160_Euclid__induct,axiom,
    ! [P: nat > nat > $o,A: nat,B: nat] :
      ( ! [A5: nat,B4: nat] :
          ( ( P @ A5 @ B4 )
          = ( P @ B4 @ A5 ) )
     => ( ! [A5: nat] : ( P @ A5 @ zero_zero_nat )
       => ( ! [A5: nat,B4: nat] :
              ( ( P @ A5 @ B4 )
             => ( P @ A5 @ ( plus_plus_nat @ A5 @ B4 ) ) )
         => ( P @ A @ B ) ) ) ) ).

% Euclid_induct
thf(fact_3161_of__bool__eq__iff,axiom,
    ! [P5: $o,Q4: $o] :
      ( ( ( zero_n2687167440665602831ol_nat @ P5 )
        = ( zero_n2687167440665602831ol_nat @ Q4 ) )
      = ( P5 = Q4 ) ) ).

% of_bool_eq_iff
thf(fact_3162_of__bool__eq__iff,axiom,
    ! [P5: $o,Q4: $o] :
      ( ( ( zero_n2684676970156552555ol_int @ P5 )
        = ( zero_n2684676970156552555ol_int @ Q4 ) )
      = ( P5 = Q4 ) ) ).

% of_bool_eq_iff
thf(fact_3163_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
      = ( plus_p5714425477246183910nteger @ A @ ( plus_p5714425477246183910nteger @ B @ C ) ) ) ).

% ab_semigroup_add_class.add_ac(1)
thf(fact_3164_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( plus_plus_rat @ ( plus_plus_rat @ A @ B ) @ C )
      = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).

% ab_semigroup_add_class.add_ac(1)
thf(fact_3165_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
      = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).

% ab_semigroup_add_class.add_ac(1)
thf(fact_3166_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: int,B: int,C: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
      = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).

% ab_semigroup_add_class.add_ac(1)
thf(fact_3167_is__num__normalize_I1_J,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
      = ( plus_p5714425477246183910nteger @ A @ ( plus_p5714425477246183910nteger @ B @ C ) ) ) ).

% is_num_normalize(1)
thf(fact_3168_is__num__normalize_I1_J,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( plus_plus_rat @ ( plus_plus_rat @ A @ B ) @ C )
      = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).

% is_num_normalize(1)
thf(fact_3169_is__num__normalize_I1_J,axiom,
    ! [A: int,B: int,C: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
      = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).

% is_num_normalize(1)
thf(fact_3170_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I2: code_integer,J: code_integer,K: code_integer,L: code_integer] :
      ( ( ( I2 = J )
        & ( K = L ) )
     => ( ( plus_p5714425477246183910nteger @ I2 @ K )
        = ( plus_p5714425477246183910nteger @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_3171_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I2: rat,J: rat,K: rat,L: rat] :
      ( ( ( I2 = J )
        & ( K = L ) )
     => ( ( plus_plus_rat @ I2 @ K )
        = ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_3172_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I2: nat,J: nat,K: nat,L: nat] :
      ( ( ( I2 = J )
        & ( K = L ) )
     => ( ( plus_plus_nat @ I2 @ K )
        = ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_3173_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I2: int,J: int,K: int,L: int] :
      ( ( ( I2 = J )
        & ( K = L ) )
     => ( ( plus_plus_int @ I2 @ K )
        = ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_3174_group__cancel_Oadd1,axiom,
    ! [A3: code_integer,K: code_integer,A: code_integer,B: code_integer] :
      ( ( A3
        = ( plus_p5714425477246183910nteger @ K @ A ) )
     => ( ( plus_p5714425477246183910nteger @ A3 @ B )
        = ( plus_p5714425477246183910nteger @ K @ ( plus_p5714425477246183910nteger @ A @ B ) ) ) ) ).

% group_cancel.add1
thf(fact_3175_group__cancel_Oadd1,axiom,
    ! [A3: rat,K: rat,A: rat,B: rat] :
      ( ( A3
        = ( plus_plus_rat @ K @ A ) )
     => ( ( plus_plus_rat @ A3 @ B )
        = ( plus_plus_rat @ K @ ( plus_plus_rat @ A @ B ) ) ) ) ).

% group_cancel.add1
thf(fact_3176_group__cancel_Oadd1,axiom,
    ! [A3: nat,K: nat,A: nat,B: nat] :
      ( ( A3
        = ( plus_plus_nat @ K @ A ) )
     => ( ( plus_plus_nat @ A3 @ B )
        = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% group_cancel.add1
thf(fact_3177_group__cancel_Oadd1,axiom,
    ! [A3: int,K: int,A: int,B: int] :
      ( ( A3
        = ( plus_plus_int @ K @ A ) )
     => ( ( plus_plus_int @ A3 @ B )
        = ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).

% group_cancel.add1
thf(fact_3178_group__cancel_Oadd2,axiom,
    ! [B5: code_integer,K: code_integer,B: code_integer,A: code_integer] :
      ( ( B5
        = ( plus_p5714425477246183910nteger @ K @ B ) )
     => ( ( plus_p5714425477246183910nteger @ A @ B5 )
        = ( plus_p5714425477246183910nteger @ K @ ( plus_p5714425477246183910nteger @ A @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_3179_group__cancel_Oadd2,axiom,
    ! [B5: rat,K: rat,B: rat,A: rat] :
      ( ( B5
        = ( plus_plus_rat @ K @ B ) )
     => ( ( plus_plus_rat @ A @ B5 )
        = ( plus_plus_rat @ K @ ( plus_plus_rat @ A @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_3180_group__cancel_Oadd2,axiom,
    ! [B5: nat,K: nat,B: nat,A: nat] :
      ( ( B5
        = ( plus_plus_nat @ K @ B ) )
     => ( ( plus_plus_nat @ A @ B5 )
        = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_3181_group__cancel_Oadd2,axiom,
    ! [B5: int,K: int,B: int,A: int] :
      ( ( B5
        = ( plus_plus_int @ K @ B ) )
     => ( ( plus_plus_int @ A @ B5 )
        = ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_3182_add_Oassoc,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
      = ( plus_p5714425477246183910nteger @ A @ ( plus_p5714425477246183910nteger @ B @ C ) ) ) ).

% add.assoc
thf(fact_3183_add_Oassoc,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( plus_plus_rat @ ( plus_plus_rat @ A @ B ) @ C )
      = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).

% add.assoc
thf(fact_3184_add_Oassoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
      = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).

% add.assoc
thf(fact_3185_add_Oassoc,axiom,
    ! [A: int,B: int,C: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
      = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).

% add.assoc
thf(fact_3186_add_Oleft__cancel,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( ( plus_p5714425477246183910nteger @ A @ B )
        = ( plus_p5714425477246183910nteger @ A @ C ) )
      = ( B = C ) ) ).

% add.left_cancel
thf(fact_3187_add_Oleft__cancel,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ( plus_plus_rat @ A @ B )
        = ( plus_plus_rat @ A @ C ) )
      = ( B = C ) ) ).

% add.left_cancel
thf(fact_3188_add_Oleft__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ( plus_plus_int @ A @ B )
        = ( plus_plus_int @ A @ C ) )
      = ( B = C ) ) ).

% add.left_cancel
thf(fact_3189_add_Oright__cancel,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( ( plus_p5714425477246183910nteger @ B @ A )
        = ( plus_p5714425477246183910nteger @ C @ A ) )
      = ( B = C ) ) ).

% add.right_cancel
thf(fact_3190_add_Oright__cancel,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ( plus_plus_rat @ B @ A )
        = ( plus_plus_rat @ C @ A ) )
      = ( B = C ) ) ).

% add.right_cancel
thf(fact_3191_add_Oright__cancel,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ( plus_plus_int @ B @ A )
        = ( plus_plus_int @ C @ A ) )
      = ( B = C ) ) ).

% add.right_cancel
thf(fact_3192_add_Ocommute,axiom,
    ( plus_p5714425477246183910nteger
    = ( ^ [A2: code_integer,B2: code_integer] : ( plus_p5714425477246183910nteger @ B2 @ A2 ) ) ) ).

% add.commute
thf(fact_3193_add_Ocommute,axiom,
    ( plus_plus_rat
    = ( ^ [A2: rat,B2: rat] : ( plus_plus_rat @ B2 @ A2 ) ) ) ).

% add.commute
thf(fact_3194_add_Ocommute,axiom,
    ( plus_plus_nat
    = ( ^ [A2: nat,B2: nat] : ( plus_plus_nat @ B2 @ A2 ) ) ) ).

% add.commute
thf(fact_3195_add_Ocommute,axiom,
    ( plus_plus_int
    = ( ^ [A2: int,B2: int] : ( plus_plus_int @ B2 @ A2 ) ) ) ).

% add.commute
thf(fact_3196_add_Oleft__commute,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( plus_p5714425477246183910nteger @ B @ ( plus_p5714425477246183910nteger @ A @ C ) )
      = ( plus_p5714425477246183910nteger @ A @ ( plus_p5714425477246183910nteger @ B @ C ) ) ) ).

% add.left_commute
thf(fact_3197_add_Oleft__commute,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( plus_plus_rat @ B @ ( plus_plus_rat @ A @ C ) )
      = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).

% add.left_commute
thf(fact_3198_add_Oleft__commute,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
      = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).

% add.left_commute
thf(fact_3199_add_Oleft__commute,axiom,
    ! [B: int,A: int,C: int] :
      ( ( plus_plus_int @ B @ ( plus_plus_int @ A @ C ) )
      = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).

% add.left_commute
thf(fact_3200_add__left__imp__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( ( plus_p5714425477246183910nteger @ A @ B )
        = ( plus_p5714425477246183910nteger @ A @ C ) )
     => ( B = C ) ) ).

% add_left_imp_eq
thf(fact_3201_add__left__imp__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ( plus_plus_rat @ A @ B )
        = ( plus_plus_rat @ A @ C ) )
     => ( B = C ) ) ).

% add_left_imp_eq
thf(fact_3202_add__left__imp__eq,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( plus_plus_nat @ A @ B )
        = ( plus_plus_nat @ A @ C ) )
     => ( B = C ) ) ).

% add_left_imp_eq
thf(fact_3203_add__left__imp__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ( plus_plus_int @ A @ B )
        = ( plus_plus_int @ A @ C ) )
     => ( B = C ) ) ).

% add_left_imp_eq
thf(fact_3204_add__right__imp__eq,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( ( plus_p5714425477246183910nteger @ B @ A )
        = ( plus_p5714425477246183910nteger @ C @ A ) )
     => ( B = C ) ) ).

% add_right_imp_eq
thf(fact_3205_add__right__imp__eq,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ( plus_plus_rat @ B @ A )
        = ( plus_plus_rat @ C @ A ) )
     => ( B = C ) ) ).

% add_right_imp_eq
thf(fact_3206_add__right__imp__eq,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ( plus_plus_nat @ B @ A )
        = ( plus_plus_nat @ C @ A ) )
     => ( B = C ) ) ).

% add_right_imp_eq
thf(fact_3207_add__right__imp__eq,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ( plus_plus_int @ B @ A )
        = ( plus_plus_int @ C @ A ) )
     => ( B = C ) ) ).

% add_right_imp_eq
thf(fact_3208_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_3209_nat__int__add,axiom,
    ! [A: nat,B: nat] :
      ( ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) )
      = ( plus_plus_nat @ A @ B ) ) ).

% nat_int_add
thf(fact_3210_ceiling__add__le,axiom,
    ! [X: rat,Y3: rat] : ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X @ Y3 ) ) @ ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X ) @ ( archim2889992004027027881ng_rat @ Y3 ) ) ) ).

% ceiling_add_le
thf(fact_3211_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_3212_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_3213_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_3214_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_3215_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I2: code_integer,J: code_integer,K: code_integer,L: code_integer] :
      ( ( ( ord_le3102999989581377725nteger @ I2 @ J )
        & ( K = L ) )
     => ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ I2 @ K ) @ ( plus_p5714425477246183910nteger @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_3216_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I2: rat,J: rat,K: rat,L: rat] :
      ( ( ( ord_less_eq_rat @ I2 @ J )
        & ( K = L ) )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ I2 @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_3217_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I2: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_eq_nat @ I2 @ J )
        & ( K = L ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_3218_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I2: int,J: int,K: int,L: int] :
      ( ( ( ord_less_eq_int @ I2 @ J )
        & ( K = L ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_3219_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I2: code_integer,J: code_integer,K: code_integer,L: code_integer] :
      ( ( ( I2 = J )
        & ( ord_le3102999989581377725nteger @ K @ L ) )
     => ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ I2 @ K ) @ ( plus_p5714425477246183910nteger @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_3220_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I2: rat,J: rat,K: rat,L: rat] :
      ( ( ( I2 = J )
        & ( ord_less_eq_rat @ K @ L ) )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ I2 @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_3221_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I2: nat,J: nat,K: nat,L: nat] :
      ( ( ( I2 = J )
        & ( ord_less_eq_nat @ K @ L ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_3222_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I2: int,J: int,K: int,L: int] :
      ( ( ( I2 = J )
        & ( ord_less_eq_int @ K @ L ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_3223_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I2: code_integer,J: code_integer,K: code_integer,L: code_integer] :
      ( ( ( ord_le3102999989581377725nteger @ I2 @ J )
        & ( ord_le3102999989581377725nteger @ K @ L ) )
     => ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ I2 @ K ) @ ( plus_p5714425477246183910nteger @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_3224_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I2: rat,J: rat,K: rat,L: rat] :
      ( ( ( ord_less_eq_rat @ I2 @ J )
        & ( ord_less_eq_rat @ K @ L ) )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ I2 @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_3225_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I2: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_eq_nat @ I2 @ J )
        & ( ord_less_eq_nat @ K @ L ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_3226_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I2: int,J: int,K: int,L: int] :
      ( ( ( ord_less_eq_int @ I2 @ J )
        & ( ord_less_eq_int @ K @ L ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_3227_add__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ B )
     => ( ( ord_le3102999989581377725nteger @ C @ D )
       => ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ A @ C ) @ ( plus_p5714425477246183910nteger @ B @ D ) ) ) ) ).

% add_mono
thf(fact_3228_add__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ C @ D )
       => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).

% add_mono
thf(fact_3229_add__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ D )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).

% add_mono
thf(fact_3230_add__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ C @ D )
       => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).

% add_mono
thf(fact_3231_add__left__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ B )
     => ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ C @ A ) @ ( plus_p5714425477246183910nteger @ C @ B ) ) ) ).

% add_left_mono
thf(fact_3232_add__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) ) ) ).

% add_left_mono
thf(fact_3233_add__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).

% add_left_mono
thf(fact_3234_add__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).

% add_left_mono
thf(fact_3235_less__eqE,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ~ ! [C2: nat] :
            ( B
           != ( plus_plus_nat @ A @ C2 ) ) ) ).

% less_eqE
thf(fact_3236_add__right__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ B )
     => ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ A @ C ) @ ( plus_p5714425477246183910nteger @ B @ C ) ) ) ).

% add_right_mono
thf(fact_3237_add__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) ) ) ).

% add_right_mono
thf(fact_3238_add__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).

% add_right_mono
thf(fact_3239_add__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).

% add_right_mono
thf(fact_3240_le__iff__add,axiom,
    ( ord_less_eq_nat
    = ( ^ [A2: nat,B2: nat] :
        ? [C3: nat] :
          ( B2
          = ( plus_plus_nat @ A2 @ C3 ) ) ) ) ).

% le_iff_add
thf(fact_3241_add__le__imp__le__left,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ C @ A ) @ ( plus_p5714425477246183910nteger @ C @ B ) )
     => ( ord_le3102999989581377725nteger @ A @ B ) ) ).

% add_le_imp_le_left
thf(fact_3242_add__le__imp__le__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
     => ( ord_less_eq_rat @ A @ B ) ) ).

% add_le_imp_le_left
thf(fact_3243_add__le__imp__le__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
     => ( ord_less_eq_nat @ A @ B ) ) ).

% add_le_imp_le_left
thf(fact_3244_add__le__imp__le__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
     => ( ord_less_eq_int @ A @ B ) ) ).

% add_le_imp_le_left
thf(fact_3245_add__le__imp__le__right,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ A @ C ) @ ( plus_p5714425477246183910nteger @ B @ C ) )
     => ( ord_le3102999989581377725nteger @ A @ B ) ) ).

% add_le_imp_le_right
thf(fact_3246_add__le__imp__le__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
     => ( ord_less_eq_rat @ A @ B ) ) ).

% add_le_imp_le_right
thf(fact_3247_add__le__imp__le__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
     => ( ord_less_eq_nat @ A @ B ) ) ).

% add_le_imp_le_right
thf(fact_3248_add__le__imp__le__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
     => ( ord_less_eq_int @ A @ B ) ) ).

% add_le_imp_le_right
thf(fact_3249_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_3250_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_3251_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: code_integer] :
      ( ( plus_p5714425477246183910nteger @ zero_z3403309356797280102nteger @ A )
      = A ) ).

% comm_monoid_add_class.add_0
thf(fact_3252_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_3253_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: code_natural] :
      ( ( plus_p4538020629002901425atural @ zero_z2226904508553997617atural @ A )
      = A ) ).

% comm_monoid_add_class.add_0
thf(fact_3254_add_Ocomm__neutral,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% add.comm_neutral
thf(fact_3255_add_Ocomm__neutral,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ zero_zero_int )
      = A ) ).

% add.comm_neutral
thf(fact_3256_add_Ocomm__neutral,axiom,
    ! [A: code_integer] :
      ( ( plus_p5714425477246183910nteger @ A @ zero_z3403309356797280102nteger )
      = A ) ).

% add.comm_neutral
thf(fact_3257_add_Ocomm__neutral,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ A @ zero_zero_rat )
      = A ) ).

% add.comm_neutral
thf(fact_3258_add_Ocomm__neutral,axiom,
    ! [A: code_natural] :
      ( ( plus_p4538020629002901425atural @ A @ zero_z2226904508553997617atural )
      = A ) ).

% add.comm_neutral
thf(fact_3259_add_Ogroup__left__neutral,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ zero_zero_int @ A )
      = A ) ).

% add.group_left_neutral
thf(fact_3260_add_Ogroup__left__neutral,axiom,
    ! [A: code_integer] :
      ( ( plus_p5714425477246183910nteger @ zero_z3403309356797280102nteger @ A )
      = A ) ).

% add.group_left_neutral
thf(fact_3261_add_Ogroup__left__neutral,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ zero_zero_rat @ A )
      = A ) ).

% add.group_left_neutral
thf(fact_3262_verit__sum__simplify,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% verit_sum_simplify
thf(fact_3263_verit__sum__simplify,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ zero_zero_int )
      = A ) ).

% verit_sum_simplify
thf(fact_3264_verit__sum__simplify,axiom,
    ! [A: code_integer] :
      ( ( plus_p5714425477246183910nteger @ A @ zero_z3403309356797280102nteger )
      = A ) ).

% verit_sum_simplify
thf(fact_3265_verit__sum__simplify,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ A @ zero_zero_rat )
      = A ) ).

% verit_sum_simplify
thf(fact_3266_verit__sum__simplify,axiom,
    ! [A: code_natural] :
      ( ( plus_p4538020629002901425atural @ A @ zero_z2226904508553997617atural )
      = A ) ).

% verit_sum_simplify
thf(fact_3267_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I2: code_integer,J: code_integer,K: code_integer,L: code_integer] :
      ( ( ( ord_le6747313008572928689nteger @ I2 @ J )
        & ( ord_le6747313008572928689nteger @ K @ L ) )
     => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ I2 @ K ) @ ( plus_p5714425477246183910nteger @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_3268_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I2: rat,J: rat,K: rat,L: rat] :
      ( ( ( ord_less_rat @ I2 @ J )
        & ( ord_less_rat @ K @ L ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I2 @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_3269_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I2: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_nat @ I2 @ J )
        & ( ord_less_nat @ K @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_3270_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I2: int,J: int,K: int,L: int] :
      ( ( ( ord_less_int @ I2 @ J )
        & ( ord_less_int @ K @ L ) )
     => ( ord_less_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_3271_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I2: code_integer,J: code_integer,K: code_integer,L: code_integer] :
      ( ( ( I2 = J )
        & ( ord_le6747313008572928689nteger @ K @ L ) )
     => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ I2 @ K ) @ ( plus_p5714425477246183910nteger @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_3272_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I2: rat,J: rat,K: rat,L: rat] :
      ( ( ( I2 = J )
        & ( ord_less_rat @ K @ L ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I2 @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_3273_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I2: nat,J: nat,K: nat,L: nat] :
      ( ( ( I2 = J )
        & ( ord_less_nat @ K @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_3274_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I2: int,J: int,K: int,L: int] :
      ( ( ( I2 = J )
        & ( ord_less_int @ K @ L ) )
     => ( ord_less_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_3275_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I2: code_integer,J: code_integer,K: code_integer,L: code_integer] :
      ( ( ( ord_le6747313008572928689nteger @ I2 @ J )
        & ( K = L ) )
     => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ I2 @ K ) @ ( plus_p5714425477246183910nteger @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_3276_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I2: rat,J: rat,K: rat,L: rat] :
      ( ( ( ord_less_rat @ I2 @ J )
        & ( K = L ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I2 @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_3277_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I2: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_nat @ I2 @ J )
        & ( K = L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_3278_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I2: int,J: int,K: int,L: int] :
      ( ( ( ord_less_int @ I2 @ J )
        & ( K = L ) )
     => ( ord_less_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_3279_add__strict__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ B )
     => ( ( ord_le6747313008572928689nteger @ C @ D )
       => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A @ C ) @ ( plus_p5714425477246183910nteger @ B @ D ) ) ) ) ).

% add_strict_mono
thf(fact_3280_add__strict__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ C @ D )
       => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).

% add_strict_mono
thf(fact_3281_add__strict__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ C @ D )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).

% add_strict_mono
thf(fact_3282_add__strict__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ C @ D )
       => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).

% add_strict_mono
thf(fact_3283_add__strict__left__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ B )
     => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ C @ A ) @ ( plus_p5714425477246183910nteger @ C @ B ) ) ) ).

% add_strict_left_mono
thf(fact_3284_add__strict__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) ) ) ).

% add_strict_left_mono
thf(fact_3285_add__strict__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).

% add_strict_left_mono
thf(fact_3286_add__strict__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).

% add_strict_left_mono
thf(fact_3287_add__strict__right__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ B )
     => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A @ C ) @ ( plus_p5714425477246183910nteger @ B @ C ) ) ) ).

% add_strict_right_mono
thf(fact_3288_add__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) ) ) ).

% add_strict_right_mono
thf(fact_3289_add__strict__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).

% add_strict_right_mono
thf(fact_3290_add__strict__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).

% add_strict_right_mono
thf(fact_3291_add__less__imp__less__left,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ C @ A ) @ ( plus_p5714425477246183910nteger @ C @ B ) )
     => ( ord_le6747313008572928689nteger @ A @ B ) ) ).

% add_less_imp_less_left
thf(fact_3292_add__less__imp__less__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
     => ( ord_less_rat @ A @ B ) ) ).

% add_less_imp_less_left
thf(fact_3293_add__less__imp__less__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
     => ( ord_less_nat @ A @ B ) ) ).

% add_less_imp_less_left
thf(fact_3294_add__less__imp__less__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
     => ( ord_less_int @ A @ B ) ) ).

% add_less_imp_less_left
thf(fact_3295_add__less__imp__less__right,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A @ C ) @ ( plus_p5714425477246183910nteger @ B @ C ) )
     => ( ord_le6747313008572928689nteger @ A @ B ) ) ).

% add_less_imp_less_right
thf(fact_3296_add__less__imp__less__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
     => ( ord_less_rat @ A @ B ) ) ).

% add_less_imp_less_right
thf(fact_3297_add__less__imp__less__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
     => ( ord_less_nat @ A @ B ) ) ).

% add_less_imp_less_right
thf(fact_3298_add__less__imp__less__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
     => ( ord_less_int @ A @ B ) ) ).

% add_less_imp_less_right
thf(fact_3299_combine__common__factor,axiom,
    ! [A: code_integer,E: code_integer,B: code_integer,C: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ E ) @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ E ) @ C ) )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ E ) @ C ) ) ).

% combine_common_factor
thf(fact_3300_combine__common__factor,axiom,
    ! [A: rat,E: rat,B: rat,C: rat] :
      ( ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ C ) )
      = ( plus_plus_rat @ ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ E ) @ C ) ) ).

% combine_common_factor
thf(fact_3301_combine__common__factor,axiom,
    ! [A: nat,E: nat,B: nat,C: nat] :
      ( ( plus_plus_nat @ ( times_times_nat @ A @ E ) @ ( plus_plus_nat @ ( times_times_nat @ B @ E ) @ C ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ E ) @ C ) ) ).

% combine_common_factor
thf(fact_3302_combine__common__factor,axiom,
    ! [A: int,E: int,B: int,C: int] :
      ( ( plus_plus_int @ ( times_times_int @ A @ E ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ C ) )
      = ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A @ B ) @ E ) @ C ) ) ).

% combine_common_factor
thf(fact_3303_distrib__right,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) ) ) ).

% distrib_right
thf(fact_3304_distrib__right,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C )
      = ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).

% distrib_right
thf(fact_3305_distrib__right,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
      = ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).

% distrib_right
thf(fact_3306_distrib__right,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
      = ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).

% distrib_right
thf(fact_3307_distrib__left,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( times_3573771949741848930nteger @ A @ ( plus_p5714425477246183910nteger @ B @ C ) )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ B ) @ ( times_3573771949741848930nteger @ A @ C ) ) ) ).

% distrib_left
thf(fact_3308_distrib__left,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ A @ ( plus_plus_rat @ B @ C ) )
      = ( plus_plus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).

% distrib_left
thf(fact_3309_distrib__left,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( times_times_nat @ A @ ( plus_plus_nat @ B @ C ) )
      = ( plus_plus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).

% distrib_left
thf(fact_3310_distrib__left,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
      = ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).

% distrib_left
thf(fact_3311_comm__semiring__class_Odistrib,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) ) ) ).

% comm_semiring_class.distrib
thf(fact_3312_comm__semiring__class_Odistrib,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C )
      = ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).

% comm_semiring_class.distrib
thf(fact_3313_comm__semiring__class_Odistrib,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
      = ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).

% comm_semiring_class.distrib
thf(fact_3314_comm__semiring__class_Odistrib,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
      = ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).

% comm_semiring_class.distrib
thf(fact_3315_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( times_3573771949741848930nteger @ A @ ( plus_p5714425477246183910nteger @ B @ C ) )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ B ) @ ( times_3573771949741848930nteger @ A @ C ) ) ) ).

% ring_class.ring_distribs(1)
thf(fact_3316_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ A @ ( plus_plus_rat @ B @ C ) )
      = ( plus_plus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).

% ring_class.ring_distribs(1)
thf(fact_3317_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
      = ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).

% ring_class.ring_distribs(1)
thf(fact_3318_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) ) ) ).

% ring_class.ring_distribs(2)
thf(fact_3319_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C )
      = ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).

% ring_class.ring_distribs(2)
thf(fact_3320_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
      = ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).

% ring_class.ring_distribs(2)
thf(fact_3321_group__cancel_Osub1,axiom,
    ! [A3: code_integer,K: code_integer,A: code_integer,B: code_integer] :
      ( ( A3
        = ( plus_p5714425477246183910nteger @ K @ A ) )
     => ( ( minus_8373710615458151222nteger @ A3 @ B )
        = ( plus_p5714425477246183910nteger @ K @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ) ).

% group_cancel.sub1
thf(fact_3322_group__cancel_Osub1,axiom,
    ! [A3: rat,K: rat,A: rat,B: rat] :
      ( ( A3
        = ( plus_plus_rat @ K @ A ) )
     => ( ( minus_minus_rat @ A3 @ B )
        = ( plus_plus_rat @ K @ ( minus_minus_rat @ A @ B ) ) ) ) ).

% group_cancel.sub1
thf(fact_3323_group__cancel_Osub1,axiom,
    ! [A3: int,K: int,A: int,B: int] :
      ( ( A3
        = ( plus_plus_int @ K @ A ) )
     => ( ( minus_minus_int @ A3 @ B )
        = ( plus_plus_int @ K @ ( minus_minus_int @ A @ B ) ) ) ) ).

% group_cancel.sub1
thf(fact_3324_diff__eq__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( ( minus_8373710615458151222nteger @ A @ B )
        = C )
      = ( A
        = ( plus_p5714425477246183910nteger @ C @ B ) ) ) ).

% diff_eq_eq
thf(fact_3325_diff__eq__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ( minus_minus_rat @ A @ B )
        = C )
      = ( A
        = ( plus_plus_rat @ C @ B ) ) ) ).

% diff_eq_eq
thf(fact_3326_diff__eq__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ( minus_minus_int @ A @ B )
        = C )
      = ( A
        = ( plus_plus_int @ C @ B ) ) ) ).

% diff_eq_eq
thf(fact_3327_eq__diff__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( A
        = ( minus_8373710615458151222nteger @ C @ B ) )
      = ( ( plus_p5714425477246183910nteger @ A @ B )
        = C ) ) ).

% eq_diff_eq
thf(fact_3328_eq__diff__eq,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( A
        = ( minus_minus_rat @ C @ B ) )
      = ( ( plus_plus_rat @ A @ B )
        = C ) ) ).

% eq_diff_eq
thf(fact_3329_eq__diff__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( ( A
        = ( minus_minus_int @ C @ B ) )
      = ( ( plus_plus_int @ A @ B )
        = C ) ) ).

% eq_diff_eq
thf(fact_3330_add__diff__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( plus_p5714425477246183910nteger @ A @ ( minus_8373710615458151222nteger @ B @ C ) )
      = ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C ) ) ).

% add_diff_eq
thf(fact_3331_add__diff__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( plus_plus_rat @ A @ ( minus_minus_rat @ B @ C ) )
      = ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ C ) ) ).

% add_diff_eq
thf(fact_3332_add__diff__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( plus_plus_int @ A @ ( minus_minus_int @ B @ C ) )
      = ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).

% add_diff_eq
thf(fact_3333_diff__diff__eq2,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( minus_8373710615458151222nteger @ A @ ( minus_8373710615458151222nteger @ B @ C ) )
      = ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A @ C ) @ B ) ) ).

% diff_diff_eq2
thf(fact_3334_diff__diff__eq2,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( minus_minus_rat @ A @ ( minus_minus_rat @ B @ C ) )
      = ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ).

% diff_diff_eq2
thf(fact_3335_diff__diff__eq2,axiom,
    ! [A: int,B: int,C: int] :
      ( ( minus_minus_int @ A @ ( minus_minus_int @ B @ C ) )
      = ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).

% diff_diff_eq2
thf(fact_3336_diff__add__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C )
      = ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A @ C ) @ B ) ) ).

% diff_add_eq
thf(fact_3337_diff__add__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ C )
      = ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ).

% diff_add_eq
thf(fact_3338_diff__add__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ C )
      = ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).

% diff_add_eq
thf(fact_3339_diff__add__eq__diff__diff__swap,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( minus_8373710615458151222nteger @ A @ ( plus_p5714425477246183910nteger @ B @ C ) )
      = ( minus_8373710615458151222nteger @ ( minus_8373710615458151222nteger @ A @ C ) @ B ) ) ).

% diff_add_eq_diff_diff_swap
thf(fact_3340_diff__add__eq__diff__diff__swap,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( minus_minus_rat @ A @ ( plus_plus_rat @ B @ C ) )
      = ( minus_minus_rat @ ( minus_minus_rat @ A @ C ) @ B ) ) ).

% diff_add_eq_diff_diff_swap
thf(fact_3341_diff__add__eq__diff__diff__swap,axiom,
    ! [A: int,B: int,C: int] :
      ( ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) )
      = ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B ) ) ).

% diff_add_eq_diff_diff_swap
thf(fact_3342_add__implies__diff,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( ( ( plus_p5714425477246183910nteger @ C @ B )
        = A )
     => ( C
        = ( minus_8373710615458151222nteger @ A @ B ) ) ) ).

% add_implies_diff
thf(fact_3343_add__implies__diff,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ( plus_plus_rat @ C @ B )
        = A )
     => ( C
        = ( minus_minus_rat @ A @ B ) ) ) ).

% add_implies_diff
thf(fact_3344_add__implies__diff,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( ( plus_plus_nat @ C @ B )
        = A )
     => ( C
        = ( minus_minus_nat @ A @ B ) ) ) ).

% add_implies_diff
thf(fact_3345_add__implies__diff,axiom,
    ! [C: int,B: int,A: int] :
      ( ( ( plus_plus_int @ C @ B )
        = A )
     => ( C
        = ( minus_minus_int @ A @ B ) ) ) ).

% add_implies_diff
thf(fact_3346_diff__diff__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( minus_8373710615458151222nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C )
      = ( minus_8373710615458151222nteger @ A @ ( plus_p5714425477246183910nteger @ B @ C ) ) ) ).

% diff_diff_eq
thf(fact_3347_diff__diff__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( minus_minus_rat @ ( minus_minus_rat @ A @ B ) @ C )
      = ( minus_minus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).

% diff_diff_eq
thf(fact_3348_diff__diff__eq,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
      = ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).

% diff_diff_eq
thf(fact_3349_diff__diff__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C )
      = ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).

% diff_diff_eq
thf(fact_3350_is__num__normalize_I8_J,axiom,
    ! [A: int,B: int] :
      ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
      = ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).

% is_num_normalize(8)
thf(fact_3351_is__num__normalize_I8_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A @ B ) )
      = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).

% is_num_normalize(8)
thf(fact_3352_is__num__normalize_I8_J,axiom,
    ! [A: rat,B: rat] :
      ( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B ) )
      = ( plus_plus_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).

% is_num_normalize(8)
thf(fact_3353_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_3354_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_3355_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_3356_add_Oinverse__distrib__swap,axiom,
    ! [A: int,B: int] :
      ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
      = ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).

% add.inverse_distrib_swap
thf(fact_3357_add_Oinverse__distrib__swap,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A @ B ) )
      = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).

% add.inverse_distrib_swap
thf(fact_3358_add_Oinverse__distrib__swap,axiom,
    ! [A: rat,B: rat] :
      ( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B ) )
      = ( plus_plus_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).

% add.inverse_distrib_swap
thf(fact_3359_dvd__add,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ B )
     => ( ( dvd_dvd_Code_integer @ A @ C )
       => ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ C ) ) ) ) ).

% dvd_add
thf(fact_3360_dvd__add,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( dvd_dvd_rat @ A @ B )
     => ( ( dvd_dvd_rat @ A @ C )
       => ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ) ).

% dvd_add
thf(fact_3361_dvd__add,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( ( dvd_dvd_nat @ A @ C )
       => ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ) ).

% dvd_add
thf(fact_3362_dvd__add,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ A @ B )
     => ( ( dvd_dvd_int @ A @ C )
       => ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C ) ) ) ) ).

% dvd_add
thf(fact_3363_dvd__add__left__iff,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ C )
     => ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ C ) )
        = ( dvd_dvd_Code_integer @ A @ B ) ) ) ).

% dvd_add_left_iff
thf(fact_3364_dvd__add__left__iff,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( dvd_dvd_rat @ A @ C )
     => ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ C ) )
        = ( dvd_dvd_rat @ A @ B ) ) ) ).

% dvd_add_left_iff
thf(fact_3365_dvd__add__left__iff,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ C )
     => ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C ) )
        = ( dvd_dvd_nat @ A @ B ) ) ) ).

% dvd_add_left_iff
thf(fact_3366_dvd__add__left__iff,axiom,
    ! [A: int,C: int,B: int] :
      ( ( dvd_dvd_int @ A @ C )
     => ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C ) )
        = ( dvd_dvd_int @ A @ B ) ) ) ).

% dvd_add_left_iff
thf(fact_3367_dvd__add__right__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ B )
     => ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ C ) )
        = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).

% dvd_add_right_iff
thf(fact_3368_dvd__add__right__iff,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( dvd_dvd_rat @ A @ B )
     => ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ C ) )
        = ( dvd_dvd_rat @ A @ C ) ) ) ).

% dvd_add_right_iff
thf(fact_3369_dvd__add__right__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C ) )
        = ( dvd_dvd_nat @ A @ C ) ) ) ).

% dvd_add_right_iff
thf(fact_3370_dvd__add__right__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ A @ B )
     => ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C ) )
        = ( dvd_dvd_int @ A @ C ) ) ) ).

% dvd_add_right_iff
thf(fact_3371_add__divide__distrib,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( divide_divide_rat @ ( plus_plus_rat @ A @ B ) @ C )
      = ( plus_plus_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ).

% add_divide_distrib
thf(fact_3372_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_3373_add__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( plus_plus_nat @ ( suc @ M ) @ N )
      = ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).

% add_Suc
thf(fact_3374_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_3375_plus__nat_Oadd__0,axiom,
    ! [N: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ N )
      = N ) ).

% plus_nat.add_0
thf(fact_3376_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_3377_mod__add__eq,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B @ C ) ) @ C )
      = ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C ) ) ).

% mod_add_eq
thf(fact_3378_mod__add__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B @ C ) ) @ C )
      = ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).

% mod_add_eq
thf(fact_3379_mod__add__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
      = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C ) ) ).

% mod_add_eq
thf(fact_3380_mod__add__eq,axiom,
    ! [A: code_natural,C: code_natural,B: code_natural] :
      ( ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ ( modulo8411746178871703098atural @ A @ C ) @ ( modulo8411746178871703098atural @ B @ C ) ) @ C )
      = ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ A @ B ) @ C ) ) ).

% mod_add_eq
thf(fact_3381_mod__add__cong,axiom,
    ! [A: nat,C: nat,A4: nat,B: nat,B3: nat] :
      ( ( ( modulo_modulo_nat @ A @ C )
        = ( modulo_modulo_nat @ A4 @ C ) )
     => ( ( ( modulo_modulo_nat @ B @ C )
          = ( modulo_modulo_nat @ B3 @ C ) )
       => ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C )
          = ( modulo_modulo_nat @ ( plus_plus_nat @ A4 @ B3 ) @ C ) ) ) ) ).

% mod_add_cong
thf(fact_3382_mod__add__cong,axiom,
    ! [A: int,C: int,A4: int,B: int,B3: int] :
      ( ( ( modulo_modulo_int @ A @ C )
        = ( modulo_modulo_int @ A4 @ C ) )
     => ( ( ( modulo_modulo_int @ B @ C )
          = ( modulo_modulo_int @ B3 @ C ) )
       => ( ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C )
          = ( modulo_modulo_int @ ( plus_plus_int @ A4 @ B3 ) @ C ) ) ) ) ).

% mod_add_cong
thf(fact_3383_mod__add__cong,axiom,
    ! [A: code_integer,C: code_integer,A4: code_integer,B: code_integer,B3: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ C )
        = ( modulo364778990260209775nteger @ A4 @ C ) )
     => ( ( ( modulo364778990260209775nteger @ B @ C )
          = ( modulo364778990260209775nteger @ B3 @ C ) )
       => ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
          = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A4 @ B3 ) @ C ) ) ) ) ).

% mod_add_cong
thf(fact_3384_mod__add__cong,axiom,
    ! [A: code_natural,C: code_natural,A4: code_natural,B: code_natural,B3: code_natural] :
      ( ( ( modulo8411746178871703098atural @ A @ C )
        = ( modulo8411746178871703098atural @ A4 @ C ) )
     => ( ( ( modulo8411746178871703098atural @ B @ C )
          = ( modulo8411746178871703098atural @ B3 @ C ) )
       => ( ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ A @ B ) @ C )
          = ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ A4 @ B3 ) @ C ) ) ) ) ).

% mod_add_cong
thf(fact_3385_mod__add__left__eq,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ B ) @ C )
      = ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C ) ) ).

% mod_add_left_eq
thf(fact_3386_mod__add__left__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ B ) @ C )
      = ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).

% mod_add_left_eq
thf(fact_3387_mod__add__left__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ C ) @ B ) @ C )
      = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C ) ) ).

% mod_add_left_eq
thf(fact_3388_mod__add__left__eq,axiom,
    ! [A: code_natural,C: code_natural,B: code_natural] :
      ( ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ ( modulo8411746178871703098atural @ A @ C ) @ B ) @ C )
      = ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ A @ B ) @ C ) ) ).

% mod_add_left_eq
thf(fact_3389_mod__add__right__eq,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( modulo_modulo_nat @ B @ C ) ) @ C )
      = ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C ) ) ).

% mod_add_right_eq
thf(fact_3390_mod__add__right__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( modulo_modulo_int @ B @ C ) ) @ C )
      = ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).

% mod_add_right_eq
thf(fact_3391_mod__add__right__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
      = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C ) ) ).

% mod_add_right_eq
thf(fact_3392_mod__add__right__eq,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ A @ ( modulo8411746178871703098atural @ B @ C ) ) @ C )
      = ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ A @ B ) @ C ) ) ).

% mod_add_right_eq
thf(fact_3393_plus__int__code_I2_J,axiom,
    ! [L: int] :
      ( ( plus_plus_int @ zero_zero_int @ L )
      = L ) ).

% plus_int_code(2)
thf(fact_3394_plus__int__code_I1_J,axiom,
    ! [K: int] :
      ( ( plus_plus_int @ K @ zero_zero_int )
      = K ) ).

% plus_int_code(1)
thf(fact_3395_add__lessD1,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ I2 @ J ) @ K )
     => ( ord_less_nat @ I2 @ K ) ) ).

% add_lessD1
thf(fact_3396_add__less__mono,axiom,
    ! [I2: nat,J: nat,K: nat,L: nat] :
      ( ( ord_less_nat @ I2 @ J )
     => ( ( ord_less_nat @ K @ L )
       => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).

% add_less_mono
thf(fact_3397_not__add__less1,axiom,
    ! [I2: nat,J: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ I2 @ J ) @ I2 ) ).

% not_add_less1
thf(fact_3398_not__add__less2,axiom,
    ! [J: nat,I2: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ J @ I2 ) @ I2 ) ).

% not_add_less2
thf(fact_3399_add__less__mono1,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ I2 @ J )
     => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).

% add_less_mono1
thf(fact_3400_trans__less__add1,axiom,
    ! [I2: nat,J: nat,M: nat] :
      ( ( ord_less_nat @ I2 @ J )
     => ( ord_less_nat @ I2 @ ( plus_plus_nat @ J @ M ) ) ) ).

% trans_less_add1
thf(fact_3401_trans__less__add2,axiom,
    ! [I2: nat,J: nat,M: nat] :
      ( ( ord_less_nat @ I2 @ J )
     => ( ord_less_nat @ I2 @ ( plus_plus_nat @ M @ J ) ) ) ).

% trans_less_add2
thf(fact_3402_less__add__eq__less,axiom,
    ! [K: nat,L: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ K @ L )
     => ( ( ( plus_plus_nat @ M @ L )
          = ( plus_plus_nat @ K @ N ) )
       => ( ord_less_nat @ M @ N ) ) ) ).

% less_add_eq_less
thf(fact_3403_nat__le__iff__add,axiom,
    ( ord_less_eq_nat
    = ( ^ [M2: nat,N3: nat] :
        ? [K3: nat] :
          ( N3
          = ( plus_plus_nat @ M2 @ K3 ) ) ) ) ).

% nat_le_iff_add
thf(fact_3404_trans__le__add2,axiom,
    ! [I2: nat,J: nat,M: nat] :
      ( ( ord_less_eq_nat @ I2 @ J )
     => ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ M @ J ) ) ) ).

% trans_le_add2
thf(fact_3405_trans__le__add1,axiom,
    ! [I2: nat,J: nat,M: nat] :
      ( ( ord_less_eq_nat @ I2 @ J )
     => ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ J @ M ) ) ) ).

% trans_le_add1
thf(fact_3406_add__le__mono1,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I2 @ J )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).

% add_le_mono1
thf(fact_3407_add__le__mono,axiom,
    ! [I2: nat,J: nat,K: nat,L: nat] :
      ( ( ord_less_eq_nat @ I2 @ J )
     => ( ( ord_less_eq_nat @ K @ L )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).

% add_le_mono
thf(fact_3408_le__Suc__ex,axiom,
    ! [K: nat,L: nat] :
      ( ( ord_less_eq_nat @ K @ L )
     => ? [N4: nat] :
          ( L
          = ( plus_plus_nat @ K @ N4 ) ) ) ).

% le_Suc_ex
thf(fact_3409_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_3410_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_3411_le__add2,axiom,
    ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).

% le_add2
thf(fact_3412_le__add1,axiom,
    ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).

% le_add1
thf(fact_3413_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_3414_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_3415_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_3416_diff__add__inverse,axiom,
    ! [N: nat,M: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
      = M ) ).

% diff_add_inverse
thf(fact_3417_diff__add__inverse2,axiom,
    ! [M: nat,N: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
      = M ) ).

% diff_add_inverse2
thf(fact_3418_left__add__mult__distrib,axiom,
    ! [I2: nat,U: nat,J: nat,K: nat] :
      ( ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ K ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I2 @ J ) @ U ) @ K ) ) ).

% left_add_mult_distrib
thf(fact_3419_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_3420_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_3421_int__distrib_I2_J,axiom,
    ! [W2: int,Z1: int,Z22: int] :
      ( ( times_times_int @ W2 @ ( plus_plus_int @ Z1 @ Z22 ) )
      = ( plus_plus_int @ ( times_times_int @ W2 @ Z1 ) @ ( times_times_int @ W2 @ Z22 ) ) ) ).

% int_distrib(2)
thf(fact_3422_int__distrib_I1_J,axiom,
    ! [Z1: int,Z22: int,W2: int] :
      ( ( times_times_int @ ( plus_plus_int @ Z1 @ Z22 ) @ W2 )
      = ( plus_plus_int @ ( times_times_int @ Z1 @ W2 ) @ ( times_times_int @ Z22 @ W2 ) ) ) ).

% int_distrib(1)
thf(fact_3423_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_3424_same__sgn__sgn__add,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ( sgn_sgn_Code_integer @ B )
        = ( sgn_sgn_Code_integer @ A ) )
     => ( ( sgn_sgn_Code_integer @ ( plus_p5714425477246183910nteger @ A @ B ) )
        = ( sgn_sgn_Code_integer @ A ) ) ) ).

% same_sgn_sgn_add
thf(fact_3425_same__sgn__sgn__add,axiom,
    ! [B: rat,A: rat] :
      ( ( ( sgn_sgn_rat @ B )
        = ( sgn_sgn_rat @ A ) )
     => ( ( sgn_sgn_rat @ ( plus_plus_rat @ A @ B ) )
        = ( sgn_sgn_rat @ A ) ) ) ).

% same_sgn_sgn_add
thf(fact_3426_same__sgn__sgn__add,axiom,
    ! [B: int,A: int] :
      ( ( ( sgn_sgn_int @ B )
        = ( sgn_sgn_int @ A ) )
     => ( ( sgn_sgn_int @ ( plus_plus_int @ A @ B ) )
        = ( sgn_sgn_int @ A ) ) ) ).

% same_sgn_sgn_add
thf(fact_3427_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_3428_bezout__lemma__nat,axiom,
    ! [D: nat,A: nat,B: nat,X: nat,Y3: nat] :
      ( ( dvd_dvd_nat @ D @ A )
     => ( ( dvd_dvd_nat @ D @ B )
       => ( ( ( ( times_times_nat @ A @ X )
              = ( plus_plus_nat @ ( times_times_nat @ B @ Y3 ) @ D ) )
            | ( ( times_times_nat @ B @ X )
              = ( 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 @ B ) )
              & ( ( ( times_times_nat @ A @ X3 )
                  = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ Y4 ) @ D ) )
                | ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ X3 )
                  = ( plus_plus_nat @ ( times_times_nat @ A @ Y4 ) @ D ) ) ) ) ) ) ) ).

% bezout_lemma_nat
thf(fact_3429_bezout__add__nat,axiom,
    ! [A: nat,B: nat] :
    ? [D5: nat,X3: nat,Y4: nat] :
      ( ( dvd_dvd_nat @ D5 @ A )
      & ( dvd_dvd_nat @ D5 @ B )
      & ( ( ( times_times_nat @ A @ X3 )
          = ( plus_plus_nat @ ( times_times_nat @ B @ Y4 ) @ D5 ) )
        | ( ( times_times_nat @ B @ X3 )
          = ( plus_plus_nat @ ( times_times_nat @ A @ Y4 ) @ D5 ) ) ) ) ).

% bezout_add_nat
thf(fact_3430_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_3431_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_3432_nat__add__distrib,axiom,
    ! [Z2: int,Z5: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ Z5 )
       => ( ( nat2 @ ( plus_plus_int @ Z2 @ Z5 ) )
          = ( plus_plus_nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z5 ) ) ) ) ) ).

% nat_add_distrib
thf(fact_3433_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_3434_zero__less__eq__of__bool,axiom,
    ! [P: $o] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( zero_n356916108424825756nteger @ P ) ) ).

% zero_less_eq_of_bool
thf(fact_3435_zero__less__eq__of__bool,axiom,
    ! [P: $o] : ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ ( zero_n8403883297036319079atural @ P ) ) ).

% zero_less_eq_of_bool
thf(fact_3436_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_3437_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_3438_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_3439_of__bool__less__eq__one,axiom,
    ! [P: $o] : ( ord_le3102999989581377725nteger @ ( zero_n356916108424825756nteger @ P ) @ one_one_Code_integer ) ).

% of_bool_less_eq_one
thf(fact_3440_of__bool__less__eq__one,axiom,
    ! [P: $o] : ( ord_le1926595141338095240atural @ ( zero_n8403883297036319079atural @ P ) @ one_one_Code_natural ) ).

% of_bool_less_eq_one
thf(fact_3441_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_3442_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_3443_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_3444_of__bool__def,axiom,
    ( zero_n356916108424825756nteger
    = ( ^ [P6: $o] : ( if_Code_integer @ P6 @ one_one_Code_integer @ zero_z3403309356797280102nteger ) ) ) ).

% of_bool_def
thf(fact_3445_of__bool__def,axiom,
    ( zero_n2052037380579107095ol_rat
    = ( ^ [P6: $o] : ( if_rat @ P6 @ one_one_rat @ zero_zero_rat ) ) ) ).

% of_bool_def
thf(fact_3446_of__bool__def,axiom,
    ( zero_n8403883297036319079atural
    = ( ^ [P6: $o] : ( if_Code_natural @ P6 @ one_one_Code_natural @ zero_z2226904508553997617atural ) ) ) ).

% of_bool_def
thf(fact_3447_of__bool__def,axiom,
    ( zero_n2687167440665602831ol_nat
    = ( ^ [P6: $o] : ( if_nat @ P6 @ one_one_nat @ zero_zero_nat ) ) ) ).

% of_bool_def
thf(fact_3448_of__bool__def,axiom,
    ( zero_n2684676970156552555ol_int
    = ( ^ [P6: $o] : ( if_int @ P6 @ one_one_int @ zero_zero_int ) ) ) ).

% of_bool_def
thf(fact_3449_split__of__bool,axiom,
    ! [P: code_integer > $o,P5: $o] :
      ( ( P @ ( zero_n356916108424825756nteger @ P5 ) )
      = ( ( P5
         => ( P @ one_one_Code_integer ) )
        & ( ~ P5
         => ( P @ zero_z3403309356797280102nteger ) ) ) ) ).

% split_of_bool
thf(fact_3450_split__of__bool,axiom,
    ! [P: rat > $o,P5: $o] :
      ( ( P @ ( zero_n2052037380579107095ol_rat @ P5 ) )
      = ( ( P5
         => ( P @ one_one_rat ) )
        & ( ~ P5
         => ( P @ zero_zero_rat ) ) ) ) ).

% split_of_bool
thf(fact_3451_split__of__bool,axiom,
    ! [P: code_natural > $o,P5: $o] :
      ( ( P @ ( zero_n8403883297036319079atural @ P5 ) )
      = ( ( P5
         => ( P @ one_one_Code_natural ) )
        & ( ~ P5
         => ( P @ zero_z2226904508553997617atural ) ) ) ) ).

% split_of_bool
thf(fact_3452_split__of__bool,axiom,
    ! [P: nat > $o,P5: $o] :
      ( ( P @ ( zero_n2687167440665602831ol_nat @ P5 ) )
      = ( ( P5
         => ( P @ one_one_nat ) )
        & ( ~ P5
         => ( P @ zero_zero_nat ) ) ) ) ).

% split_of_bool
thf(fact_3453_split__of__bool,axiom,
    ! [P: int > $o,P5: $o] :
      ( ( P @ ( zero_n2684676970156552555ol_int @ P5 ) )
      = ( ( P5
         => ( P @ one_one_int ) )
        & ( ~ P5
         => ( P @ zero_zero_int ) ) ) ) ).

% split_of_bool
thf(fact_3454_split__of__bool__asm,axiom,
    ! [P: code_integer > $o,P5: $o] :
      ( ( P @ ( zero_n356916108424825756nteger @ P5 ) )
      = ( ~ ( ( P5
              & ~ ( P @ one_one_Code_integer ) )
            | ( ~ P5
              & ~ ( P @ zero_z3403309356797280102nteger ) ) ) ) ) ).

% split_of_bool_asm
thf(fact_3455_split__of__bool__asm,axiom,
    ! [P: rat > $o,P5: $o] :
      ( ( P @ ( zero_n2052037380579107095ol_rat @ P5 ) )
      = ( ~ ( ( P5
              & ~ ( P @ one_one_rat ) )
            | ( ~ P5
              & ~ ( P @ zero_zero_rat ) ) ) ) ) ).

% split_of_bool_asm
thf(fact_3456_split__of__bool__asm,axiom,
    ! [P: code_natural > $o,P5: $o] :
      ( ( P @ ( zero_n8403883297036319079atural @ P5 ) )
      = ( ~ ( ( P5
              & ~ ( P @ one_one_Code_natural ) )
            | ( ~ P5
              & ~ ( P @ zero_z2226904508553997617atural ) ) ) ) ) ).

% split_of_bool_asm
thf(fact_3457_split__of__bool__asm,axiom,
    ! [P: nat > $o,P5: $o] :
      ( ( P @ ( zero_n2687167440665602831ol_nat @ P5 ) )
      = ( ~ ( ( P5
              & ~ ( P @ one_one_nat ) )
            | ( ~ P5
              & ~ ( P @ zero_zero_nat ) ) ) ) ) ).

% split_of_bool_asm
thf(fact_3458_split__of__bool__asm,axiom,
    ! [P: int > $o,P5: $o] :
      ( ( P @ ( zero_n2684676970156552555ol_int @ P5 ) )
      = ( ~ ( ( P5
              & ~ ( P @ one_one_int ) )
            | ( ~ P5
              & ~ ( P @ zero_zero_int ) ) ) ) ) ).

% split_of_bool_asm
thf(fact_3459_add__decreasing,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger )
     => ( ( ord_le3102999989581377725nteger @ C @ B )
       => ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ A @ C ) @ B ) ) ) ).

% add_decreasing
thf(fact_3460_add__decreasing,axiom,
    ! [A: code_natural,C: code_natural,B: code_natural] :
      ( ( ord_le1926595141338095240atural @ A @ zero_z2226904508553997617atural )
     => ( ( ord_le1926595141338095240atural @ C @ B )
       => ( ord_le1926595141338095240atural @ ( plus_p4538020629002901425atural @ A @ C ) @ B ) ) ) ).

% add_decreasing
thf(fact_3461_add__decreasing,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ C @ B )
       => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ) ).

% add_decreasing
thf(fact_3462_add__decreasing,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ C @ B )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).

% add_decreasing
thf(fact_3463_add__decreasing,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( ord_less_eq_int @ C @ B )
       => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).

% add_decreasing
thf(fact_3464_add__increasing,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le3102999989581377725nteger @ B @ C )
       => ( ord_le3102999989581377725nteger @ B @ ( plus_p5714425477246183910nteger @ A @ C ) ) ) ) ).

% add_increasing
thf(fact_3465_add__increasing,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ A )
     => ( ( ord_le1926595141338095240atural @ B @ C )
       => ( ord_le1926595141338095240atural @ B @ ( plus_p4538020629002901425atural @ A @ C ) ) ) ) ).

% add_increasing
thf(fact_3466_add__increasing,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ord_less_eq_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).

% add_increasing
thf(fact_3467_add__increasing,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).

% add_increasing
thf(fact_3468_add__increasing,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ B @ C )
       => ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).

% add_increasing
thf(fact_3469_add__decreasing2,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ C @ zero_z3403309356797280102nteger )
     => ( ( ord_le3102999989581377725nteger @ A @ B )
       => ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ A @ C ) @ B ) ) ) ).

% add_decreasing2
thf(fact_3470_add__decreasing2,axiom,
    ! [C: code_natural,A: code_natural,B: code_natural] :
      ( ( ord_le1926595141338095240atural @ C @ zero_z2226904508553997617atural )
     => ( ( ord_le1926595141338095240atural @ A @ B )
       => ( ord_le1926595141338095240atural @ ( plus_p4538020629002901425atural @ A @ C ) @ B ) ) ) ).

% add_decreasing2
thf(fact_3471_add__decreasing2,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_eq_rat @ C @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ A @ B )
       => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ) ).

% add_decreasing2
thf(fact_3472_add__decreasing2,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ C @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ A @ B )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).

% add_decreasing2
thf(fact_3473_add__decreasing2,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ C @ zero_zero_int )
     => ( ( ord_less_eq_int @ A @ B )
       => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).

% add_decreasing2
thf(fact_3474_add__increasing2,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
     => ( ( ord_le3102999989581377725nteger @ B @ A )
       => ( ord_le3102999989581377725nteger @ B @ ( plus_p5714425477246183910nteger @ A @ C ) ) ) ) ).

% add_increasing2
thf(fact_3475_add__increasing2,axiom,
    ! [C: code_natural,B: code_natural,A: code_natural] :
      ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ C )
     => ( ( ord_le1926595141338095240atural @ B @ A )
       => ( ord_le1926595141338095240atural @ B @ ( plus_p4538020629002901425atural @ A @ C ) ) ) ) ).

% add_increasing2
thf(fact_3476_add__increasing2,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ C )
     => ( ( ord_less_eq_rat @ B @ A )
       => ( ord_less_eq_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).

% add_increasing2
thf(fact_3477_add__increasing2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ C )
     => ( ( ord_less_eq_nat @ B @ A )
       => ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).

% add_increasing2
thf(fact_3478_add__increasing2,axiom,
    ! [C: int,B: int,A: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ C )
     => ( ( ord_less_eq_int @ B @ A )
       => ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).

% add_increasing2
thf(fact_3479_add__nonneg__nonneg,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
       => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ A @ B ) ) ) ) ).

% add_nonneg_nonneg
thf(fact_3480_add__nonneg__nonneg,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ A )
     => ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ B )
       => ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ ( plus_p4538020629002901425atural @ A @ B ) ) ) ) ).

% add_nonneg_nonneg
thf(fact_3481_add__nonneg__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).

% add_nonneg_nonneg
thf(fact_3482_add__nonneg__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% add_nonneg_nonneg
thf(fact_3483_add__nonneg__nonneg,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ zero_zero_int @ B )
       => ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).

% add_nonneg_nonneg
thf(fact_3484_add__nonpos__nonpos,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger )
     => ( ( ord_le3102999989581377725nteger @ B @ zero_z3403309356797280102nteger )
       => ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ zero_z3403309356797280102nteger ) ) ) ).

% add_nonpos_nonpos
thf(fact_3485_add__nonpos__nonpos,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( ord_le1926595141338095240atural @ A @ zero_z2226904508553997617atural )
     => ( ( ord_le1926595141338095240atural @ B @ zero_z2226904508553997617atural )
       => ( ord_le1926595141338095240atural @ ( plus_p4538020629002901425atural @ A @ B ) @ zero_z2226904508553997617atural ) ) ) ).

% add_nonpos_nonpos
thf(fact_3486_add__nonpos__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).

% add_nonpos_nonpos
thf(fact_3487_add__nonpos__nonpos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% add_nonpos_nonpos
thf(fact_3488_add__nonpos__nonpos,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( ord_less_eq_int @ B @ zero_zero_int )
       => ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).

% add_nonpos_nonpos
thf(fact_3489_add__nonneg__eq__0__iff,axiom,
    ! [X: code_integer,Y3: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ Y3 )
       => ( ( ( plus_p5714425477246183910nteger @ X @ Y3 )
            = zero_z3403309356797280102nteger )
          = ( ( X = zero_z3403309356797280102nteger )
            & ( Y3 = zero_z3403309356797280102nteger ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_3490_add__nonneg__eq__0__iff,axiom,
    ! [X: code_natural,Y3: code_natural] :
      ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ X )
     => ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ Y3 )
       => ( ( ( plus_p4538020629002901425atural @ X @ Y3 )
            = zero_z2226904508553997617atural )
          = ( ( X = zero_z2226904508553997617atural )
            & ( Y3 = zero_z2226904508553997617atural ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_3491_add__nonneg__eq__0__iff,axiom,
    ! [X: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
       => ( ( ( plus_plus_rat @ X @ Y3 )
            = zero_zero_rat )
          = ( ( X = zero_zero_rat )
            & ( Y3 = zero_zero_rat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_3492_add__nonneg__eq__0__iff,axiom,
    ! [X: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ X )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ Y3 )
       => ( ( ( plus_plus_nat @ X @ Y3 )
            = zero_zero_nat )
          = ( ( X = zero_zero_nat )
            & ( Y3 = zero_zero_nat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_3493_add__nonneg__eq__0__iff,axiom,
    ! [X: int,Y3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
       => ( ( ( plus_plus_int @ X @ Y3 )
            = zero_zero_int )
          = ( ( X = zero_zero_int )
            & ( Y3 = zero_zero_int ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_3494_add__nonpos__eq__0__iff,axiom,
    ! [X: code_integer,Y3: code_integer] :
      ( ( ord_le3102999989581377725nteger @ X @ zero_z3403309356797280102nteger )
     => ( ( ord_le3102999989581377725nteger @ Y3 @ zero_z3403309356797280102nteger )
       => ( ( ( plus_p5714425477246183910nteger @ X @ Y3 )
            = zero_z3403309356797280102nteger )
          = ( ( X = zero_z3403309356797280102nteger )
            & ( Y3 = zero_z3403309356797280102nteger ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_3495_add__nonpos__eq__0__iff,axiom,
    ! [X: code_natural,Y3: code_natural] :
      ( ( ord_le1926595141338095240atural @ X @ zero_z2226904508553997617atural )
     => ( ( ord_le1926595141338095240atural @ Y3 @ zero_z2226904508553997617atural )
       => ( ( ( plus_p4538020629002901425atural @ X @ Y3 )
            = zero_z2226904508553997617atural )
          = ( ( X = zero_z2226904508553997617atural )
            & ( Y3 = zero_z2226904508553997617atural ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_3496_add__nonpos__eq__0__iff,axiom,
    ! [X: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ X @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ Y3 @ zero_zero_rat )
       => ( ( ( plus_plus_rat @ X @ Y3 )
            = zero_zero_rat )
          = ( ( X = zero_zero_rat )
            & ( Y3 = zero_zero_rat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_3497_add__nonpos__eq__0__iff,axiom,
    ! [X: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ X @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ Y3 @ zero_zero_nat )
       => ( ( ( plus_plus_nat @ X @ Y3 )
            = zero_zero_nat )
          = ( ( X = zero_zero_nat )
            & ( Y3 = zero_zero_nat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_3498_add__nonpos__eq__0__iff,axiom,
    ! [X: int,Y3: int] :
      ( ( ord_less_eq_int @ X @ zero_zero_int )
     => ( ( ord_less_eq_int @ Y3 @ zero_zero_int )
       => ( ( ( plus_plus_int @ X @ Y3 )
            = zero_zero_int )
          = ( ( X = zero_zero_int )
            & ( Y3 = zero_zero_int ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_3499_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I2: code_integer,J: code_integer,K: code_integer,L: code_integer] :
      ( ( ( ord_le3102999989581377725nteger @ I2 @ J )
        & ( ord_le6747313008572928689nteger @ K @ L ) )
     => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ I2 @ K ) @ ( plus_p5714425477246183910nteger @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_3500_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I2: rat,J: rat,K: rat,L: rat] :
      ( ( ( ord_less_eq_rat @ I2 @ J )
        & ( ord_less_rat @ K @ L ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I2 @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_3501_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I2: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_eq_nat @ I2 @ J )
        & ( ord_less_nat @ K @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_3502_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I2: int,J: int,K: int,L: int] :
      ( ( ( ord_less_eq_int @ I2 @ J )
        & ( ord_less_int @ K @ L ) )
     => ( ord_less_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_3503_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I2: code_integer,J: code_integer,K: code_integer,L: code_integer] :
      ( ( ( ord_le6747313008572928689nteger @ I2 @ J )
        & ( ord_le3102999989581377725nteger @ K @ L ) )
     => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ I2 @ K ) @ ( plus_p5714425477246183910nteger @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_3504_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I2: rat,J: rat,K: rat,L: rat] :
      ( ( ( ord_less_rat @ I2 @ J )
        & ( ord_less_eq_rat @ K @ L ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I2 @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_3505_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I2: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_nat @ I2 @ J )
        & ( ord_less_eq_nat @ K @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_3506_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I2: int,J: int,K: int,L: int] :
      ( ( ( ord_less_int @ I2 @ J )
        & ( ord_less_eq_int @ K @ L ) )
     => ( ord_less_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_3507_add__le__less__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ B )
     => ( ( ord_le6747313008572928689nteger @ C @ D )
       => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A @ C ) @ ( plus_p5714425477246183910nteger @ B @ D ) ) ) ) ).

% add_le_less_mono
thf(fact_3508_add__le__less__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_rat @ C @ D )
       => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).

% add_le_less_mono
thf(fact_3509_add__le__less__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ C @ D )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).

% add_le_less_mono
thf(fact_3510_add__le__less__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_int @ C @ D )
       => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).

% add_le_less_mono
thf(fact_3511_add__less__le__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ B )
     => ( ( ord_le3102999989581377725nteger @ C @ D )
       => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A @ C ) @ ( plus_p5714425477246183910nteger @ B @ D ) ) ) ) ).

% add_less_le_mono
thf(fact_3512_add__less__le__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_eq_rat @ C @ D )
       => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).

% add_less_le_mono
thf(fact_3513_add__less__le__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ D )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).

% add_less_le_mono
thf(fact_3514_add__less__le__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_eq_int @ C @ D )
       => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).

% add_less_le_mono
thf(fact_3515_add__neg__neg,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( ord_le5570908160329646204atural @ A @ zero_z2226904508553997617atural )
     => ( ( ord_le5570908160329646204atural @ B @ zero_z2226904508553997617atural )
       => ( ord_le5570908160329646204atural @ ( plus_p4538020629002901425atural @ A @ B ) @ zero_z2226904508553997617atural ) ) ) ).

% add_neg_neg
thf(fact_3516_add__neg__neg,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger )
     => ( ( ord_le6747313008572928689nteger @ B @ zero_z3403309356797280102nteger )
       => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ zero_z3403309356797280102nteger ) ) ) ).

% add_neg_neg
thf(fact_3517_add__neg__neg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( ord_less_rat @ B @ zero_zero_rat )
       => ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).

% add_neg_neg
thf(fact_3518_add__neg__neg,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ zero_zero_nat )
     => ( ( ord_less_nat @ B @ zero_zero_nat )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% add_neg_neg
thf(fact_3519_add__neg__neg,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ B @ zero_zero_int )
       => ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).

% add_neg_neg
thf(fact_3520_add__pos__pos,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ A )
     => ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ B )
       => ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ ( plus_p4538020629002901425atural @ A @ B ) ) ) ) ).

% add_pos_pos
thf(fact_3521_add__pos__pos,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
       => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ A @ B ) ) ) ) ).

% add_pos_pos
thf(fact_3522_add__pos__pos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ zero_zero_rat @ B )
       => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).

% add_pos_pos
thf(fact_3523_add__pos__pos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ B )
       => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% add_pos_pos
thf(fact_3524_add__pos__pos,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).

% add_pos_pos
thf(fact_3525_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ! [C2: nat] :
            ( ( B
              = ( plus_plus_nat @ A @ C2 ) )
           => ( C2 = zero_zero_nat ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
thf(fact_3526_pos__add__strict,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ A )
     => ( ( ord_le5570908160329646204atural @ B @ C )
       => ( ord_le5570908160329646204atural @ B @ ( plus_p4538020629002901425atural @ A @ C ) ) ) ) ).

% pos_add_strict
thf(fact_3527_pos__add__strict,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le6747313008572928689nteger @ B @ C )
       => ( ord_le6747313008572928689nteger @ B @ ( plus_p5714425477246183910nteger @ A @ C ) ) ) ) ).

% pos_add_strict
thf(fact_3528_pos__add__strict,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ B @ C )
       => ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).

% pos_add_strict
thf(fact_3529_pos__add__strict,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ B @ C )
       => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).

% pos_add_strict
thf(fact_3530_pos__add__strict,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ B @ C )
       => ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).

% pos_add_strict
thf(fact_3531_add__less__zeroD,axiom,
    ! [X: code_integer,Y3: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ X @ Y3 ) @ zero_z3403309356797280102nteger )
     => ( ( ord_le6747313008572928689nteger @ X @ zero_z3403309356797280102nteger )
        | ( ord_le6747313008572928689nteger @ Y3 @ zero_z3403309356797280102nteger ) ) ) ).

% add_less_zeroD
thf(fact_3532_add__less__zeroD,axiom,
    ! [X: rat,Y3: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ X @ Y3 ) @ zero_zero_rat )
     => ( ( ord_less_rat @ X @ zero_zero_rat )
        | ( ord_less_rat @ Y3 @ zero_zero_rat ) ) ) ).

% add_less_zeroD
thf(fact_3533_add__less__zeroD,axiom,
    ! [X: int,Y3: int] :
      ( ( ord_less_int @ ( plus_plus_int @ X @ Y3 ) @ zero_zero_int )
     => ( ( ord_less_int @ X @ zero_zero_int )
        | ( ord_less_int @ Y3 @ zero_zero_int ) ) ) ).

% add_less_zeroD
thf(fact_3534_add__le__imp__le__diff,axiom,
    ! [I2: code_integer,K: code_integer,N: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ I2 @ K ) @ N )
     => ( ord_le3102999989581377725nteger @ I2 @ ( minus_8373710615458151222nteger @ N @ K ) ) ) ).

% add_le_imp_le_diff
thf(fact_3535_add__le__imp__le__diff,axiom,
    ! [I2: rat,K: rat,N: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ I2 @ K ) @ N )
     => ( ord_less_eq_rat @ I2 @ ( minus_minus_rat @ N @ K ) ) ) ).

% add_le_imp_le_diff
thf(fact_3536_add__le__imp__le__diff,axiom,
    ! [I2: nat,K: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ N )
     => ( ord_less_eq_nat @ I2 @ ( minus_minus_nat @ N @ K ) ) ) ).

% add_le_imp_le_diff
thf(fact_3537_add__le__imp__le__diff,axiom,
    ! [I2: int,K: int,N: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K ) @ N )
     => ( ord_less_eq_int @ I2 @ ( minus_minus_int @ N @ K ) ) ) ).

% add_le_imp_le_diff
thf(fact_3538_add__le__add__imp__diff__le,axiom,
    ! [I2: code_integer,K: code_integer,N: code_integer,J: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ I2 @ K ) @ N )
     => ( ( ord_le3102999989581377725nteger @ N @ ( plus_p5714425477246183910nteger @ J @ K ) )
       => ( ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ I2 @ K ) @ N )
         => ( ( ord_le3102999989581377725nteger @ N @ ( plus_p5714425477246183910nteger @ J @ K ) )
           => ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ N @ K ) @ J ) ) ) ) ) ).

% add_le_add_imp_diff_le
thf(fact_3539_add__le__add__imp__diff__le,axiom,
    ! [I2: rat,K: rat,N: rat,J: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ I2 @ K ) @ N )
     => ( ( ord_less_eq_rat @ N @ ( plus_plus_rat @ J @ K ) )
       => ( ( ord_less_eq_rat @ ( plus_plus_rat @ I2 @ 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_3540_add__le__add__imp__diff__le,axiom,
    ! [I2: nat,K: nat,N: nat,J: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ N )
     => ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
       => ( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ 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_3541_add__le__add__imp__diff__le,axiom,
    ! [I2: int,K: int,N: int,J: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K ) @ N )
     => ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
       => ( ( ord_less_eq_int @ ( plus_plus_int @ I2 @ 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_3542_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ A @ B )
       => ( ( ( minus_minus_nat @ B @ A )
            = C )
          = ( B
            = ( plus_plus_nat @ C @ A ) ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_3543_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
        = B ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_3544_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_3545_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A )
        = ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_3546_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C )
        = ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_3547_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A )
        = ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_3548_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_3549_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A ) )
        = ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_3550_le__add__diff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).

% le_add_diff
thf(fact_3551_diff__add,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
        = B ) ) ).

% diff_add
thf(fact_3552_le__diff__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ ( minus_8373710615458151222nteger @ C @ B ) )
      = ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C ) ) ).

% le_diff_eq
thf(fact_3553_le__diff__eq,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ ( minus_minus_rat @ C @ B ) )
      = ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B ) @ C ) ) ).

% le_diff_eq
thf(fact_3554_le__diff__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_eq_int @ A @ ( minus_minus_int @ C @ B ) )
      = ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).

% le_diff_eq
thf(fact_3555_diff__le__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C )
      = ( ord_le3102999989581377725nteger @ A @ ( plus_p5714425477246183910nteger @ C @ B ) ) ) ).

% diff_le_eq
thf(fact_3556_diff__le__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ ( minus_minus_rat @ A @ B ) @ C )
      = ( ord_less_eq_rat @ A @ ( plus_plus_rat @ C @ B ) ) ) ).

% diff_le_eq
thf(fact_3557_diff__le__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ ( minus_minus_int @ A @ B ) @ C )
      = ( ord_less_eq_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).

% diff_le_eq
thf(fact_3558_less__add__one,axiom,
    ! [A: code_natural] : ( ord_le5570908160329646204atural @ A @ ( plus_p4538020629002901425atural @ A @ one_one_Code_natural ) ) ).

% less_add_one
thf(fact_3559_less__add__one,axiom,
    ! [A: code_integer] : ( ord_le6747313008572928689nteger @ A @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) ) ).

% less_add_one
thf(fact_3560_less__add__one,axiom,
    ! [A: rat] : ( ord_less_rat @ A @ ( plus_plus_rat @ A @ one_one_rat ) ) ).

% less_add_one
thf(fact_3561_less__add__one,axiom,
    ! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).

% less_add_one
thf(fact_3562_less__add__one,axiom,
    ! [A: int] : ( ord_less_int @ A @ ( plus_plus_int @ A @ one_one_int ) ) ).

% less_add_one
thf(fact_3563_add__mono1,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( ord_le5570908160329646204atural @ A @ B )
     => ( ord_le5570908160329646204atural @ ( plus_p4538020629002901425atural @ A @ one_one_Code_natural ) @ ( plus_p4538020629002901425atural @ B @ one_one_Code_natural ) ) ) ).

% add_mono1
thf(fact_3564_add__mono1,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ B )
     => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) @ ( plus_p5714425477246183910nteger @ B @ one_one_Code_integer ) ) ) ).

% add_mono1
thf(fact_3565_add__mono1,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ord_less_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( plus_plus_rat @ B @ one_one_rat ) ) ) ).

% add_mono1
thf(fact_3566_add__mono1,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ord_less_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).

% add_mono1
thf(fact_3567_add__mono1,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ( ord_less_int @ ( plus_plus_int @ A @ one_one_int ) @ ( plus_plus_int @ B @ one_one_int ) ) ) ).

% add_mono1
thf(fact_3568_diff__less__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C )
      = ( ord_le6747313008572928689nteger @ A @ ( plus_p5714425477246183910nteger @ C @ B ) ) ) ).

% diff_less_eq
thf(fact_3569_diff__less__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ ( minus_minus_rat @ A @ B ) @ C )
      = ( ord_less_rat @ A @ ( plus_plus_rat @ C @ B ) ) ) ).

% diff_less_eq
thf(fact_3570_diff__less__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ ( minus_minus_int @ A @ B ) @ C )
      = ( ord_less_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).

% diff_less_eq
thf(fact_3571_less__diff__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ ( minus_8373710615458151222nteger @ C @ B ) )
      = ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C ) ) ).

% less_diff_eq
thf(fact_3572_less__diff__eq,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_rat @ A @ ( minus_minus_rat @ C @ B ) )
      = ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ C ) ) ).

% less_diff_eq
thf(fact_3573_less__diff__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_int @ A @ ( minus_minus_int @ C @ B ) )
      = ( ord_less_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).

% less_diff_eq
thf(fact_3574_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ~ ( ord_le6747313008572928689nteger @ A @ B )
     => ( ( plus_p5714425477246183910nteger @ B @ ( minus_8373710615458151222nteger @ A @ B ) )
        = A ) ) ).

% linordered_semidom_class.add_diff_inverse
thf(fact_3575_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: rat,B: rat] :
      ( ~ ( ord_less_rat @ A @ B )
     => ( ( plus_plus_rat @ B @ ( minus_minus_rat @ A @ B ) )
        = A ) ) ).

% linordered_semidom_class.add_diff_inverse
thf(fact_3576_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: nat,B: nat] :
      ( ~ ( ord_less_nat @ A @ B )
     => ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
        = A ) ) ).

% linordered_semidom_class.add_diff_inverse
thf(fact_3577_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: int,B: int] :
      ( ~ ( ord_less_int @ A @ B )
     => ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
        = A ) ) ).

% linordered_semidom_class.add_diff_inverse
thf(fact_3578_add__eq__0__iff,axiom,
    ! [A: int,B: int] :
      ( ( ( plus_plus_int @ A @ B )
        = zero_zero_int )
      = ( B
        = ( uminus_uminus_int @ A ) ) ) ).

% add_eq_0_iff
thf(fact_3579_add__eq__0__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( plus_p5714425477246183910nteger @ A @ B )
        = zero_z3403309356797280102nteger )
      = ( B
        = ( uminus1351360451143612070nteger @ A ) ) ) ).

% add_eq_0_iff
thf(fact_3580_add__eq__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ( plus_plus_rat @ A @ B )
        = zero_zero_rat )
      = ( B
        = ( uminus_uminus_rat @ A ) ) ) ).

% add_eq_0_iff
thf(fact_3581_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_3582_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_3583_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_3584_add_Oinverse__unique,axiom,
    ! [A: int,B: int] :
      ( ( ( plus_plus_int @ A @ B )
        = zero_zero_int )
     => ( ( uminus_uminus_int @ A )
        = B ) ) ).

% add.inverse_unique
thf(fact_3585_add_Oinverse__unique,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( plus_p5714425477246183910nteger @ A @ B )
        = zero_z3403309356797280102nteger )
     => ( ( uminus1351360451143612070nteger @ A )
        = B ) ) ).

% add.inverse_unique
thf(fact_3586_add_Oinverse__unique,axiom,
    ! [A: rat,B: rat] :
      ( ( ( plus_plus_rat @ A @ B )
        = zero_zero_rat )
     => ( ( uminus_uminus_rat @ A )
        = B ) ) ).

% add.inverse_unique
thf(fact_3587_eq__neg__iff__add__eq__0,axiom,
    ! [A: int,B: int] :
      ( ( A
        = ( uminus_uminus_int @ B ) )
      = ( ( plus_plus_int @ A @ B )
        = zero_zero_int ) ) ).

% eq_neg_iff_add_eq_0
thf(fact_3588_eq__neg__iff__add__eq__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A
        = ( uminus1351360451143612070nteger @ B ) )
      = ( ( plus_p5714425477246183910nteger @ A @ B )
        = zero_z3403309356797280102nteger ) ) ).

% eq_neg_iff_add_eq_0
thf(fact_3589_eq__neg__iff__add__eq__0,axiom,
    ! [A: rat,B: rat] :
      ( ( A
        = ( uminus_uminus_rat @ B ) )
      = ( ( plus_plus_rat @ A @ B )
        = zero_zero_rat ) ) ).

% eq_neg_iff_add_eq_0
thf(fact_3590_neg__eq__iff__add__eq__0,axiom,
    ! [A: int,B: int] :
      ( ( ( uminus_uminus_int @ A )
        = B )
      = ( ( plus_plus_int @ A @ B )
        = zero_zero_int ) ) ).

% neg_eq_iff_add_eq_0
thf(fact_3591_neg__eq__iff__add__eq__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( uminus1351360451143612070nteger @ A )
        = B )
      = ( ( plus_p5714425477246183910nteger @ A @ B )
        = zero_z3403309356797280102nteger ) ) ).

% neg_eq_iff_add_eq_0
thf(fact_3592_neg__eq__iff__add__eq__0,axiom,
    ! [A: rat,B: rat] :
      ( ( ( uminus_uminus_rat @ A )
        = B )
      = ( ( plus_plus_rat @ A @ B )
        = zero_zero_rat ) ) ).

% neg_eq_iff_add_eq_0
thf(fact_3593_square__diff__square__factored,axiom,
    ! [X: code_integer,Y3: code_integer] :
      ( ( minus_8373710615458151222nteger @ ( times_3573771949741848930nteger @ X @ X ) @ ( times_3573771949741848930nteger @ Y3 @ Y3 ) )
      = ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ X @ Y3 ) @ ( minus_8373710615458151222nteger @ X @ Y3 ) ) ) ).

% square_diff_square_factored
thf(fact_3594_square__diff__square__factored,axiom,
    ! [X: rat,Y3: rat] :
      ( ( minus_minus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y3 @ Y3 ) )
      = ( times_times_rat @ ( plus_plus_rat @ X @ Y3 ) @ ( minus_minus_rat @ X @ Y3 ) ) ) ).

% square_diff_square_factored
thf(fact_3595_square__diff__square__factored,axiom,
    ! [X: int,Y3: int] :
      ( ( minus_minus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y3 @ Y3 ) )
      = ( times_times_int @ ( plus_plus_int @ X @ Y3 ) @ ( minus_minus_int @ X @ Y3 ) ) ) ).

% square_diff_square_factored
thf(fact_3596_eq__add__iff2,axiom,
    ! [A: code_integer,E: code_integer,C: code_integer,B: code_integer,D: code_integer] :
      ( ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ E ) @ C )
        = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ E ) @ D ) )
      = ( C
        = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ B @ A ) @ E ) @ D ) ) ) ).

% eq_add_iff2
thf(fact_3597_eq__add__iff2,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
      ( ( ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C )
        = ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D ) )
      = ( C
        = ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E ) @ D ) ) ) ).

% eq_add_iff2
thf(fact_3598_eq__add__iff2,axiom,
    ! [A: int,E: int,C: int,B: int,D: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ A @ E ) @ C )
        = ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
      = ( C
        = ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E ) @ D ) ) ) ).

% eq_add_iff2
thf(fact_3599_eq__add__iff1,axiom,
    ! [A: code_integer,E: code_integer,C: code_integer,B: code_integer,D: code_integer] :
      ( ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ E ) @ C )
        = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ E ) @ D ) )
      = ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ E ) @ C )
        = D ) ) ).

% eq_add_iff1
thf(fact_3600_eq__add__iff1,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
      ( ( ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C )
        = ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D ) )
      = ( ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E ) @ C )
        = D ) ) ).

% eq_add_iff1
thf(fact_3601_eq__add__iff1,axiom,
    ! [A: int,E: int,C: int,B: int,D: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ A @ E ) @ C )
        = ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
      = ( ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E ) @ C )
        = D ) ) ).

% eq_add_iff1
thf(fact_3602_pinf_I9_J,axiom,
    ! [D: code_integer,S: code_integer] :
    ? [Z3: code_integer] :
    ! [X4: code_integer] :
      ( ( ord_le6747313008572928689nteger @ Z3 @ X4 )
     => ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X4 @ S ) )
        = ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X4 @ S ) ) ) ) ).

% pinf(9)
thf(fact_3603_pinf_I9_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z3: rat] :
    ! [X4: rat] :
      ( ( ord_less_rat @ Z3 @ X4 )
     => ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X4 @ S ) )
        = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X4 @ S ) ) ) ) ).

% pinf(9)
thf(fact_3604_pinf_I9_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z3: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ Z3 @ X4 )
     => ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X4 @ S ) )
        = ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X4 @ S ) ) ) ) ).

% pinf(9)
thf(fact_3605_pinf_I9_J,axiom,
    ! [D: int,S: int] :
    ? [Z3: int] :
    ! [X4: int] :
      ( ( ord_less_int @ Z3 @ X4 )
     => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X4 @ S ) )
        = ( dvd_dvd_int @ D @ ( plus_plus_int @ X4 @ S ) ) ) ) ).

% pinf(9)
thf(fact_3606_pinf_I10_J,axiom,
    ! [D: code_integer,S: code_integer] :
    ? [Z3: code_integer] :
    ! [X4: code_integer] :
      ( ( ord_le6747313008572928689nteger @ Z3 @ X4 )
     => ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X4 @ S ) ) )
        = ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X4 @ S ) ) ) ) ) ).

% pinf(10)
thf(fact_3607_pinf_I10_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z3: rat] :
    ! [X4: rat] :
      ( ( ord_less_rat @ Z3 @ X4 )
     => ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X4 @ S ) ) )
        = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X4 @ S ) ) ) ) ) ).

% pinf(10)
thf(fact_3608_pinf_I10_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z3: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ Z3 @ X4 )
     => ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X4 @ S ) ) )
        = ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X4 @ S ) ) ) ) ) ).

% pinf(10)
thf(fact_3609_pinf_I10_J,axiom,
    ! [D: int,S: int] :
    ? [Z3: int] :
    ! [X4: int] :
      ( ( ord_less_int @ Z3 @ X4 )
     => ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X4 @ S ) ) )
        = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X4 @ S ) ) ) ) ) ).

% pinf(10)
thf(fact_3610_minf_I9_J,axiom,
    ! [D: code_integer,S: code_integer] :
    ? [Z3: code_integer] :
    ! [X4: code_integer] :
      ( ( ord_le6747313008572928689nteger @ X4 @ Z3 )
     => ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X4 @ S ) )
        = ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X4 @ S ) ) ) ) ).

% minf(9)
thf(fact_3611_minf_I9_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z3: rat] :
    ! [X4: rat] :
      ( ( ord_less_rat @ X4 @ Z3 )
     => ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X4 @ S ) )
        = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X4 @ S ) ) ) ) ).

% minf(9)
thf(fact_3612_minf_I9_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z3: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ X4 @ Z3 )
     => ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X4 @ S ) )
        = ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X4 @ S ) ) ) ) ).

% minf(9)
thf(fact_3613_minf_I9_J,axiom,
    ! [D: int,S: int] :
    ? [Z3: int] :
    ! [X4: int] :
      ( ( ord_less_int @ X4 @ Z3 )
     => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X4 @ S ) )
        = ( dvd_dvd_int @ D @ ( plus_plus_int @ X4 @ S ) ) ) ) ).

% minf(9)
thf(fact_3614_minf_I10_J,axiom,
    ! [D: code_integer,S: code_integer] :
    ? [Z3: code_integer] :
    ! [X4: code_integer] :
      ( ( ord_le6747313008572928689nteger @ X4 @ Z3 )
     => ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X4 @ S ) ) )
        = ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X4 @ S ) ) ) ) ) ).

% minf(10)
thf(fact_3615_minf_I10_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z3: rat] :
    ! [X4: rat] :
      ( ( ord_less_rat @ X4 @ Z3 )
     => ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X4 @ S ) ) )
        = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X4 @ S ) ) ) ) ) ).

% minf(10)
thf(fact_3616_minf_I10_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z3: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ X4 @ Z3 )
     => ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X4 @ S ) ) )
        = ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X4 @ S ) ) ) ) ) ).

% minf(10)
thf(fact_3617_minf_I10_J,axiom,
    ! [D: int,S: int] :
    ? [Z3: int] :
    ! [X4: int] :
      ( ( ord_less_int @ X4 @ Z3 )
     => ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X4 @ S ) ) )
        = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X4 @ S ) ) ) ) ) ).

% minf(10)
thf(fact_3618_group__cancel_Osub2,axiom,
    ! [B5: int,K: int,B: int,A: int] :
      ( ( B5
        = ( plus_plus_int @ K @ B ) )
     => ( ( minus_minus_int @ A @ B5 )
        = ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( minus_minus_int @ A @ B ) ) ) ) ).

% group_cancel.sub2
thf(fact_3619_group__cancel_Osub2,axiom,
    ! [B5: code_integer,K: code_integer,B: code_integer,A: code_integer] :
      ( ( B5
        = ( plus_p5714425477246183910nteger @ K @ B ) )
     => ( ( minus_8373710615458151222nteger @ A @ B5 )
        = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ K ) @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ) ).

% group_cancel.sub2
thf(fact_3620_group__cancel_Osub2,axiom,
    ! [B5: rat,K: rat,B: rat,A: rat] :
      ( ( B5
        = ( plus_plus_rat @ K @ B ) )
     => ( ( minus_minus_rat @ A @ B5 )
        = ( plus_plus_rat @ ( uminus_uminus_rat @ K ) @ ( minus_minus_rat @ A @ B ) ) ) ) ).

% group_cancel.sub2
thf(fact_3621_diff__conv__add__uminus,axiom,
    ( minus_minus_int
    = ( ^ [A2: int,B2: int] : ( plus_plus_int @ A2 @ ( uminus_uminus_int @ B2 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_3622_diff__conv__add__uminus,axiom,
    ( minus_8373710615458151222nteger
    = ( ^ [A2: code_integer,B2: code_integer] : ( plus_p5714425477246183910nteger @ A2 @ ( uminus1351360451143612070nteger @ B2 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_3623_diff__conv__add__uminus,axiom,
    ( minus_minus_rat
    = ( ^ [A2: rat,B2: rat] : ( plus_plus_rat @ A2 @ ( uminus_uminus_rat @ B2 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_3624_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_minus_int
    = ( ^ [A2: int,B2: int] : ( plus_plus_int @ A2 @ ( uminus_uminus_int @ B2 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_3625_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_8373710615458151222nteger
    = ( ^ [A2: code_integer,B2: code_integer] : ( plus_p5714425477246183910nteger @ A2 @ ( uminus1351360451143612070nteger @ B2 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_3626_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_minus_rat
    = ( ^ [A2: rat,B2: rat] : ( plus_plus_rat @ A2 @ ( uminus_uminus_rat @ B2 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_3627_div__plus__div__distrib__dvd__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( dvd_dvd_nat @ C @ A )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
        = ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) ) ) ).

% div_plus_div_distrib_dvd_left
thf(fact_3628_div__plus__div__distrib__dvd__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( dvd_dvd_int @ C @ A )
     => ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
        = ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ).

% div_plus_div_distrib_dvd_left
thf(fact_3629_div__plus__div__distrib__dvd__left,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ A )
     => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
        = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) ) ) ).

% div_plus_div_distrib_dvd_left
thf(fact_3630_div__plus__div__distrib__dvd__left,axiom,
    ! [C: code_natural,A: code_natural,B: code_natural] :
      ( ( dvd_dvd_Code_natural @ C @ A )
     => ( ( divide5121882707175180666atural @ ( plus_p4538020629002901425atural @ A @ B ) @ C )
        = ( plus_p4538020629002901425atural @ ( divide5121882707175180666atural @ A @ C ) @ ( divide5121882707175180666atural @ B @ C ) ) ) ) ).

% div_plus_div_distrib_dvd_left
thf(fact_3631_div__plus__div__distrib__dvd__right,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( dvd_dvd_nat @ C @ B )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
        = ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) ) ) ).

% div_plus_div_distrib_dvd_right
thf(fact_3632_div__plus__div__distrib__dvd__right,axiom,
    ! [C: int,B: int,A: int] :
      ( ( dvd_dvd_int @ C @ B )
     => ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
        = ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ).

% div_plus_div_distrib_dvd_right
thf(fact_3633_div__plus__div__distrib__dvd__right,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ B )
     => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
        = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) ) ) ).

% div_plus_div_distrib_dvd_right
thf(fact_3634_div__plus__div__distrib__dvd__right,axiom,
    ! [C: code_natural,B: code_natural,A: code_natural] :
      ( ( dvd_dvd_Code_natural @ C @ B )
     => ( ( divide5121882707175180666atural @ ( plus_p4538020629002901425atural @ A @ B ) @ C )
        = ( plus_p4538020629002901425atural @ ( divide5121882707175180666atural @ A @ C ) @ ( divide5121882707175180666atural @ B @ C ) ) ) ) ).

% div_plus_div_distrib_dvd_right
thf(fact_3635_mod__eqE,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ( modulo_modulo_int @ A @ C )
        = ( modulo_modulo_int @ B @ C ) )
     => ~ ! [D5: int] :
            ( B
           != ( plus_plus_int @ A @ ( times_times_int @ C @ D5 ) ) ) ) ).

% mod_eqE
thf(fact_3636_mod__eqE,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ C )
        = ( modulo364778990260209775nteger @ B @ C ) )
     => ~ ! [D5: code_integer] :
            ( B
           != ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ C @ D5 ) ) ) ) ).

% mod_eqE
thf(fact_3637_div__add1__eq,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
      = ( plus_plus_nat @ ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) @ ( divide_divide_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B @ C ) ) @ C ) ) ) ).

% div_add1_eq
thf(fact_3638_div__add1__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
      = ( plus_plus_int @ ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) @ ( divide_divide_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B @ C ) ) @ C ) ) ) ).

% div_add1_eq
thf(fact_3639_div__add1__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
      = ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) @ ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B @ C ) ) @ C ) ) ) ).

% div_add1_eq
thf(fact_3640_div__add1__eq,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( ( divide5121882707175180666atural @ ( plus_p4538020629002901425atural @ A @ B ) @ C )
      = ( plus_p4538020629002901425atural @ ( plus_p4538020629002901425atural @ ( divide5121882707175180666atural @ A @ C ) @ ( divide5121882707175180666atural @ B @ C ) ) @ ( divide5121882707175180666atural @ ( plus_p4538020629002901425atural @ ( modulo8411746178871703098atural @ A @ C ) @ ( modulo8411746178871703098atural @ B @ C ) ) @ C ) ) ) ).

% div_add1_eq
thf(fact_3641_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_3642_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_3643_less__natE,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ~ ! [Q5: nat] :
            ( N
           != ( suc @ ( plus_plus_nat @ M @ Q5 ) ) ) ) ).

% less_natE
thf(fact_3644_less__add__Suc1,axiom,
    ! [I2: nat,M: nat] : ( ord_less_nat @ I2 @ ( suc @ ( plus_plus_nat @ I2 @ M ) ) ) ).

% less_add_Suc1
thf(fact_3645_less__add__Suc2,axiom,
    ! [I2: nat,M: nat] : ( ord_less_nat @ I2 @ ( suc @ ( plus_plus_nat @ M @ I2 ) ) ) ).

% less_add_Suc2
thf(fact_3646_less__iff__Suc__add,axiom,
    ( ord_less_nat
    = ( ^ [M2: nat,N3: nat] :
        ? [K3: nat] :
          ( N3
          = ( suc @ ( plus_plus_nat @ M2 @ K3 ) ) ) ) ) ).

% less_iff_Suc_add
thf(fact_3647_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_3648_less__imp__add__positive,axiom,
    ! [I2: nat,J: nat] :
      ( ( ord_less_nat @ I2 @ J )
     => ? [K2: nat] :
          ( ( ord_less_nat @ zero_zero_nat @ K2 )
          & ( ( plus_plus_nat @ I2 @ K2 )
            = J ) ) ) ).

% less_imp_add_positive
thf(fact_3649_mono__nat__linear__lb,axiom,
    ! [F: nat > nat,M: nat,K: nat] :
      ( ! [M6: nat,N4: nat] :
          ( ( ord_less_nat @ M6 @ N4 )
         => ( ord_less_nat @ ( F @ M6 ) @ ( F @ N4 ) ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).

% mono_nat_linear_lb
thf(fact_3650_bezout__add__strong__nat,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ? [D5: nat,X3: nat,Y4: nat] :
          ( ( dvd_dvd_nat @ D5 @ A )
          & ( dvd_dvd_nat @ D5 @ B )
          & ( ( times_times_nat @ A @ X3 )
            = ( plus_plus_nat @ ( times_times_nat @ B @ Y4 ) @ D5 ) ) ) ) ).

% bezout_add_strong_nat
thf(fact_3651_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_3652_odd__nonzero,axiom,
    ! [Z2: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z2 ) @ Z2 )
     != zero_zero_int ) ).

% odd_nonzero
thf(fact_3653_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_3654_less__diff__conv,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
      = ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ J ) ) ).

% less_diff_conv
thf(fact_3655_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_3656_Suc__eq__plus1__left,axiom,
    ( suc
    = ( plus_plus_nat @ one_one_nat ) ) ).

% Suc_eq_plus1_left
thf(fact_3657_plus__1__eq__Suc,axiom,
    ( ( plus_plus_nat @ one_one_nat )
    = suc ) ).

% plus_1_eq_Suc
thf(fact_3658_Suc__eq__plus1,axiom,
    ( suc
    = ( ^ [N3: nat] : ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ).

% Suc_eq_plus1
thf(fact_3659_Nat_Ole__imp__diff__is__add,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I2 @ J )
     => ( ( ( minus_minus_nat @ J @ I2 )
          = K )
        = ( J
          = ( plus_plus_nat @ K @ I2 ) ) ) ) ).

% Nat.le_imp_diff_is_add
thf(fact_3660_Nat_Odiff__add__assoc2,axiom,
    ! [K: nat,J: nat,I2: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I2 ) @ K )
        = ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I2 ) ) ) ).

% Nat.diff_add_assoc2
thf(fact_3661_Nat_Odiff__add__assoc,axiom,
    ! [K: nat,J: nat,I2: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ I2 @ J ) @ K )
        = ( plus_plus_nat @ I2 @ ( minus_minus_nat @ J @ K ) ) ) ) ).

% Nat.diff_add_assoc
thf(fact_3662_Nat_Ole__diff__conv2,axiom,
    ! [K: nat,J: nat,I2: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( ord_less_eq_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
        = ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ J ) ) ) ).

% Nat.le_diff_conv2
thf(fact_3663_le__diff__conv,axiom,
    ! [J: nat,K: nat,I2: nat] :
      ( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I2 )
      = ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I2 @ K ) ) ) ).

% le_diff_conv
thf(fact_3664_int__ge__induct,axiom,
    ! [K: int,I2: int,P: int > $o] :
      ( ( ord_less_eq_int @ K @ I2 )
     => ( ( P @ K )
       => ( ! [I3: int] :
              ( ( ord_less_eq_int @ K @ I3 )
             => ( ( P @ I3 )
               => ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
         => ( P @ I2 ) ) ) ) ).

% int_ge_induct
thf(fact_3665_zless__add1__eq,axiom,
    ! [W2: int,Z2: int] :
      ( ( ord_less_int @ W2 @ ( plus_plus_int @ Z2 @ one_one_int ) )
      = ( ( ord_less_int @ W2 @ Z2 )
        | ( W2 = Z2 ) ) ) ).

% zless_add1_eq
thf(fact_3666_int__gr__induct,axiom,
    ! [K: int,I2: int,P: int > $o] :
      ( ( ord_less_int @ K @ I2 )
     => ( ( 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 @ I2 ) ) ) ) ).

% int_gr_induct
thf(fact_3667_zle__iff__zadd,axiom,
    ( ord_less_eq_int
    = ( ^ [W3: int,Z6: int] :
        ? [N3: nat] :
          ( Z6
          = ( plus_plus_int @ W3 @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).

% zle_iff_zadd
thf(fact_3668_zdvd__period,axiom,
    ! [A: int,D: int,X: int,T: int,C: int] :
      ( ( dvd_dvd_int @ A @ D )
     => ( ( dvd_dvd_int @ A @ ( plus_plus_int @ X @ T ) )
        = ( dvd_dvd_int @ A @ ( plus_plus_int @ ( plus_plus_int @ X @ ( times_times_int @ C @ D ) ) @ T ) ) ) ) ).

% zdvd_period
thf(fact_3669_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_3670_nat__mod__eq__iff,axiom,
    ! [X: nat,N: nat,Y3: nat] :
      ( ( ( modulo_modulo_nat @ X @ N )
        = ( modulo_modulo_nat @ Y3 @ N ) )
      = ( ? [Q1: nat,Q22: nat] :
            ( ( plus_plus_nat @ X @ ( times_times_nat @ N @ Q1 ) )
            = ( plus_plus_nat @ Y3 @ ( times_times_nat @ N @ Q22 ) ) ) ) ) ).

% nat_mod_eq_iff
thf(fact_3671_frac__1__eq,axiom,
    ! [X: rat] :
      ( ( archimedean_frac_rat @ ( plus_plus_rat @ X @ one_one_rat ) )
      = ( archimedean_frac_rat @ X ) ) ).

% frac_1_eq
thf(fact_3672_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_3673_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_3674_dbl__inc__def,axiom,
    ( neg_nu5831290666863070958nteger
    = ( ^ [X2: code_integer] : ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ X2 @ X2 ) @ one_one_Code_integer ) ) ) ).

% dbl_inc_def
thf(fact_3675_dbl__inc__def,axiom,
    ( neg_nu5219082963157363817nc_rat
    = ( ^ [X2: rat] : ( plus_plus_rat @ ( plus_plus_rat @ X2 @ X2 ) @ one_one_rat ) ) ) ).

% dbl_inc_def
thf(fact_3676_dbl__inc__def,axiom,
    ( neg_nu5851722552734809277nc_int
    = ( ^ [X2: int] : ( plus_plus_int @ ( plus_plus_int @ X2 @ X2 ) @ one_one_int ) ) ) ).

% dbl_inc_def
thf(fact_3677_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_3678_field__le__epsilon,axiom,
    ! [X: rat,Y3: rat] :
      ( ! [E2: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ E2 )
         => ( ord_less_eq_rat @ X @ ( plus_plus_rat @ Y3 @ E2 ) ) )
     => ( ord_less_eq_rat @ X @ Y3 ) ) ).

% field_le_epsilon
thf(fact_3679_add__strict__increasing2,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ A )
     => ( ( ord_le5570908160329646204atural @ B @ C )
       => ( ord_le5570908160329646204atural @ B @ ( plus_p4538020629002901425atural @ A @ C ) ) ) ) ).

% add_strict_increasing2
thf(fact_3680_add__strict__increasing2,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le6747313008572928689nteger @ B @ C )
       => ( ord_le6747313008572928689nteger @ B @ ( plus_p5714425477246183910nteger @ A @ C ) ) ) ) ).

% add_strict_increasing2
thf(fact_3681_add__strict__increasing2,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ B @ C )
       => ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).

% add_strict_increasing2
thf(fact_3682_add__strict__increasing2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ B @ C )
       => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).

% add_strict_increasing2
thf(fact_3683_add__strict__increasing2,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ B @ C )
       => ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).

% add_strict_increasing2
thf(fact_3684_add__strict__increasing,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ A )
     => ( ( ord_le1926595141338095240atural @ B @ C )
       => ( ord_le5570908160329646204atural @ B @ ( plus_p4538020629002901425atural @ A @ C ) ) ) ) ).

% add_strict_increasing
thf(fact_3685_add__strict__increasing,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le3102999989581377725nteger @ B @ C )
       => ( ord_le6747313008572928689nteger @ B @ ( plus_p5714425477246183910nteger @ A @ C ) ) ) ) ).

% add_strict_increasing
thf(fact_3686_add__strict__increasing,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).

% add_strict_increasing
thf(fact_3687_add__strict__increasing,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).

% add_strict_increasing
thf(fact_3688_add__strict__increasing,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ B @ C )
       => ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).

% add_strict_increasing
thf(fact_3689_add__pos__nonneg,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ A )
     => ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ B )
       => ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ ( plus_p4538020629002901425atural @ A @ B ) ) ) ) ).

% add_pos_nonneg
thf(fact_3690_add__pos__nonneg,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
       => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ A @ B ) ) ) ) ).

% add_pos_nonneg
thf(fact_3691_add__pos__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
       => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).

% add_pos_nonneg
thf(fact_3692_add__pos__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% add_pos_nonneg
thf(fact_3693_add__pos__nonneg,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ zero_zero_int @ B )
       => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).

% add_pos_nonneg
thf(fact_3694_add__nonpos__neg,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( ord_le1926595141338095240atural @ A @ zero_z2226904508553997617atural )
     => ( ( ord_le5570908160329646204atural @ B @ zero_z2226904508553997617atural )
       => ( ord_le5570908160329646204atural @ ( plus_p4538020629002901425atural @ A @ B ) @ zero_z2226904508553997617atural ) ) ) ).

% add_nonpos_neg
thf(fact_3695_add__nonpos__neg,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger )
     => ( ( ord_le6747313008572928689nteger @ B @ zero_z3403309356797280102nteger )
       => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ zero_z3403309356797280102nteger ) ) ) ).

% add_nonpos_neg
thf(fact_3696_add__nonpos__neg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ zero_zero_rat )
     => ( ( ord_less_rat @ B @ zero_zero_rat )
       => ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).

% add_nonpos_neg
thf(fact_3697_add__nonpos__neg,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( ( ord_less_nat @ B @ zero_zero_nat )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% add_nonpos_neg
thf(fact_3698_add__nonpos__neg,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ B @ zero_zero_int )
       => ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).

% add_nonpos_neg
thf(fact_3699_add__nonneg__pos,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ A )
     => ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ B )
       => ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ ( plus_p4538020629002901425atural @ A @ B ) ) ) ) ).

% add_nonneg_pos
thf(fact_3700_add__nonneg__pos,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
       => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ A @ B ) ) ) ) ).

% add_nonneg_pos
thf(fact_3701_add__nonneg__pos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ zero_zero_rat @ B )
       => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).

% add_nonneg_pos
thf(fact_3702_add__nonneg__pos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ B )
       => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% add_nonneg_pos
thf(fact_3703_add__nonneg__pos,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).

% add_nonneg_pos
thf(fact_3704_add__neg__nonpos,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( ord_le5570908160329646204atural @ A @ zero_z2226904508553997617atural )
     => ( ( ord_le1926595141338095240atural @ B @ zero_z2226904508553997617atural )
       => ( ord_le5570908160329646204atural @ ( plus_p4538020629002901425atural @ A @ B ) @ zero_z2226904508553997617atural ) ) ) ).

% add_neg_nonpos
thf(fact_3705_add__neg__nonpos,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger )
     => ( ( ord_le3102999989581377725nteger @ B @ zero_z3403309356797280102nteger )
       => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ zero_z3403309356797280102nteger ) ) ) ).

% add_neg_nonpos
thf(fact_3706_add__neg__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
       => ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).

% add_neg_nonpos
thf(fact_3707_add__neg__nonpos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% add_neg_nonpos
thf(fact_3708_add__neg__nonpos,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ( ord_less_eq_int @ B @ zero_zero_int )
       => ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).

% add_neg_nonpos
thf(fact_3709_sum__squares__ge__zero,axiom,
    ! [X: code_integer,Y3: code_integer] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ X @ X ) @ ( times_3573771949741848930nteger @ Y3 @ Y3 ) ) ) ).

% sum_squares_ge_zero
thf(fact_3710_sum__squares__ge__zero,axiom,
    ! [X: rat,Y3: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y3 @ Y3 ) ) ) ).

% sum_squares_ge_zero
thf(fact_3711_sum__squares__ge__zero,axiom,
    ! [X: int,Y3: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y3 @ Y3 ) ) ) ).

% sum_squares_ge_zero
thf(fact_3712_discrete,axiom,
    ( ord_le6747313008572928689nteger
    = ( ^ [A2: code_integer] : ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ A2 @ one_one_Code_integer ) ) ) ) ).

% discrete
thf(fact_3713_discrete,axiom,
    ( ord_less_nat
    = ( ^ [A2: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ one_one_nat ) ) ) ) ).

% discrete
thf(fact_3714_discrete,axiom,
    ( ord_less_int
    = ( ^ [A2: int] : ( ord_less_eq_int @ ( plus_plus_int @ A2 @ one_one_int ) ) ) ) ).

% discrete
thf(fact_3715_not__sum__squares__lt__zero,axiom,
    ! [X: code_integer,Y3: code_integer] :
      ~ ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ X @ X ) @ ( times_3573771949741848930nteger @ Y3 @ Y3 ) ) @ zero_z3403309356797280102nteger ) ).

% not_sum_squares_lt_zero
thf(fact_3716_not__sum__squares__lt__zero,axiom,
    ! [X: rat,Y3: rat] :
      ~ ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y3 @ Y3 ) ) @ zero_zero_rat ) ).

% not_sum_squares_lt_zero
thf(fact_3717_not__sum__squares__lt__zero,axiom,
    ! [X: int,Y3: int] :
      ~ ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y3 @ Y3 ) ) @ zero_zero_int ) ).

% not_sum_squares_lt_zero
thf(fact_3718_zero__less__two,axiom,
    ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ ( plus_p4538020629002901425atural @ one_one_Code_natural @ one_one_Code_natural ) ).

% zero_less_two
thf(fact_3719_zero__less__two,axiom,
    ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ one_one_Code_integer ) ).

% zero_less_two
thf(fact_3720_zero__less__two,axiom,
    ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ).

% zero_less_two
thf(fact_3721_zero__less__two,axiom,
    ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).

% zero_less_two
thf(fact_3722_zero__less__two,axiom,
    ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).

% zero_less_two
thf(fact_3723_le__add__iff2,axiom,
    ! [A: code_integer,E: code_integer,C: code_integer,B: code_integer,D: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ E ) @ C ) @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ E ) @ D ) )
      = ( ord_le3102999989581377725nteger @ C @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ B @ A ) @ E ) @ D ) ) ) ).

% le_add_iff2
thf(fact_3724_le__add__iff2,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D ) )
      = ( ord_less_eq_rat @ C @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E ) @ D ) ) ) ).

% le_add_iff2
thf(fact_3725_le__add__iff2,axiom,
    ! [A: int,E: int,C: int,B: int,D: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
      = ( ord_less_eq_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E ) @ D ) ) ) ).

% le_add_iff2
thf(fact_3726_le__add__iff1,axiom,
    ! [A: code_integer,E: code_integer,C: code_integer,B: code_integer,D: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ E ) @ C ) @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ E ) @ D ) )
      = ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ E ) @ C ) @ D ) ) ).

% le_add_iff1
thf(fact_3727_le__add__iff1,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D ) )
      = ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E ) @ C ) @ D ) ) ).

% le_add_iff1
thf(fact_3728_le__add__iff1,axiom,
    ! [A: int,E: int,C: int,B: int,D: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
      = ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E ) @ C ) @ D ) ) ).

% le_add_iff1
thf(fact_3729_less__add__iff2,axiom,
    ! [A: code_integer,E: code_integer,C: code_integer,B: code_integer,D: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ E ) @ C ) @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ E ) @ D ) )
      = ( ord_le6747313008572928689nteger @ C @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ B @ A ) @ E ) @ D ) ) ) ).

% less_add_iff2
thf(fact_3730_less__add__iff2,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D ) )
      = ( ord_less_rat @ C @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E ) @ D ) ) ) ).

% less_add_iff2
thf(fact_3731_less__add__iff2,axiom,
    ! [A: int,E: int,C: int,B: int,D: int] :
      ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
      = ( ord_less_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E ) @ D ) ) ) ).

% less_add_iff2
thf(fact_3732_less__add__iff1,axiom,
    ! [A: code_integer,E: code_integer,C: code_integer,B: code_integer,D: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ E ) @ C ) @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ E ) @ D ) )
      = ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ E ) @ C ) @ D ) ) ).

% less_add_iff1
thf(fact_3733_less__add__iff1,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D ) )
      = ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E ) @ C ) @ D ) ) ).

% less_add_iff1
thf(fact_3734_less__add__iff1,axiom,
    ! [A: int,E: int,C: int,B: int,D: int] :
      ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
      = ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E ) @ C ) @ D ) ) ).

% less_add_iff1
thf(fact_3735_unity__coeff__ex,axiom,
    ! [P: code_natural > $o,L: code_natural] :
      ( ( ? [X2: code_natural] : ( P @ ( times_2397367101498566445atural @ L @ X2 ) ) )
      = ( ? [X2: code_natural] :
            ( ( dvd_dvd_Code_natural @ L @ ( plus_p4538020629002901425atural @ X2 @ zero_z2226904508553997617atural ) )
            & ( P @ X2 ) ) ) ) ).

% unity_coeff_ex
thf(fact_3736_unity__coeff__ex,axiom,
    ! [P: code_integer > $o,L: code_integer] :
      ( ( ? [X2: code_integer] : ( P @ ( times_3573771949741848930nteger @ L @ X2 ) ) )
      = ( ? [X2: code_integer] :
            ( ( dvd_dvd_Code_integer @ L @ ( plus_p5714425477246183910nteger @ X2 @ zero_z3403309356797280102nteger ) )
            & ( P @ X2 ) ) ) ) ).

% unity_coeff_ex
thf(fact_3737_unity__coeff__ex,axiom,
    ! [P: rat > $o,L: rat] :
      ( ( ? [X2: rat] : ( P @ ( times_times_rat @ L @ X2 ) ) )
      = ( ? [X2: rat] :
            ( ( dvd_dvd_rat @ L @ ( plus_plus_rat @ X2 @ zero_zero_rat ) )
            & ( P @ X2 ) ) ) ) ).

% unity_coeff_ex
thf(fact_3738_unity__coeff__ex,axiom,
    ! [P: nat > $o,L: nat] :
      ( ( ? [X2: nat] : ( P @ ( times_times_nat @ L @ X2 ) ) )
      = ( ? [X2: nat] :
            ( ( dvd_dvd_nat @ L @ ( plus_plus_nat @ X2 @ zero_zero_nat ) )
            & ( P @ X2 ) ) ) ) ).

% unity_coeff_ex
thf(fact_3739_unity__coeff__ex,axiom,
    ! [P: int > $o,L: int] :
      ( ( ? [X2: int] : ( P @ ( times_times_int @ L @ X2 ) ) )
      = ( ? [X2: int] :
            ( ( dvd_dvd_int @ L @ ( plus_plus_int @ X2 @ zero_zero_int ) )
            & ( P @ X2 ) ) ) ) ).

% unity_coeff_ex
thf(fact_3740_add__divide__eq__if__simps_I2_J,axiom,
    ! [Z2: rat,A: rat,B: rat] :
      ( ( ( Z2 = zero_zero_rat )
       => ( ( plus_plus_rat @ ( divide_divide_rat @ A @ Z2 ) @ B )
          = B ) )
      & ( ( Z2 != zero_zero_rat )
       => ( ( plus_plus_rat @ ( divide_divide_rat @ A @ Z2 ) @ B )
          = ( divide_divide_rat @ ( plus_plus_rat @ A @ ( times_times_rat @ B @ Z2 ) ) @ Z2 ) ) ) ) ).

% add_divide_eq_if_simps(2)
thf(fact_3741_add__divide__eq__if__simps_I1_J,axiom,
    ! [Z2: rat,A: rat,B: rat] :
      ( ( ( Z2 = zero_zero_rat )
       => ( ( plus_plus_rat @ A @ ( divide_divide_rat @ B @ Z2 ) )
          = A ) )
      & ( ( Z2 != zero_zero_rat )
       => ( ( plus_plus_rat @ A @ ( divide_divide_rat @ B @ Z2 ) )
          = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ Z2 ) @ B ) @ Z2 ) ) ) ) ).

% add_divide_eq_if_simps(1)
thf(fact_3742_add__frac__eq,axiom,
    ! [Y3: rat,Z2: rat,X: rat,W2: rat] :
      ( ( Y3 != zero_zero_rat )
     => ( ( Z2 != zero_zero_rat )
       => ( ( plus_plus_rat @ ( divide_divide_rat @ X @ Y3 ) @ ( divide_divide_rat @ W2 @ Z2 ) )
          = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ Z2 ) @ ( times_times_rat @ W2 @ Y3 ) ) @ ( times_times_rat @ Y3 @ Z2 ) ) ) ) ) ).

% add_frac_eq
thf(fact_3743_add__frac__num,axiom,
    ! [Y3: rat,X: rat,Z2: rat] :
      ( ( Y3 != zero_zero_rat )
     => ( ( plus_plus_rat @ ( divide_divide_rat @ X @ Y3 ) @ Z2 )
        = ( divide_divide_rat @ ( plus_plus_rat @ X @ ( times_times_rat @ Z2 @ Y3 ) ) @ Y3 ) ) ) ).

% add_frac_num
thf(fact_3744_add__num__frac,axiom,
    ! [Y3: rat,Z2: rat,X: rat] :
      ( ( Y3 != zero_zero_rat )
     => ( ( plus_plus_rat @ Z2 @ ( divide_divide_rat @ X @ Y3 ) )
        = ( divide_divide_rat @ ( plus_plus_rat @ X @ ( times_times_rat @ Z2 @ Y3 ) ) @ Y3 ) ) ) ).

% add_num_frac
thf(fact_3745_add__divide__eq__iff,axiom,
    ! [Z2: rat,X: rat,Y3: rat] :
      ( ( Z2 != zero_zero_rat )
     => ( ( plus_plus_rat @ X @ ( divide_divide_rat @ Y3 @ Z2 ) )
        = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ Z2 ) @ Y3 ) @ Z2 ) ) ) ).

% add_divide_eq_iff
thf(fact_3746_divide__add__eq__iff,axiom,
    ! [Z2: rat,X: rat,Y3: rat] :
      ( ( Z2 != zero_zero_rat )
     => ( ( plus_plus_rat @ ( divide_divide_rat @ X @ Z2 ) @ Y3 )
        = ( divide_divide_rat @ ( plus_plus_rat @ X @ ( times_times_rat @ Y3 @ Z2 ) ) @ Z2 ) ) ) ).

% divide_add_eq_iff
thf(fact_3747_div__add__self1,axiom,
    ! [B: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ B @ A ) @ B )
        = ( plus_plus_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).

% div_add_self1
thf(fact_3748_div__add__self1,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ B @ A ) @ B )
        = ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).

% div_add_self1
thf(fact_3749_div__add__self1,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ B @ A ) @ B )
        = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ B ) @ one_one_Code_integer ) ) ) ).

% div_add_self1
thf(fact_3750_div__add__self1,axiom,
    ! [B: code_natural,A: code_natural] :
      ( ( B != zero_z2226904508553997617atural )
     => ( ( divide5121882707175180666atural @ ( plus_p4538020629002901425atural @ B @ A ) @ B )
        = ( plus_p4538020629002901425atural @ ( divide5121882707175180666atural @ A @ B ) @ one_one_Code_natural ) ) ) ).

% div_add_self1
thf(fact_3751_div__add__self2,axiom,
    ! [B: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ B )
        = ( plus_plus_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).

% div_add_self2
thf(fact_3752_div__add__self2,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ B )
        = ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).

% div_add_self2
thf(fact_3753_div__add__self2,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ B )
        = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ B ) @ one_one_Code_integer ) ) ) ).

% div_add_self2
thf(fact_3754_div__add__self2,axiom,
    ! [B: code_natural,A: code_natural] :
      ( ( B != zero_z2226904508553997617atural )
     => ( ( divide5121882707175180666atural @ ( plus_p4538020629002901425atural @ A @ B ) @ B )
        = ( plus_p4538020629002901425atural @ ( divide5121882707175180666atural @ A @ B ) @ one_one_Code_natural ) ) ) ).

% div_add_self2
thf(fact_3755_square__diff__one__factored,axiom,
    ! [X: code_integer] :
      ( ( minus_8373710615458151222nteger @ ( times_3573771949741848930nteger @ X @ X ) @ one_one_Code_integer )
      = ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ X @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ X @ one_one_Code_integer ) ) ) ).

% square_diff_one_factored
thf(fact_3756_square__diff__one__factored,axiom,
    ! [X: rat] :
      ( ( minus_minus_rat @ ( times_times_rat @ X @ X ) @ one_one_rat )
      = ( times_times_rat @ ( plus_plus_rat @ X @ one_one_rat ) @ ( minus_minus_rat @ X @ one_one_rat ) ) ) ).

% square_diff_one_factored
thf(fact_3757_square__diff__one__factored,axiom,
    ! [X: int] :
      ( ( minus_minus_int @ ( times_times_int @ X @ X ) @ one_one_int )
      = ( times_times_int @ ( plus_plus_int @ X @ one_one_int ) @ ( minus_minus_int @ X @ one_one_int ) ) ) ).

% square_diff_one_factored
thf(fact_3758_gt__half__sum,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ord_less_rat @ ( divide_divide_rat @ ( plus_plus_rat @ A @ B ) @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ) @ B ) ) ).

% gt_half_sum
thf(fact_3759_less__half__sum,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ord_less_rat @ A @ ( divide_divide_rat @ ( plus_plus_rat @ A @ B ) @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ) ) ) ).

% less_half_sum
thf(fact_3760_inf__period_I4_J,axiom,
    ! [D: code_integer,D3: code_integer,T: code_integer] :
      ( ( dvd_dvd_Code_integer @ D @ D3 )
     => ! [X4: code_integer,K4: code_integer] :
          ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X4 @ T ) ) )
          = ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ X4 @ ( times_3573771949741848930nteger @ K4 @ D3 ) ) @ T ) ) ) ) ) ).

% inf_period(4)
thf(fact_3761_inf__period_I4_J,axiom,
    ! [D: rat,D3: rat,T: rat] :
      ( ( dvd_dvd_rat @ D @ D3 )
     => ! [X4: rat,K4: rat] :
          ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X4 @ T ) ) )
          = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ ( minus_minus_rat @ X4 @ ( times_times_rat @ K4 @ D3 ) ) @ T ) ) ) ) ) ).

% inf_period(4)
thf(fact_3762_inf__period_I4_J,axiom,
    ! [D: int,D3: int,T: int] :
      ( ( dvd_dvd_int @ D @ D3 )
     => ! [X4: int,K4: int] :
          ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X4 @ T ) ) )
          = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X4 @ ( times_times_int @ K4 @ D3 ) ) @ T ) ) ) ) ) ).

% inf_period(4)
thf(fact_3763_inf__period_I3_J,axiom,
    ! [D: code_integer,D3: code_integer,T: code_integer] :
      ( ( dvd_dvd_Code_integer @ D @ D3 )
     => ! [X4: code_integer,K4: code_integer] :
          ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X4 @ T ) )
          = ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ X4 @ ( times_3573771949741848930nteger @ K4 @ D3 ) ) @ T ) ) ) ) ).

% inf_period(3)
thf(fact_3764_inf__period_I3_J,axiom,
    ! [D: rat,D3: rat,T: rat] :
      ( ( dvd_dvd_rat @ D @ D3 )
     => ! [X4: rat,K4: rat] :
          ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X4 @ T ) )
          = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ ( minus_minus_rat @ X4 @ ( times_times_rat @ K4 @ D3 ) ) @ T ) ) ) ) ).

% inf_period(3)
thf(fact_3765_inf__period_I3_J,axiom,
    ! [D: int,D3: int,T: int] :
      ( ( dvd_dvd_int @ D @ D3 )
     => ! [X4: int,K4: int] :
          ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X4 @ T ) )
          = ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X4 @ ( times_times_int @ K4 @ D3 ) ) @ T ) ) ) ) ).

% inf_period(3)
thf(fact_3766_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_3767_div__mult1__eq,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ C )
      = ( plus_plus_nat @ ( times_times_nat @ A @ ( divide_divide_nat @ B @ C ) ) @ ( divide_divide_nat @ ( times_times_nat @ A @ ( modulo_modulo_nat @ B @ C ) ) @ C ) ) ) ).

% div_mult1_eq
thf(fact_3768_div__mult1__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ C )
      = ( plus_plus_int @ ( times_times_int @ A @ ( divide_divide_int @ B @ C ) ) @ ( divide_divide_int @ ( times_times_int @ A @ ( modulo_modulo_int @ B @ C ) ) @ C ) ) ) ).

% div_mult1_eq
thf(fact_3769_div__mult1__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B @ C ) ) @ ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ ( modulo364778990260209775nteger @ B @ C ) ) @ C ) ) ) ).

% div_mult1_eq
thf(fact_3770_div__mult1__eq,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( ( divide5121882707175180666atural @ ( times_2397367101498566445atural @ A @ B ) @ C )
      = ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ A @ ( divide5121882707175180666atural @ B @ C ) ) @ ( divide5121882707175180666atural @ ( times_2397367101498566445atural @ A @ ( modulo8411746178871703098atural @ B @ C ) ) @ C ) ) ) ).

% div_mult1_eq
thf(fact_3771_mult__div__mod__eq,axiom,
    ! [B: nat,A: nat] :
      ( ( plus_plus_nat @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) @ ( modulo_modulo_nat @ A @ B ) )
      = A ) ).

% mult_div_mod_eq
thf(fact_3772_mult__div__mod__eq,axiom,
    ! [B: int,A: int] :
      ( ( plus_plus_int @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) @ ( modulo_modulo_int @ A @ B ) )
      = A ) ).

% mult_div_mod_eq
thf(fact_3773_mult__div__mod__eq,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) @ ( modulo364778990260209775nteger @ A @ B ) )
      = A ) ).

% mult_div_mod_eq
thf(fact_3774_mult__div__mod__eq,axiom,
    ! [B: code_natural,A: code_natural] :
      ( ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ B @ ( divide5121882707175180666atural @ A @ B ) ) @ ( modulo8411746178871703098atural @ A @ B ) )
      = A ) ).

% mult_div_mod_eq
thf(fact_3775_mod__mult__div__eq,axiom,
    ! [A: nat,B: nat] :
      ( ( plus_plus_nat @ ( modulo_modulo_nat @ A @ B ) @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) )
      = A ) ).

% mod_mult_div_eq
thf(fact_3776_mod__mult__div__eq,axiom,
    ! [A: int,B: int] :
      ( ( plus_plus_int @ ( modulo_modulo_int @ A @ B ) @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) )
      = A ) ).

% mod_mult_div_eq
thf(fact_3777_mod__mult__div__eq,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ B ) @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) )
      = A ) ).

% mod_mult_div_eq
thf(fact_3778_mod__mult__div__eq,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( plus_p4538020629002901425atural @ ( modulo8411746178871703098atural @ A @ B ) @ ( times_2397367101498566445atural @ B @ ( divide5121882707175180666atural @ A @ B ) ) )
      = A ) ).

% mod_mult_div_eq
thf(fact_3779_mod__div__mult__eq,axiom,
    ! [A: nat,B: nat] :
      ( ( plus_plus_nat @ ( modulo_modulo_nat @ A @ B ) @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) )
      = A ) ).

% mod_div_mult_eq
thf(fact_3780_mod__div__mult__eq,axiom,
    ! [A: int,B: int] :
      ( ( plus_plus_int @ ( modulo_modulo_int @ A @ B ) @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) )
      = A ) ).

% mod_div_mult_eq
thf(fact_3781_mod__div__mult__eq,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ B ) @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) )
      = A ) ).

% mod_div_mult_eq
thf(fact_3782_mod__div__mult__eq,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( plus_p4538020629002901425atural @ ( modulo8411746178871703098atural @ A @ B ) @ ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ A @ B ) @ B ) )
      = A ) ).

% mod_div_mult_eq
thf(fact_3783_div__mult__mod__eq,axiom,
    ! [A: nat,B: nat] :
      ( ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) @ ( modulo_modulo_nat @ A @ B ) )
      = A ) ).

% div_mult_mod_eq
thf(fact_3784_div__mult__mod__eq,axiom,
    ! [A: int,B: int] :
      ( ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) @ ( modulo_modulo_int @ A @ B ) )
      = A ) ).

% div_mult_mod_eq
thf(fact_3785_div__mult__mod__eq,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) @ ( modulo364778990260209775nteger @ A @ B ) )
      = A ) ).

% div_mult_mod_eq
thf(fact_3786_div__mult__mod__eq,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ A @ B ) @ B ) @ ( modulo8411746178871703098atural @ A @ B ) )
      = A ) ).

% div_mult_mod_eq
thf(fact_3787_mod__div__decomp,axiom,
    ! [A: nat,B: nat] :
      ( A
      = ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) @ ( modulo_modulo_nat @ A @ B ) ) ) ).

% mod_div_decomp
thf(fact_3788_mod__div__decomp,axiom,
    ! [A: int,B: int] :
      ( A
      = ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) @ ( modulo_modulo_int @ A @ B ) ) ) ).

% mod_div_decomp
thf(fact_3789_mod__div__decomp,axiom,
    ! [A: code_integer,B: code_integer] :
      ( A
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).

% mod_div_decomp
thf(fact_3790_mod__div__decomp,axiom,
    ! [A: code_natural,B: code_natural] :
      ( A
      = ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ A @ B ) @ B ) @ ( modulo8411746178871703098atural @ A @ B ) ) ) ).

% mod_div_decomp
thf(fact_3791_cancel__div__mod__rules_I1_J,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) @ ( modulo_modulo_nat @ A @ B ) ) @ C )
      = ( plus_plus_nat @ A @ C ) ) ).

% cancel_div_mod_rules(1)
thf(fact_3792_cancel__div__mod__rules_I1_J,axiom,
    ! [A: int,B: int,C: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) @ ( modulo_modulo_int @ A @ B ) ) @ C )
      = ( plus_plus_int @ A @ C ) ) ).

% cancel_div_mod_rules(1)
thf(fact_3793_cancel__div__mod__rules_I1_J,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) @ ( modulo364778990260209775nteger @ A @ B ) ) @ C )
      = ( plus_p5714425477246183910nteger @ A @ C ) ) ).

% cancel_div_mod_rules(1)
thf(fact_3794_cancel__div__mod__rules_I1_J,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( ( plus_p4538020629002901425atural @ ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ A @ B ) @ B ) @ ( modulo8411746178871703098atural @ A @ B ) ) @ C )
      = ( plus_p4538020629002901425atural @ A @ C ) ) ).

% cancel_div_mod_rules(1)
thf(fact_3795_cancel__div__mod__rules_I2_J,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) @ ( modulo_modulo_nat @ A @ B ) ) @ C )
      = ( plus_plus_nat @ A @ C ) ) ).

% cancel_div_mod_rules(2)
thf(fact_3796_cancel__div__mod__rules_I2_J,axiom,
    ! [B: int,A: int,C: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) @ ( modulo_modulo_int @ A @ B ) ) @ C )
      = ( plus_plus_int @ A @ C ) ) ).

% cancel_div_mod_rules(2)
thf(fact_3797_cancel__div__mod__rules_I2_J,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) @ ( modulo364778990260209775nteger @ A @ B ) ) @ C )
      = ( plus_p5714425477246183910nteger @ A @ C ) ) ).

% cancel_div_mod_rules(2)
thf(fact_3798_cancel__div__mod__rules_I2_J,axiom,
    ! [B: code_natural,A: code_natural,C: code_natural] :
      ( ( plus_p4538020629002901425atural @ ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ B @ ( divide5121882707175180666atural @ A @ B ) ) @ ( modulo8411746178871703098atural @ A @ B ) ) @ C )
      = ( plus_p4538020629002901425atural @ A @ C ) ) ).

% cancel_div_mod_rules(2)
thf(fact_3799_nat__diff__split__asm,axiom,
    ! [P: nat > $o,A: nat,B: nat] :
      ( ( P @ ( minus_minus_nat @ A @ B ) )
      = ( ~ ( ( ( ord_less_nat @ A @ B )
              & ~ ( P @ zero_zero_nat ) )
            | ? [D4: nat] :
                ( ( A
                  = ( plus_plus_nat @ B @ D4 ) )
                & ~ ( P @ D4 ) ) ) ) ) ).

% nat_diff_split_asm
thf(fact_3800_nat__diff__split,axiom,
    ! [P: nat > $o,A: nat,B: nat] :
      ( ( P @ ( minus_minus_nat @ A @ B ) )
      = ( ( ( ord_less_nat @ A @ B )
         => ( P @ zero_zero_nat ) )
        & ! [D4: nat] :
            ( ( A
              = ( plus_plus_nat @ B @ D4 ) )
           => ( P @ D4 ) ) ) ) ).

% nat_diff_split
thf(fact_3801_mod__pos__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ( ( ord_less_eq_int @ ( plus_plus_int @ K @ L ) @ zero_zero_int )
       => ( ( modulo_modulo_int @ K @ L )
          = ( plus_plus_int @ K @ L ) ) ) ) ).

% mod_pos_neg_trivial
thf(fact_3802_less__diff__conv2,axiom,
    ! [K: nat,J: nat,I2: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I2 )
        = ( ord_less_nat @ J @ ( plus_plus_nat @ I2 @ K ) ) ) ) ).

% less_diff_conv2
thf(fact_3803_odd__less__0__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z2 ) @ Z2 ) @ zero_zero_int )
      = ( ord_less_int @ Z2 @ zero_zero_int ) ) ).

% odd_less_0_iff
thf(fact_3804_zless__iff__Suc__zadd,axiom,
    ( ord_less_int
    = ( ^ [W3: int,Z6: int] :
        ? [N3: nat] :
          ( Z6
          = ( plus_plus_int @ W3 @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) ) ) ) ).

% zless_iff_Suc_zadd
thf(fact_3805_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_3806_int__Suc,axiom,
    ! [N: nat] :
      ( ( semiri1314217659103216013at_int @ ( suc @ N ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ).

% int_Suc
thf(fact_3807_zmod__zmult2__eq,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ C )
     => ( ( modulo_modulo_int @ A @ ( times_times_int @ B @ C ) )
        = ( plus_plus_int @ ( times_times_int @ B @ ( modulo_modulo_int @ ( divide_divide_int @ A @ B ) @ C ) ) @ ( modulo_modulo_int @ A @ B ) ) ) ) ).

% zmod_zmult2_eq
thf(fact_3808_zless__imp__add1__zle,axiom,
    ! [W2: int,Z2: int] :
      ( ( ord_less_int @ W2 @ Z2 )
     => ( ord_less_eq_int @ ( plus_plus_int @ W2 @ one_one_int ) @ Z2 ) ) ).

% zless_imp_add1_zle
thf(fact_3809_add1__zle__eq,axiom,
    ! [W2: int,Z2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ W2 @ one_one_int ) @ Z2 )
      = ( ord_less_int @ W2 @ Z2 ) ) ).

% add1_zle_eq
thf(fact_3810_nat__eq__add__iff1,axiom,
    ! [J: nat,I2: nat,U: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ J @ I2 )
     => ( ( ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M )
          = ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I2 @ J ) @ U ) @ M )
          = N ) ) ) ).

% nat_eq_add_iff1
thf(fact_3811_nat__eq__add__iff2,axiom,
    ! [I2: nat,J: nat,U: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ I2 @ J )
     => ( ( ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M )
          = ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( M
          = ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I2 ) @ U ) @ N ) ) ) ) ).

% nat_eq_add_iff2
thf(fact_3812_nat__le__add__iff1,axiom,
    ! [J: nat,I2: nat,U: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ J @ I2 )
     => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I2 @ J ) @ U ) @ M ) @ N ) ) ) ).

% nat_le_add_iff1
thf(fact_3813_nat__le__add__iff2,axiom,
    ! [I2: nat,J: nat,U: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ I2 @ J )
     => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ 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 @ I2 ) @ U ) @ N ) ) ) ) ).

% nat_le_add_iff2
thf(fact_3814_nat__diff__add__eq1,axiom,
    ! [J: nat,I2: nat,U: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ J @ I2 )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I2 @ J ) @ U ) @ M ) @ N ) ) ) ).

% nat_diff_add_eq1
thf(fact_3815_nat__diff__add__eq2,axiom,
    ! [I2: nat,J: nat,U: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ I2 @ J )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ 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 @ I2 ) @ U ) @ N ) ) ) ) ).

% nat_diff_add_eq2
thf(fact_3816_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_3817_fact__fact__dvd__fact,axiom,
    ! [K: nat,N: nat] : ( dvd_dvd_Code_natural @ ( times_2397367101498566445atural @ ( semiri2447717529341329178atural @ K ) @ ( semiri2447717529341329178atural @ N ) ) @ ( semiri2447717529341329178atural @ ( plus_plus_nat @ K @ N ) ) ) ).

% fact_fact_dvd_fact
thf(fact_3818_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_3819_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_3820_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_3821_int__induct,axiom,
    ! [P: int > $o,K: int,I2: 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 @ I2 ) ) ) ) ).

% int_induct
thf(fact_3822_mod__eq__nat1E,axiom,
    ! [M: nat,Q4: nat,N: nat] :
      ( ( ( modulo_modulo_nat @ M @ Q4 )
        = ( modulo_modulo_nat @ N @ Q4 ) )
     => ( ( ord_less_eq_nat @ N @ M )
       => ~ ! [S2: nat] :
              ( M
             != ( plus_plus_nat @ N @ ( times_times_nat @ Q4 @ S2 ) ) ) ) ) ).

% mod_eq_nat1E
thf(fact_3823_mod__eq__nat2E,axiom,
    ! [M: nat,Q4: nat,N: nat] :
      ( ( ( modulo_modulo_nat @ M @ Q4 )
        = ( modulo_modulo_nat @ N @ Q4 ) )
     => ( ( ord_less_eq_nat @ M @ N )
       => ~ ! [S2: nat] :
              ( N
             != ( plus_plus_nat @ M @ ( times_times_nat @ Q4 @ S2 ) ) ) ) ) ).

% mod_eq_nat2E
thf(fact_3824_nat__mod__eq__lemma,axiom,
    ! [X: nat,N: nat,Y3: nat] :
      ( ( ( modulo_modulo_nat @ X @ N )
        = ( modulo_modulo_nat @ Y3 @ N ) )
     => ( ( ord_less_eq_nat @ Y3 @ X )
       => ? [Q5: nat] :
            ( X
            = ( plus_plus_nat @ Y3 @ ( times_times_nat @ N @ Q5 ) ) ) ) ) ).

% nat_mod_eq_lemma
thf(fact_3825_mod__mult2__eq,axiom,
    ! [M: nat,N: nat,Q4: nat] :
      ( ( modulo_modulo_nat @ M @ ( times_times_nat @ N @ Q4 ) )
      = ( plus_plus_nat @ ( times_times_nat @ N @ ( modulo_modulo_nat @ ( divide_divide_nat @ M @ N ) @ Q4 ) ) @ ( modulo_modulo_nat @ M @ N ) ) ) ).

% mod_mult2_eq
thf(fact_3826_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_3827_dbl__dec__def,axiom,
    ( neg_nu7757733837767384882nteger
    = ( ^ [X2: code_integer] : ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ X2 @ X2 ) @ one_one_Code_integer ) ) ) ).

% dbl_dec_def
thf(fact_3828_dbl__dec__def,axiom,
    ( neg_nu3179335615603231917ec_rat
    = ( ^ [X2: rat] : ( minus_minus_rat @ ( plus_plus_rat @ X2 @ X2 ) @ one_one_rat ) ) ) ).

% dbl_dec_def
thf(fact_3829_dbl__dec__def,axiom,
    ( neg_nu3811975205180677377ec_int
    = ( ^ [X2: int] : ( minus_minus_int @ ( plus_plus_int @ X2 @ X2 ) @ one_one_int ) ) ) ).

% dbl_dec_def
thf(fact_3830_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_3831_convex__bound__le,axiom,
    ! [X: code_integer,A: code_integer,Y3: code_integer,U: code_integer,V: code_integer] :
      ( ( ord_le3102999989581377725nteger @ X @ A )
     => ( ( ord_le3102999989581377725nteger @ Y3 @ A )
       => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ U )
         => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ V )
           => ( ( ( plus_p5714425477246183910nteger @ U @ V )
                = one_one_Code_integer )
             => ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ U @ X ) @ ( times_3573771949741848930nteger @ V @ Y3 ) ) @ A ) ) ) ) ) ) ).

% convex_bound_le
thf(fact_3832_convex__bound__le,axiom,
    ! [X: rat,A: rat,Y3: rat,U: rat,V: rat] :
      ( ( ord_less_eq_rat @ X @ 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 @ X ) @ ( times_times_rat @ V @ Y3 ) ) @ A ) ) ) ) ) ) ).

% convex_bound_le
thf(fact_3833_convex__bound__le,axiom,
    ! [X: int,A: int,Y3: int,U: int,V: int] :
      ( ( ord_less_eq_int @ X @ 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 @ X ) @ ( times_times_int @ V @ Y3 ) ) @ A ) ) ) ) ) ) ).

% convex_bound_le
thf(fact_3834_divide__int__unfold,axiom,
    ! [L: int,K: int,N: nat,M: nat] :
      ( ( ( ( ( sgn_sgn_int @ L )
            = 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 @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
          = zero_zero_int ) )
      & ( ~ ( ( ( sgn_sgn_int @ L )
              = zero_zero_int )
            | ( ( sgn_sgn_int @ K )
              = zero_zero_int )
            | ( N = zero_zero_nat ) )
       => ( ( ( ( sgn_sgn_int @ K )
              = ( sgn_sgn_int @ L ) )
           => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
              = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N ) ) ) )
          & ( ( ( sgn_sgn_int @ K )
             != ( sgn_sgn_int @ L ) )
           => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( 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_3835_add__divide__eq__if__simps_I3_J,axiom,
    ! [Z2: rat,A: rat,B: rat] :
      ( ( ( Z2 = zero_zero_rat )
       => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z2 ) ) @ B )
          = B ) )
      & ( ( Z2 != zero_zero_rat )
       => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z2 ) ) @ B )
          = ( divide_divide_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_rat @ B @ Z2 ) ) @ Z2 ) ) ) ) ).

% add_divide_eq_if_simps(3)
thf(fact_3836_minus__divide__add__eq__iff,axiom,
    ! [Z2: rat,X: rat,Y3: rat] :
      ( ( Z2 != zero_zero_rat )
     => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ X @ Z2 ) ) @ Y3 )
        = ( divide_divide_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ X ) @ ( times_times_rat @ Y3 @ Z2 ) ) @ Z2 ) ) ) ).

% minus_divide_add_eq_iff
thf(fact_3837_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_3838_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_3839_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_3840_mod__mult2__eq_H,axiom,
    ! [A: code_natural,M: nat,N: nat] :
      ( ( modulo8411746178871703098atural @ A @ ( times_2397367101498566445atural @ ( semiri3763490453095760265atural @ M ) @ ( semiri3763490453095760265atural @ N ) ) )
      = ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ ( semiri3763490453095760265atural @ M ) @ ( modulo8411746178871703098atural @ ( divide5121882707175180666atural @ A @ ( semiri3763490453095760265atural @ M ) ) @ ( semiri3763490453095760265atural @ N ) ) ) @ ( modulo8411746178871703098atural @ A @ ( semiri3763490453095760265atural @ M ) ) ) ) ).

% mod_mult2_eq'
thf(fact_3841_add__eq__if,axiom,
    ( plus_plus_nat
    = ( ^ [M2: nat,N3: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ N3 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N3 ) ) ) ) ) ).

% add_eq_if
thf(fact_3842_le__imp__0__less,axiom,
    ! [Z2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
     => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z2 ) ) ) ).

% le_imp_0_less
thf(fact_3843_verit__le__mono__div__int,axiom,
    ! [A3: int,B5: int,N: int] :
      ( ( ord_less_int @ A3 @ B5 )
     => ( ( ord_less_int @ zero_zero_int @ N )
       => ( ord_less_eq_int
          @ ( plus_plus_int @ ( divide_divide_int @ A3 @ N )
            @ ( if_int
              @ ( ( modulo_modulo_int @ B5 @ N )
                = zero_zero_int )
              @ one_one_int
              @ zero_zero_int ) )
          @ ( divide_divide_int @ B5 @ N ) ) ) ) ).

% verit_le_mono_div_int
thf(fact_3844_nat__less__add__iff1,axiom,
    ! [J: nat,I2: nat,U: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ J @ I2 )
     => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I2 @ J ) @ U ) @ M ) @ N ) ) ) ).

% nat_less_add_iff1
thf(fact_3845_nat__less__add__iff2,axiom,
    ! [I2: nat,J: nat,U: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ I2 @ J )
     => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ 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 @ I2 ) @ U ) @ N ) ) ) ) ).

% nat_less_add_iff2
thf(fact_3846_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 )
         => ! [X4: int] :
              ( ( P @ X4 )
             => ( P @ ( plus_plus_int @ X4 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).

% incr_mult_lemma
thf(fact_3847_unique__quotient__lemma__neg,axiom,
    ! [B: int,Q6: int,R4: int,Q4: int,R: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B @ Q6 ) @ R4 ) @ ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R ) )
     => ( ( ord_less_eq_int @ R @ zero_zero_int )
       => ( ( ord_less_int @ B @ R )
         => ( ( ord_less_int @ B @ R4 )
           => ( ord_less_eq_int @ Q4 @ Q6 ) ) ) ) ) ).

% unique_quotient_lemma_neg
thf(fact_3848_unique__quotient__lemma,axiom,
    ! [B: int,Q6: int,R4: int,Q4: int,R: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B @ Q6 ) @ R4 ) @ ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ R4 )
       => ( ( ord_less_int @ R4 @ B )
         => ( ( ord_less_int @ R @ B )
           => ( ord_less_eq_int @ Q6 @ Q4 ) ) ) ) ) ).

% unique_quotient_lemma
thf(fact_3849_zdiv__mono2__neg__lemma,axiom,
    ! [B: int,Q4: int,R: int,B3: int,Q6: int,R4: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R )
        = ( plus_plus_int @ ( times_times_int @ B3 @ Q6 ) @ R4 ) )
     => ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ B3 @ Q6 ) @ R4 ) @ zero_zero_int )
       => ( ( ord_less_int @ R @ B )
         => ( ( ord_less_eq_int @ zero_zero_int @ R4 )
           => ( ( ord_less_int @ zero_zero_int @ B3 )
             => ( ( ord_less_eq_int @ B3 @ B )
               => ( ord_less_eq_int @ Q6 @ Q4 ) ) ) ) ) ) ) ).

% zdiv_mono2_neg_lemma
thf(fact_3850_zdiv__mono2__lemma,axiom,
    ! [B: int,Q4: int,R: int,B3: int,Q6: int,R4: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R )
        = ( plus_plus_int @ ( times_times_int @ B3 @ Q6 ) @ R4 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B3 @ Q6 ) @ R4 ) )
       => ( ( ord_less_int @ R4 @ B3 )
         => ( ( ord_less_eq_int @ zero_zero_int @ R )
           => ( ( ord_less_int @ zero_zero_int @ B3 )
             => ( ( ord_less_eq_int @ B3 @ B )
               => ( ord_less_eq_int @ Q4 @ Q6 ) ) ) ) ) ) ) ).

% zdiv_mono2_lemma
thf(fact_3851_q__pos__lemma,axiom,
    ! [B3: int,Q6: int,R4: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B3 @ Q6 ) @ R4 ) )
     => ( ( ord_less_int @ R4 @ B3 )
       => ( ( ord_less_int @ zero_zero_int @ B3 )
         => ( ord_less_eq_int @ zero_zero_int @ Q6 ) ) ) ) ).

% q_pos_lemma
thf(fact_3852_int__div__pos__eq,axiom,
    ! [A: int,B: int,Q4: int,R: int] :
      ( ( A
        = ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ R )
       => ( ( ord_less_int @ R @ B )
         => ( ( divide_divide_int @ A @ B )
            = Q4 ) ) ) ) ).

% int_div_pos_eq
thf(fact_3853_int__div__neg__eq,axiom,
    ! [A: int,B: int,Q4: int,R: int] :
      ( ( A
        = ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R ) )
     => ( ( ord_less_eq_int @ R @ zero_zero_int )
       => ( ( ord_less_int @ B @ R )
         => ( ( divide_divide_int @ A @ B )
            = Q4 ) ) ) ) ).

% int_div_neg_eq
thf(fact_3854_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 )
         => ! [I4: int,J3: int] :
              ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
                & ( ord_less_int @ J3 @ K )
                & ( N
                  = ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
             => ( P @ I4 ) ) )
        & ( ( ord_less_int @ K @ zero_zero_int )
         => ! [I4: int,J3: int] :
              ( ( ( ord_less_int @ K @ J3 )
                & ( ord_less_eq_int @ J3 @ zero_zero_int )
                & ( N
                  = ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
             => ( P @ I4 ) ) ) ) ) ).

% split_zdiv
thf(fact_3855_int__mod__pos__eq,axiom,
    ! [A: int,B: int,Q4: int,R: int] :
      ( ( A
        = ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ R )
       => ( ( ord_less_int @ R @ B )
         => ( ( modulo_modulo_int @ A @ B )
            = R ) ) ) ) ).

% int_mod_pos_eq
thf(fact_3856_int__mod__neg__eq,axiom,
    ! [A: int,B: int,Q4: int,R: int] :
      ( ( A
        = ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R ) )
     => ( ( ord_less_eq_int @ R @ zero_zero_int )
       => ( ( ord_less_int @ B @ R )
         => ( ( modulo_modulo_int @ A @ B )
            = R ) ) ) ) ).

% int_mod_neg_eq
thf(fact_3857_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 )
         => ! [I4: int,J3: int] :
              ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
                & ( ord_less_int @ J3 @ K )
                & ( N
                  = ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
             => ( P @ J3 ) ) )
        & ( ( ord_less_int @ K @ zero_zero_int )
         => ! [I4: int,J3: int] :
              ( ( ( ord_less_int @ K @ J3 )
                & ( ord_less_eq_int @ J3 @ zero_zero_int )
                & ( N
                  = ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
             => ( P @ J3 ) ) ) ) ) ).

% split_zmod
thf(fact_3858_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 ) )
        = ( ! [I4: int,J3: int] :
              ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
                & ( ord_less_int @ J3 @ K )
                & ( N
                  = ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
             => ( P @ I4 @ J3 ) ) ) ) ) ).

% split_pos_lemma
thf(fact_3859_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 ) )
        = ( ! [I4: int,J3: int] :
              ( ( ( ord_less_int @ K @ J3 )
                & ( ord_less_eq_int @ J3 @ zero_zero_int )
                & ( N
                  = ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
             => ( P @ I4 @ J3 ) ) ) ) ) ).

% split_neg_lemma
thf(fact_3860_frac__add,axiom,
    ! [X: rat,Y3: rat] :
      ( ( ( ord_less_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y3 ) ) @ one_one_rat )
       => ( ( archimedean_frac_rat @ ( plus_plus_rat @ X @ Y3 ) )
          = ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y3 ) ) ) )
      & ( ~ ( ord_less_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y3 ) ) @ one_one_rat )
       => ( ( archimedean_frac_rat @ ( plus_plus_rat @ X @ Y3 ) )
          = ( minus_minus_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y3 ) ) @ one_one_rat ) ) ) ) ).

% frac_add
thf(fact_3861_mult__eq__if,axiom,
    ( times_times_nat
    = ( ^ [M2: nat,N3: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N3 @ ( times_times_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N3 ) ) ) ) ) ).

% mult_eq_if
thf(fact_3862_Suc__as__int,axiom,
    ( suc
    = ( ^ [A2: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A2 ) @ one_one_int ) ) ) ) ).

% Suc_as_int
thf(fact_3863_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 )
         => ! [I4: nat,J3: nat] :
              ( ( ord_less_nat @ J3 @ N )
             => ( ( M
                  = ( plus_plus_nat @ ( times_times_nat @ N @ I4 ) @ J3 ) )
               => ( P @ J3 ) ) ) ) ) ) ).

% split_mod
thf(fact_3864_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_3865_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_3866_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 )
         => ! [I4: nat,J3: nat] :
              ( ( ord_less_nat @ J3 @ N )
             => ( ( M
                  = ( plus_plus_nat @ ( times_times_nat @ N @ I4 ) @ J3 ) )
               => ( P @ I4 ) ) ) ) ) ) ).

% split_div
thf(fact_3867_dvd__minus__add,axiom,
    ! [Q4: nat,N: nat,R: nat,M: nat] :
      ( ( ord_less_eq_nat @ Q4 @ N )
     => ( ( ord_less_eq_nat @ Q4 @ ( times_times_nat @ R @ M ) )
       => ( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ Q4 ) )
          = ( dvd_dvd_nat @ M @ ( plus_plus_nat @ N @ ( minus_minus_nat @ ( times_times_nat @ R @ M ) @ Q4 ) ) ) ) ) ) ).

% dvd_minus_add
thf(fact_3868_convex__bound__lt,axiom,
    ! [X: code_integer,A: code_integer,Y3: code_integer,U: code_integer,V: code_integer] :
      ( ( ord_le6747313008572928689nteger @ X @ A )
     => ( ( ord_le6747313008572928689nteger @ Y3 @ A )
       => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ U )
         => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ V )
           => ( ( ( plus_p5714425477246183910nteger @ U @ V )
                = one_one_Code_integer )
             => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ U @ X ) @ ( times_3573771949741848930nteger @ V @ Y3 ) ) @ A ) ) ) ) ) ) ).

% convex_bound_lt
thf(fact_3869_convex__bound__lt,axiom,
    ! [X: rat,A: rat,Y3: rat,U: rat,V: rat] :
      ( ( ord_less_rat @ X @ 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 @ X ) @ ( times_times_rat @ V @ Y3 ) ) @ A ) ) ) ) ) ) ).

% convex_bound_lt
thf(fact_3870_convex__bound__lt,axiom,
    ! [X: int,A: int,Y3: int,U: int,V: int] :
      ( ( ord_less_int @ X @ 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 @ X ) @ ( times_times_int @ V @ Y3 ) ) @ A ) ) ) ) ) ) ).

% convex_bound_lt
thf(fact_3871_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_3872_scaling__mono,axiom,
    ! [U: rat,V: rat,R: rat,S: rat] :
      ( ( ord_less_eq_rat @ U @ V )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ R )
       => ( ( ord_less_eq_rat @ R @ S )
         => ( ord_less_eq_rat @ ( plus_plus_rat @ U @ ( divide_divide_rat @ ( times_times_rat @ R @ ( minus_minus_rat @ V @ U ) ) @ S ) ) @ V ) ) ) ) ).

% scaling_mono
thf(fact_3873_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_3874_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
     => ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
        = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) @ ( modulo364778990260209775nteger @ A @ B ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
thf(fact_3875_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ C )
     => ( ( modulo_modulo_nat @ A @ ( times_times_nat @ B @ C ) )
        = ( plus_plus_nat @ ( times_times_nat @ B @ ( modulo_modulo_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) @ ( modulo_modulo_nat @ A @ B ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
thf(fact_3876_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ C )
     => ( ( modulo_modulo_int @ A @ ( times_times_int @ B @ C ) )
        = ( plus_plus_int @ ( times_times_int @ B @ ( modulo_modulo_int @ ( divide_divide_int @ A @ B ) @ C ) ) @ ( modulo_modulo_int @ A @ B ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
thf(fact_3877_Suc__nat__eq__nat__zadd1,axiom,
    ! [Z2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
     => ( ( suc @ ( nat2 @ Z2 ) )
        = ( nat2 @ ( plus_plus_int @ one_one_int @ Z2 ) ) ) ) ).

% Suc_nat_eq_nat_zadd1
thf(fact_3878_div__pos__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ( ( ord_less_eq_int @ ( plus_plus_int @ K @ L ) @ zero_zero_int )
       => ( ( divide_divide_int @ K @ L )
          = ( uminus_uminus_int @ one_one_int ) ) ) ) ).

% div_pos_neg_trivial
thf(fact_3879_div__pos__geq,axiom,
    ! [L: int,K: int] :
      ( ( ord_less_int @ zero_zero_int @ L )
     => ( ( ord_less_eq_int @ L @ K )
       => ( ( divide_divide_int @ K @ L )
          = ( plus_plus_int @ ( divide_divide_int @ ( minus_minus_int @ K @ L ) @ L ) @ one_one_int ) ) ) ) ).

% div_pos_geq
thf(fact_3880_div__eqI,axiom,
    ! [B: nat,R: nat,Q4: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( ( ( euclid3398187327856392827nt_nat @ R )
          = ( euclid3398187327856392827nt_nat @ B ) )
       => ( ( ord_less_nat @ ( euclid4777050414544973029ze_nat @ R ) @ ( euclid4777050414544973029ze_nat @ B ) )
         => ( ( ( plus_plus_nat @ ( times_times_nat @ Q4 @ B ) @ R )
              = A )
           => ( ( divide_divide_nat @ A @ B )
              = Q4 ) ) ) ) ) ).

% div_eqI
thf(fact_3881_div__eqI,axiom,
    ! [B: int,R: int,Q4: int,A: int] :
      ( ( B != zero_zero_int )
     => ( ( ( euclid3395696857347342551nt_int @ R )
          = ( euclid3395696857347342551nt_int @ B ) )
       => ( ( ord_less_nat @ ( euclid4774559944035922753ze_int @ R ) @ ( euclid4774559944035922753ze_int @ B ) )
         => ( ( ( plus_plus_int @ ( times_times_int @ Q4 @ B ) @ R )
              = A )
           => ( ( divide_divide_int @ A @ B )
              = Q4 ) ) ) ) ) ).

% div_eqI
thf(fact_3882_div__eqI,axiom,
    ! [B: code_integer,R: code_integer,Q4: code_integer,A: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( ( ( euclid6289375185220004616nteger @ R )
          = ( euclid6289375185220004616nteger @ B ) )
       => ( ( ord_less_nat @ ( euclid6377331345833325938nteger @ R ) @ ( euclid6377331345833325938nteger @ B ) )
         => ( ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ Q4 @ B ) @ R )
              = A )
           => ( ( divide6298287555418463151nteger @ A @ B )
              = Q4 ) ) ) ) ) ).

% div_eqI
thf(fact_3883_div__eqI,axiom,
    ! [B: code_natural,R: code_natural,Q4: code_natural,A: code_natural] :
      ( ( B != zero_z2226904508553997617atural )
     => ( ( ( euclid5112970336976722131atural @ R )
          = ( euclid5112970336976722131atural @ B ) )
       => ( ( ord_less_nat @ ( euclid5200926497590043453atural @ R ) @ ( euclid5200926497590043453atural @ B ) )
         => ( ( ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ Q4 @ B ) @ R )
              = A )
           => ( ( divide5121882707175180666atural @ A @ B )
              = Q4 ) ) ) ) ) ).

% div_eqI
thf(fact_3884_div__bounded,axiom,
    ! [B: nat,R: nat,Q4: nat] :
      ( ( B != zero_zero_nat )
     => ( ( ( euclid3398187327856392827nt_nat @ R )
          = ( euclid3398187327856392827nt_nat @ B ) )
       => ( ( ord_less_nat @ ( euclid4777050414544973029ze_nat @ R ) @ ( euclid4777050414544973029ze_nat @ B ) )
         => ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ Q4 @ B ) @ R ) @ B )
            = Q4 ) ) ) ) ).

% div_bounded
thf(fact_3885_div__bounded,axiom,
    ! [B: int,R: int,Q4: int] :
      ( ( B != zero_zero_int )
     => ( ( ( euclid3395696857347342551nt_int @ R )
          = ( euclid3395696857347342551nt_int @ B ) )
       => ( ( ord_less_nat @ ( euclid4774559944035922753ze_int @ R ) @ ( euclid4774559944035922753ze_int @ B ) )
         => ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ Q4 @ B ) @ R ) @ B )
            = Q4 ) ) ) ) ).

% div_bounded
thf(fact_3886_div__bounded,axiom,
    ! [B: code_integer,R: code_integer,Q4: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( ( ( euclid6289375185220004616nteger @ R )
          = ( euclid6289375185220004616nteger @ B ) )
       => ( ( ord_less_nat @ ( euclid6377331345833325938nteger @ R ) @ ( euclid6377331345833325938nteger @ B ) )
         => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ Q4 @ B ) @ R ) @ B )
            = Q4 ) ) ) ) ).

% div_bounded
thf(fact_3887_div__bounded,axiom,
    ! [B: code_natural,R: code_natural,Q4: code_natural] :
      ( ( B != zero_z2226904508553997617atural )
     => ( ( ( euclid5112970336976722131atural @ R )
          = ( euclid5112970336976722131atural @ B ) )
       => ( ( ord_less_nat @ ( euclid5200926497590043453atural @ R ) @ ( euclid5200926497590043453atural @ B ) )
         => ( ( divide5121882707175180666atural @ ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ Q4 @ B ) @ R ) @ B )
            = Q4 ) ) ) ) ).

% div_bounded
thf(fact_3888_mod__eqI,axiom,
    ! [B: nat,R: nat,Q4: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( ( ( euclid3398187327856392827nt_nat @ R )
          = ( euclid3398187327856392827nt_nat @ B ) )
       => ( ( ord_less_nat @ ( euclid4777050414544973029ze_nat @ R ) @ ( euclid4777050414544973029ze_nat @ B ) )
         => ( ( ( plus_plus_nat @ ( times_times_nat @ Q4 @ B ) @ R )
              = A )
           => ( ( modulo_modulo_nat @ A @ B )
              = R ) ) ) ) ) ).

% mod_eqI
thf(fact_3889_mod__eqI,axiom,
    ! [B: int,R: int,Q4: int,A: int] :
      ( ( B != zero_zero_int )
     => ( ( ( euclid3395696857347342551nt_int @ R )
          = ( euclid3395696857347342551nt_int @ B ) )
       => ( ( ord_less_nat @ ( euclid4774559944035922753ze_int @ R ) @ ( euclid4774559944035922753ze_int @ B ) )
         => ( ( ( plus_plus_int @ ( times_times_int @ Q4 @ B ) @ R )
              = A )
           => ( ( modulo_modulo_int @ A @ B )
              = R ) ) ) ) ) ).

% mod_eqI
thf(fact_3890_mod__eqI,axiom,
    ! [B: code_integer,R: code_integer,Q4: code_integer,A: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( ( ( euclid6289375185220004616nteger @ R )
          = ( euclid6289375185220004616nteger @ B ) )
       => ( ( ord_less_nat @ ( euclid6377331345833325938nteger @ R ) @ ( euclid6377331345833325938nteger @ B ) )
         => ( ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ Q4 @ B ) @ R )
              = A )
           => ( ( modulo364778990260209775nteger @ A @ B )
              = R ) ) ) ) ) ).

% mod_eqI
thf(fact_3891_mod__eqI,axiom,
    ! [B: code_natural,R: code_natural,Q4: code_natural,A: code_natural] :
      ( ( B != zero_z2226904508553997617atural )
     => ( ( ( euclid5112970336976722131atural @ R )
          = ( euclid5112970336976722131atural @ B ) )
       => ( ( ord_less_nat @ ( euclid5200926497590043453atural @ R ) @ ( euclid5200926497590043453atural @ B ) )
         => ( ( ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ Q4 @ B ) @ R )
              = A )
           => ( ( modulo8411746178871703098atural @ A @ B )
              = R ) ) ) ) ) ).

% mod_eqI
thf(fact_3892_verit__le__mono__div,axiom,
    ! [A3: nat,B5: nat,N: nat] :
      ( ( ord_less_nat @ A3 @ B5 )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_eq_nat
          @ ( plus_plus_nat @ ( divide_divide_nat @ A3 @ N )
            @ ( if_nat
              @ ( ( modulo_modulo_nat @ B5 @ N )
                = zero_zero_nat )
              @ one_one_nat
              @ zero_zero_nat ) )
          @ ( divide_divide_nat @ B5 @ N ) ) ) ) ).

% verit_le_mono_div
thf(fact_3893_division__decomp,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) )
     => ? [B6: nat,C4: nat] :
          ( ( A
            = ( times_times_nat @ B6 @ C4 ) )
          & ( dvd_dvd_nat @ B6 @ B )
          & ( dvd_dvd_nat @ C4 @ C ) ) ) ).

% division_decomp
thf(fact_3894_division__decomp,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) )
     => ? [B6: int,C4: int] :
          ( ( A
            = ( times_times_int @ B6 @ C4 ) )
          & ( dvd_dvd_int @ B6 @ B )
          & ( dvd_dvd_int @ C4 @ C ) ) ) ).

% division_decomp
thf(fact_3895_dvd__productE,axiom,
    ! [P5: nat,A: nat,B: nat] :
      ( ( dvd_dvd_nat @ P5 @ ( times_times_nat @ A @ B ) )
     => ~ ! [X3: nat,Y4: nat] :
            ( ( P5
              = ( times_times_nat @ X3 @ Y4 ) )
           => ( ( dvd_dvd_nat @ X3 @ A )
             => ~ ( dvd_dvd_nat @ Y4 @ B ) ) ) ) ).

% dvd_productE
thf(fact_3896_dvd__productE,axiom,
    ! [P5: int,A: int,B: int] :
      ( ( dvd_dvd_int @ P5 @ ( times_times_int @ A @ B ) )
     => ~ ! [X3: int,Y4: int] :
            ( ( P5
              = ( times_times_int @ X3 @ Y4 ) )
           => ( ( dvd_dvd_int @ X3 @ A )
             => ~ ( dvd_dvd_int @ Y4 @ B ) ) ) ) ).

% dvd_productE
thf(fact_3897_gcd__nat_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ zero_zero_nat @ A )
     => ( A = zero_zero_nat ) ) ).

% gcd_nat.extremum_uniqueI
thf(fact_3898_gcd__nat_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
      = ( ( dvd_dvd_nat @ A @ zero_zero_nat )
        & ( A != zero_zero_nat ) ) ) ).

% gcd_nat.not_eq_extremum
thf(fact_3899_gcd__nat_Oextremum__unique,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ zero_zero_nat @ A )
      = ( A = zero_zero_nat ) ) ).

% gcd_nat.extremum_unique
thf(fact_3900_gcd__nat_Oextremum__strict,axiom,
    ! [A: nat] :
      ~ ( ( dvd_dvd_nat @ zero_zero_nat @ A )
        & ( zero_zero_nat != A ) ) ).

% gcd_nat.extremum_strict
thf(fact_3901_sum__squares__gt__zero__iff,axiom,
    ! [X: code_integer,Y3: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ X @ X ) @ ( times_3573771949741848930nteger @ Y3 @ Y3 ) ) )
      = ( ( X != zero_z3403309356797280102nteger )
        | ( Y3 != zero_z3403309356797280102nteger ) ) ) ).

% sum_squares_gt_zero_iff
thf(fact_3902_sum__squares__gt__zero__iff,axiom,
    ! [X: rat,Y3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y3 @ Y3 ) ) )
      = ( ( X != zero_zero_rat )
        | ( Y3 != zero_zero_rat ) ) ) ).

% sum_squares_gt_zero_iff
thf(fact_3903_sum__squares__gt__zero__iff,axiom,
    ! [X: int,Y3: int] :
      ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y3 @ Y3 ) ) )
      = ( ( X != zero_zero_int )
        | ( Y3 != zero_zero_int ) ) ) ).

% sum_squares_gt_zero_iff
thf(fact_3904_sum__squares__le__zero__iff,axiom,
    ! [X: code_integer,Y3: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ X @ X ) @ ( times_3573771949741848930nteger @ Y3 @ Y3 ) ) @ zero_z3403309356797280102nteger )
      = ( ( X = zero_z3403309356797280102nteger )
        & ( Y3 = zero_z3403309356797280102nteger ) ) ) ).

% sum_squares_le_zero_iff
thf(fact_3905_sum__squares__le__zero__iff,axiom,
    ! [X: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y3 @ Y3 ) ) @ zero_zero_rat )
      = ( ( X = zero_zero_rat )
        & ( Y3 = zero_zero_rat ) ) ) ).

% sum_squares_le_zero_iff
thf(fact_3906_sum__squares__le__zero__iff,axiom,
    ! [X: int,Y3: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y3 @ Y3 ) ) @ zero_zero_int )
      = ( ( X = zero_zero_int )
        & ( Y3 = zero_zero_int ) ) ) ).

% sum_squares_le_zero_iff
thf(fact_3907_sum__squares__eq__zero__iff,axiom,
    ! [X: code_integer,Y3: code_integer] :
      ( ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ X @ X ) @ ( times_3573771949741848930nteger @ Y3 @ Y3 ) )
        = zero_z3403309356797280102nteger )
      = ( ( X = zero_z3403309356797280102nteger )
        & ( Y3 = zero_z3403309356797280102nteger ) ) ) ).

% sum_squares_eq_zero_iff
thf(fact_3908_sum__squares__eq__zero__iff,axiom,
    ! [X: rat,Y3: rat] :
      ( ( ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y3 @ Y3 ) )
        = zero_zero_rat )
      = ( ( X = zero_zero_rat )
        & ( Y3 = zero_zero_rat ) ) ) ).

% sum_squares_eq_zero_iff
thf(fact_3909_sum__squares__eq__zero__iff,axiom,
    ! [X: int,Y3: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y3 @ Y3 ) )
        = zero_zero_int )
      = ( ( X = zero_zero_int )
        & ( Y3 = zero_zero_int ) ) ) ).

% sum_squares_eq_zero_iff
thf(fact_3910_add__scale__eq__noteq,axiom,
    ! [R: code_integer,A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
      ( ( R != zero_z3403309356797280102nteger )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ R @ C ) )
         != ( plus_p5714425477246183910nteger @ B @ ( times_3573771949741848930nteger @ R @ D ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_3911_add__scale__eq__noteq,axiom,
    ! [R: rat,A: rat,B: rat,C: rat,D: rat] :
      ( ( R != zero_zero_rat )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( ( plus_plus_rat @ A @ ( times_times_rat @ R @ C ) )
         != ( plus_plus_rat @ B @ ( times_times_rat @ R @ D ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_3912_add__scale__eq__noteq,axiom,
    ! [R: nat,A: nat,B: nat,C: nat,D: nat] :
      ( ( R != zero_zero_nat )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( ( plus_plus_nat @ A @ ( times_times_nat @ R @ C ) )
         != ( plus_plus_nat @ B @ ( times_times_nat @ R @ D ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_3913_add__scale__eq__noteq,axiom,
    ! [R: int,A: int,B: int,C: int,D: int] :
      ( ( R != zero_zero_int )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( ( plus_plus_int @ A @ ( times_times_int @ R @ C ) )
         != ( plus_plus_int @ B @ ( times_times_int @ R @ D ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_3914_divide__int__def,axiom,
    ( divide_divide_int
    = ( ^ [K3: int,L3: int] :
          ( if_int @ ( L3 = zero_zero_int ) @ zero_zero_int
          @ ( if_int
            @ ( ( sgn_sgn_int @ K3 )
              = ( sgn_sgn_int @ L3 ) )
            @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K3 ) ) @ ( nat2 @ ( abs_abs_int @ L3 ) ) ) )
            @ ( uminus_uminus_int
              @ ( semiri1314217659103216013at_int
                @ ( plus_plus_nat @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K3 ) ) @ ( nat2 @ ( abs_abs_int @ L3 ) ) )
                  @ ( zero_n2687167440665602831ol_nat
                    @ ~ ( dvd_dvd_int @ L3 @ K3 ) ) ) ) ) ) ) ) ) ).

% divide_int_def
thf(fact_3915_gbinomial__pochhammer_H,axiom,
    ( gbinomial_rat
    = ( ^ [A2: rat,K3: nat] : ( divide_divide_rat @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ A2 @ ( semiri681578069525770553at_rat @ K3 ) ) @ one_one_rat ) @ K3 ) @ ( semiri773545260158071498ct_rat @ K3 ) ) ) ) ).

% gbinomial_pochhammer'
thf(fact_3916_abs__idempotent,axiom,
    ! [A: int] :
      ( ( abs_abs_int @ ( abs_abs_int @ A ) )
      = ( abs_abs_int @ A ) ) ).

% abs_idempotent
thf(fact_3917_abs__idempotent,axiom,
    ! [A: code_integer] :
      ( ( abs_abs_Code_integer @ ( abs_abs_Code_integer @ A ) )
      = ( abs_abs_Code_integer @ A ) ) ).

% abs_idempotent
thf(fact_3918_abs__idempotent,axiom,
    ! [A: rat] :
      ( ( abs_abs_rat @ ( abs_abs_rat @ A ) )
      = ( abs_abs_rat @ A ) ) ).

% abs_idempotent
thf(fact_3919_abs__abs,axiom,
    ! [A: int] :
      ( ( abs_abs_int @ ( abs_abs_int @ A ) )
      = ( abs_abs_int @ A ) ) ).

% abs_abs
thf(fact_3920_abs__abs,axiom,
    ! [A: code_integer] :
      ( ( abs_abs_Code_integer @ ( abs_abs_Code_integer @ A ) )
      = ( abs_abs_Code_integer @ A ) ) ).

% abs_abs
thf(fact_3921_abs__abs,axiom,
    ! [A: rat] :
      ( ( abs_abs_rat @ ( abs_abs_rat @ A ) )
      = ( abs_abs_rat @ A ) ) ).

% abs_abs
thf(fact_3922_abs__0,axiom,
    ( ( abs_abs_int @ zero_zero_int )
    = zero_zero_int ) ).

% abs_0
thf(fact_3923_abs__0,axiom,
    ( ( abs_abs_Code_integer @ zero_z3403309356797280102nteger )
    = zero_z3403309356797280102nteger ) ).

% abs_0
thf(fact_3924_abs__0,axiom,
    ( ( abs_abs_rat @ zero_zero_rat )
    = zero_zero_rat ) ).

% abs_0
thf(fact_3925_abs__zero,axiom,
    ( ( abs_abs_int @ zero_zero_int )
    = zero_zero_int ) ).

% abs_zero
thf(fact_3926_abs__zero,axiom,
    ( ( abs_abs_Code_integer @ zero_z3403309356797280102nteger )
    = zero_z3403309356797280102nteger ) ).

% abs_zero
thf(fact_3927_abs__zero,axiom,
    ( ( abs_abs_rat @ zero_zero_rat )
    = zero_zero_rat ) ).

% abs_zero
thf(fact_3928_abs__eq__0,axiom,
    ! [A: int] :
      ( ( ( abs_abs_int @ A )
        = zero_zero_int )
      = ( A = zero_zero_int ) ) ).

% abs_eq_0
thf(fact_3929_abs__eq__0,axiom,
    ! [A: code_integer] :
      ( ( ( abs_abs_Code_integer @ A )
        = zero_z3403309356797280102nteger )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% abs_eq_0
thf(fact_3930_abs__eq__0,axiom,
    ! [A: rat] :
      ( ( ( abs_abs_rat @ A )
        = zero_zero_rat )
      = ( A = zero_zero_rat ) ) ).

% abs_eq_0
thf(fact_3931_abs__0__eq,axiom,
    ! [A: int] :
      ( ( zero_zero_int
        = ( abs_abs_int @ A ) )
      = ( A = zero_zero_int ) ) ).

% abs_0_eq
thf(fact_3932_abs__0__eq,axiom,
    ! [A: code_integer] :
      ( ( zero_z3403309356797280102nteger
        = ( abs_abs_Code_integer @ A ) )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% abs_0_eq
thf(fact_3933_abs__0__eq,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat
        = ( abs_abs_rat @ A ) )
      = ( A = zero_zero_rat ) ) ).

% abs_0_eq
thf(fact_3934_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_3935_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_3936_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_3937_abs__add__abs,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( abs_abs_Code_integer @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) )
      = ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ).

% abs_add_abs
thf(fact_3938_abs__add__abs,axiom,
    ! [A: rat,B: rat] :
      ( ( abs_abs_rat @ ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) )
      = ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).

% abs_add_abs
thf(fact_3939_abs__add__abs,axiom,
    ! [A: int,B: int] :
      ( ( abs_abs_int @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) )
      = ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).

% abs_add_abs
thf(fact_3940_abs__1,axiom,
    ( ( abs_abs_int @ one_one_int )
    = one_one_int ) ).

% abs_1
thf(fact_3941_abs__1,axiom,
    ( ( abs_abs_Code_integer @ one_one_Code_integer )
    = one_one_Code_integer ) ).

% abs_1
thf(fact_3942_abs__1,axiom,
    ( ( abs_abs_rat @ one_one_rat )
    = one_one_rat ) ).

% abs_1
thf(fact_3943_abs__minus,axiom,
    ! [A: int] :
      ( ( abs_abs_int @ ( uminus_uminus_int @ A ) )
      = ( abs_abs_int @ A ) ) ).

% abs_minus
thf(fact_3944_abs__minus,axiom,
    ! [A: code_integer] :
      ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ A ) )
      = ( abs_abs_Code_integer @ A ) ) ).

% abs_minus
thf(fact_3945_abs__minus,axiom,
    ! [A: rat] :
      ( ( abs_abs_rat @ ( uminus_uminus_rat @ A ) )
      = ( abs_abs_rat @ A ) ) ).

% abs_minus
thf(fact_3946_abs__minus__cancel,axiom,
    ! [A: int] :
      ( ( abs_abs_int @ ( uminus_uminus_int @ A ) )
      = ( abs_abs_int @ A ) ) ).

% abs_minus_cancel
thf(fact_3947_abs__minus__cancel,axiom,
    ! [A: code_integer] :
      ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ A ) )
      = ( abs_abs_Code_integer @ A ) ) ).

% abs_minus_cancel
thf(fact_3948_abs__minus__cancel,axiom,
    ! [A: rat] :
      ( ( abs_abs_rat @ ( uminus_uminus_rat @ A ) )
      = ( abs_abs_rat @ A ) ) ).

% abs_minus_cancel
thf(fact_3949_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_3950_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_3951_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_3952_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_3953_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_3954_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_3955_abs__divide,axiom,
    ! [A: rat,B: rat] :
      ( ( abs_abs_rat @ ( divide_divide_rat @ A @ B ) )
      = ( divide_divide_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).

% abs_divide
thf(fact_3956_abs__of__nat,axiom,
    ! [N: nat] :
      ( ( abs_abs_int @ ( semiri1314217659103216013at_int @ N ) )
      = ( semiri1314217659103216013at_int @ N ) ) ).

% abs_of_nat
thf(fact_3957_abs__of__nat,axiom,
    ! [N: nat] :
      ( ( abs_abs_Code_integer @ ( semiri4939895301339042750nteger @ N ) )
      = ( semiri4939895301339042750nteger @ N ) ) ).

% abs_of_nat
thf(fact_3958_abs__of__nat,axiom,
    ! [N: nat] :
      ( ( abs_abs_rat @ ( semiri681578069525770553at_rat @ N ) )
      = ( semiri681578069525770553at_rat @ N ) ) ).

% abs_of_nat
thf(fact_3959_abs__bool__eq,axiom,
    ! [P: $o] :
      ( ( abs_abs_Code_integer @ ( zero_n356916108424825756nteger @ P ) )
      = ( zero_n356916108424825756nteger @ P ) ) ).

% abs_bool_eq
thf(fact_3960_abs__bool__eq,axiom,
    ! [P: $o] :
      ( ( abs_abs_rat @ ( zero_n2052037380579107095ol_rat @ P ) )
      = ( zero_n2052037380579107095ol_rat @ P ) ) ).

% abs_bool_eq
thf(fact_3961_abs__bool__eq,axiom,
    ! [P: $o] :
      ( ( abs_abs_int @ ( zero_n2684676970156552555ol_int @ P ) )
      = ( zero_n2684676970156552555ol_int @ P ) ) ).

% abs_bool_eq
thf(fact_3962_nat__of__bool,axiom,
    ! [P: $o] :
      ( ( nat2 @ ( zero_n2684676970156552555ol_int @ P ) )
      = ( zero_n2687167440665602831ol_nat @ P ) ) ).

% nat_of_bool
thf(fact_3963_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_3964_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_3965_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_3966_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_3967_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_3968_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_3969_abs__of__nonneg,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( abs_abs_Code_integer @ A )
        = A ) ) ).

% abs_of_nonneg
thf(fact_3970_abs__of__nonneg,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( abs_abs_rat @ A )
        = A ) ) ).

% abs_of_nonneg
thf(fact_3971_abs__of__nonneg,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( abs_abs_int @ A )
        = A ) ) ).

% abs_of_nonneg
thf(fact_3972_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_3973_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_3974_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_3975_abs__neg__one,axiom,
    ( ( abs_abs_int @ ( uminus_uminus_int @ one_one_int ) )
    = one_one_int ) ).

% abs_neg_one
thf(fact_3976_abs__neg__one,axiom,
    ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = one_one_Code_integer ) ).

% abs_neg_one
thf(fact_3977_abs__neg__one,axiom,
    ( ( abs_abs_rat @ ( uminus_uminus_rat @ one_one_rat ) )
    = one_one_rat ) ).

% abs_neg_one
thf(fact_3978_pochhammer__0,axiom,
    ! [A: nat] :
      ( ( comm_s4663373288045622133er_nat @ A @ zero_zero_nat )
      = one_one_nat ) ).

% pochhammer_0
thf(fact_3979_pochhammer__0,axiom,
    ! [A: int] :
      ( ( comm_s4660882817536571857er_int @ A @ zero_zero_nat )
      = one_one_int ) ).

% pochhammer_0
thf(fact_3980_pochhammer__0,axiom,
    ! [A: code_integer] :
      ( ( comm_s8582702949713902594nteger @ A @ zero_zero_nat )
      = one_one_Code_integer ) ).

% pochhammer_0
thf(fact_3981_pochhammer__0,axiom,
    ! [A: rat] :
      ( ( comm_s4028243227959126397er_rat @ A @ zero_zero_nat )
      = one_one_rat ) ).

% pochhammer_0
thf(fact_3982_pochhammer__0,axiom,
    ! [A: code_natural] :
      ( ( comm_s7406298101470620109atural @ A @ zero_zero_nat )
      = one_one_Code_natural ) ).

% pochhammer_0
thf(fact_3983_zdvd1__eq,axiom,
    ! [X: int] :
      ( ( dvd_dvd_int @ X @ one_one_int )
      = ( ( abs_abs_int @ X )
        = one_one_int ) ) ).

% zdvd1_eq
thf(fact_3984_abs__division__segment,axiom,
    ! [K: int] :
      ( ( abs_abs_int @ ( euclid3395696857347342551nt_int @ K ) )
      = one_one_int ) ).

% abs_division_segment
thf(fact_3985_abs__of__nonpos,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger )
     => ( ( abs_abs_Code_integer @ A )
        = ( uminus1351360451143612070nteger @ A ) ) ) ).

% abs_of_nonpos
thf(fact_3986_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_3987_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_3988_zero__le__divide__abs__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ ( abs_abs_rat @ B ) ) )
      = ( ( ord_less_eq_rat @ zero_zero_rat @ A )
        | ( B = zero_zero_rat ) ) ) ).

% zero_le_divide_abs_iff
thf(fact_3989_divide__le__0__abs__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ ( abs_abs_rat @ B ) ) @ zero_zero_rat )
      = ( ( ord_less_eq_rat @ A @ zero_zero_rat )
        | ( B = zero_zero_rat ) ) ) ).

% divide_le_0_abs_iff
thf(fact_3990_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_3991_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_3992_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_3993_zabs__less__one__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_int @ ( abs_abs_int @ Z2 ) @ one_one_int )
      = ( Z2 = zero_zero_int ) ) ).

% zabs_less_one_iff
thf(fact_3994_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_3995_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_3996_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_3997_sgn__abs,axiom,
    ! [A: code_integer] :
      ( ( abs_abs_Code_integer @ ( sgn_sgn_Code_integer @ A ) )
      = ( zero_n356916108424825756nteger @ ( A != zero_z3403309356797280102nteger ) ) ) ).

% sgn_abs
thf(fact_3998_sgn__abs,axiom,
    ! [A: rat] :
      ( ( abs_abs_rat @ ( sgn_sgn_rat @ A ) )
      = ( zero_n2052037380579107095ol_rat @ ( A != zero_zero_rat ) ) ) ).

% sgn_abs
thf(fact_3999_sgn__abs,axiom,
    ! [A: int] :
      ( ( abs_abs_int @ ( sgn_sgn_int @ A ) )
      = ( zero_n2684676970156552555ol_int @ ( A != zero_zero_int ) ) ) ).

% sgn_abs
thf(fact_4000_nat__abs__dvd__iff,axiom,
    ! [K: int,N: nat] :
      ( ( dvd_dvd_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ N )
      = ( dvd_dvd_int @ K @ ( semiri1314217659103216013at_int @ N ) ) ) ).

% nat_abs_dvd_iff
thf(fact_4001_dvd__nat__abs__iff,axiom,
    ! [N: nat,K: int] :
      ( ( dvd_dvd_nat @ N @ ( nat2 @ ( abs_abs_int @ K ) ) )
      = ( dvd_dvd_int @ ( semiri1314217659103216013at_int @ N ) @ K ) ) ).

% dvd_nat_abs_iff
thf(fact_4002_pochhammer__of__nat,axiom,
    ! [X: nat,N: nat] :
      ( ( comm_s4660882817536571857er_int @ ( semiri1314217659103216013at_int @ X ) @ N )
      = ( semiri1314217659103216013at_int @ ( comm_s4663373288045622133er_nat @ X @ N ) ) ) ).

% pochhammer_of_nat
thf(fact_4003_pochhammer__of__nat,axiom,
    ! [X: nat,N: nat] :
      ( ( comm_s8582702949713902594nteger @ ( semiri4939895301339042750nteger @ X ) @ N )
      = ( semiri4939895301339042750nteger @ ( comm_s4663373288045622133er_nat @ X @ N ) ) ) ).

% pochhammer_of_nat
thf(fact_4004_pochhammer__of__nat,axiom,
    ! [X: nat,N: nat] :
      ( ( comm_s4663373288045622133er_nat @ ( semiri1316708129612266289at_nat @ X ) @ N )
      = ( semiri1316708129612266289at_nat @ ( comm_s4663373288045622133er_nat @ X @ N ) ) ) ).

% pochhammer_of_nat
thf(fact_4005_pochhammer__of__nat,axiom,
    ! [X: nat,N: nat] :
      ( ( comm_s4028243227959126397er_rat @ ( semiri681578069525770553at_rat @ X ) @ N )
      = ( semiri681578069525770553at_rat @ ( comm_s4663373288045622133er_nat @ X @ N ) ) ) ).

% pochhammer_of_nat
thf(fact_4006_pochhammer__of__nat,axiom,
    ! [X: nat,N: nat] :
      ( ( comm_s7406298101470620109atural @ ( semiri3763490453095760265atural @ X ) @ N )
      = ( semiri3763490453095760265atural @ ( comm_s4663373288045622133er_nat @ X @ N ) ) ) ).

% pochhammer_of_nat
thf(fact_4007_abs__ge__self,axiom,
    ! [A: code_integer] : ( ord_le3102999989581377725nteger @ A @ ( abs_abs_Code_integer @ A ) ) ).

% abs_ge_self
thf(fact_4008_abs__ge__self,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ A @ ( abs_abs_rat @ A ) ) ).

% abs_ge_self
thf(fact_4009_abs__ge__self,axiom,
    ! [A: int] : ( ord_less_eq_int @ A @ ( abs_abs_int @ A ) ) ).

% abs_ge_self
thf(fact_4010_abs__le__D1,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B )
     => ( ord_le3102999989581377725nteger @ A @ B ) ) ).

% abs_le_D1
thf(fact_4011_abs__le__D1,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B )
     => ( ord_less_eq_rat @ A @ B ) ) ).

% abs_le_D1
thf(fact_4012_abs__le__D1,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
     => ( ord_less_eq_int @ A @ B ) ) ).

% abs_le_D1
thf(fact_4013_abs__eq__0__iff,axiom,
    ! [A: int] :
      ( ( ( abs_abs_int @ A )
        = zero_zero_int )
      = ( A = zero_zero_int ) ) ).

% abs_eq_0_iff
thf(fact_4014_abs__eq__0__iff,axiom,
    ! [A: code_integer] :
      ( ( ( abs_abs_Code_integer @ A )
        = zero_z3403309356797280102nteger )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% abs_eq_0_iff
thf(fact_4015_abs__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( ( abs_abs_rat @ A )
        = zero_zero_rat )
      = ( A = zero_zero_rat ) ) ).

% abs_eq_0_iff
thf(fact_4016_abs__mult,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) )
      = ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ).

% abs_mult
thf(fact_4017_abs__mult,axiom,
    ! [A: rat,B: rat] :
      ( ( abs_abs_rat @ ( times_times_rat @ A @ B ) )
      = ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).

% abs_mult
thf(fact_4018_abs__mult,axiom,
    ! [A: int,B: int] :
      ( ( abs_abs_int @ ( times_times_int @ A @ B ) )
      = ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).

% abs_mult
thf(fact_4019_abs__one,axiom,
    ( ( abs_abs_int @ one_one_int )
    = one_one_int ) ).

% abs_one
thf(fact_4020_abs__one,axiom,
    ( ( abs_abs_Code_integer @ one_one_Code_integer )
    = one_one_Code_integer ) ).

% abs_one
thf(fact_4021_abs__one,axiom,
    ( ( abs_abs_rat @ one_one_rat )
    = one_one_rat ) ).

% abs_one
thf(fact_4022_abs__minus__commute,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B ) )
      = ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B @ A ) ) ) ).

% abs_minus_commute
thf(fact_4023_abs__minus__commute,axiom,
    ! [A: rat,B: rat] :
      ( ( abs_abs_rat @ ( minus_minus_rat @ A @ B ) )
      = ( abs_abs_rat @ ( minus_minus_rat @ B @ A ) ) ) ).

% abs_minus_commute
thf(fact_4024_abs__minus__commute,axiom,
    ! [A: int,B: int] :
      ( ( abs_abs_int @ ( minus_minus_int @ A @ B ) )
      = ( abs_abs_int @ ( minus_minus_int @ B @ A ) ) ) ).

% abs_minus_commute
thf(fact_4025_abs__eq__iff,axiom,
    ! [X: int,Y3: int] :
      ( ( ( abs_abs_int @ X )
        = ( abs_abs_int @ Y3 ) )
      = ( ( X = Y3 )
        | ( X
          = ( uminus_uminus_int @ Y3 ) ) ) ) ).

% abs_eq_iff
thf(fact_4026_abs__eq__iff,axiom,
    ! [X: code_integer,Y3: code_integer] :
      ( ( ( abs_abs_Code_integer @ X )
        = ( abs_abs_Code_integer @ Y3 ) )
      = ( ( X = Y3 )
        | ( X
          = ( uminus1351360451143612070nteger @ Y3 ) ) ) ) ).

% abs_eq_iff
thf(fact_4027_abs__eq__iff,axiom,
    ! [X: rat,Y3: rat] :
      ( ( ( abs_abs_rat @ X )
        = ( abs_abs_rat @ Y3 ) )
      = ( ( X = Y3 )
        | ( X
          = ( uminus_uminus_rat @ Y3 ) ) ) ) ).

% abs_eq_iff
thf(fact_4028_dvd__if__abs__eq,axiom,
    ! [L: int,K: int] :
      ( ( ( abs_abs_int @ L )
        = ( abs_abs_int @ K ) )
     => ( dvd_dvd_int @ L @ K ) ) ).

% dvd_if_abs_eq
thf(fact_4029_dvd__if__abs__eq,axiom,
    ! [L: code_integer,K: code_integer] :
      ( ( ( abs_abs_Code_integer @ L )
        = ( abs_abs_Code_integer @ K ) )
     => ( dvd_dvd_Code_integer @ L @ K ) ) ).

% dvd_if_abs_eq
thf(fact_4030_dvd__if__abs__eq,axiom,
    ! [L: rat,K: rat] :
      ( ( ( abs_abs_rat @ L )
        = ( abs_abs_rat @ K ) )
     => ( dvd_dvd_rat @ L @ K ) ) ).

% dvd_if_abs_eq
thf(fact_4031_zdvd__antisym__abs,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ B )
     => ( ( dvd_dvd_int @ B @ A )
       => ( ( abs_abs_int @ A )
          = ( abs_abs_int @ B ) ) ) ) ).

% zdvd_antisym_abs
thf(fact_4032_abs__ge__zero,axiom,
    ! [A: code_integer] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ A ) ) ).

% abs_ge_zero
thf(fact_4033_abs__ge__zero,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( abs_abs_rat @ A ) ) ).

% abs_ge_zero
thf(fact_4034_abs__ge__zero,axiom,
    ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( abs_abs_int @ A ) ) ).

% abs_ge_zero
thf(fact_4035_abs__not__less__zero,axiom,
    ! [A: code_integer] :
      ~ ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ zero_z3403309356797280102nteger ) ).

% abs_not_less_zero
thf(fact_4036_abs__not__less__zero,axiom,
    ! [A: rat] :
      ~ ( ord_less_rat @ ( abs_abs_rat @ A ) @ zero_zero_rat ) ).

% abs_not_less_zero
thf(fact_4037_abs__not__less__zero,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ ( abs_abs_int @ A ) @ zero_zero_int ) ).

% abs_not_less_zero
thf(fact_4038_abs__of__pos,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( abs_abs_Code_integer @ A )
        = A ) ) ).

% abs_of_pos
thf(fact_4039_abs__of__pos,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( abs_abs_rat @ A )
        = A ) ) ).

% abs_of_pos
thf(fact_4040_abs__of__pos,axiom,
    ! [A: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( abs_abs_int @ A )
        = A ) ) ).

% abs_of_pos
thf(fact_4041_abs__triangle__ineq,axiom,
    ! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( plus_p5714425477246183910nteger @ A @ B ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ).

% abs_triangle_ineq
thf(fact_4042_abs__triangle__ineq,axiom,
    ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( plus_plus_rat @ A @ B ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).

% abs_triangle_ineq
thf(fact_4043_abs__triangle__ineq,axiom,
    ! [A: int,B: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( plus_plus_int @ A @ B ) ) @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).

% abs_triangle_ineq
thf(fact_4044_abs__triangle__ineq2,axiom,
    ! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ).

% abs_triangle_ineq2
thf(fact_4045_abs__triangle__ineq2,axiom,
    ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ A @ B ) ) ) ).

% abs_triangle_ineq2
thf(fact_4046_abs__triangle__ineq2,axiom,
    ! [A: int,B: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) @ ( abs_abs_int @ ( minus_minus_int @ A @ B ) ) ) ).

% abs_triangle_ineq2
thf(fact_4047_abs__triangle__ineq3,axiom,
    ! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ).

% abs_triangle_ineq3
thf(fact_4048_abs__triangle__ineq3,axiom,
    ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ A @ B ) ) ) ).

% abs_triangle_ineq3
thf(fact_4049_abs__triangle__ineq3,axiom,
    ! [A: int,B: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) @ ( abs_abs_int @ ( minus_minus_int @ A @ B ) ) ) ).

% abs_triangle_ineq3
thf(fact_4050_abs__triangle__ineq2__sym,axiom,
    ! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B @ A ) ) ) ).

% abs_triangle_ineq2_sym
thf(fact_4051_abs__triangle__ineq2__sym,axiom,
    ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ B @ A ) ) ) ).

% abs_triangle_ineq2_sym
thf(fact_4052_abs__triangle__ineq2__sym,axiom,
    ! [A: int,B: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) @ ( abs_abs_int @ ( minus_minus_int @ B @ A ) ) ) ).

% abs_triangle_ineq2_sym
thf(fact_4053_abs__mult__less,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer,D: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ C )
     => ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ B ) @ D )
       => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) @ ( times_3573771949741848930nteger @ C @ D ) ) ) ) ).

% abs_mult_less
thf(fact_4054_abs__mult__less,axiom,
    ! [A: rat,C: rat,B: rat,D: rat] :
      ( ( ord_less_rat @ ( abs_abs_rat @ A ) @ C )
     => ( ( ord_less_rat @ ( abs_abs_rat @ B ) @ D )
       => ( ord_less_rat @ ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) @ ( times_times_rat @ C @ D ) ) ) ) ).

% abs_mult_less
thf(fact_4055_abs__mult__less,axiom,
    ! [A: int,C: int,B: int,D: int] :
      ( ( ord_less_int @ ( abs_abs_int @ A ) @ C )
     => ( ( ord_less_int @ ( abs_abs_int @ B ) @ D )
       => ( ord_less_int @ ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) @ ( times_times_int @ C @ D ) ) ) ) ).

% abs_mult_less
thf(fact_4056_abs__ge__minus__self,axiom,
    ! [A: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ ( abs_abs_Code_integer @ A ) ) ).

% abs_ge_minus_self
thf(fact_4057_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_4058_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_4059_abs__le__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B )
      = ( ( ord_le3102999989581377725nteger @ A @ B )
        & ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ) ).

% abs_le_iff
thf(fact_4060_abs__le__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B )
      = ( ( ord_less_eq_rat @ A @ B )
        & ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ) ).

% abs_le_iff
thf(fact_4061_abs__le__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
      = ( ( ord_less_eq_int @ A @ B )
        & ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).

% abs_le_iff
thf(fact_4062_abs__le__D2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B )
     => ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).

% abs_le_D2
thf(fact_4063_abs__le__D2,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B )
     => ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ).

% abs_le_D2
thf(fact_4064_abs__le__D2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
     => ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B ) ) ).

% abs_le_D2
thf(fact_4065_abs__leI,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ B )
     => ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
       => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B ) ) ) ).

% abs_leI
thf(fact_4066_abs__leI,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B )
       => ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B ) ) ) ).

% abs_leI
thf(fact_4067_abs__leI,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B )
       => ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B ) ) ) ).

% abs_leI
thf(fact_4068_abs__less__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ ( abs_abs_int @ A ) @ B )
      = ( ( ord_less_int @ A @ B )
        & ( ord_less_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).

% abs_less_iff
thf(fact_4069_abs__less__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ B )
      = ( ( ord_le6747313008572928689nteger @ A @ B )
        & ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ) ).

% abs_less_iff
thf(fact_4070_abs__less__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ ( abs_abs_rat @ A ) @ B )
      = ( ( ord_less_rat @ A @ B )
        & ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ) ).

% abs_less_iff
thf(fact_4071_nonzero__abs__divide,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( ( abs_abs_rat @ ( divide_divide_rat @ A @ B ) )
        = ( divide_divide_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ) ).

% nonzero_abs_divide
thf(fact_4072_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_4073_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_4074_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_4075_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_4076_abs__mult__sgn,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ ( abs_abs_rat @ A ) @ ( sgn_sgn_rat @ A ) )
      = A ) ).

% abs_mult_sgn
thf(fact_4077_abs__mult__sgn,axiom,
    ! [A: int] :
      ( ( times_times_int @ ( abs_abs_int @ A ) @ ( sgn_sgn_int @ A ) )
      = A ) ).

% abs_mult_sgn
thf(fact_4078_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_4079_sgn__mult__abs,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ ( sgn_sgn_rat @ A ) @ ( abs_abs_rat @ A ) )
      = A ) ).

% sgn_mult_abs
thf(fact_4080_sgn__mult__abs,axiom,
    ! [A: int] :
      ( ( times_times_int @ ( sgn_sgn_int @ A ) @ ( abs_abs_int @ A ) )
      = A ) ).

% sgn_mult_abs
thf(fact_4081_mult__sgn__abs,axiom,
    ! [X: code_integer] :
      ( ( times_3573771949741848930nteger @ ( sgn_sgn_Code_integer @ X ) @ ( abs_abs_Code_integer @ X ) )
      = X ) ).

% mult_sgn_abs
thf(fact_4082_mult__sgn__abs,axiom,
    ! [X: rat] :
      ( ( times_times_rat @ ( sgn_sgn_rat @ X ) @ ( abs_abs_rat @ X ) )
      = X ) ).

% mult_sgn_abs
thf(fact_4083_mult__sgn__abs,axiom,
    ! [X: int] :
      ( ( times_times_int @ ( sgn_sgn_int @ X ) @ ( abs_abs_int @ X ) )
      = X ) ).

% mult_sgn_abs
thf(fact_4084_same__sgn__abs__add,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ( sgn_sgn_Code_integer @ B )
        = ( sgn_sgn_Code_integer @ A ) )
     => ( ( abs_abs_Code_integer @ ( plus_p5714425477246183910nteger @ A @ B ) )
        = ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ) ).

% same_sgn_abs_add
thf(fact_4085_same__sgn__abs__add,axiom,
    ! [B: rat,A: rat] :
      ( ( ( sgn_sgn_rat @ B )
        = ( sgn_sgn_rat @ A ) )
     => ( ( abs_abs_rat @ ( plus_plus_rat @ A @ B ) )
        = ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ) ).

% same_sgn_abs_add
thf(fact_4086_same__sgn__abs__add,axiom,
    ! [B: int,A: int] :
      ( ( ( sgn_sgn_int @ B )
        = ( sgn_sgn_int @ A ) )
     => ( ( abs_abs_int @ ( plus_plus_int @ A @ B ) )
        = ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ) ).

% same_sgn_abs_add
thf(fact_4087_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_4088_nat__abs__triangle__ineq,axiom,
    ! [K: int,L: int] : ( ord_less_eq_nat @ ( nat2 @ ( abs_abs_int @ ( plus_plus_int @ K @ L ) ) ) @ ( plus_plus_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) ) ).

% nat_abs_triangle_ineq
thf(fact_4089_pochhammer__pos,axiom,
    ! [X: code_natural,N: nat] :
      ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ X )
     => ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ ( comm_s7406298101470620109atural @ X @ N ) ) ) ).

% pochhammer_pos
thf(fact_4090_pochhammer__pos,axiom,
    ! [X: code_integer,N: nat] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ X )
     => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( comm_s8582702949713902594nteger @ X @ N ) ) ) ).

% pochhammer_pos
thf(fact_4091_pochhammer__pos,axiom,
    ! [X: rat,N: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ X )
     => ( ord_less_rat @ zero_zero_rat @ ( comm_s4028243227959126397er_rat @ X @ N ) ) ) ).

% pochhammer_pos
thf(fact_4092_pochhammer__pos,axiom,
    ! [X: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ X )
     => ( ord_less_nat @ zero_zero_nat @ ( comm_s4663373288045622133er_nat @ X @ N ) ) ) ).

% pochhammer_pos
thf(fact_4093_pochhammer__pos,axiom,
    ! [X: int,N: nat] :
      ( ( ord_less_int @ zero_zero_int @ X )
     => ( ord_less_int @ zero_zero_int @ ( comm_s4660882817536571857er_int @ X @ N ) ) ) ).

% pochhammer_pos
thf(fact_4094_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_4095_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_4096_pochhammer__fact,axiom,
    ( semiri1408675320244567234ct_nat
    = ( comm_s4663373288045622133er_nat @ one_one_nat ) ) ).

% pochhammer_fact
thf(fact_4097_pochhammer__fact,axiom,
    ( semiri2447717529341329178atural
    = ( comm_s7406298101470620109atural @ one_one_Code_natural ) ) ).

% pochhammer_fact
thf(fact_4098_pochhammer__fact,axiom,
    ( semiri773545260158071498ct_rat
    = ( comm_s4028243227959126397er_rat @ one_one_rat ) ) ).

% pochhammer_fact
thf(fact_4099_pochhammer__fact,axiom,
    ( semiri3624122377584611663nteger
    = ( comm_s8582702949713902594nteger @ one_one_Code_integer ) ) ).

% pochhammer_fact
thf(fact_4100_pochhammer__fact,axiom,
    ( semiri1406184849735516958ct_int
    = ( comm_s4660882817536571857er_int @ one_one_int ) ) ).

% pochhammer_fact
thf(fact_4101_dense__eq0__I,axiom,
    ! [X: rat] :
      ( ! [E2: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ E2 )
         => ( ord_less_eq_rat @ ( abs_abs_rat @ X ) @ E2 ) )
     => ( X = zero_zero_rat ) ) ).

% dense_eq0_I
thf(fact_4102_abs__eq__mult,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
          | ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) )
        & ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
          | ( ord_le3102999989581377725nteger @ B @ zero_z3403309356797280102nteger ) ) )
     => ( ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) )
        = ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ) ).

% abs_eq_mult
thf(fact_4103_abs__eq__mult,axiom,
    ! [A: rat,B: rat] :
      ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
          | ( ord_less_eq_rat @ A @ zero_zero_rat ) )
        & ( ( ord_less_eq_rat @ zero_zero_rat @ B )
          | ( ord_less_eq_rat @ B @ zero_zero_rat ) ) )
     => ( ( abs_abs_rat @ ( times_times_rat @ A @ B ) )
        = ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ) ).

% abs_eq_mult
thf(fact_4104_abs__eq__mult,axiom,
    ! [A: int,B: int] :
      ( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
          | ( ord_less_eq_int @ A @ zero_zero_int ) )
        & ( ( ord_less_eq_int @ zero_zero_int @ B )
          | ( ord_less_eq_int @ B @ zero_zero_int ) ) )
     => ( ( abs_abs_int @ ( times_times_int @ A @ B ) )
        = ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ) ).

% abs_eq_mult
thf(fact_4105_abs__mult__pos,axiom,
    ! [X: code_integer,Y3: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X )
     => ( ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ Y3 ) @ X )
        = ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ Y3 @ X ) ) ) ) ).

% abs_mult_pos
thf(fact_4106_abs__mult__pos,axiom,
    ! [X: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X )
     => ( ( times_times_rat @ ( abs_abs_rat @ Y3 ) @ X )
        = ( abs_abs_rat @ ( times_times_rat @ Y3 @ X ) ) ) ) ).

% abs_mult_pos
thf(fact_4107_abs__mult__pos,axiom,
    ! [X: int,Y3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( times_times_int @ ( abs_abs_int @ Y3 ) @ X )
        = ( abs_abs_int @ ( times_times_int @ Y3 @ X ) ) ) ) ).

% abs_mult_pos
thf(fact_4108_abs__minus__le__zero,axiom,
    ! [A: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( abs_abs_Code_integer @ A ) ) @ zero_z3403309356797280102nteger ) ).

% abs_minus_le_zero
thf(fact_4109_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_4110_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_4111_eq__abs__iff_H,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A
        = ( abs_abs_Code_integer @ B ) )
      = ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
        & ( ( B = A )
          | ( B
            = ( uminus1351360451143612070nteger @ A ) ) ) ) ) ).

% eq_abs_iff'
thf(fact_4112_eq__abs__iff_H,axiom,
    ! [A: rat,B: rat] :
      ( ( A
        = ( abs_abs_rat @ B ) )
      = ( ( ord_less_eq_rat @ zero_zero_rat @ A )
        & ( ( B = A )
          | ( B
            = ( uminus_uminus_rat @ A ) ) ) ) ) ).

% eq_abs_iff'
thf(fact_4113_eq__abs__iff_H,axiom,
    ! [A: int,B: int] :
      ( ( A
        = ( abs_abs_int @ B ) )
      = ( ( ord_less_eq_int @ zero_zero_int @ A )
        & ( ( B = A )
          | ( B
            = ( uminus_uminus_int @ A ) ) ) ) ) ).

% eq_abs_iff'
thf(fact_4114_abs__eq__iff_H,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( abs_abs_Code_integer @ A )
        = B )
      = ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
        & ( ( A = B )
          | ( A
            = ( uminus1351360451143612070nteger @ B ) ) ) ) ) ).

% abs_eq_iff'
thf(fact_4115_abs__eq__iff_H,axiom,
    ! [A: rat,B: rat] :
      ( ( ( abs_abs_rat @ A )
        = B )
      = ( ( ord_less_eq_rat @ zero_zero_rat @ B )
        & ( ( A = B )
          | ( A
            = ( uminus_uminus_rat @ B ) ) ) ) ) ).

% abs_eq_iff'
thf(fact_4116_abs__eq__iff_H,axiom,
    ! [A: int,B: int] :
      ( ( ( abs_abs_int @ A )
        = B )
      = ( ( ord_less_eq_int @ zero_zero_int @ B )
        & ( ( A = B )
          | ( A
            = ( uminus_uminus_int @ B ) ) ) ) ) ).

% abs_eq_iff'
thf(fact_4117_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_4118_abs__of__neg,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger )
     => ( ( abs_abs_Code_integer @ A )
        = ( uminus1351360451143612070nteger @ A ) ) ) ).

% abs_of_neg
thf(fact_4119_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_4120_abs__if__raw,axiom,
    ( abs_abs_int
    = ( ^ [A2: int] : ( if_int @ ( ord_less_int @ A2 @ zero_zero_int ) @ ( uminus_uminus_int @ A2 ) @ A2 ) ) ) ).

% abs_if_raw
thf(fact_4121_abs__if__raw,axiom,
    ( abs_abs_Code_integer
    = ( ^ [A2: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ A2 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ A2 ) @ A2 ) ) ) ).

% abs_if_raw
thf(fact_4122_abs__if__raw,axiom,
    ( abs_abs_rat
    = ( ^ [A2: rat] : ( if_rat @ ( ord_less_rat @ A2 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A2 ) @ A2 ) ) ) ).

% abs_if_raw
thf(fact_4123_abs__if,axiom,
    ( abs_abs_int
    = ( ^ [A2: int] : ( if_int @ ( ord_less_int @ A2 @ zero_zero_int ) @ ( uminus_uminus_int @ A2 ) @ A2 ) ) ) ).

% abs_if
thf(fact_4124_abs__if,axiom,
    ( abs_abs_Code_integer
    = ( ^ [A2: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ A2 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ A2 ) @ A2 ) ) ) ).

% abs_if
thf(fact_4125_abs__if,axiom,
    ( abs_abs_rat
    = ( ^ [A2: rat] : ( if_rat @ ( ord_less_rat @ A2 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A2 ) @ A2 ) ) ) ).

% abs_if
thf(fact_4126_abs__diff__le__iff,axiom,
    ! [X: code_integer,A: code_integer,R: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X @ A ) ) @ R )
      = ( ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ A @ R ) @ X )
        & ( ord_le3102999989581377725nteger @ X @ ( plus_p5714425477246183910nteger @ A @ R ) ) ) ) ).

% abs_diff_le_iff
thf(fact_4127_abs__diff__le__iff,axiom,
    ! [X: rat,A: rat,R: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X @ A ) ) @ R )
      = ( ( ord_less_eq_rat @ ( minus_minus_rat @ A @ R ) @ X )
        & ( ord_less_eq_rat @ X @ ( plus_plus_rat @ A @ R ) ) ) ) ).

% abs_diff_le_iff
thf(fact_4128_abs__diff__le__iff,axiom,
    ! [X: int,A: int,R: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ X @ A ) ) @ R )
      = ( ( ord_less_eq_int @ ( minus_minus_int @ A @ R ) @ X )
        & ( ord_less_eq_int @ X @ ( plus_plus_int @ A @ R ) ) ) ) ).

% abs_diff_le_iff
thf(fact_4129_abs__diff__triangle__ineq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ ( plus_p5714425477246183910nteger @ C @ D ) ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ C ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B @ D ) ) ) ) ).

% abs_diff_triangle_ineq
thf(fact_4130_abs__diff__triangle__ineq,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ ( plus_plus_rat @ C @ D ) ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ ( minus_minus_rat @ A @ C ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ B @ D ) ) ) ) ).

% abs_diff_triangle_ineq
thf(fact_4131_abs__diff__triangle__ineq,axiom,
    ! [A: int,B: int,C: int,D: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ ( plus_plus_int @ C @ D ) ) ) @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ A @ C ) ) @ ( abs_abs_int @ ( minus_minus_int @ B @ D ) ) ) ) ).

% abs_diff_triangle_ineq
thf(fact_4132_abs__triangle__ineq4,axiom,
    ! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ).

% abs_triangle_ineq4
thf(fact_4133_abs__triangle__ineq4,axiom,
    ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ A @ B ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).

% abs_triangle_ineq4
thf(fact_4134_abs__triangle__ineq4,axiom,
    ! [A: int,B: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ A @ B ) ) @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).

% abs_triangle_ineq4
thf(fact_4135_abs__div__pos,axiom,
    ! [Y3: rat,X: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Y3 )
     => ( ( divide_divide_rat @ ( abs_abs_rat @ X ) @ Y3 )
        = ( abs_abs_rat @ ( divide_divide_rat @ X @ Y3 ) ) ) ) ).

% abs_div_pos
thf(fact_4136_abs__diff__less__iff,axiom,
    ! [X: code_integer,A: code_integer,R: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X @ A ) ) @ R )
      = ( ( ord_le6747313008572928689nteger @ ( minus_8373710615458151222nteger @ A @ R ) @ X )
        & ( ord_le6747313008572928689nteger @ X @ ( plus_p5714425477246183910nteger @ A @ R ) ) ) ) ).

% abs_diff_less_iff
thf(fact_4137_abs__diff__less__iff,axiom,
    ! [X: rat,A: rat,R: rat] :
      ( ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X @ A ) ) @ R )
      = ( ( ord_less_rat @ ( minus_minus_rat @ A @ R ) @ X )
        & ( ord_less_rat @ X @ ( plus_plus_rat @ A @ R ) ) ) ) ).

% abs_diff_less_iff
thf(fact_4138_abs__diff__less__iff,axiom,
    ! [X: int,A: int,R: int] :
      ( ( ord_less_int @ ( abs_abs_int @ ( minus_minus_int @ X @ A ) ) @ R )
      = ( ( ord_less_int @ ( minus_minus_int @ A @ R ) @ X )
        & ( ord_less_int @ X @ ( plus_plus_int @ A @ R ) ) ) ) ).

% abs_diff_less_iff
thf(fact_4139_div__noneq__sgn__abs,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero_int )
     => ( ( ( sgn_sgn_int @ K )
         != ( sgn_sgn_int @ L ) )
       => ( ( divide_divide_int @ K @ L )
          = ( minus_minus_int @ ( uminus_uminus_int @ ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) ) )
            @ ( zero_n2684676970156552555ol_int
              @ ~ ( dvd_dvd_int @ L @ K ) ) ) ) ) ) ).

% div_noneq_sgn_abs
thf(fact_4140_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_4141_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_4142_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_4143_zabs__def,axiom,
    ( abs_abs_int
    = ( ^ [I4: int] : ( if_int @ ( ord_less_int @ I4 @ zero_zero_int ) @ ( uminus_uminus_int @ I4 ) @ I4 ) ) ) ).

% zabs_def
thf(fact_4144_dvd__imp__le__int,axiom,
    ! [I2: int,D: int] :
      ( ( I2 != zero_zero_int )
     => ( ( dvd_dvd_int @ D @ I2 )
       => ( ord_less_eq_int @ ( abs_abs_int @ D ) @ ( abs_abs_int @ I2 ) ) ) ) ).

% dvd_imp_le_int
thf(fact_4145_nat__abs__mult__distrib,axiom,
    ! [W2: int,Z2: int] :
      ( ( nat2 @ ( abs_abs_int @ ( times_times_int @ W2 @ Z2 ) ) )
      = ( times_times_nat @ ( nat2 @ ( abs_abs_int @ W2 ) ) @ ( nat2 @ ( abs_abs_int @ Z2 ) ) ) ) ).

% nat_abs_mult_distrib
thf(fact_4146_abs__mod__less,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero_int )
     => ( ord_less_int @ ( abs_abs_int @ ( modulo_modulo_int @ K @ L ) ) @ ( abs_abs_int @ L ) ) ) ).

% abs_mod_less
thf(fact_4147_pochhammer__nonneg,axiom,
    ! [X: code_natural,N: nat] :
      ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ X )
     => ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ ( comm_s7406298101470620109atural @ X @ N ) ) ) ).

% pochhammer_nonneg
thf(fact_4148_pochhammer__nonneg,axiom,
    ! [X: code_integer,N: nat] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ X )
     => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( comm_s8582702949713902594nteger @ X @ N ) ) ) ).

% pochhammer_nonneg
thf(fact_4149_pochhammer__nonneg,axiom,
    ! [X: rat,N: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ X )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( comm_s4028243227959126397er_rat @ X @ N ) ) ) ).

% pochhammer_nonneg
thf(fact_4150_pochhammer__nonneg,axiom,
    ! [X: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ X )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( comm_s4663373288045622133er_nat @ X @ N ) ) ) ).

% pochhammer_nonneg
thf(fact_4151_pochhammer__nonneg,axiom,
    ! [X: int,N: nat] :
      ( ( ord_less_int @ zero_zero_int @ X )
     => ( ord_less_eq_int @ zero_zero_int @ ( comm_s4660882817536571857er_int @ X @ N ) ) ) ).

% pochhammer_nonneg
thf(fact_4152_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_4153_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_4154_pochhammer__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( comm_s8582702949713902594nteger @ zero_z3403309356797280102nteger @ N )
          = one_one_Code_integer ) )
      & ( ( N != zero_zero_nat )
       => ( ( comm_s8582702949713902594nteger @ zero_z3403309356797280102nteger @ N )
          = zero_z3403309356797280102nteger ) ) ) ).

% pochhammer_0_left
thf(fact_4155_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_4156_pochhammer__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( comm_s7406298101470620109atural @ zero_z2226904508553997617atural @ N )
          = one_one_Code_natural ) )
      & ( ( N != zero_zero_nat )
       => ( ( comm_s7406298101470620109atural @ zero_z2226904508553997617atural @ N )
          = zero_z2226904508553997617atural ) ) ) ).

% pochhammer_0_left
thf(fact_4157_pochhammer__product_H,axiom,
    ! [Z2: int,N: nat,M: nat] :
      ( ( comm_s4660882817536571857er_int @ Z2 @ ( plus_plus_nat @ N @ M ) )
      = ( times_times_int @ ( comm_s4660882817536571857er_int @ Z2 @ N ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ Z2 @ ( semiri1314217659103216013at_int @ N ) ) @ M ) ) ) ).

% pochhammer_product'
thf(fact_4158_pochhammer__product_H,axiom,
    ! [Z2: code_integer,N: nat,M: nat] :
      ( ( comm_s8582702949713902594nteger @ Z2 @ ( plus_plus_nat @ N @ M ) )
      = ( times_3573771949741848930nteger @ ( comm_s8582702949713902594nteger @ Z2 @ N ) @ ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ Z2 @ ( semiri4939895301339042750nteger @ N ) ) @ M ) ) ) ).

% pochhammer_product'
thf(fact_4159_pochhammer__product_H,axiom,
    ! [Z2: nat,N: nat,M: nat] :
      ( ( comm_s4663373288045622133er_nat @ Z2 @ ( plus_plus_nat @ N @ M ) )
      = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ Z2 @ N ) @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ Z2 @ ( semiri1316708129612266289at_nat @ N ) ) @ M ) ) ) ).

% pochhammer_product'
thf(fact_4160_pochhammer__product_H,axiom,
    ! [Z2: rat,N: nat,M: nat] :
      ( ( comm_s4028243227959126397er_rat @ Z2 @ ( plus_plus_nat @ N @ M ) )
      = ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z2 @ N ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z2 @ ( semiri681578069525770553at_rat @ N ) ) @ M ) ) ) ).

% pochhammer_product'
thf(fact_4161_pochhammer__product_H,axiom,
    ! [Z2: code_natural,N: nat,M: nat] :
      ( ( comm_s7406298101470620109atural @ Z2 @ ( plus_plus_nat @ N @ M ) )
      = ( times_2397367101498566445atural @ ( comm_s7406298101470620109atural @ Z2 @ N ) @ ( comm_s7406298101470620109atural @ ( plus_p4538020629002901425atural @ Z2 @ ( semiri3763490453095760265atural @ N ) ) @ M ) ) ) ).

% pochhammer_product'
thf(fact_4162_abs__add__one__gt__zero,axiom,
    ! [X: code_integer] : ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( abs_abs_Code_integer @ X ) ) ) ).

% abs_add_one_gt_zero
thf(fact_4163_abs__add__one__gt__zero,axiom,
    ! [X: rat] : ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ ( abs_abs_rat @ X ) ) ) ).

% abs_add_one_gt_zero
thf(fact_4164_abs__add__one__gt__zero,axiom,
    ! [X: int] : ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ ( abs_abs_int @ X ) ) ) ).

% abs_add_one_gt_zero
thf(fact_4165_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_4166_div__sgn__abs__cancel,axiom,
    ! [V: int,K: int,L: 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 @ L ) ) )
        = ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) ) ) ) ).

% div_sgn_abs_cancel
thf(fact_4167_div__abs__eq__div__nat,axiom,
    ! [K: int,L: int] :
      ( ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) )
      = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) ) ) ).

% div_abs_eq_div_nat
thf(fact_4168_mod__abs__eq__div__nat,axiom,
    ! [K: int,L: int] :
      ( ( modulo_modulo_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) )
      = ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) ) ) ).

% mod_abs_eq_div_nat
thf(fact_4169_pochhammer__rec,axiom,
    ! [A: code_natural,N: nat] :
      ( ( comm_s7406298101470620109atural @ A @ ( suc @ N ) )
      = ( times_2397367101498566445atural @ A @ ( comm_s7406298101470620109atural @ ( plus_p4538020629002901425atural @ A @ one_one_Code_natural ) @ N ) ) ) ).

% pochhammer_rec
thf(fact_4170_pochhammer__rec,axiom,
    ! [A: code_integer,N: nat] :
      ( ( comm_s8582702949713902594nteger @ A @ ( suc @ N ) )
      = ( times_3573771949741848930nteger @ A @ ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) @ N ) ) ) ).

% pochhammer_rec
thf(fact_4171_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_4172_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_4173_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_4174_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_4175_pochhammer__Suc,axiom,
    ! [A: code_integer,N: nat] :
      ( ( comm_s8582702949713902594nteger @ A @ ( suc @ N ) )
      = ( times_3573771949741848930nteger @ ( comm_s8582702949713902594nteger @ A @ N ) @ ( plus_p5714425477246183910nteger @ A @ ( semiri4939895301339042750nteger @ N ) ) ) ) ).

% pochhammer_Suc
thf(fact_4176_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_4177_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_4178_pochhammer__Suc,axiom,
    ! [A: code_natural,N: nat] :
      ( ( comm_s7406298101470620109atural @ A @ ( suc @ N ) )
      = ( times_2397367101498566445atural @ ( comm_s7406298101470620109atural @ A @ N ) @ ( plus_p4538020629002901425atural @ A @ ( semiri3763490453095760265atural @ N ) ) ) ) ).

% pochhammer_Suc
thf(fact_4179_pochhammer__rec_H,axiom,
    ! [Z2: int,N: nat] :
      ( ( comm_s4660882817536571857er_int @ Z2 @ ( suc @ N ) )
      = ( times_times_int @ ( plus_plus_int @ Z2 @ ( semiri1314217659103216013at_int @ N ) ) @ ( comm_s4660882817536571857er_int @ Z2 @ N ) ) ) ).

% pochhammer_rec'
thf(fact_4180_pochhammer__rec_H,axiom,
    ! [Z2: code_integer,N: nat] :
      ( ( comm_s8582702949713902594nteger @ Z2 @ ( suc @ N ) )
      = ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ Z2 @ ( semiri4939895301339042750nteger @ N ) ) @ ( comm_s8582702949713902594nteger @ Z2 @ N ) ) ) ).

% pochhammer_rec'
thf(fact_4181_pochhammer__rec_H,axiom,
    ! [Z2: nat,N: nat] :
      ( ( comm_s4663373288045622133er_nat @ Z2 @ ( suc @ N ) )
      = ( times_times_nat @ ( plus_plus_nat @ Z2 @ ( semiri1316708129612266289at_nat @ N ) ) @ ( comm_s4663373288045622133er_nat @ Z2 @ N ) ) ) ).

% pochhammer_rec'
thf(fact_4182_pochhammer__rec_H,axiom,
    ! [Z2: rat,N: nat] :
      ( ( comm_s4028243227959126397er_rat @ Z2 @ ( suc @ N ) )
      = ( times_times_rat @ ( plus_plus_rat @ Z2 @ ( semiri681578069525770553at_rat @ N ) ) @ ( comm_s4028243227959126397er_rat @ Z2 @ N ) ) ) ).

% pochhammer_rec'
thf(fact_4183_pochhammer__rec_H,axiom,
    ! [Z2: code_natural,N: nat] :
      ( ( comm_s7406298101470620109atural @ Z2 @ ( suc @ N ) )
      = ( times_2397367101498566445atural @ ( plus_p4538020629002901425atural @ Z2 @ ( semiri3763490453095760265atural @ N ) ) @ ( comm_s7406298101470620109atural @ Z2 @ N ) ) ) ).

% pochhammer_rec'
thf(fact_4184_div__dvd__sgn__abs,axiom,
    ! [L: int,K: int] :
      ( ( dvd_dvd_int @ L @ K )
     => ( ( divide_divide_int @ K @ L )
        = ( times_times_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( sgn_sgn_int @ L ) ) @ ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) ) ) ) ) ).

% div_dvd_sgn_abs
thf(fact_4185_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_4186_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_4187_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_4188_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_4189_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_4190_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_4191_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_4192_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_4193_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_4194_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_4195_modulo__int__def,axiom,
    ( modulo_modulo_int
    = ( ^ [K3: int,L3: int] :
          ( if_int @ ( L3 = zero_zero_int ) @ K3
          @ ( if_int
            @ ( ( sgn_sgn_int @ K3 )
              = ( sgn_sgn_int @ L3 ) )
            @ ( times_times_int @ ( sgn_sgn_int @ L3 ) @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K3 ) ) @ ( nat2 @ ( abs_abs_int @ L3 ) ) ) ) )
            @ ( times_times_int @ ( sgn_sgn_int @ L3 )
              @ ( minus_minus_int
                @ ( times_times_int @ ( abs_abs_int @ L3 )
                  @ ( zero_n2684676970156552555ol_int
                    @ ~ ( dvd_dvd_int @ L3 @ K3 ) ) )
                @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K3 ) ) @ ( nat2 @ ( abs_abs_int @ L3 ) ) ) ) ) ) ) ) ) ) ).

% modulo_int_def
thf(fact_4196_incr__lemma,axiom,
    ! [D: int,Z2: int,X: int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ord_less_int @ Z2 @ ( plus_plus_int @ X @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Z2 ) ) @ one_one_int ) @ D ) ) ) ) ).

% incr_lemma
thf(fact_4197_decr__lemma,axiom,
    ! [D: int,X: int,Z2: int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ord_less_int @ ( minus_minus_int @ X @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Z2 ) ) @ one_one_int ) @ D ) ) @ Z2 ) ) ).

% decr_lemma
thf(fact_4198_nat__abs__int__diff,axiom,
    ! [A: nat,B: nat] :
      ( ( ( ord_less_eq_nat @ A @ B )
       => ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) )
          = ( minus_minus_nat @ B @ A ) ) )
      & ( ~ ( ord_less_eq_nat @ A @ B )
       => ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) )
          = ( minus_minus_nat @ A @ B ) ) ) ) ).

% nat_abs_int_diff
thf(fact_4199_pochhammer__product,axiom,
    ! [M: nat,N: nat,Z2: int] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( comm_s4660882817536571857er_int @ Z2 @ N )
        = ( times_times_int @ ( comm_s4660882817536571857er_int @ Z2 @ M ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ Z2 @ ( semiri1314217659103216013at_int @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).

% pochhammer_product
thf(fact_4200_pochhammer__product,axiom,
    ! [M: nat,N: nat,Z2: code_integer] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( comm_s8582702949713902594nteger @ Z2 @ N )
        = ( times_3573771949741848930nteger @ ( comm_s8582702949713902594nteger @ Z2 @ M ) @ ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ Z2 @ ( semiri4939895301339042750nteger @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).

% pochhammer_product
thf(fact_4201_pochhammer__product,axiom,
    ! [M: nat,N: nat,Z2: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( comm_s4663373288045622133er_nat @ Z2 @ N )
        = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ Z2 @ M ) @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ Z2 @ ( semiri1316708129612266289at_nat @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).

% pochhammer_product
thf(fact_4202_pochhammer__product,axiom,
    ! [M: nat,N: nat,Z2: rat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( comm_s4028243227959126397er_rat @ Z2 @ N )
        = ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z2 @ M ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z2 @ ( semiri681578069525770553at_rat @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).

% pochhammer_product
thf(fact_4203_pochhammer__product,axiom,
    ! [M: nat,N: nat,Z2: code_natural] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( comm_s7406298101470620109atural @ Z2 @ N )
        = ( times_2397367101498566445atural @ ( comm_s7406298101470620109atural @ Z2 @ M ) @ ( comm_s7406298101470620109atural @ ( plus_p4538020629002901425atural @ Z2 @ ( semiri3763490453095760265atural @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).

% pochhammer_product
thf(fact_4204_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_4205_pochhammer__absorb__comp,axiom,
    ! [R: int,K: nat] :
      ( ( times_times_int @ ( minus_minus_int @ R @ ( semiri1314217659103216013at_int @ K ) ) @ ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ R ) @ K ) )
      = ( times_times_int @ R @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( uminus_uminus_int @ R ) @ one_one_int ) @ K ) ) ) ).

% pochhammer_absorb_comp
thf(fact_4206_pochhammer__absorb__comp,axiom,
    ! [R: code_integer,K: nat] :
      ( ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ R @ ( semiri4939895301339042750nteger @ K ) ) @ ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ R ) @ K ) )
      = ( times_3573771949741848930nteger @ R @ ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ R ) @ one_one_Code_integer ) @ K ) ) ) ).

% pochhammer_absorb_comp
thf(fact_4207_pochhammer__absorb__comp,axiom,
    ! [R: rat,K: nat] :
      ( ( times_times_rat @ ( minus_minus_rat @ R @ ( semiri681578069525770553at_rat @ K ) ) @ ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ R ) @ K ) )
      = ( times_times_rat @ R @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ R ) @ one_one_rat ) @ K ) ) ) ).

% pochhammer_absorb_comp
thf(fact_4208_add__0__iff,axiom,
    ! [B: nat,A: nat] :
      ( ( B
        = ( plus_plus_nat @ B @ A ) )
      = ( A = zero_zero_nat ) ) ).

% add_0_iff
thf(fact_4209_add__0__iff,axiom,
    ! [B: int,A: int] :
      ( ( B
        = ( plus_plus_int @ B @ A ) )
      = ( A = zero_zero_int ) ) ).

% add_0_iff
thf(fact_4210_add__0__iff,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( B
        = ( plus_p5714425477246183910nteger @ B @ A ) )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% add_0_iff
thf(fact_4211_add__0__iff,axiom,
    ! [B: rat,A: rat] :
      ( ( B
        = ( plus_plus_rat @ B @ A ) )
      = ( A = zero_zero_rat ) ) ).

% add_0_iff
thf(fact_4212_crossproduct__noteq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
      ( ( ( A != B )
        & ( C != D ) )
      = ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ D ) )
       != ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ D ) @ ( times_3573771949741848930nteger @ B @ C ) ) ) ) ).

% crossproduct_noteq
thf(fact_4213_crossproduct__noteq,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ( A != B )
        & ( C != D ) )
      = ( ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) )
       != ( plus_plus_rat @ ( times_times_rat @ A @ D ) @ ( times_times_rat @ B @ C ) ) ) ) ).

% crossproduct_noteq
thf(fact_4214_crossproduct__noteq,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ( A != B )
        & ( C != D ) )
      = ( ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) )
       != ( plus_plus_nat @ ( times_times_nat @ A @ D ) @ ( times_times_nat @ B @ C ) ) ) ) ).

% crossproduct_noteq
thf(fact_4215_crossproduct__noteq,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ( A != B )
        & ( C != D ) )
      = ( ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) )
       != ( plus_plus_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ B @ C ) ) ) ) ).

% crossproduct_noteq
thf(fact_4216_crossproduct__eq,axiom,
    ! [W2: code_integer,Y3: code_integer,X: code_integer,Z2: code_integer] :
      ( ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ W2 @ Y3 ) @ ( times_3573771949741848930nteger @ X @ Z2 ) )
        = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ W2 @ Z2 ) @ ( times_3573771949741848930nteger @ X @ Y3 ) ) )
      = ( ( W2 = X )
        | ( Y3 = Z2 ) ) ) ).

% crossproduct_eq
thf(fact_4217_crossproduct__eq,axiom,
    ! [W2: rat,Y3: rat,X: rat,Z2: rat] :
      ( ( ( plus_plus_rat @ ( times_times_rat @ W2 @ Y3 ) @ ( times_times_rat @ X @ Z2 ) )
        = ( plus_plus_rat @ ( times_times_rat @ W2 @ Z2 ) @ ( times_times_rat @ X @ Y3 ) ) )
      = ( ( W2 = X )
        | ( Y3 = Z2 ) ) ) ).

% crossproduct_eq
thf(fact_4218_crossproduct__eq,axiom,
    ! [W2: nat,Y3: nat,X: nat,Z2: nat] :
      ( ( ( plus_plus_nat @ ( times_times_nat @ W2 @ Y3 ) @ ( times_times_nat @ X @ Z2 ) )
        = ( plus_plus_nat @ ( times_times_nat @ W2 @ Z2 ) @ ( times_times_nat @ X @ Y3 ) ) )
      = ( ( W2 = X )
        | ( Y3 = Z2 ) ) ) ).

% crossproduct_eq
thf(fact_4219_crossproduct__eq,axiom,
    ! [W2: int,Y3: int,X: int,Z2: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ W2 @ Y3 ) @ ( times_times_int @ X @ Z2 ) )
        = ( plus_plus_int @ ( times_times_int @ W2 @ Z2 ) @ ( times_times_int @ X @ Y3 ) ) )
      = ( ( W2 = X )
        | ( Y3 = Z2 ) ) ) ).

% crossproduct_eq
thf(fact_4220_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_4221_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_4222_gbinomial__pochhammer,axiom,
    ( gbinomial_rat
    = ( ^ [A2: 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 @ A2 ) @ K3 ) ) @ ( semiri773545260158071498ct_rat @ K3 ) ) ) ) ).

% gbinomial_pochhammer
thf(fact_4223_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_4224_pochhammer__minus,axiom,
    ! [B: int,K: nat] :
      ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ B ) @ K )
      = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ K ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( minus_minus_int @ B @ ( semiri1314217659103216013at_int @ K ) ) @ one_one_int ) @ K ) ) ) ).

% pochhammer_minus
thf(fact_4225_pochhammer__minus,axiom,
    ! [B: code_integer,K: nat] :
      ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ B ) @ K )
      = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ K ) @ ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ B @ ( semiri4939895301339042750nteger @ K ) ) @ one_one_Code_integer ) @ K ) ) ) ).

% pochhammer_minus
thf(fact_4226_pochhammer__minus,axiom,
    ! [B: rat,K: nat] :
      ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ B ) @ K )
      = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ B @ ( semiri681578069525770553at_rat @ K ) ) @ one_one_rat ) @ K ) ) ) ).

% pochhammer_minus
thf(fact_4227_pochhammer__minus_H,axiom,
    ! [B: int,K: nat] :
      ( ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( minus_minus_int @ B @ ( 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 @ B ) @ K ) ) ) ).

% pochhammer_minus'
thf(fact_4228_pochhammer__minus_H,axiom,
    ! [B: code_integer,K: nat] :
      ( ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ B @ ( semiri4939895301339042750nteger @ K ) ) @ one_one_Code_integer ) @ K )
      = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ K ) @ ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ B ) @ K ) ) ) ).

% pochhammer_minus'
thf(fact_4229_pochhammer__minus_H,axiom,
    ! [B: rat,K: nat] :
      ( ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ B @ ( 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 @ B ) @ K ) ) ) ).

% pochhammer_minus'
thf(fact_4230_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_4231_triangle__Suc,axiom,
    ! [N: nat] :
      ( ( nat_triangle @ ( suc @ N ) )
      = ( plus_plus_nat @ ( nat_triangle @ N ) @ ( suc @ N ) ) ) ).

% triangle_Suc
thf(fact_4232_frac__unique__iff,axiom,
    ! [X: rat,A: rat] :
      ( ( ( archimedean_frac_rat @ X )
        = A )
      = ( ( member_rat @ ( minus_minus_rat @ X @ 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_4233_power__Suc__0,axiom,
    ! [N: nat] :
      ( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N )
      = ( suc @ zero_zero_nat ) ) ).

% power_Suc_0
thf(fact_4234_nat__power__eq__Suc__0__iff,axiom,
    ! [X: nat,M: nat] :
      ( ( ( power_power_nat @ X @ M )
        = ( suc @ zero_zero_nat ) )
      = ( ( M = zero_zero_nat )
        | ( X
          = ( suc @ zero_zero_nat ) ) ) ) ).

% nat_power_eq_Suc_0_iff
thf(fact_4235_power__one,axiom,
    ! [N: nat] :
      ( ( power_8256067586552552935nteger @ one_one_Code_integer @ N )
      = one_one_Code_integer ) ).

% power_one
thf(fact_4236_power__one,axiom,
    ! [N: nat] :
      ( ( power_power_rat @ one_one_rat @ N )
      = one_one_rat ) ).

% power_one
thf(fact_4237_power__one,axiom,
    ! [N: nat] :
      ( ( power_7079662738309270450atural @ one_one_Code_natural @ N )
      = one_one_Code_natural ) ).

% power_one
thf(fact_4238_power__one,axiom,
    ! [N: nat] :
      ( ( power_power_int @ one_one_int @ N )
      = one_one_int ) ).

% power_one
thf(fact_4239_power__one,axiom,
    ! [N: nat] :
      ( ( power_power_nat @ one_one_nat @ N )
      = one_one_nat ) ).

% power_one
thf(fact_4240_nat__zero__less__power__iff,axiom,
    ! [X: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X )
        | ( N = zero_zero_nat ) ) ) ).

% nat_zero_less_power_iff
thf(fact_4241_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_4242_of__nat__power,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri4939895301339042750nteger @ ( power_power_nat @ M @ N ) )
      = ( power_8256067586552552935nteger @ ( semiri4939895301339042750nteger @ M ) @ N ) ) ).

% of_nat_power
thf(fact_4243_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_4244_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_4245_of__nat__power,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri3763490453095760265atural @ ( power_power_nat @ M @ N ) )
      = ( power_7079662738309270450atural @ ( semiri3763490453095760265atural @ M ) @ N ) ) ).

% of_nat_power
thf(fact_4246_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X: nat] :
      ( ( ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W2 )
        = ( semiri1314217659103216013at_int @ X ) )
      = ( ( power_power_nat @ B @ W2 )
        = X ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
thf(fact_4247_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X: nat] :
      ( ( ( power_8256067586552552935nteger @ ( semiri4939895301339042750nteger @ B ) @ W2 )
        = ( semiri4939895301339042750nteger @ X ) )
      = ( ( power_power_nat @ B @ W2 )
        = X ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
thf(fact_4248_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X: nat] :
      ( ( ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W2 )
        = ( semiri1316708129612266289at_nat @ X ) )
      = ( ( power_power_nat @ B @ W2 )
        = X ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
thf(fact_4249_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X: nat] :
      ( ( ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W2 )
        = ( semiri681578069525770553at_rat @ X ) )
      = ( ( power_power_nat @ B @ W2 )
        = X ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
thf(fact_4250_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X: nat] :
      ( ( ( power_7079662738309270450atural @ ( semiri3763490453095760265atural @ B ) @ W2 )
        = ( semiri3763490453095760265atural @ X ) )
      = ( ( power_power_nat @ B @ W2 )
        = X ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
thf(fact_4251_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W2: nat] :
      ( ( ( semiri1314217659103216013at_int @ X )
        = ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W2 ) )
      = ( X
        = ( power_power_nat @ B @ W2 ) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
thf(fact_4252_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W2: nat] :
      ( ( ( semiri4939895301339042750nteger @ X )
        = ( power_8256067586552552935nteger @ ( semiri4939895301339042750nteger @ B ) @ W2 ) )
      = ( X
        = ( power_power_nat @ B @ W2 ) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
thf(fact_4253_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W2: nat] :
      ( ( ( semiri1316708129612266289at_nat @ X )
        = ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W2 ) )
      = ( X
        = ( power_power_nat @ B @ W2 ) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
thf(fact_4254_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W2: nat] :
      ( ( ( semiri681578069525770553at_rat @ X )
        = ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W2 ) )
      = ( X
        = ( power_power_nat @ B @ W2 ) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
thf(fact_4255_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W2: nat] :
      ( ( ( semiri3763490453095760265atural @ X )
        = ( power_7079662738309270450atural @ ( semiri3763490453095760265atural @ B ) @ W2 ) )
      = ( X
        = ( power_power_nat @ B @ W2 ) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
thf(fact_4256_power__one__right,axiom,
    ! [A: int] :
      ( ( power_power_int @ A @ one_one_nat )
      = A ) ).

% power_one_right
thf(fact_4257_power__one__right,axiom,
    ! [A: nat] :
      ( ( power_power_nat @ A @ one_one_nat )
      = A ) ).

% power_one_right
thf(fact_4258_triangle__0,axiom,
    ( ( nat_triangle @ zero_zero_nat )
    = zero_zero_nat ) ).

% triangle_0
thf(fact_4259_power__inject__exp,axiom,
    ! [A: code_natural,M: nat,N: nat] :
      ( ( ord_le5570908160329646204atural @ one_one_Code_natural @ A )
     => ( ( ( power_7079662738309270450atural @ A @ M )
          = ( power_7079662738309270450atural @ A @ N ) )
        = ( M = N ) ) ) ).

% power_inject_exp
thf(fact_4260_power__inject__exp,axiom,
    ! [A: code_integer,M: nat,N: nat] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A )
     => ( ( ( power_8256067586552552935nteger @ A @ M )
          = ( power_8256067586552552935nteger @ A @ N ) )
        = ( M = N ) ) ) ).

% power_inject_exp
thf(fact_4261_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_4262_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_4263_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_4264_power__0__Suc,axiom,
    ! [N: nat] :
      ( ( power_8256067586552552935nteger @ zero_z3403309356797280102nteger @ ( suc @ N ) )
      = zero_z3403309356797280102nteger ) ).

% power_0_Suc
thf(fact_4265_power__0__Suc,axiom,
    ! [N: nat] :
      ( ( power_power_rat @ zero_zero_rat @ ( suc @ N ) )
      = zero_zero_rat ) ).

% power_0_Suc
thf(fact_4266_power__0__Suc,axiom,
    ! [N: nat] :
      ( ( power_7079662738309270450atural @ zero_z2226904508553997617atural @ ( suc @ N ) )
      = zero_z2226904508553997617atural ) ).

% power_0_Suc
thf(fact_4267_power__0__Suc,axiom,
    ! [N: nat] :
      ( ( power_power_int @ zero_zero_int @ ( suc @ N ) )
      = zero_zero_int ) ).

% power_0_Suc
thf(fact_4268_power__0__Suc,axiom,
    ! [N: nat] :
      ( ( power_power_nat @ zero_zero_nat @ ( suc @ N ) )
      = zero_zero_nat ) ).

% power_0_Suc
thf(fact_4269_power__Suc0__right,axiom,
    ! [A: int] :
      ( ( power_power_int @ A @ ( suc @ zero_zero_nat ) )
      = A ) ).

% power_Suc0_right
thf(fact_4270_power__Suc0__right,axiom,
    ! [A: nat] :
      ( ( power_power_nat @ A @ ( suc @ zero_zero_nat ) )
      = A ) ).

% power_Suc0_right
thf(fact_4271_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_4272_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_4273_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_4274_frac__eq__0__iff,axiom,
    ! [X: rat] :
      ( ( ( archimedean_frac_rat @ X )
        = zero_zero_rat )
      = ( member_rat @ X @ ring_1_Ints_rat ) ) ).

% frac_eq_0_iff
thf(fact_4275_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_4276_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_4277_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_4278_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_4279_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_4280_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_4281_power__strict__increasing__iff,axiom,
    ! [B: code_natural,X: nat,Y3: nat] :
      ( ( ord_le5570908160329646204atural @ one_one_Code_natural @ B )
     => ( ( ord_le5570908160329646204atural @ ( power_7079662738309270450atural @ B @ X ) @ ( power_7079662738309270450atural @ B @ Y3 ) )
        = ( ord_less_nat @ X @ Y3 ) ) ) ).

% power_strict_increasing_iff
thf(fact_4282_power__strict__increasing__iff,axiom,
    ! [B: code_integer,X: nat,Y3: nat] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ B )
     => ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ B @ X ) @ ( power_8256067586552552935nteger @ B @ Y3 ) )
        = ( ord_less_nat @ X @ Y3 ) ) ) ).

% power_strict_increasing_iff
thf(fact_4283_power__strict__increasing__iff,axiom,
    ! [B: rat,X: nat,Y3: nat] :
      ( ( ord_less_rat @ one_one_rat @ B )
     => ( ( ord_less_rat @ ( power_power_rat @ B @ X ) @ ( power_power_rat @ B @ Y3 ) )
        = ( ord_less_nat @ X @ Y3 ) ) ) ).

% power_strict_increasing_iff
thf(fact_4284_power__strict__increasing__iff,axiom,
    ! [B: nat,X: nat,Y3: nat] :
      ( ( ord_less_nat @ one_one_nat @ B )
     => ( ( ord_less_nat @ ( power_power_nat @ B @ X ) @ ( power_power_nat @ B @ Y3 ) )
        = ( ord_less_nat @ X @ Y3 ) ) ) ).

% power_strict_increasing_iff
thf(fact_4285_power__strict__increasing__iff,axiom,
    ! [B: int,X: nat,Y3: nat] :
      ( ( ord_less_int @ one_one_int @ B )
     => ( ( ord_less_int @ ( power_power_int @ B @ X ) @ ( power_power_int @ B @ Y3 ) )
        = ( ord_less_nat @ X @ Y3 ) ) ) ).

% power_strict_increasing_iff
thf(fact_4286_power__eq__0__iff,axiom,
    ! [A: code_integer,N: nat] :
      ( ( ( power_8256067586552552935nteger @ A @ N )
        = zero_z3403309356797280102nteger )
      = ( ( A = zero_z3403309356797280102nteger )
        & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% power_eq_0_iff
thf(fact_4287_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_4288_power__eq__0__iff,axiom,
    ! [A: code_natural,N: nat] :
      ( ( ( power_7079662738309270450atural @ A @ N )
        = zero_z2226904508553997617atural )
      = ( ( A = zero_z2226904508553997617atural )
        & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% power_eq_0_iff
thf(fact_4289_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_4290_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_4291_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W2: nat] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W2 ) )
      = ( ord_less_nat @ X @ ( power_power_nat @ B @ W2 ) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
thf(fact_4292_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W2: nat] :
      ( ( ord_le6747313008572928689nteger @ ( semiri4939895301339042750nteger @ X ) @ ( power_8256067586552552935nteger @ ( semiri4939895301339042750nteger @ B ) @ W2 ) )
      = ( ord_less_nat @ X @ ( power_power_nat @ B @ W2 ) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
thf(fact_4293_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W2: nat] :
      ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W2 ) )
      = ( ord_less_nat @ X @ ( power_power_nat @ B @ W2 ) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
thf(fact_4294_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W2: nat] :
      ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W2 ) )
      = ( ord_less_nat @ X @ ( power_power_nat @ B @ W2 ) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
thf(fact_4295_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W2: nat] :
      ( ( ord_le5570908160329646204atural @ ( semiri3763490453095760265atural @ X ) @ ( power_7079662738309270450atural @ ( semiri3763490453095760265atural @ B ) @ W2 ) )
      = ( ord_less_nat @ X @ ( power_power_nat @ B @ W2 ) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
thf(fact_4296_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X: nat] :
      ( ( ord_less_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W2 ) @ ( semiri1314217659103216013at_int @ X ) )
      = ( ord_less_nat @ ( power_power_nat @ B @ W2 ) @ X ) ) ).

% of_nat_less_of_nat_power_cancel_iff
thf(fact_4297_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X: nat] :
      ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ ( semiri4939895301339042750nteger @ B ) @ W2 ) @ ( semiri4939895301339042750nteger @ X ) )
      = ( ord_less_nat @ ( power_power_nat @ B @ W2 ) @ X ) ) ).

% of_nat_less_of_nat_power_cancel_iff
thf(fact_4298_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W2 ) @ ( semiri1316708129612266289at_nat @ X ) )
      = ( ord_less_nat @ ( power_power_nat @ B @ W2 ) @ X ) ) ).

% of_nat_less_of_nat_power_cancel_iff
thf(fact_4299_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X: nat] :
      ( ( ord_less_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W2 ) @ ( semiri681578069525770553at_rat @ X ) )
      = ( ord_less_nat @ ( power_power_nat @ B @ W2 ) @ X ) ) ).

% of_nat_less_of_nat_power_cancel_iff
thf(fact_4300_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X: nat] :
      ( ( ord_le5570908160329646204atural @ ( power_7079662738309270450atural @ ( semiri3763490453095760265atural @ B ) @ W2 ) @ ( semiri3763490453095760265atural @ X ) )
      = ( ord_less_nat @ ( power_power_nat @ B @ W2 ) @ X ) ) ).

% of_nat_less_of_nat_power_cancel_iff
thf(fact_4301_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W2: nat] :
      ( ( ord_le3102999989581377725nteger @ ( semiri4939895301339042750nteger @ X ) @ ( power_8256067586552552935nteger @ ( semiri4939895301339042750nteger @ B ) @ W2 ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W2 ) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
thf(fact_4302_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W2: nat] :
      ( ( ord_le1926595141338095240atural @ ( semiri3763490453095760265atural @ X ) @ ( power_7079662738309270450atural @ ( semiri3763490453095760265atural @ B ) @ W2 ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W2 ) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
thf(fact_4303_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W2: nat] :
      ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W2 ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W2 ) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
thf(fact_4304_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W2: nat] :
      ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W2 ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W2 ) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
thf(fact_4305_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W2: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W2 ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W2 ) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
thf(fact_4306_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X: nat] :
      ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ ( semiri4939895301339042750nteger @ B ) @ W2 ) @ ( semiri4939895301339042750nteger @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ B @ W2 ) @ X ) ) ).

% of_nat_le_of_nat_power_cancel_iff
thf(fact_4307_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X: nat] :
      ( ( ord_le1926595141338095240atural @ ( power_7079662738309270450atural @ ( semiri3763490453095760265atural @ B ) @ W2 ) @ ( semiri3763490453095760265atural @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ B @ W2 ) @ X ) ) ).

% of_nat_le_of_nat_power_cancel_iff
thf(fact_4308_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X: nat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W2 ) @ ( semiri681578069525770553at_rat @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ B @ W2 ) @ X ) ) ).

% of_nat_le_of_nat_power_cancel_iff
thf(fact_4309_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W2 ) @ ( semiri1316708129612266289at_nat @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ B @ W2 ) @ X ) ) ).

% of_nat_le_of_nat_power_cancel_iff
thf(fact_4310_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X: nat] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W2 ) @ ( semiri1314217659103216013at_int @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ B @ W2 ) @ X ) ) ).

% of_nat_le_of_nat_power_cancel_iff
thf(fact_4311_pow__divides__pow__iff,axiom,
    ! [N: nat,A: int,B: int] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
        = ( dvd_dvd_int @ A @ B ) ) ) ).

% pow_divides_pow_iff
thf(fact_4312_pow__divides__pow__iff,axiom,
    ! [N: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
        = ( dvd_dvd_nat @ A @ B ) ) ) ).

% pow_divides_pow_iff
thf(fact_4313_frac__gt__0__iff,axiom,
    ! [X: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( archimedean_frac_rat @ X ) )
      = ( ~ ( member_rat @ X @ ring_1_Ints_rat ) ) ) ).

% frac_gt_0_iff
thf(fact_4314_power__strict__decreasing__iff,axiom,
    ! [B: code_natural,M: nat,N: nat] :
      ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ B )
     => ( ( ord_le5570908160329646204atural @ B @ one_one_Code_natural )
       => ( ( ord_le5570908160329646204atural @ ( power_7079662738309270450atural @ B @ M ) @ ( power_7079662738309270450atural @ B @ N ) )
          = ( ord_less_nat @ N @ M ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_4315_power__strict__decreasing__iff,axiom,
    ! [B: code_integer,M: nat,N: nat] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
     => ( ( ord_le6747313008572928689nteger @ B @ one_one_Code_integer )
       => ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ B @ M ) @ ( power_8256067586552552935nteger @ B @ N ) )
          = ( ord_less_nat @ N @ M ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_4316_power__strict__decreasing__iff,axiom,
    ! [B: rat,M: nat,N: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ B )
     => ( ( ord_less_rat @ B @ one_one_rat )
       => ( ( ord_less_rat @ ( power_power_rat @ B @ M ) @ ( power_power_rat @ B @ N ) )
          = ( ord_less_nat @ N @ M ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_4317_power__strict__decreasing__iff,axiom,
    ! [B: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B )
     => ( ( ord_less_nat @ B @ one_one_nat )
       => ( ( ord_less_nat @ ( power_power_nat @ B @ M ) @ ( power_power_nat @ B @ N ) )
          = ( ord_less_nat @ N @ M ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_4318_power__strict__decreasing__iff,axiom,
    ! [B: int,M: nat,N: nat] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_int @ B @ one_one_int )
       => ( ( ord_less_int @ ( power_power_int @ B @ M ) @ ( power_power_int @ B @ N ) )
          = ( ord_less_nat @ N @ M ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_4319_power__increasing__iff,axiom,
    ! [B: code_natural,X: nat,Y3: nat] :
      ( ( ord_le5570908160329646204atural @ one_one_Code_natural @ B )
     => ( ( ord_le1926595141338095240atural @ ( power_7079662738309270450atural @ B @ X ) @ ( power_7079662738309270450atural @ B @ Y3 ) )
        = ( ord_less_eq_nat @ X @ Y3 ) ) ) ).

% power_increasing_iff
thf(fact_4320_power__increasing__iff,axiom,
    ! [B: code_integer,X: nat,Y3: nat] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ B )
     => ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ B @ X ) @ ( power_8256067586552552935nteger @ B @ Y3 ) )
        = ( ord_less_eq_nat @ X @ Y3 ) ) ) ).

% power_increasing_iff
thf(fact_4321_power__increasing__iff,axiom,
    ! [B: rat,X: nat,Y3: nat] :
      ( ( ord_less_rat @ one_one_rat @ B )
     => ( ( ord_less_eq_rat @ ( power_power_rat @ B @ X ) @ ( power_power_rat @ B @ Y3 ) )
        = ( ord_less_eq_nat @ X @ Y3 ) ) ) ).

% power_increasing_iff
thf(fact_4322_power__increasing__iff,axiom,
    ! [B: nat,X: nat,Y3: nat] :
      ( ( ord_less_nat @ one_one_nat @ B )
     => ( ( ord_less_eq_nat @ ( power_power_nat @ B @ X ) @ ( power_power_nat @ B @ Y3 ) )
        = ( ord_less_eq_nat @ X @ Y3 ) ) ) ).

% power_increasing_iff
thf(fact_4323_power__increasing__iff,axiom,
    ! [B: int,X: nat,Y3: nat] :
      ( ( ord_less_int @ one_one_int @ B )
     => ( ( ord_less_eq_int @ ( power_power_int @ B @ X ) @ ( power_power_int @ B @ Y3 ) )
        = ( ord_less_eq_nat @ X @ Y3 ) ) ) ).

% power_increasing_iff
thf(fact_4324_power__mono__iff,axiom,
    ! [A: code_integer,B: code_integer,N: nat] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
       => ( ( ord_less_nat @ zero_zero_nat @ N )
         => ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A @ N ) @ ( power_8256067586552552935nteger @ B @ N ) )
            = ( ord_le3102999989581377725nteger @ A @ B ) ) ) ) ) ).

% power_mono_iff
thf(fact_4325_power__mono__iff,axiom,
    ! [A: code_natural,B: code_natural,N: nat] :
      ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ A )
     => ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ B )
       => ( ( ord_less_nat @ zero_zero_nat @ N )
         => ( ( ord_le1926595141338095240atural @ ( power_7079662738309270450atural @ A @ N ) @ ( power_7079662738309270450atural @ B @ N ) )
            = ( ord_le1926595141338095240atural @ A @ B ) ) ) ) ) ).

% power_mono_iff
thf(fact_4326_power__mono__iff,axiom,
    ! [A: rat,B: rat,N: nat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
       => ( ( ord_less_nat @ zero_zero_nat @ N )
         => ( ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) )
            = ( ord_less_eq_rat @ A @ B ) ) ) ) ) ).

% power_mono_iff
thf(fact_4327_power__mono__iff,axiom,
    ! [A: nat,B: nat,N: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ( ord_less_nat @ zero_zero_nat @ N )
         => ( ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
            = ( ord_less_eq_nat @ A @ B ) ) ) ) ) ).

% power_mono_iff
thf(fact_4328_power__mono__iff,axiom,
    ! [A: int,B: int,N: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ zero_zero_int @ B )
       => ( ( ord_less_nat @ zero_zero_nat @ N )
         => ( ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
            = ( ord_less_eq_int @ A @ B ) ) ) ) ) ).

% power_mono_iff
thf(fact_4329_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_4330_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_4331_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_4332_power__decreasing__iff,axiom,
    ! [B: code_natural,M: nat,N: nat] :
      ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ B )
     => ( ( ord_le5570908160329646204atural @ B @ one_one_Code_natural )
       => ( ( ord_le1926595141338095240atural @ ( power_7079662738309270450atural @ B @ M ) @ ( power_7079662738309270450atural @ B @ N ) )
          = ( ord_less_eq_nat @ N @ M ) ) ) ) ).

% power_decreasing_iff
thf(fact_4333_power__decreasing__iff,axiom,
    ! [B: code_integer,M: nat,N: nat] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
     => ( ( ord_le6747313008572928689nteger @ B @ one_one_Code_integer )
       => ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ B @ M ) @ ( power_8256067586552552935nteger @ B @ N ) )
          = ( ord_less_eq_nat @ N @ M ) ) ) ) ).

% power_decreasing_iff
thf(fact_4334_power__decreasing__iff,axiom,
    ! [B: rat,M: nat,N: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ B )
     => ( ( ord_less_rat @ B @ one_one_rat )
       => ( ( ord_less_eq_rat @ ( power_power_rat @ B @ M ) @ ( power_power_rat @ B @ N ) )
          = ( ord_less_eq_nat @ N @ M ) ) ) ) ).

% power_decreasing_iff
thf(fact_4335_power__decreasing__iff,axiom,
    ! [B: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B )
     => ( ( ord_less_nat @ B @ one_one_nat )
       => ( ( ord_less_eq_nat @ ( power_power_nat @ B @ M ) @ ( power_power_nat @ B @ N ) )
          = ( ord_less_eq_nat @ N @ M ) ) ) ) ).

% power_decreasing_iff
thf(fact_4336_power__decreasing__iff,axiom,
    ! [B: int,M: nat,N: nat] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_int @ B @ one_one_int )
       => ( ( ord_less_eq_int @ ( power_power_int @ B @ M ) @ ( power_power_int @ B @ N ) )
          = ( ord_less_eq_nat @ N @ M ) ) ) ) ).

% power_decreasing_iff
thf(fact_4337_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,N: nat] :
      ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ X ) @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X )
        | ( N = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
thf(fact_4338_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,N: nat] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ ( semiri4939895301339042750nteger @ X ) @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X )
        | ( N = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
thf(fact_4339_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ X ) @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X )
        | ( N = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
thf(fact_4340_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,N: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ X ) @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X )
        | ( N = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
thf(fact_4341_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,N: nat] :
      ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ ( power_7079662738309270450atural @ ( semiri3763490453095760265atural @ X ) @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X )
        | ( N = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
thf(fact_4342_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_4343_power__not__zero,axiom,
    ! [A: code_integer,N: nat] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( power_8256067586552552935nteger @ A @ N )
       != zero_z3403309356797280102nteger ) ) ).

% power_not_zero
thf(fact_4344_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_4345_power__not__zero,axiom,
    ! [A: code_natural,N: nat] :
      ( ( A != zero_z2226904508553997617atural )
     => ( ( power_7079662738309270450atural @ A @ N )
       != zero_z2226904508553997617atural ) ) ).

% power_not_zero
thf(fact_4346_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_4347_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_4348_power__commutes,axiom,
    ! [A: code_integer,N: nat] :
      ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ A @ N ) @ A )
      = ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ A @ N ) ) ) ).

% power_commutes
thf(fact_4349_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_4350_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_4351_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_4352_power__mult__distrib,axiom,
    ! [A: code_integer,B: code_integer,N: nat] :
      ( ( power_8256067586552552935nteger @ ( times_3573771949741848930nteger @ A @ B ) @ N )
      = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ A @ N ) @ ( power_8256067586552552935nteger @ B @ N ) ) ) ).

% power_mult_distrib
thf(fact_4353_power__mult__distrib,axiom,
    ! [A: rat,B: rat,N: nat] :
      ( ( power_power_rat @ ( times_times_rat @ A @ B ) @ N )
      = ( times_times_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ).

% power_mult_distrib
thf(fact_4354_power__mult__distrib,axiom,
    ! [A: nat,B: nat,N: nat] :
      ( ( power_power_nat @ ( times_times_nat @ A @ B ) @ N )
      = ( times_times_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ).

% power_mult_distrib
thf(fact_4355_power__mult__distrib,axiom,
    ! [A: int,B: int,N: nat] :
      ( ( power_power_int @ ( times_times_int @ A @ B ) @ N )
      = ( times_times_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ).

% power_mult_distrib
thf(fact_4356_power__commuting__commutes,axiom,
    ! [X: code_integer,Y3: code_integer,N: nat] :
      ( ( ( times_3573771949741848930nteger @ X @ Y3 )
        = ( times_3573771949741848930nteger @ Y3 @ X ) )
     => ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ X @ N ) @ Y3 )
        = ( times_3573771949741848930nteger @ Y3 @ ( power_8256067586552552935nteger @ X @ N ) ) ) ) ).

% power_commuting_commutes
thf(fact_4357_power__commuting__commutes,axiom,
    ! [X: rat,Y3: rat,N: nat] :
      ( ( ( times_times_rat @ X @ Y3 )
        = ( times_times_rat @ Y3 @ X ) )
     => ( ( times_times_rat @ ( power_power_rat @ X @ N ) @ Y3 )
        = ( times_times_rat @ Y3 @ ( power_power_rat @ X @ N ) ) ) ) ).

% power_commuting_commutes
thf(fact_4358_power__commuting__commutes,axiom,
    ! [X: nat,Y3: nat,N: nat] :
      ( ( ( times_times_nat @ X @ Y3 )
        = ( times_times_nat @ Y3 @ X ) )
     => ( ( times_times_nat @ ( power_power_nat @ X @ N ) @ Y3 )
        = ( times_times_nat @ Y3 @ ( power_power_nat @ X @ N ) ) ) ) ).

% power_commuting_commutes
thf(fact_4359_power__commuting__commutes,axiom,
    ! [X: int,Y3: int,N: nat] :
      ( ( ( times_times_int @ X @ Y3 )
        = ( times_times_int @ Y3 @ X ) )
     => ( ( times_times_int @ ( power_power_int @ X @ N ) @ Y3 )
        = ( times_times_int @ Y3 @ ( power_power_int @ X @ N ) ) ) ) ).

% power_commuting_commutes
thf(fact_4360_power__mod,axiom,
    ! [A: nat,B: nat,N: nat] :
      ( ( modulo_modulo_nat @ ( power_power_nat @ ( modulo_modulo_nat @ A @ B ) @ N ) @ B )
      = ( modulo_modulo_nat @ ( power_power_nat @ A @ N ) @ B ) ) ).

% power_mod
thf(fact_4361_power__mod,axiom,
    ! [A: int,B: int,N: nat] :
      ( ( modulo_modulo_int @ ( power_power_int @ ( modulo_modulo_int @ A @ B ) @ N ) @ B )
      = ( modulo_modulo_int @ ( power_power_int @ A @ N ) @ B ) ) ).

% power_mod
thf(fact_4362_power__mod,axiom,
    ! [A: code_integer,B: code_integer,N: nat] :
      ( ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ ( modulo364778990260209775nteger @ A @ B ) @ N ) @ B )
      = ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ A @ N ) @ B ) ) ).

% power_mod
thf(fact_4363_power__mod,axiom,
    ! [A: code_natural,B: code_natural,N: nat] :
      ( ( modulo8411746178871703098atural @ ( power_7079662738309270450atural @ ( modulo8411746178871703098atural @ A @ B ) @ N ) @ B )
      = ( modulo8411746178871703098atural @ ( power_7079662738309270450atural @ A @ N ) @ B ) ) ).

% power_mod
thf(fact_4364_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_4365_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_4366_Ints__0,axiom,
    member_int @ zero_zero_int @ ring_1_Ints_int ).

% Ints_0
thf(fact_4367_Ints__0,axiom,
    member_Code_integer @ zero_z3403309356797280102nteger @ ring_11222124179247155820nteger ).

% Ints_0
thf(fact_4368_Ints__0,axiom,
    member_rat @ zero_zero_rat @ ring_1_Ints_rat ).

% Ints_0
thf(fact_4369_Ints__mult,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( member_Code_integer @ A @ ring_11222124179247155820nteger )
     => ( ( member_Code_integer @ B @ ring_11222124179247155820nteger )
       => ( member_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ ring_11222124179247155820nteger ) ) ) ).

% Ints_mult
thf(fact_4370_Ints__mult,axiom,
    ! [A: rat,B: rat] :
      ( ( member_rat @ A @ ring_1_Ints_rat )
     => ( ( member_rat @ B @ ring_1_Ints_rat )
       => ( member_rat @ ( times_times_rat @ A @ B ) @ ring_1_Ints_rat ) ) ) ).

% Ints_mult
thf(fact_4371_Ints__mult,axiom,
    ! [A: int,B: int] :
      ( ( member_int @ A @ ring_1_Ints_int )
     => ( ( member_int @ B @ ring_1_Ints_int )
       => ( member_int @ ( times_times_int @ A @ B ) @ ring_1_Ints_int ) ) ) ).

% Ints_mult
thf(fact_4372_Ints__add,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( member_Code_integer @ A @ ring_11222124179247155820nteger )
     => ( ( member_Code_integer @ B @ ring_11222124179247155820nteger )
       => ( member_Code_integer @ ( plus_p5714425477246183910nteger @ A @ B ) @ ring_11222124179247155820nteger ) ) ) ).

% Ints_add
thf(fact_4373_Ints__add,axiom,
    ! [A: rat,B: rat] :
      ( ( member_rat @ A @ ring_1_Ints_rat )
     => ( ( member_rat @ B @ ring_1_Ints_rat )
       => ( member_rat @ ( plus_plus_rat @ A @ B ) @ ring_1_Ints_rat ) ) ) ).

% Ints_add
thf(fact_4374_Ints__add,axiom,
    ! [A: int,B: int] :
      ( ( member_int @ A @ ring_1_Ints_int )
     => ( ( member_int @ B @ ring_1_Ints_int )
       => ( member_int @ ( plus_plus_int @ A @ B ) @ ring_1_Ints_int ) ) ) ).

% Ints_add
thf(fact_4375_Ints__1,axiom,
    member_int @ one_one_int @ ring_1_Ints_int ).

% Ints_1
thf(fact_4376_Ints__1,axiom,
    member_Code_integer @ one_one_Code_integer @ ring_11222124179247155820nteger ).

% Ints_1
thf(fact_4377_Ints__1,axiom,
    member_rat @ one_one_rat @ ring_1_Ints_rat ).

% Ints_1
thf(fact_4378_Ints__diff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( member_Code_integer @ A @ ring_11222124179247155820nteger )
     => ( ( member_Code_integer @ B @ ring_11222124179247155820nteger )
       => ( member_Code_integer @ ( minus_8373710615458151222nteger @ A @ B ) @ ring_11222124179247155820nteger ) ) ) ).

% Ints_diff
thf(fact_4379_Ints__diff,axiom,
    ! [A: rat,B: rat] :
      ( ( member_rat @ A @ ring_1_Ints_rat )
     => ( ( member_rat @ B @ ring_1_Ints_rat )
       => ( member_rat @ ( minus_minus_rat @ A @ B ) @ ring_1_Ints_rat ) ) ) ).

% Ints_diff
thf(fact_4380_Ints__diff,axiom,
    ! [A: int,B: int] :
      ( ( member_int @ A @ ring_1_Ints_int )
     => ( ( member_int @ B @ ring_1_Ints_int )
       => ( member_int @ ( minus_minus_int @ A @ B ) @ ring_1_Ints_int ) ) ) ).

% Ints_diff
thf(fact_4381_minus__in__Ints__iff,axiom,
    ! [X: int] :
      ( ( member_int @ ( uminus_uminus_int @ X ) @ ring_1_Ints_int )
      = ( member_int @ X @ ring_1_Ints_int ) ) ).

% minus_in_Ints_iff
thf(fact_4382_minus__in__Ints__iff,axiom,
    ! [X: code_integer] :
      ( ( member_Code_integer @ ( uminus1351360451143612070nteger @ X ) @ ring_11222124179247155820nteger )
      = ( member_Code_integer @ X @ ring_11222124179247155820nteger ) ) ).

% minus_in_Ints_iff
thf(fact_4383_minus__in__Ints__iff,axiom,
    ! [X: rat] :
      ( ( member_rat @ ( uminus_uminus_rat @ X ) @ ring_1_Ints_rat )
      = ( member_rat @ X @ ring_1_Ints_rat ) ) ).

% minus_in_Ints_iff
thf(fact_4384_Ints__minus,axiom,
    ! [A: int] :
      ( ( member_int @ A @ ring_1_Ints_int )
     => ( member_int @ ( uminus_uminus_int @ A ) @ ring_1_Ints_int ) ) ).

% Ints_minus
thf(fact_4385_Ints__minus,axiom,
    ! [A: code_integer] :
      ( ( member_Code_integer @ A @ ring_11222124179247155820nteger )
     => ( member_Code_integer @ ( uminus1351360451143612070nteger @ A ) @ ring_11222124179247155820nteger ) ) ).

% Ints_minus
thf(fact_4386_Ints__minus,axiom,
    ! [A: rat] :
      ( ( member_rat @ A @ ring_1_Ints_rat )
     => ( member_rat @ ( uminus_uminus_rat @ A ) @ ring_1_Ints_rat ) ) ).

% Ints_minus
thf(fact_4387_Ints__of__nat,axiom,
    ! [N: nat] : ( member_int @ ( semiri1314217659103216013at_int @ N ) @ ring_1_Ints_int ) ).

% Ints_of_nat
thf(fact_4388_Ints__of__nat,axiom,
    ! [N: nat] : ( member_Code_integer @ ( semiri4939895301339042750nteger @ N ) @ ring_11222124179247155820nteger ) ).

% Ints_of_nat
thf(fact_4389_Ints__of__nat,axiom,
    ! [N: nat] : ( member_rat @ ( semiri681578069525770553at_rat @ N ) @ ring_1_Ints_rat ) ).

% Ints_of_nat
thf(fact_4390_Ints__abs,axiom,
    ! [A: int] :
      ( ( member_int @ A @ ring_1_Ints_int )
     => ( member_int @ ( abs_abs_int @ A ) @ ring_1_Ints_int ) ) ).

% Ints_abs
thf(fact_4391_Ints__abs,axiom,
    ! [A: code_integer] :
      ( ( member_Code_integer @ A @ ring_11222124179247155820nteger )
     => ( member_Code_integer @ ( abs_abs_Code_integer @ A ) @ ring_11222124179247155820nteger ) ) ).

% Ints_abs
thf(fact_4392_Ints__abs,axiom,
    ! [A: rat] :
      ( ( member_rat @ A @ ring_1_Ints_rat )
     => ( member_rat @ ( abs_abs_rat @ A ) @ ring_1_Ints_rat ) ) ).

% Ints_abs
thf(fact_4393_fact__in__Ints,axiom,
    ! [N: nat] : ( member_rat @ ( semiri773545260158071498ct_rat @ N ) @ ring_1_Ints_rat ) ).

% fact_in_Ints
thf(fact_4394_fact__in__Ints,axiom,
    ! [N: nat] : ( member_Code_integer @ ( semiri3624122377584611663nteger @ N ) @ ring_11222124179247155820nteger ) ).

% fact_in_Ints
thf(fact_4395_fact__in__Ints,axiom,
    ! [N: nat] : ( member_int @ ( semiri1406184849735516958ct_int @ N ) @ ring_1_Ints_int ) ).

% fact_in_Ints
thf(fact_4396_nat__power__eq,axiom,
    ! [Z2: int,N: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
     => ( ( nat2 @ ( power_power_int @ Z2 @ N ) )
        = ( power_power_nat @ ( nat2 @ Z2 ) @ N ) ) ) ).

% nat_power_eq
thf(fact_4397_power__mono,axiom,
    ! [A: code_integer,B: code_integer,N: nat] :
      ( ( ord_le3102999989581377725nteger @ A @ B )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
       => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A @ N ) @ ( power_8256067586552552935nteger @ B @ N ) ) ) ) ).

% power_mono
thf(fact_4398_power__mono,axiom,
    ! [A: code_natural,B: code_natural,N: nat] :
      ( ( ord_le1926595141338095240atural @ A @ B )
     => ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ A )
       => ( ord_le1926595141338095240atural @ ( power_7079662738309270450atural @ A @ N ) @ ( power_7079662738309270450atural @ B @ N ) ) ) ) ).

% power_mono
thf(fact_4399_power__mono,axiom,
    ! [A: rat,B: rat,N: nat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
       => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ) ).

% power_mono
thf(fact_4400_power__mono,axiom,
    ! [A: nat,B: nat,N: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ).

% power_mono
thf(fact_4401_power__mono,axiom,
    ! [A: int,B: int,N: nat] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).

% power_mono
thf(fact_4402_zero__le__power,axiom,
    ! [A: code_integer,N: nat] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ A @ N ) ) ) ).

% zero_le_power
thf(fact_4403_zero__le__power,axiom,
    ! [A: code_natural,N: nat] :
      ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ A )
     => ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ ( power_7079662738309270450atural @ A @ N ) ) ) ).

% zero_le_power
thf(fact_4404_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_4405_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_4406_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_4407_zero__less__power,axiom,
    ! [A: code_natural,N: nat] :
      ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ A )
     => ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ ( power_7079662738309270450atural @ A @ N ) ) ) ).

% zero_less_power
thf(fact_4408_zero__less__power,axiom,
    ! [A: code_integer,N: nat] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
     => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ A @ N ) ) ) ).

% zero_less_power
thf(fact_4409_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_4410_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_4411_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_4412_one__le__power,axiom,
    ! [A: code_integer,N: nat] :
      ( ( ord_le3102999989581377725nteger @ one_one_Code_integer @ A )
     => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) ) ) ).

% one_le_power
thf(fact_4413_one__le__power,axiom,
    ! [A: code_natural,N: nat] :
      ( ( ord_le1926595141338095240atural @ one_one_Code_natural @ A )
     => ( ord_le1926595141338095240atural @ one_one_Code_natural @ ( power_7079662738309270450atural @ A @ N ) ) ) ).

% one_le_power
thf(fact_4414_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_4415_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_4416_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_4417_left__right__inverse__power,axiom,
    ! [X: code_natural,Y3: code_natural,N: nat] :
      ( ( ( times_2397367101498566445atural @ X @ Y3 )
        = one_one_Code_natural )
     => ( ( times_2397367101498566445atural @ ( power_7079662738309270450atural @ X @ N ) @ ( power_7079662738309270450atural @ Y3 @ N ) )
        = one_one_Code_natural ) ) ).

% left_right_inverse_power
thf(fact_4418_left__right__inverse__power,axiom,
    ! [X: code_integer,Y3: code_integer,N: nat] :
      ( ( ( times_3573771949741848930nteger @ X @ Y3 )
        = one_one_Code_integer )
     => ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ X @ N ) @ ( power_8256067586552552935nteger @ Y3 @ N ) )
        = one_one_Code_integer ) ) ).

% left_right_inverse_power
thf(fact_4419_left__right__inverse__power,axiom,
    ! [X: rat,Y3: rat,N: nat] :
      ( ( ( times_times_rat @ X @ Y3 )
        = one_one_rat )
     => ( ( times_times_rat @ ( power_power_rat @ X @ N ) @ ( power_power_rat @ Y3 @ N ) )
        = one_one_rat ) ) ).

% left_right_inverse_power
thf(fact_4420_left__right__inverse__power,axiom,
    ! [X: nat,Y3: nat,N: nat] :
      ( ( ( times_times_nat @ X @ Y3 )
        = one_one_nat )
     => ( ( times_times_nat @ ( power_power_nat @ X @ N ) @ ( power_power_nat @ Y3 @ N ) )
        = one_one_nat ) ) ).

% left_right_inverse_power
thf(fact_4421_left__right__inverse__power,axiom,
    ! [X: int,Y3: int,N: nat] :
      ( ( ( times_times_int @ X @ Y3 )
        = one_one_int )
     => ( ( times_times_int @ ( power_power_int @ X @ N ) @ ( power_power_int @ Y3 @ N ) )
        = one_one_int ) ) ).

% left_right_inverse_power
thf(fact_4422_power__Suc2,axiom,
    ! [A: code_integer,N: nat] :
      ( ( power_8256067586552552935nteger @ A @ ( suc @ N ) )
      = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ A @ N ) @ A ) ) ).

% power_Suc2
thf(fact_4423_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_4424_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_4425_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_4426_power__Suc,axiom,
    ! [A: code_integer,N: nat] :
      ( ( power_8256067586552552935nteger @ A @ ( suc @ N ) )
      = ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ A @ N ) ) ) ).

% power_Suc
thf(fact_4427_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_4428_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_4429_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_4430_power__0,axiom,
    ! [A: code_integer] :
      ( ( power_8256067586552552935nteger @ A @ zero_zero_nat )
      = one_one_Code_integer ) ).

% power_0
thf(fact_4431_power__0,axiom,
    ! [A: rat] :
      ( ( power_power_rat @ A @ zero_zero_nat )
      = one_one_rat ) ).

% power_0
thf(fact_4432_power__0,axiom,
    ! [A: code_natural] :
      ( ( power_7079662738309270450atural @ A @ zero_zero_nat )
      = one_one_Code_natural ) ).

% power_0
thf(fact_4433_power__0,axiom,
    ! [A: int] :
      ( ( power_power_int @ A @ zero_zero_nat )
      = one_one_int ) ).

% power_0
thf(fact_4434_power__0,axiom,
    ! [A: nat] :
      ( ( power_power_nat @ A @ zero_zero_nat )
      = one_one_nat ) ).

% power_0
thf(fact_4435_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_4436_power__add,axiom,
    ! [A: code_integer,M: nat,N: nat] :
      ( ( power_8256067586552552935nteger @ A @ ( plus_plus_nat @ M @ N ) )
      = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ A @ M ) @ ( power_8256067586552552935nteger @ A @ N ) ) ) ).

% power_add
thf(fact_4437_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_4438_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_4439_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_4440_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_4441_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_4442_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_4443_power__le__dvd,axiom,
    ! [A: code_integer,N: nat,B: code_integer,M: nat] :
      ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) @ B )
     => ( ( ord_less_eq_nat @ M @ N )
       => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ M ) @ B ) ) ) ).

% power_le_dvd
thf(fact_4444_power__le__dvd,axiom,
    ! [A: int,N: nat,B: int,M: nat] :
      ( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ B )
     => ( ( ord_less_eq_nat @ M @ N )
       => ( dvd_dvd_int @ ( power_power_int @ A @ M ) @ B ) ) ) ).

% power_le_dvd
thf(fact_4445_power__le__dvd,axiom,
    ! [A: nat,N: nat,B: nat,M: nat] :
      ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ B )
     => ( ( ord_less_eq_nat @ M @ N )
       => ( dvd_dvd_nat @ ( power_power_nat @ A @ M ) @ B ) ) ) ).

% power_le_dvd
thf(fact_4446_dvd__power__le,axiom,
    ! [X: code_integer,Y3: code_integer,N: nat,M: nat] :
      ( ( dvd_dvd_Code_integer @ X @ Y3 )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X @ N ) @ ( power_8256067586552552935nteger @ Y3 @ M ) ) ) ) ).

% dvd_power_le
thf(fact_4447_dvd__power__le,axiom,
    ! [X: int,Y3: int,N: nat,M: nat] :
      ( ( dvd_dvd_int @ X @ Y3 )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( dvd_dvd_int @ ( power_power_int @ X @ N ) @ ( power_power_int @ Y3 @ M ) ) ) ) ).

% dvd_power_le
thf(fact_4448_dvd__power__le,axiom,
    ! [X: nat,Y3: nat,N: nat,M: nat] :
      ( ( dvd_dvd_nat @ X @ Y3 )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( dvd_dvd_nat @ ( power_power_nat @ X @ N ) @ ( power_power_nat @ Y3 @ M ) ) ) ) ).

% dvd_power_le
thf(fact_4449_nat__power__less__imp__less,axiom,
    ! [I2: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ I2 )
     => ( ( ord_less_nat @ ( power_power_nat @ I2 @ M ) @ ( power_power_nat @ I2 @ N ) )
       => ( ord_less_nat @ M @ N ) ) ) ).

% nat_power_less_imp_less
thf(fact_4450_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_4451_Ints__double__eq__0__iff,axiom,
    ! [A: code_integer] :
      ( ( member_Code_integer @ A @ ring_11222124179247155820nteger )
     => ( ( ( plus_p5714425477246183910nteger @ A @ A )
          = zero_z3403309356797280102nteger )
        = ( A = zero_z3403309356797280102nteger ) ) ) ).

% Ints_double_eq_0_iff
thf(fact_4452_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_4453_power__less__imp__less__base,axiom,
    ! [A: code_natural,N: nat,B: code_natural] :
      ( ( ord_le5570908160329646204atural @ ( power_7079662738309270450atural @ A @ N ) @ ( power_7079662738309270450atural @ B @ N ) )
     => ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ B )
       => ( ord_le5570908160329646204atural @ A @ B ) ) ) ).

% power_less_imp_less_base
thf(fact_4454_power__less__imp__less__base,axiom,
    ! [A: code_integer,N: nat,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A @ N ) @ ( power_8256067586552552935nteger @ B @ N ) )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
       => ( ord_le6747313008572928689nteger @ A @ B ) ) ) ).

% power_less_imp_less_base
thf(fact_4455_power__less__imp__less__base,axiom,
    ! [A: rat,N: nat,B: rat] :
      ( ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
       => ( ord_less_rat @ A @ B ) ) ) ).

% power_less_imp_less_base
thf(fact_4456_power__less__imp__less__base,axiom,
    ! [A: nat,N: nat,B: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ord_less_nat @ A @ B ) ) ) ).

% power_less_imp_less_base
thf(fact_4457_power__less__imp__less__base,axiom,
    ! [A: int,N: nat,B: int] :
      ( ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ B )
       => ( ord_less_int @ A @ B ) ) ) ).

% power_less_imp_less_base
thf(fact_4458_power__le__one,axiom,
    ! [A: code_integer,N: nat] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le3102999989581377725nteger @ A @ one_one_Code_integer )
       => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A @ N ) @ one_one_Code_integer ) ) ) ).

% power_le_one
thf(fact_4459_power__le__one,axiom,
    ! [A: code_natural,N: nat] :
      ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ A )
     => ( ( ord_le1926595141338095240atural @ A @ one_one_Code_natural )
       => ( ord_le1926595141338095240atural @ ( power_7079662738309270450atural @ A @ N ) @ one_one_Code_natural ) ) ) ).

% power_le_one
thf(fact_4460_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_4461_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_4462_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_4463_power__inject__base,axiom,
    ! [A: code_integer,N: nat,B: code_integer] :
      ( ( ( power_8256067586552552935nteger @ A @ ( suc @ N ) )
        = ( power_8256067586552552935nteger @ B @ ( suc @ N ) ) )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
       => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
         => ( A = B ) ) ) ) ).

% power_inject_base
thf(fact_4464_power__inject__base,axiom,
    ! [A: code_natural,N: nat,B: code_natural] :
      ( ( ( power_7079662738309270450atural @ A @ ( suc @ N ) )
        = ( power_7079662738309270450atural @ B @ ( suc @ N ) ) )
     => ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ A )
       => ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ B )
         => ( A = B ) ) ) ) ).

% power_inject_base
thf(fact_4465_power__inject__base,axiom,
    ! [A: rat,N: nat,B: rat] :
      ( ( ( power_power_rat @ A @ ( suc @ N ) )
        = ( power_power_rat @ B @ ( suc @ N ) ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
         => ( A = B ) ) ) ) ).

% power_inject_base
thf(fact_4466_power__inject__base,axiom,
    ! [A: nat,N: nat,B: nat] :
      ( ( ( power_power_nat @ A @ ( suc @ N ) )
        = ( power_power_nat @ B @ ( suc @ N ) ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
         => ( A = B ) ) ) ) ).

% power_inject_base
thf(fact_4467_power__inject__base,axiom,
    ! [A: int,N: nat,B: int] :
      ( ( ( power_power_int @ A @ ( suc @ N ) )
        = ( power_power_int @ B @ ( suc @ N ) ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ( ord_less_eq_int @ zero_zero_int @ B )
         => ( A = B ) ) ) ) ).

% power_inject_base
thf(fact_4468_power__le__imp__le__base,axiom,
    ! [A: code_integer,N: nat,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A @ ( suc @ N ) ) @ ( power_8256067586552552935nteger @ B @ ( suc @ N ) ) )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
       => ( ord_le3102999989581377725nteger @ A @ B ) ) ) ).

% power_le_imp_le_base
thf(fact_4469_power__le__imp__le__base,axiom,
    ! [A: code_natural,N: nat,B: code_natural] :
      ( ( ord_le1926595141338095240atural @ ( power_7079662738309270450atural @ A @ ( suc @ N ) ) @ ( power_7079662738309270450atural @ B @ ( suc @ N ) ) )
     => ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ B )
       => ( ord_le1926595141338095240atural @ A @ B ) ) ) ).

% power_le_imp_le_base
thf(fact_4470_power__le__imp__le__base,axiom,
    ! [A: rat,N: nat,B: rat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( suc @ N ) ) @ ( power_power_rat @ B @ ( suc @ N ) ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
       => ( ord_less_eq_rat @ A @ B ) ) ) ).

% power_le_imp_le_base
thf(fact_4471_power__le__imp__le__base,axiom,
    ! [A: nat,N: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ ( power_power_nat @ B @ ( suc @ N ) ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ord_less_eq_nat @ A @ B ) ) ) ).

% power_le_imp_le_base
thf(fact_4472_power__le__imp__le__base,axiom,
    ! [A: int,N: nat,B: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N ) ) @ ( power_power_int @ B @ ( suc @ N ) ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ B )
       => ( ord_less_eq_int @ A @ B ) ) ) ).

% power_le_imp_le_base
thf(fact_4473_power__gt1__lemma,axiom,
    ! [A: code_natural,N: nat] :
      ( ( ord_le5570908160329646204atural @ one_one_Code_natural @ A )
     => ( ord_le5570908160329646204atural @ one_one_Code_natural @ ( times_2397367101498566445atural @ A @ ( power_7079662738309270450atural @ A @ N ) ) ) ) ).

% power_gt1_lemma
thf(fact_4474_power__gt1__lemma,axiom,
    ! [A: code_integer,N: nat] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A )
     => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ A @ N ) ) ) ) ).

% power_gt1_lemma
thf(fact_4475_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_4476_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_4477_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_4478_power__less__power__Suc,axiom,
    ! [A: code_natural,N: nat] :
      ( ( ord_le5570908160329646204atural @ one_one_Code_natural @ A )
     => ( ord_le5570908160329646204atural @ ( power_7079662738309270450atural @ A @ N ) @ ( times_2397367101498566445atural @ A @ ( power_7079662738309270450atural @ A @ N ) ) ) ) ).

% power_less_power_Suc
thf(fact_4479_power__less__power__Suc,axiom,
    ! [A: code_integer,N: nat] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A )
     => ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A @ N ) @ ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ A @ N ) ) ) ) ).

% power_less_power_Suc
thf(fact_4480_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_4481_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_4482_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_4483_power__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( power_8256067586552552935nteger @ zero_z3403309356797280102nteger @ N )
          = one_one_Code_integer ) )
      & ( ( N != zero_zero_nat )
       => ( ( power_8256067586552552935nteger @ zero_z3403309356797280102nteger @ N )
          = zero_z3403309356797280102nteger ) ) ) ).

% power_0_left
thf(fact_4484_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_4485_power__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( power_7079662738309270450atural @ zero_z2226904508553997617atural @ N )
          = one_one_Code_natural ) )
      & ( ( N != zero_zero_nat )
       => ( ( power_7079662738309270450atural @ zero_z2226904508553997617atural @ N )
          = zero_z2226904508553997617atural ) ) ) ).

% power_0_left
thf(fact_4486_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_4487_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_4488_power__gt1,axiom,
    ! [A: code_natural,N: nat] :
      ( ( ord_le5570908160329646204atural @ one_one_Code_natural @ A )
     => ( ord_le5570908160329646204atural @ one_one_Code_natural @ ( power_7079662738309270450atural @ A @ ( suc @ N ) ) ) ) ).

% power_gt1
thf(fact_4489_power__gt1,axiom,
    ! [A: code_integer,N: nat] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A )
     => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ A @ ( suc @ N ) ) ) ) ).

% power_gt1
thf(fact_4490_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_4491_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_4492_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_4493_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_4494_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_4495_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_4496_power__less__imp__less__exp,axiom,
    ! [A: code_natural,M: nat,N: nat] :
      ( ( ord_le5570908160329646204atural @ one_one_Code_natural @ A )
     => ( ( ord_le5570908160329646204atural @ ( power_7079662738309270450atural @ A @ M ) @ ( power_7079662738309270450atural @ A @ N ) )
       => ( ord_less_nat @ M @ N ) ) ) ).

% power_less_imp_less_exp
thf(fact_4497_power__less__imp__less__exp,axiom,
    ! [A: code_integer,M: nat,N: nat] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A )
     => ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A @ M ) @ ( power_8256067586552552935nteger @ A @ N ) )
       => ( ord_less_nat @ M @ N ) ) ) ).

% power_less_imp_less_exp
thf(fact_4498_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_4499_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_4500_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_4501_power__strict__increasing,axiom,
    ! [N: nat,N5: nat,A: code_natural] :
      ( ( ord_less_nat @ N @ N5 )
     => ( ( ord_le5570908160329646204atural @ one_one_Code_natural @ A )
       => ( ord_le5570908160329646204atural @ ( power_7079662738309270450atural @ A @ N ) @ ( power_7079662738309270450atural @ A @ N5 ) ) ) ) ).

% power_strict_increasing
thf(fact_4502_power__strict__increasing,axiom,
    ! [N: nat,N5: nat,A: code_integer] :
      ( ( ord_less_nat @ N @ N5 )
     => ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A )
       => ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A @ N ) @ ( power_8256067586552552935nteger @ A @ N5 ) ) ) ) ).

% power_strict_increasing
thf(fact_4503_power__strict__increasing,axiom,
    ! [N: nat,N5: nat,A: rat] :
      ( ( ord_less_nat @ N @ N5 )
     => ( ( ord_less_rat @ one_one_rat @ A )
       => ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ A @ N5 ) ) ) ) ).

% power_strict_increasing
thf(fact_4504_power__strict__increasing,axiom,
    ! [N: nat,N5: nat,A: nat] :
      ( ( ord_less_nat @ N @ N5 )
     => ( ( ord_less_nat @ one_one_nat @ A )
       => ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N5 ) ) ) ) ).

% power_strict_increasing
thf(fact_4505_power__strict__increasing,axiom,
    ! [N: nat,N5: nat,A: int] :
      ( ( ord_less_nat @ N @ N5 )
     => ( ( ord_less_int @ one_one_int @ A )
       => ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N5 ) ) ) ) ).

% power_strict_increasing
thf(fact_4506_power__increasing,axiom,
    ! [N: nat,N5: nat,A: code_integer] :
      ( ( ord_less_eq_nat @ N @ N5 )
     => ( ( ord_le3102999989581377725nteger @ one_one_Code_integer @ A )
       => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A @ N ) @ ( power_8256067586552552935nteger @ A @ N5 ) ) ) ) ).

% power_increasing
thf(fact_4507_power__increasing,axiom,
    ! [N: nat,N5: nat,A: code_natural] :
      ( ( ord_less_eq_nat @ N @ N5 )
     => ( ( ord_le1926595141338095240atural @ one_one_Code_natural @ A )
       => ( ord_le1926595141338095240atural @ ( power_7079662738309270450atural @ A @ N ) @ ( power_7079662738309270450atural @ A @ N5 ) ) ) ) ).

% power_increasing
thf(fact_4508_power__increasing,axiom,
    ! [N: nat,N5: nat,A: rat] :
      ( ( ord_less_eq_nat @ N @ N5 )
     => ( ( ord_less_eq_rat @ one_one_rat @ A )
       => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ A @ N5 ) ) ) ) ).

% power_increasing
thf(fact_4509_power__increasing,axiom,
    ! [N: nat,N5: nat,A: nat] :
      ( ( ord_less_eq_nat @ N @ N5 )
     => ( ( ord_less_eq_nat @ one_one_nat @ A )
       => ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N5 ) ) ) ) ).

% power_increasing
thf(fact_4510_power__increasing,axiom,
    ! [N: nat,N5: nat,A: int] :
      ( ( ord_less_eq_nat @ N @ N5 )
     => ( ( ord_less_eq_int @ one_one_int @ A )
       => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N5 ) ) ) ) ).

% power_increasing
thf(fact_4511_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_4512_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_4513_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_4514_zero__power,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( power_8256067586552552935nteger @ zero_z3403309356797280102nteger @ N )
        = zero_z3403309356797280102nteger ) ) ).

% zero_power
thf(fact_4515_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_4516_zero__power,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( power_7079662738309270450atural @ zero_z2226904508553997617atural @ N )
        = zero_z2226904508553997617atural ) ) ).

% zero_power
thf(fact_4517_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_4518_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_4519_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_4520_is__unit__power__iff,axiom,
    ! [A: code_natural,N: nat] :
      ( ( dvd_dvd_Code_natural @ ( power_7079662738309270450atural @ A @ N ) @ one_one_Code_natural )
      = ( ( dvd_dvd_Code_natural @ A @ one_one_Code_natural )
        | ( N = zero_zero_nat ) ) ) ).

% is_unit_power_iff
thf(fact_4521_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_4522_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_4523_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_4524_nat__one__le__power,axiom,
    ! [I2: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ I2 )
     => ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( power_power_nat @ I2 @ N ) ) ) ).

% nat_one_le_power
thf(fact_4525_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_4526_fact__le__power,axiom,
    ! [N: nat] : ( ord_le1926595141338095240atural @ ( semiri2447717529341329178atural @ N ) @ ( semiri3763490453095760265atural @ ( power_power_nat @ N @ N ) ) ) ).

% fact_le_power
thf(fact_4527_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_4528_fact__le__power,axiom,
    ! [N: nat] : ( ord_le3102999989581377725nteger @ ( semiri3624122377584611663nteger @ N ) @ ( semiri4939895301339042750nteger @ ( power_power_nat @ N @ N ) ) ) ).

% fact_le_power
thf(fact_4529_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_4530_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_4531_Ints__odd__nonzero,axiom,
    ! [A: code_integer] :
      ( ( member_Code_integer @ A @ ring_11222124179247155820nteger )
     => ( ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ A )
       != zero_z3403309356797280102nteger ) ) ).

% Ints_odd_nonzero
thf(fact_4532_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_4533_bounded__Max__nat,axiom,
    ! [P: nat > $o,X: nat,M7: nat] :
      ( ( P @ X )
     => ( ! [X3: nat] :
            ( ( P @ X3 )
           => ( ord_less_eq_nat @ X3 @ M7 ) )
       => ~ ! [M6: nat] :
              ( ( P @ M6 )
             => ~ ! [X4: nat] :
                    ( ( P @ X4 )
                   => ( ord_less_eq_nat @ X4 @ M6 ) ) ) ) ) ).

% bounded_Max_nat
thf(fact_4534_power__Suc__less,axiom,
    ! [A: code_natural,N: nat] :
      ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ A )
     => ( ( ord_le5570908160329646204atural @ A @ one_one_Code_natural )
       => ( ord_le5570908160329646204atural @ ( times_2397367101498566445atural @ A @ ( power_7079662738309270450atural @ A @ N ) ) @ ( power_7079662738309270450atural @ A @ N ) ) ) ) ).

% power_Suc_less
thf(fact_4535_power__Suc__less,axiom,
    ! [A: code_integer,N: nat] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le6747313008572928689nteger @ A @ one_one_Code_integer )
       => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ A @ N ) ) @ ( power_8256067586552552935nteger @ A @ N ) ) ) ) ).

% power_Suc_less
thf(fact_4536_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_4537_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_4538_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_4539_power__Suc__le__self,axiom,
    ! [A: code_integer,N: nat] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le3102999989581377725nteger @ A @ one_one_Code_integer )
       => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A @ ( suc @ N ) ) @ A ) ) ) ).

% power_Suc_le_self
thf(fact_4540_power__Suc__le__self,axiom,
    ! [A: code_natural,N: nat] :
      ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ A )
     => ( ( ord_le1926595141338095240atural @ A @ one_one_Code_natural )
       => ( ord_le1926595141338095240atural @ ( power_7079662738309270450atural @ A @ ( suc @ N ) ) @ A ) ) ) ).

% power_Suc_le_self
thf(fact_4541_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_4542_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_4543_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_4544_power__Suc__less__one,axiom,
    ! [A: code_natural,N: nat] :
      ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ A )
     => ( ( ord_le5570908160329646204atural @ A @ one_one_Code_natural )
       => ( ord_le5570908160329646204atural @ ( power_7079662738309270450atural @ A @ ( suc @ N ) ) @ one_one_Code_natural ) ) ) ).

% power_Suc_less_one
thf(fact_4545_power__Suc__less__one,axiom,
    ! [A: code_integer,N: nat] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le6747313008572928689nteger @ A @ one_one_Code_integer )
       => ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A @ ( suc @ N ) ) @ one_one_Code_integer ) ) ) ).

% power_Suc_less_one
thf(fact_4546_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_4547_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_4548_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_4549_power__strict__decreasing,axiom,
    ! [N: nat,N5: nat,A: code_natural] :
      ( ( ord_less_nat @ N @ N5 )
     => ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ A )
       => ( ( ord_le5570908160329646204atural @ A @ one_one_Code_natural )
         => ( ord_le5570908160329646204atural @ ( power_7079662738309270450atural @ A @ N5 ) @ ( power_7079662738309270450atural @ A @ N ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_4550_power__strict__decreasing,axiom,
    ! [N: nat,N5: nat,A: code_integer] :
      ( ( ord_less_nat @ N @ N5 )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
       => ( ( ord_le6747313008572928689nteger @ A @ one_one_Code_integer )
         => ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A @ N5 ) @ ( power_8256067586552552935nteger @ A @ N ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_4551_power__strict__decreasing,axiom,
    ! [N: nat,N5: nat,A: rat] :
      ( ( ord_less_nat @ N @ N5 )
     => ( ( ord_less_rat @ zero_zero_rat @ A )
       => ( ( ord_less_rat @ A @ one_one_rat )
         => ( ord_less_rat @ ( power_power_rat @ A @ N5 ) @ ( power_power_rat @ A @ N ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_4552_power__strict__decreasing,axiom,
    ! [N: nat,N5: nat,A: nat] :
      ( ( ord_less_nat @ N @ N5 )
     => ( ( ord_less_nat @ zero_zero_nat @ A )
       => ( ( ord_less_nat @ A @ one_one_nat )
         => ( ord_less_nat @ ( power_power_nat @ A @ N5 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_4553_power__strict__decreasing,axiom,
    ! [N: nat,N5: nat,A: int] :
      ( ( ord_less_nat @ N @ N5 )
     => ( ( ord_less_int @ zero_zero_int @ A )
       => ( ( ord_less_int @ A @ one_one_int )
         => ( ord_less_int @ ( power_power_int @ A @ N5 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_4554_power__decreasing,axiom,
    ! [N: nat,N5: nat,A: code_integer] :
      ( ( ord_less_eq_nat @ N @ N5 )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
       => ( ( ord_le3102999989581377725nteger @ A @ one_one_Code_integer )
         => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A @ N5 ) @ ( power_8256067586552552935nteger @ A @ N ) ) ) ) ) ).

% power_decreasing
thf(fact_4555_power__decreasing,axiom,
    ! [N: nat,N5: nat,A: code_natural] :
      ( ( ord_less_eq_nat @ N @ N5 )
     => ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ A )
       => ( ( ord_le1926595141338095240atural @ A @ one_one_Code_natural )
         => ( ord_le1926595141338095240atural @ ( power_7079662738309270450atural @ A @ N5 ) @ ( power_7079662738309270450atural @ A @ N ) ) ) ) ) ).

% power_decreasing
thf(fact_4556_power__decreasing,axiom,
    ! [N: nat,N5: nat,A: rat] :
      ( ( ord_less_eq_nat @ N @ N5 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
       => ( ( ord_less_eq_rat @ A @ one_one_rat )
         => ( ord_less_eq_rat @ ( power_power_rat @ A @ N5 ) @ ( power_power_rat @ A @ N ) ) ) ) ) ).

% power_decreasing
thf(fact_4557_power__decreasing,axiom,
    ! [N: nat,N5: nat,A: nat] :
      ( ( ord_less_eq_nat @ N @ N5 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ( ord_less_eq_nat @ A @ one_one_nat )
         => ( ord_less_eq_nat @ ( power_power_nat @ A @ N5 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).

% power_decreasing
thf(fact_4558_power__decreasing,axiom,
    ! [N: nat,N5: nat,A: int] :
      ( ( ord_less_eq_nat @ N @ N5 )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ( ord_less_eq_int @ A @ one_one_int )
         => ( ord_less_eq_int @ ( power_power_int @ A @ N5 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).

% power_decreasing
thf(fact_4559_power__le__imp__le__exp,axiom,
    ! [A: code_natural,M: nat,N: nat] :
      ( ( ord_le5570908160329646204atural @ one_one_Code_natural @ A )
     => ( ( ord_le1926595141338095240atural @ ( power_7079662738309270450atural @ A @ M ) @ ( power_7079662738309270450atural @ A @ N ) )
       => ( ord_less_eq_nat @ M @ N ) ) ) ).

% power_le_imp_le_exp
thf(fact_4560_power__le__imp__le__exp,axiom,
    ! [A: code_integer,M: nat,N: nat] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A )
     => ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A @ M ) @ ( power_8256067586552552935nteger @ A @ N ) )
       => ( ord_less_eq_nat @ M @ N ) ) ) ).

% power_le_imp_le_exp
thf(fact_4561_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_4562_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_4563_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_4564_power__eq__imp__eq__base,axiom,
    ! [A: code_integer,N: nat,B: code_integer] :
      ( ( ( power_8256067586552552935nteger @ A @ N )
        = ( power_8256067586552552935nteger @ B @ N ) )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
       => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
         => ( ( ord_less_nat @ zero_zero_nat @ N )
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
thf(fact_4565_power__eq__imp__eq__base,axiom,
    ! [A: code_natural,N: nat,B: code_natural] :
      ( ( ( power_7079662738309270450atural @ A @ N )
        = ( power_7079662738309270450atural @ B @ N ) )
     => ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ A )
       => ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ B )
         => ( ( ord_less_nat @ zero_zero_nat @ N )
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
thf(fact_4566_power__eq__imp__eq__base,axiom,
    ! [A: rat,N: nat,B: rat] :
      ( ( ( power_power_rat @ A @ N )
        = ( power_power_rat @ B @ N ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
         => ( ( ord_less_nat @ zero_zero_nat @ N )
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
thf(fact_4567_power__eq__imp__eq__base,axiom,
    ! [A: nat,N: nat,B: nat] :
      ( ( ( power_power_nat @ A @ N )
        = ( power_power_nat @ B @ N ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
         => ( ( ord_less_nat @ zero_zero_nat @ N )
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
thf(fact_4568_power__eq__imp__eq__base,axiom,
    ! [A: int,N: nat,B: int] :
      ( ( ( power_power_int @ A @ N )
        = ( power_power_int @ B @ N ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ( ord_less_eq_int @ zero_zero_int @ B )
         => ( ( ord_less_nat @ zero_zero_nat @ N )
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
thf(fact_4569_power__eq__iff__eq__base,axiom,
    ! [N: nat,A: code_integer,B: code_integer] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
       => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
         => ( ( ( power_8256067586552552935nteger @ A @ N )
              = ( power_8256067586552552935nteger @ B @ N ) )
            = ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
thf(fact_4570_power__eq__iff__eq__base,axiom,
    ! [N: nat,A: code_natural,B: code_natural] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ A )
       => ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ B )
         => ( ( ( power_7079662738309270450atural @ A @ N )
              = ( power_7079662738309270450atural @ B @ N ) )
            = ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
thf(fact_4571_power__eq__iff__eq__base,axiom,
    ! [N: nat,A: rat,B: rat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
         => ( ( ( power_power_rat @ A @ N )
              = ( power_power_rat @ B @ N ) )
            = ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
thf(fact_4572_power__eq__iff__eq__base,axiom,
    ! [N: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
         => ( ( ( power_power_nat @ A @ N )
              = ( power_power_nat @ B @ N ) )
            = ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
thf(fact_4573_power__eq__iff__eq__base,axiom,
    ! [N: nat,A: int,B: int] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ( ord_less_eq_int @ zero_zero_int @ B )
         => ( ( ( power_power_int @ A @ N )
              = ( power_power_int @ B @ N ) )
            = ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
thf(fact_4574_self__le__power,axiom,
    ! [A: code_integer,N: nat] :
      ( ( ord_le3102999989581377725nteger @ one_one_Code_integer @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_le3102999989581377725nteger @ A @ ( power_8256067586552552935nteger @ A @ N ) ) ) ) ).

% self_le_power
thf(fact_4575_self__le__power,axiom,
    ! [A: code_natural,N: nat] :
      ( ( ord_le1926595141338095240atural @ one_one_Code_natural @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_le1926595141338095240atural @ A @ ( power_7079662738309270450atural @ A @ N ) ) ) ) ).

% self_le_power
thf(fact_4576_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_4577_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_4578_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_4579_one__less__power,axiom,
    ! [A: code_natural,N: nat] :
      ( ( ord_le5570908160329646204atural @ one_one_Code_natural @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_le5570908160329646204atural @ one_one_Code_natural @ ( power_7079662738309270450atural @ A @ N ) ) ) ) ).

% one_less_power
thf(fact_4580_one__less__power,axiom,
    ! [A: code_integer,N: nat] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) ) ) ) ).

% one_less_power
thf(fact_4581_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_4582_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_4583_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_4584_dvd__power__iff,axiom,
    ! [X: code_integer,M: nat,N: nat] :
      ( ( X != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X @ M ) @ ( power_8256067586552552935nteger @ X @ N ) )
        = ( ( dvd_dvd_Code_integer @ X @ one_one_Code_integer )
          | ( ord_less_eq_nat @ M @ N ) ) ) ) ).

% dvd_power_iff
thf(fact_4585_dvd__power__iff,axiom,
    ! [X: code_natural,M: nat,N: nat] :
      ( ( X != zero_z2226904508553997617atural )
     => ( ( dvd_dvd_Code_natural @ ( power_7079662738309270450atural @ X @ M ) @ ( power_7079662738309270450atural @ X @ N ) )
        = ( ( dvd_dvd_Code_natural @ X @ one_one_Code_natural )
          | ( ord_less_eq_nat @ M @ N ) ) ) ) ).

% dvd_power_iff
thf(fact_4586_dvd__power__iff,axiom,
    ! [X: int,M: nat,N: nat] :
      ( ( X != zero_zero_int )
     => ( ( dvd_dvd_int @ ( power_power_int @ X @ M ) @ ( power_power_int @ X @ N ) )
        = ( ( dvd_dvd_int @ X @ one_one_int )
          | ( ord_less_eq_nat @ M @ N ) ) ) ) ).

% dvd_power_iff
thf(fact_4587_dvd__power__iff,axiom,
    ! [X: nat,M: nat,N: nat] :
      ( ( X != zero_zero_nat )
     => ( ( dvd_dvd_nat @ ( power_power_nat @ X @ M ) @ ( power_power_nat @ X @ N ) )
        = ( ( dvd_dvd_nat @ X @ one_one_nat )
          | ( ord_less_eq_nat @ M @ N ) ) ) ) ).

% dvd_power_iff
thf(fact_4588_dvd__power,axiom,
    ! [N: nat,X: code_integer] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N )
        | ( X = one_one_Code_integer ) )
     => ( dvd_dvd_Code_integer @ X @ ( power_8256067586552552935nteger @ X @ N ) ) ) ).

% dvd_power
thf(fact_4589_dvd__power,axiom,
    ! [N: nat,X: rat] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N )
        | ( X = one_one_rat ) )
     => ( dvd_dvd_rat @ X @ ( power_power_rat @ X @ N ) ) ) ).

% dvd_power
thf(fact_4590_dvd__power,axiom,
    ! [N: nat,X: code_natural] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N )
        | ( X = one_one_Code_natural ) )
     => ( dvd_dvd_Code_natural @ X @ ( power_7079662738309270450atural @ X @ N ) ) ) ).

% dvd_power
thf(fact_4591_dvd__power,axiom,
    ! [N: nat,X: int] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N )
        | ( X = one_one_int ) )
     => ( dvd_dvd_int @ X @ ( power_power_int @ X @ N ) ) ) ).

% dvd_power
thf(fact_4592_dvd__power,axiom,
    ! [N: nat,X: nat] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N )
        | ( X = one_one_nat ) )
     => ( dvd_dvd_nat @ X @ ( power_power_nat @ X @ N ) ) ) ).

% dvd_power
thf(fact_4593_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_4594_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_4595_power__diff,axiom,
    ! [A: code_integer,N: nat,M: nat] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( ( power_8256067586552552935nteger @ A @ ( minus_minus_nat @ M @ N ) )
          = ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ A @ M ) @ ( power_8256067586552552935nteger @ A @ N ) ) ) ) ) ).

% power_diff
thf(fact_4596_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_4597_power__diff,axiom,
    ! [A: code_natural,N: nat,M: nat] :
      ( ( A != zero_z2226904508553997617atural )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( ( power_7079662738309270450atural @ A @ ( minus_minus_nat @ M @ N ) )
          = ( divide5121882707175180666atural @ ( power_7079662738309270450atural @ A @ M ) @ ( power_7079662738309270450atural @ A @ N ) ) ) ) ) ).

% power_diff
thf(fact_4598_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_4599_power__dvd__imp__le,axiom,
    ! [I2: nat,M: nat,N: nat] :
      ( ( dvd_dvd_nat @ ( power_power_nat @ I2 @ M ) @ ( power_power_nat @ I2 @ N ) )
     => ( ( ord_less_nat @ one_one_nat @ I2 )
       => ( ord_less_eq_nat @ M @ N ) ) ) ).

% power_dvd_imp_le
thf(fact_4600_Ints__odd__less__0,axiom,
    ! [A: code_integer] :
      ( ( member_Code_integer @ A @ ring_11222124179247155820nteger )
     => ( ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ A ) @ zero_z3403309356797280102nteger )
        = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ) ).

% Ints_odd_less_0
thf(fact_4601_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_4602_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_4603_fact__div__fact__le__pow,axiom,
    ! [R: nat,N: nat] :
      ( ( ord_less_eq_nat @ R @ N )
     => ( ord_less_eq_nat @ ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ R ) ) ) @ ( power_power_nat @ N @ R ) ) ) ).

% fact_div_fact_le_pow
thf(fact_4604_Ints__nonzero__abs__ge1,axiom,
    ! [X: code_integer] :
      ( ( member_Code_integer @ X @ ring_11222124179247155820nteger )
     => ( ( X != zero_z3403309356797280102nteger )
       => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( abs_abs_Code_integer @ X ) ) ) ) ).

% Ints_nonzero_abs_ge1
thf(fact_4605_Ints__nonzero__abs__ge1,axiom,
    ! [X: rat] :
      ( ( member_rat @ X @ ring_1_Ints_rat )
     => ( ( X != zero_zero_rat )
       => ( ord_less_eq_rat @ one_one_rat @ ( abs_abs_rat @ X ) ) ) ) ).

% Ints_nonzero_abs_ge1
thf(fact_4606_Ints__nonzero__abs__ge1,axiom,
    ! [X: int] :
      ( ( member_int @ X @ ring_1_Ints_int )
     => ( ( X != zero_zero_int )
       => ( ord_less_eq_int @ one_one_int @ ( abs_abs_int @ X ) ) ) ) ).

% Ints_nonzero_abs_ge1
thf(fact_4607_Ints__nonzero__abs__less1,axiom,
    ! [X: code_integer] :
      ( ( member_Code_integer @ X @ ring_11222124179247155820nteger )
     => ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ X ) @ one_one_Code_integer )
       => ( X = zero_z3403309356797280102nteger ) ) ) ).

% Ints_nonzero_abs_less1
thf(fact_4608_Ints__nonzero__abs__less1,axiom,
    ! [X: rat] :
      ( ( member_rat @ X @ ring_1_Ints_rat )
     => ( ( ord_less_rat @ ( abs_abs_rat @ X ) @ one_one_rat )
       => ( X = zero_zero_rat ) ) ) ).

% Ints_nonzero_abs_less1
thf(fact_4609_Ints__nonzero__abs__less1,axiom,
    ! [X: int] :
      ( ( member_int @ X @ ring_1_Ints_int )
     => ( ( ord_less_int @ ( abs_abs_int @ X ) @ one_one_int )
       => ( X = zero_zero_int ) ) ) ).

% Ints_nonzero_abs_less1
thf(fact_4610_Ints__eq__abs__less1,axiom,
    ! [X: code_integer,Y3: code_integer] :
      ( ( member_Code_integer @ X @ ring_11222124179247155820nteger )
     => ( ( member_Code_integer @ Y3 @ ring_11222124179247155820nteger )
       => ( ( X = Y3 )
          = ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X @ Y3 ) ) @ one_one_Code_integer ) ) ) ) ).

% Ints_eq_abs_less1
thf(fact_4611_Ints__eq__abs__less1,axiom,
    ! [X: rat,Y3: rat] :
      ( ( member_rat @ X @ ring_1_Ints_rat )
     => ( ( member_rat @ Y3 @ ring_1_Ints_rat )
       => ( ( X = Y3 )
          = ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X @ Y3 ) ) @ one_one_rat ) ) ) ) ).

% Ints_eq_abs_less1
thf(fact_4612_Ints__eq__abs__less1,axiom,
    ! [X: int,Y3: int] :
      ( ( member_int @ X @ ring_1_Ints_int )
     => ( ( member_int @ Y3 @ ring_1_Ints_int )
       => ( ( X = Y3 )
          = ( ord_less_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Y3 ) ) @ one_one_int ) ) ) ) ).

% Ints_eq_abs_less1
thf(fact_4613_power__strict__mono,axiom,
    ! [A: code_natural,B: code_natural,N: nat] :
      ( ( ord_le5570908160329646204atural @ A @ B )
     => ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ A )
       => ( ( ord_less_nat @ zero_zero_nat @ N )
         => ( ord_le5570908160329646204atural @ ( power_7079662738309270450atural @ A @ N ) @ ( power_7079662738309270450atural @ B @ N ) ) ) ) ) ).

% power_strict_mono
thf(fact_4614_power__strict__mono,axiom,
    ! [A: code_integer,B: code_integer,N: nat] :
      ( ( ord_le6747313008572928689nteger @ A @ B )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
       => ( ( ord_less_nat @ zero_zero_nat @ N )
         => ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A @ N ) @ ( power_8256067586552552935nteger @ B @ N ) ) ) ) ) ).

% power_strict_mono
thf(fact_4615_power__strict__mono,axiom,
    ! [A: rat,B: rat,N: nat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( 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 @ B @ N ) ) ) ) ) ).

% power_strict_mono
thf(fact_4616_power__strict__mono,axiom,
    ! [A: nat,B: nat,N: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( 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 @ B @ N ) ) ) ) ) ).

% power_strict_mono
thf(fact_4617_power__strict__mono,axiom,
    ! [A: int,B: int,N: nat] :
      ( ( ord_less_int @ A @ B )
     => ( ( 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 @ B @ N ) ) ) ) ) ).

% power_strict_mono
thf(fact_4618_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_4619_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_4620_power__diff__power__eq,axiom,
    ! [A: code_integer,N: nat,M: nat] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( ( ord_less_eq_nat @ N @ M )
         => ( ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ A @ M ) @ ( power_8256067586552552935nteger @ A @ N ) )
            = ( power_8256067586552552935nteger @ A @ ( minus_minus_nat @ M @ N ) ) ) )
        & ( ~ ( ord_less_eq_nat @ N @ M )
         => ( ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ A @ M ) @ ( power_8256067586552552935nteger @ A @ N ) )
            = ( divide6298287555418463151nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ A @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).

% power_diff_power_eq
thf(fact_4621_power__diff__power__eq,axiom,
    ! [A: code_natural,N: nat,M: nat] :
      ( ( A != zero_z2226904508553997617atural )
     => ( ( ( ord_less_eq_nat @ N @ M )
         => ( ( divide5121882707175180666atural @ ( power_7079662738309270450atural @ A @ M ) @ ( power_7079662738309270450atural @ A @ N ) )
            = ( power_7079662738309270450atural @ A @ ( minus_minus_nat @ M @ N ) ) ) )
        & ( ~ ( ord_less_eq_nat @ N @ M )
         => ( ( divide5121882707175180666atural @ ( power_7079662738309270450atural @ A @ M ) @ ( power_7079662738309270450atural @ A @ N ) )
            = ( divide5121882707175180666atural @ one_one_Code_natural @ ( power_7079662738309270450atural @ A @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).

% power_diff_power_eq
thf(fact_4622_power__eq__if,axiom,
    ( power_7079662738309270450atural
    = ( ^ [P6: code_natural,M2: nat] : ( if_Code_natural @ ( M2 = zero_zero_nat ) @ one_one_Code_natural @ ( times_2397367101498566445atural @ P6 @ ( power_7079662738309270450atural @ P6 @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_4623_power__eq__if,axiom,
    ( power_8256067586552552935nteger
    = ( ^ [P6: code_integer,M2: nat] : ( if_Code_integer @ ( M2 = zero_zero_nat ) @ one_one_Code_integer @ ( times_3573771949741848930nteger @ P6 @ ( power_8256067586552552935nteger @ P6 @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_4624_power__eq__if,axiom,
    ( power_power_rat
    = ( ^ [P6: rat,M2: nat] : ( if_rat @ ( M2 = zero_zero_nat ) @ one_one_rat @ ( times_times_rat @ P6 @ ( power_power_rat @ P6 @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_4625_power__eq__if,axiom,
    ( power_power_nat
    = ( ^ [P6: nat,M2: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ P6 @ ( power_power_nat @ P6 @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_4626_power__eq__if,axiom,
    ( power_power_int
    = ( ^ [P6: int,M2: nat] : ( if_int @ ( M2 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ P6 @ ( power_power_int @ P6 @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_4627_power__minus__mult,axiom,
    ! [N: nat,A: code_integer] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
        = ( power_8256067586552552935nteger @ A @ N ) ) ) ).

% power_minus_mult
thf(fact_4628_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_4629_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_4630_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_4631_frac__neg,axiom,
    ! [X: rat] :
      ( ( ( member_rat @ X @ ring_1_Ints_rat )
       => ( ( archimedean_frac_rat @ ( uminus_uminus_rat @ X ) )
          = zero_zero_rat ) )
      & ( ~ ( member_rat @ X @ ring_1_Ints_rat )
       => ( ( archimedean_frac_rat @ ( uminus_uminus_rat @ X ) )
          = ( minus_minus_rat @ one_one_rat @ ( archimedean_frac_rat @ X ) ) ) ) ) ).

% frac_neg
thf(fact_4632_gbinomial__negated__upper,axiom,
    ( gbinomial_rat
    = ( ^ [A2: 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 ) @ A2 ) @ one_one_rat ) @ K3 ) ) ) ) ).

% gbinomial_negated_upper
thf(fact_4633_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_4634_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_4635_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_4636_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_4637_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_4638_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_4639_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_4640_power__minus_H,axiom,
    ! [X: int,N: nat] :
      ( ( nO_MATCH_int_int @ one_one_int @ X )
     => ( ( power_power_int @ ( uminus_uminus_int @ X ) @ N )
        = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( power_power_int @ X @ N ) ) ) ) ).

% power_minus'
thf(fact_4641_power__minus_H,axiom,
    ! [X: code_integer,N: nat] :
      ( ( nO_MAT8252062027627875367nteger @ one_one_Code_integer @ X )
     => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ X ) @ N )
        = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ ( power_8256067586552552935nteger @ X @ N ) ) ) ) ).

% power_minus'
thf(fact_4642_power__minus_H,axiom,
    ! [X: rat,N: nat] :
      ( ( nO_MATCH_rat_rat @ one_one_rat @ X )
     => ( ( power_power_rat @ ( uminus_uminus_rat @ X ) @ N )
        = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( power_power_rat @ X @ N ) ) ) ) ).

% power_minus'
thf(fact_4643_ceiling__divide__lower,axiom,
    ! [Q4: rat,P5: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Q4 )
     => ( ord_less_rat @ ( times_times_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ ( divide_divide_rat @ P5 @ Q4 ) ) ) @ one_one_rat ) @ Q4 ) @ P5 ) ) ).

% ceiling_divide_lower
thf(fact_4644_le__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ ( divide_divide_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) @ B ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ zero_zero_rat ) ) ) ) ) ) ).

% le_divide_eq_numeral(2)
thf(fact_4645_divide__le__eq__numeral_I2_J,axiom,
    ! [B: rat,C: rat,W2: num] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ 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 @ W2 ) ) @ C ) @ B ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ) ) ) ) ).

% divide_le_eq_numeral(2)
thf(fact_4646_mult__ceiling__le__Ints,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( member_rat @ A @ ring_1_Ints_rat )
       => ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ ( archim2889992004027027881ng_rat @ ( times_times_rat @ A @ B ) ) ) @ ( ring_18347121197199848620nteger @ ( times_times_int @ ( archim2889992004027027881ng_rat @ A ) @ ( archim2889992004027027881ng_rat @ B ) ) ) ) ) ) ).

% mult_ceiling_le_Ints
thf(fact_4647_mult__ceiling__le__Ints,axiom,
    ! [A: rat,B: 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 @ B ) ) ) @ ( ring_1_of_int_rat @ ( times_times_int @ ( archim2889992004027027881ng_rat @ A ) @ ( archim2889992004027027881ng_rat @ B ) ) ) ) ) ) ).

% mult_ceiling_le_Ints
thf(fact_4648_mult__ceiling__le__Ints,axiom,
    ! [A: rat,B: 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 @ B ) ) ) @ ( ring_1_of_int_int @ ( times_times_int @ ( archim2889992004027027881ng_rat @ A ) @ ( archim2889992004027027881ng_rat @ B ) ) ) ) ) ) ).

% mult_ceiling_le_Ints
thf(fact_4649_numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( ( numeral_numeral_int @ M )
        = ( numeral_numeral_int @ N ) )
      = ( M = N ) ) ).

% numeral_eq_iff
thf(fact_4650_numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( ( numeral_numeral_nat @ M )
        = ( numeral_numeral_nat @ N ) )
      = ( M = N ) ) ).

% numeral_eq_iff
thf(fact_4651_numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( ( numera6620942414471956472nteger @ M )
        = ( numera6620942414471956472nteger @ N ) )
      = ( M = N ) ) ).

% numeral_eq_iff
thf(fact_4652_numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( ( numeral_numeral_rat @ M )
        = ( numeral_numeral_rat @ N ) )
      = ( M = N ) ) ).

% numeral_eq_iff
thf(fact_4653_numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( ( numera5444537566228673987atural @ M )
        = ( numera5444537566228673987atural @ N ) )
      = ( M = N ) ) ).

% numeral_eq_iff
thf(fact_4654_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_4655_of__int__eq__iff,axiom,
    ! [W2: int,Z2: int] :
      ( ( ( ring_18347121197199848620nteger @ W2 )
        = ( ring_18347121197199848620nteger @ Z2 ) )
      = ( W2 = Z2 ) ) ).

% of_int_eq_iff
thf(fact_4656_of__int__eq__iff,axiom,
    ! [W2: int,Z2: int] :
      ( ( ( ring_1_of_int_rat @ W2 )
        = ( ring_1_of_int_rat @ Z2 ) )
      = ( W2 = Z2 ) ) ).

% of_int_eq_iff
thf(fact_4657_of__int__eq__iff,axiom,
    ! [W2: int,Z2: int] :
      ( ( ( ring_1_of_int_int @ W2 )
        = ( ring_1_of_int_int @ Z2 ) )
      = ( W2 = Z2 ) ) ).

% of_int_eq_iff
thf(fact_4658_nat__numeral,axiom,
    ! [K: num] :
      ( ( nat2 @ ( numeral_numeral_int @ K ) )
      = ( numeral_numeral_nat @ K ) ) ).

% nat_numeral
thf(fact_4659_numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ N ) )
      = ( ord_less_eq_num @ M @ N ) ) ).

% numeral_le_iff
thf(fact_4660_numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_le1926595141338095240atural @ ( numera5444537566228673987atural @ M ) @ ( numera5444537566228673987atural @ N ) )
      = ( ord_less_eq_num @ M @ N ) ) ).

% numeral_le_iff
thf(fact_4661_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_4662_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_4663_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_4664_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_4665_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_4666_numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ N ) )
      = ( ord_less_num @ M @ N ) ) ).

% numeral_less_iff
thf(fact_4667_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_4668_numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_le5570908160329646204atural @ ( numera5444537566228673987atural @ M ) @ ( numera5444537566228673987atural @ N ) )
      = ( ord_less_num @ M @ N ) ) ).

% numeral_less_iff
thf(fact_4669_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W2: num,Z2: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( times_times_int @ ( numeral_numeral_int @ W2 ) @ Z2 ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W2 ) ) @ Z2 ) ) ).

% mult_numeral_left_semiring_numeral
thf(fact_4670_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W2: num,Z2: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( times_times_nat @ ( numeral_numeral_nat @ W2 ) @ Z2 ) )
      = ( times_times_nat @ ( numeral_numeral_nat @ ( times_times_num @ V @ W2 ) ) @ Z2 ) ) ).

% mult_numeral_left_semiring_numeral
thf(fact_4671_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W2: num,Z2: code_integer] :
      ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ V ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ W2 ) @ Z2 ) )
      = ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V @ W2 ) ) @ Z2 ) ) ).

% mult_numeral_left_semiring_numeral
thf(fact_4672_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W2: num,Z2: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ Z2 ) )
      = ( times_times_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W2 ) ) @ Z2 ) ) ).

% mult_numeral_left_semiring_numeral
thf(fact_4673_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W2: num,Z2: code_natural] :
      ( ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ V ) @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ W2 ) @ Z2 ) )
      = ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( times_times_num @ V @ W2 ) ) @ Z2 ) ) ).

% mult_numeral_left_semiring_numeral
thf(fact_4674_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_4675_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_4676_numeral__times__numeral,axiom,
    ! [M: num,N: num] :
      ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ N ) )
      = ( numera6620942414471956472nteger @ ( times_times_num @ M @ N ) ) ) ).

% numeral_times_numeral
thf(fact_4677_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_4678_numeral__times__numeral,axiom,
    ! [M: num,N: num] :
      ( ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ M ) @ ( numera5444537566228673987atural @ N ) )
      = ( numera5444537566228673987atural @ ( times_times_num @ M @ N ) ) ) ).

% numeral_times_numeral
thf(fact_4679_add__numeral__left,axiom,
    ! [V: num,W2: num,Z2: int] :
      ( ( plus_plus_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ ( numeral_numeral_int @ W2 ) @ Z2 ) )
      = ( plus_plus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W2 ) ) @ Z2 ) ) ).

% add_numeral_left
thf(fact_4680_add__numeral__left,axiom,
    ! [V: num,W2: num,Z2: nat] :
      ( ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ ( numeral_numeral_nat @ W2 ) @ Z2 ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ W2 ) ) @ Z2 ) ) ).

% add_numeral_left
thf(fact_4681_add__numeral__left,axiom,
    ! [V: num,W2: num,Z2: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ V ) @ ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ W2 ) @ Z2 ) )
      = ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ V @ W2 ) ) @ Z2 ) ) ).

% add_numeral_left
thf(fact_4682_add__numeral__left,axiom,
    ! [V: num,W2: num,Z2: rat] :
      ( ( plus_plus_rat @ ( numeral_numeral_rat @ V ) @ ( plus_plus_rat @ ( numeral_numeral_rat @ W2 ) @ Z2 ) )
      = ( plus_plus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ V @ W2 ) ) @ Z2 ) ) ).

% add_numeral_left
thf(fact_4683_add__numeral__left,axiom,
    ! [V: num,W2: num,Z2: code_natural] :
      ( ( plus_p4538020629002901425atural @ ( numera5444537566228673987atural @ V ) @ ( plus_p4538020629002901425atural @ ( numera5444537566228673987atural @ W2 ) @ Z2 ) )
      = ( plus_p4538020629002901425atural @ ( numera5444537566228673987atural @ ( plus_plus_num @ V @ W2 ) ) @ Z2 ) ) ).

% add_numeral_left
thf(fact_4684_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_4685_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_4686_numeral__plus__numeral,axiom,
    ! [M: num,N: num] :
      ( ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ N ) )
      = ( numera6620942414471956472nteger @ ( plus_plus_num @ M @ N ) ) ) ).

% numeral_plus_numeral
thf(fact_4687_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_4688_numeral__plus__numeral,axiom,
    ! [M: num,N: num] :
      ( ( plus_p4538020629002901425atural @ ( numera5444537566228673987atural @ M ) @ ( numera5444537566228673987atural @ N ) )
      = ( numera5444537566228673987atural @ ( plus_plus_num @ M @ N ) ) ) ).

% numeral_plus_numeral
thf(fact_4689_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_4690_neg__numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) )
        = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
      = ( M = N ) ) ).

% neg_numeral_eq_iff
thf(fact_4691_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_4692_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_8256067586552552935nteger @ zero_z3403309356797280102nteger @ ( numeral_numeral_nat @ K ) )
      = zero_z3403309356797280102nteger ) ).

% power_zero_numeral
thf(fact_4693_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_power_rat @ zero_zero_rat @ ( numeral_numeral_nat @ K ) )
      = zero_zero_rat ) ).

% power_zero_numeral
thf(fact_4694_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_7079662738309270450atural @ zero_z2226904508553997617atural @ ( numeral_numeral_nat @ K ) )
      = zero_z2226904508553997617atural ) ).

% power_zero_numeral
thf(fact_4695_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ K ) )
      = zero_zero_int ) ).

% power_zero_numeral
thf(fact_4696_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ K ) )
      = zero_zero_nat ) ).

% power_zero_numeral
thf(fact_4697_power__add__numeral,axiom,
    ! [A: code_integer,M: num,N: num] :
      ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ N ) ) )
      = ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).

% power_add_numeral
thf(fact_4698_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_4699_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_4700_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_4701_power__add__numeral2,axiom,
    ! [A: code_integer,M: num,N: num,B: code_integer] :
      ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
      = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B ) ) ).

% power_add_numeral2
thf(fact_4702_power__add__numeral2,axiom,
    ! [A: rat,M: num,N: num,B: rat] :
      ( ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
      = ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B ) ) ).

% power_add_numeral2
thf(fact_4703_power__add__numeral2,axiom,
    ! [A: nat,M: num,N: num,B: nat] :
      ( ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
      = ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B ) ) ).

% power_add_numeral2
thf(fact_4704_power__add__numeral2,axiom,
    ! [A: int,M: num,N: num,B: int] :
      ( ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
      = ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B ) ) ).

% power_add_numeral2
thf(fact_4705_of__nat__numeral,axiom,
    ! [N: num] :
      ( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N ) )
      = ( numeral_numeral_int @ N ) ) ).

% of_nat_numeral
thf(fact_4706_of__nat__numeral,axiom,
    ! [N: num] :
      ( ( semiri4939895301339042750nteger @ ( numeral_numeral_nat @ N ) )
      = ( numera6620942414471956472nteger @ N ) ) ).

% of_nat_numeral
thf(fact_4707_of__nat__numeral,axiom,
    ! [N: num] :
      ( ( semiri1316708129612266289at_nat @ ( numeral_numeral_nat @ N ) )
      = ( numeral_numeral_nat @ N ) ) ).

% of_nat_numeral
thf(fact_4708_of__nat__numeral,axiom,
    ! [N: num] :
      ( ( semiri681578069525770553at_rat @ ( numeral_numeral_nat @ N ) )
      = ( numeral_numeral_rat @ N ) ) ).

% of_nat_numeral
thf(fact_4709_of__nat__numeral,axiom,
    ! [N: num] :
      ( ( semiri3763490453095760265atural @ ( numeral_numeral_nat @ N ) )
      = ( numera5444537566228673987atural @ N ) ) ).

% of_nat_numeral
thf(fact_4710_abs__numeral,axiom,
    ! [N: num] :
      ( ( abs_abs_int @ ( numeral_numeral_int @ N ) )
      = ( numeral_numeral_int @ N ) ) ).

% abs_numeral
thf(fact_4711_abs__numeral,axiom,
    ! [N: num] :
      ( ( abs_abs_Code_integer @ ( numera6620942414471956472nteger @ N ) )
      = ( numera6620942414471956472nteger @ N ) ) ).

% abs_numeral
thf(fact_4712_abs__numeral,axiom,
    ! [N: num] :
      ( ( abs_abs_rat @ ( numeral_numeral_rat @ N ) )
      = ( numeral_numeral_rat @ N ) ) ).

% abs_numeral
thf(fact_4713_of__int__eq__numeral__iff,axiom,
    ! [Z2: int,N: num] :
      ( ( ( ring_1_of_int_int @ Z2 )
        = ( numeral_numeral_int @ N ) )
      = ( Z2
        = ( numeral_numeral_int @ N ) ) ) ).

% of_int_eq_numeral_iff
thf(fact_4714_of__int__eq__numeral__iff,axiom,
    ! [Z2: int,N: num] :
      ( ( ( ring_18347121197199848620nteger @ Z2 )
        = ( numera6620942414471956472nteger @ N ) )
      = ( Z2
        = ( numeral_numeral_int @ N ) ) ) ).

% of_int_eq_numeral_iff
thf(fact_4715_of__int__eq__numeral__iff,axiom,
    ! [Z2: int,N: num] :
      ( ( ( ring_1_of_int_rat @ Z2 )
        = ( numeral_numeral_rat @ N ) )
      = ( Z2
        = ( numeral_numeral_int @ N ) ) ) ).

% of_int_eq_numeral_iff
thf(fact_4716_of__int__numeral,axiom,
    ! [K: num] :
      ( ( ring_1_of_int_int @ ( numeral_numeral_int @ K ) )
      = ( numeral_numeral_int @ K ) ) ).

% of_int_numeral
thf(fact_4717_of__int__numeral,axiom,
    ! [K: num] :
      ( ( ring_18347121197199848620nteger @ ( numeral_numeral_int @ K ) )
      = ( numera6620942414471956472nteger @ K ) ) ).

% of_int_numeral
thf(fact_4718_of__int__numeral,axiom,
    ! [K: num] :
      ( ( ring_1_of_int_rat @ ( numeral_numeral_int @ K ) )
      = ( numeral_numeral_rat @ K ) ) ).

% of_int_numeral
thf(fact_4719_ceiling__numeral,axiom,
    ! [V: num] :
      ( ( archim2889992004027027881ng_rat @ ( numeral_numeral_rat @ V ) )
      = ( numeral_numeral_int @ V ) ) ).

% ceiling_numeral
thf(fact_4720_euclidean__size__numeral,axiom,
    ! [K: num] :
      ( ( euclid6377331345833325938nteger @ ( numera6620942414471956472nteger @ K ) )
      = ( numeral_numeral_nat @ K ) ) ).

% euclidean_size_numeral
thf(fact_4721_euclidean__size__numeral,axiom,
    ! [K: num] :
      ( ( euclid4774559944035922753ze_int @ ( numeral_numeral_int @ K ) )
      = ( numeral_numeral_nat @ K ) ) ).

% euclidean_size_numeral
thf(fact_4722_euclidean__size__numeral,axiom,
    ! [K: num] :
      ( ( euclid4777050414544973029ze_nat @ ( numeral_numeral_nat @ K ) )
      = ( numeral_numeral_nat @ K ) ) ).

% euclidean_size_numeral
thf(fact_4723_euclidean__size__numeral,axiom,
    ! [K: num] :
      ( ( euclid5200926497590043453atural @ ( numera5444537566228673987atural @ K ) )
      = ( numeral_numeral_nat @ K ) ) ).

% euclidean_size_numeral
thf(fact_4724_of__int__fact,axiom,
    ! [N: nat] :
      ( ( ring_1_of_int_rat @ ( semiri1406184849735516958ct_int @ N ) )
      = ( semiri773545260158071498ct_rat @ N ) ) ).

% of_int_fact
thf(fact_4725_of__int__fact,axiom,
    ! [N: nat] :
      ( ( ring_18347121197199848620nteger @ ( semiri1406184849735516958ct_int @ N ) )
      = ( semiri3624122377584611663nteger @ N ) ) ).

% of_int_fact
thf(fact_4726_of__int__fact,axiom,
    ! [N: nat] :
      ( ( ring_1_of_int_int @ ( semiri1406184849735516958ct_int @ N ) )
      = ( semiri1406184849735516958ct_int @ N ) ) ).

% of_int_fact
thf(fact_4727_ceiling__of__int,axiom,
    ! [Z2: int] :
      ( ( archim2889992004027027881ng_rat @ ( ring_1_of_int_rat @ Z2 ) )
      = Z2 ) ).

% ceiling_of_int
thf(fact_4728_round__numeral,axiom,
    ! [N: num] :
      ( ( archim7778729529865785530nd_rat @ ( numeral_numeral_rat @ N ) )
      = ( numeral_numeral_int @ N ) ) ).

% round_numeral
thf(fact_4729_round__of__int,axiom,
    ! [N: int] :
      ( ( archim7778729529865785530nd_rat @ ( ring_1_of_int_rat @ N ) )
      = N ) ).

% round_of_int
thf(fact_4730_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_4731_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_4732_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_4733_distrib__left__numeral,axiom,
    ! [V: num,B: int,C: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ B @ C ) )
      = ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ V ) @ B ) @ ( times_times_int @ ( numeral_numeral_int @ V ) @ C ) ) ) ).

% distrib_left_numeral
thf(fact_4734_distrib__left__numeral,axiom,
    ! [V: num,B: nat,C: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ B @ C ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ B ) @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ C ) ) ) ).

% distrib_left_numeral
thf(fact_4735_distrib__left__numeral,axiom,
    ! [V: num,B: code_integer,C: code_integer] :
      ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ V ) @ ( plus_p5714425477246183910nteger @ B @ C ) )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ V ) @ B ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ V ) @ C ) ) ) ).

% distrib_left_numeral
thf(fact_4736_distrib__left__numeral,axiom,
    ! [V: num,B: rat,C: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( plus_plus_rat @ B @ C ) )
      = ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ B ) @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ C ) ) ) ).

% distrib_left_numeral
thf(fact_4737_distrib__left__numeral,axiom,
    ! [V: num,B: code_natural,C: code_natural] :
      ( ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ V ) @ ( plus_p4538020629002901425atural @ B @ C ) )
      = ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ V ) @ B ) @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ V ) @ C ) ) ) ).

% distrib_left_numeral
thf(fact_4738_distrib__right__numeral,axiom,
    ! [A: int,B: int,V: num] :
      ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ ( numeral_numeral_int @ V ) )
      = ( plus_plus_int @ ( times_times_int @ A @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ B @ ( numeral_numeral_int @ V ) ) ) ) ).

% distrib_right_numeral
thf(fact_4739_distrib__right__numeral,axiom,
    ! [A: nat,B: nat,V: num] :
      ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ ( numeral_numeral_nat @ V ) )
      = ( plus_plus_nat @ ( times_times_nat @ A @ ( numeral_numeral_nat @ V ) ) @ ( times_times_nat @ B @ ( numeral_numeral_nat @ V ) ) ) ) ).

% distrib_right_numeral
thf(fact_4740_distrib__right__numeral,axiom,
    ! [A: code_integer,B: code_integer,V: num] :
      ( ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ ( numera6620942414471956472nteger @ V ) )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ ( numera6620942414471956472nteger @ V ) ) @ ( times_3573771949741848930nteger @ B @ ( numera6620942414471956472nteger @ V ) ) ) ) ).

% distrib_right_numeral
thf(fact_4741_distrib__right__numeral,axiom,
    ! [A: rat,B: rat,V: num] :
      ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ ( numeral_numeral_rat @ V ) )
      = ( plus_plus_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ B @ ( numeral_numeral_rat @ V ) ) ) ) ).

% distrib_right_numeral
thf(fact_4742_distrib__right__numeral,axiom,
    ! [A: code_natural,B: code_natural,V: num] :
      ( ( times_2397367101498566445atural @ ( plus_p4538020629002901425atural @ A @ B ) @ ( numera5444537566228673987atural @ V ) )
      = ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ A @ ( numera5444537566228673987atural @ V ) ) @ ( times_2397367101498566445atural @ B @ ( numera5444537566228673987atural @ V ) ) ) ) ).

% distrib_right_numeral
thf(fact_4743_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_4744_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_4745_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_4746_right__diff__distrib__numeral,axiom,
    ! [V: num,B: int,C: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( minus_minus_int @ B @ C ) )
      = ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ V ) @ B ) @ ( times_times_int @ ( numeral_numeral_int @ V ) @ C ) ) ) ).

% right_diff_distrib_numeral
thf(fact_4747_right__diff__distrib__numeral,axiom,
    ! [V: num,B: code_integer,C: code_integer] :
      ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ V ) @ ( minus_8373710615458151222nteger @ B @ C ) )
      = ( minus_8373710615458151222nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ V ) @ B ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ V ) @ C ) ) ) ).

% right_diff_distrib_numeral
thf(fact_4748_right__diff__distrib__numeral,axiom,
    ! [V: num,B: rat,C: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( minus_minus_rat @ B @ C ) )
      = ( minus_minus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ B ) @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ C ) ) ) ).

% right_diff_distrib_numeral
thf(fact_4749_left__diff__distrib__numeral,axiom,
    ! [A: int,B: int,V: num] :
      ( ( times_times_int @ ( minus_minus_int @ A @ B ) @ ( numeral_numeral_int @ V ) )
      = ( minus_minus_int @ ( times_times_int @ A @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ B @ ( numeral_numeral_int @ V ) ) ) ) ).

% left_diff_distrib_numeral
thf(fact_4750_left__diff__distrib__numeral,axiom,
    ! [A: code_integer,B: code_integer,V: num] :
      ( ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ ( numera6620942414471956472nteger @ V ) )
      = ( minus_8373710615458151222nteger @ ( times_3573771949741848930nteger @ A @ ( numera6620942414471956472nteger @ V ) ) @ ( times_3573771949741848930nteger @ B @ ( numera6620942414471956472nteger @ V ) ) ) ) ).

% left_diff_distrib_numeral
thf(fact_4751_left__diff__distrib__numeral,axiom,
    ! [A: rat,B: rat,V: num] :
      ( ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ ( numeral_numeral_rat @ V ) )
      = ( minus_minus_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ B @ ( numeral_numeral_rat @ V ) ) ) ) ).

% left_diff_distrib_numeral
thf(fact_4752_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_4753_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_4754_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_4755_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_4756_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_4757_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_4758_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_4759_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_4760_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_4761_semiring__norm_I172_J,axiom,
    ! [V: num,W2: num,Y3: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W2 ) ) @ Y3 ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W2 ) ) @ Y3 ) ) ).

% semiring_norm(172)
thf(fact_4762_semiring__norm_I172_J,axiom,
    ! [V: num,W2: num,Y3: code_integer] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) @ ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W2 ) ) @ Y3 ) )
      = ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V @ W2 ) ) @ Y3 ) ) ).

% semiring_norm(172)
thf(fact_4763_semiring__norm_I172_J,axiom,
    ! [V: num,W2: num,Y3: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ Y3 ) )
      = ( times_times_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W2 ) ) @ Y3 ) ) ).

% semiring_norm(172)
thf(fact_4764_semiring__norm_I171_J,axiom,
    ! [V: num,W2: num,Y3: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W2 ) ) @ Y3 ) )
      = ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W2 ) ) ) @ Y3 ) ) ).

% semiring_norm(171)
thf(fact_4765_semiring__norm_I171_J,axiom,
    ! [V: num,W2: num,Y3: code_integer] :
      ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ V ) @ ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W2 ) ) @ Y3 ) )
      = ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V @ W2 ) ) ) @ Y3 ) ) ).

% semiring_norm(171)
thf(fact_4766_semiring__norm_I171_J,axiom,
    ! [V: num,W2: num,Y3: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ Y3 ) )
      = ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W2 ) ) ) @ Y3 ) ) ).

% semiring_norm(171)
thf(fact_4767_semiring__norm_I170_J,axiom,
    ! [V: num,W2: num,Y3: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ ( numeral_numeral_int @ W2 ) @ Y3 ) )
      = ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W2 ) ) ) @ Y3 ) ) ).

% semiring_norm(170)
thf(fact_4768_semiring__norm_I170_J,axiom,
    ! [V: num,W2: num,Y3: code_integer] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ W2 ) @ Y3 ) )
      = ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V @ W2 ) ) ) @ Y3 ) ) ).

% semiring_norm(170)
thf(fact_4769_semiring__norm_I170_J,axiom,
    ! [V: num,W2: num,Y3: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ Y3 ) )
      = ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W2 ) ) ) @ Y3 ) ) ).

% semiring_norm(170)
thf(fact_4770_semiring__norm_I168_J,axiom,
    ! [V: num,W2: num,Y3: int] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W2 ) ) @ Y3 ) )
      = ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W2 ) ) ) @ Y3 ) ) ).

% semiring_norm(168)
thf(fact_4771_semiring__norm_I168_J,axiom,
    ! [V: num,W2: num,Y3: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W2 ) ) @ Y3 ) )
      = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ V @ W2 ) ) ) @ Y3 ) ) ).

% semiring_norm(168)
thf(fact_4772_semiring__norm_I168_J,axiom,
    ! [V: num,W2: num,Y3: rat] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ Y3 ) )
      = ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ V @ W2 ) ) ) @ Y3 ) ) ).

% semiring_norm(168)
thf(fact_4773_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_4774_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_4775_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_4776_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_4777_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_4778_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_4779_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_4780_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_4781_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_4782_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_4783_abs__neg__numeral,axiom,
    ! [N: num] :
      ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
      = ( numera6620942414471956472nteger @ N ) ) ).

% abs_neg_numeral
thf(fact_4784_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_4785_of__int__0,axiom,
    ( ( ring_18347121197199848620nteger @ zero_zero_int )
    = zero_z3403309356797280102nteger ) ).

% of_int_0
thf(fact_4786_of__int__0,axiom,
    ( ( ring_1_of_int_rat @ zero_zero_int )
    = zero_zero_rat ) ).

% of_int_0
thf(fact_4787_of__int__0,axiom,
    ( ( ring_1_of_int_int @ zero_zero_int )
    = zero_zero_int ) ).

% of_int_0
thf(fact_4788_of__int__0__eq__iff,axiom,
    ! [Z2: int] :
      ( ( zero_z3403309356797280102nteger
        = ( ring_18347121197199848620nteger @ Z2 ) )
      = ( Z2 = zero_zero_int ) ) ).

% of_int_0_eq_iff
thf(fact_4789_of__int__0__eq__iff,axiom,
    ! [Z2: int] :
      ( ( zero_zero_rat
        = ( ring_1_of_int_rat @ Z2 ) )
      = ( Z2 = zero_zero_int ) ) ).

% of_int_0_eq_iff
thf(fact_4790_of__int__0__eq__iff,axiom,
    ! [Z2: int] :
      ( ( zero_zero_int
        = ( ring_1_of_int_int @ Z2 ) )
      = ( Z2 = zero_zero_int ) ) ).

% of_int_0_eq_iff
thf(fact_4791_of__int__eq__0__iff,axiom,
    ! [Z2: int] :
      ( ( ( ring_18347121197199848620nteger @ Z2 )
        = zero_z3403309356797280102nteger )
      = ( Z2 = zero_zero_int ) ) ).

% of_int_eq_0_iff
thf(fact_4792_of__int__eq__0__iff,axiom,
    ! [Z2: int] :
      ( ( ( ring_1_of_int_rat @ Z2 )
        = zero_zero_rat )
      = ( Z2 = zero_zero_int ) ) ).

% of_int_eq_0_iff
thf(fact_4793_of__int__eq__0__iff,axiom,
    ! [Z2: int] :
      ( ( ( ring_1_of_int_int @ Z2 )
        = zero_zero_int )
      = ( Z2 = zero_zero_int ) ) ).

% of_int_eq_0_iff
thf(fact_4794_of__int__le__iff,axiom,
    ! [W2: int,Z2: int] :
      ( ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ W2 ) @ ( ring_18347121197199848620nteger @ Z2 ) )
      = ( ord_less_eq_int @ W2 @ Z2 ) ) ).

% of_int_le_iff
thf(fact_4795_of__int__le__iff,axiom,
    ! [W2: int,Z2: int] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ W2 ) @ ( ring_1_of_int_rat @ Z2 ) )
      = ( ord_less_eq_int @ W2 @ Z2 ) ) ).

% of_int_le_iff
thf(fact_4796_of__int__le__iff,axiom,
    ! [W2: int,Z2: int] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ W2 ) @ ( ring_1_of_int_int @ Z2 ) )
      = ( ord_less_eq_int @ W2 @ Z2 ) ) ).

% of_int_le_iff
thf(fact_4797_of__int__less__iff,axiom,
    ! [W2: int,Z2: int] :
      ( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ W2 ) @ ( ring_18347121197199848620nteger @ Z2 ) )
      = ( ord_less_int @ W2 @ Z2 ) ) ).

% of_int_less_iff
thf(fact_4798_of__int__less__iff,axiom,
    ! [W2: int,Z2: int] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ W2 ) @ ( ring_1_of_int_rat @ Z2 ) )
      = ( ord_less_int @ W2 @ Z2 ) ) ).

% of_int_less_iff
thf(fact_4799_of__int__less__iff,axiom,
    ! [W2: int,Z2: int] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ W2 ) @ ( ring_1_of_int_int @ Z2 ) )
      = ( ord_less_int @ W2 @ Z2 ) ) ).

% of_int_less_iff
thf(fact_4800_of__int__eq__1__iff,axiom,
    ! [Z2: int] :
      ( ( ( ring_18347121197199848620nteger @ Z2 )
        = one_one_Code_integer )
      = ( Z2 = one_one_int ) ) ).

% of_int_eq_1_iff
thf(fact_4801_of__int__eq__1__iff,axiom,
    ! [Z2: int] :
      ( ( ( ring_1_of_int_rat @ Z2 )
        = one_one_rat )
      = ( Z2 = one_one_int ) ) ).

% of_int_eq_1_iff
thf(fact_4802_of__int__eq__1__iff,axiom,
    ! [Z2: int] :
      ( ( ( ring_1_of_int_int @ Z2 )
        = one_one_int )
      = ( Z2 = one_one_int ) ) ).

% of_int_eq_1_iff
thf(fact_4803_of__int__1,axiom,
    ( ( ring_18347121197199848620nteger @ one_one_int )
    = one_one_Code_integer ) ).

% of_int_1
thf(fact_4804_of__int__1,axiom,
    ( ( ring_1_of_int_rat @ one_one_int )
    = one_one_rat ) ).

% of_int_1
thf(fact_4805_of__int__1,axiom,
    ( ( ring_1_of_int_int @ one_one_int )
    = one_one_int ) ).

% of_int_1
thf(fact_4806_of__int__mult,axiom,
    ! [W2: int,Z2: int] :
      ( ( ring_18347121197199848620nteger @ ( times_times_int @ W2 @ Z2 ) )
      = ( times_3573771949741848930nteger @ ( ring_18347121197199848620nteger @ W2 ) @ ( ring_18347121197199848620nteger @ Z2 ) ) ) ).

% of_int_mult
thf(fact_4807_of__int__mult,axiom,
    ! [W2: int,Z2: int] :
      ( ( ring_1_of_int_rat @ ( times_times_int @ W2 @ Z2 ) )
      = ( times_times_rat @ ( ring_1_of_int_rat @ W2 ) @ ( ring_1_of_int_rat @ Z2 ) ) ) ).

% of_int_mult
thf(fact_4808_of__int__mult,axiom,
    ! [W2: int,Z2: int] :
      ( ( ring_1_of_int_int @ ( times_times_int @ W2 @ Z2 ) )
      = ( times_times_int @ ( ring_1_of_int_int @ W2 ) @ ( ring_1_of_int_int @ Z2 ) ) ) ).

% of_int_mult
thf(fact_4809_of__int__add,axiom,
    ! [W2: int,Z2: int] :
      ( ( ring_18347121197199848620nteger @ ( plus_plus_int @ W2 @ Z2 ) )
      = ( plus_p5714425477246183910nteger @ ( ring_18347121197199848620nteger @ W2 ) @ ( ring_18347121197199848620nteger @ Z2 ) ) ) ).

% of_int_add
thf(fact_4810_of__int__add,axiom,
    ! [W2: int,Z2: int] :
      ( ( ring_1_of_int_rat @ ( plus_plus_int @ W2 @ Z2 ) )
      = ( plus_plus_rat @ ( ring_1_of_int_rat @ W2 ) @ ( ring_1_of_int_rat @ Z2 ) ) ) ).

% of_int_add
thf(fact_4811_of__int__add,axiom,
    ! [W2: int,Z2: int] :
      ( ( ring_1_of_int_int @ ( plus_plus_int @ W2 @ Z2 ) )
      = ( plus_plus_int @ ( ring_1_of_int_int @ W2 ) @ ( ring_1_of_int_int @ Z2 ) ) ) ).

% of_int_add
thf(fact_4812_of__int__minus,axiom,
    ! [Z2: int] :
      ( ( ring_1_of_int_int @ ( uminus_uminus_int @ Z2 ) )
      = ( uminus_uminus_int @ ( ring_1_of_int_int @ Z2 ) ) ) ).

% of_int_minus
thf(fact_4813_of__int__minus,axiom,
    ! [Z2: int] :
      ( ( ring_18347121197199848620nteger @ ( uminus_uminus_int @ Z2 ) )
      = ( uminus1351360451143612070nteger @ ( ring_18347121197199848620nteger @ Z2 ) ) ) ).

% of_int_minus
thf(fact_4814_of__int__minus,axiom,
    ! [Z2: int] :
      ( ( ring_1_of_int_rat @ ( uminus_uminus_int @ Z2 ) )
      = ( uminus_uminus_rat @ ( ring_1_of_int_rat @ Z2 ) ) ) ).

% of_int_minus
thf(fact_4815_nat__neg__numeral,axiom,
    ! [K: num] :
      ( ( nat2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
      = zero_zero_nat ) ).

% nat_neg_numeral
thf(fact_4816_of__int__diff,axiom,
    ! [W2: int,Z2: int] :
      ( ( ring_18347121197199848620nteger @ ( minus_minus_int @ W2 @ Z2 ) )
      = ( minus_8373710615458151222nteger @ ( ring_18347121197199848620nteger @ W2 ) @ ( ring_18347121197199848620nteger @ Z2 ) ) ) ).

% of_int_diff
thf(fact_4817_of__int__diff,axiom,
    ! [W2: int,Z2: int] :
      ( ( ring_1_of_int_rat @ ( minus_minus_int @ W2 @ Z2 ) )
      = ( minus_minus_rat @ ( ring_1_of_int_rat @ W2 ) @ ( ring_1_of_int_rat @ Z2 ) ) ) ).

% of_int_diff
thf(fact_4818_of__int__diff,axiom,
    ! [W2: int,Z2: int] :
      ( ( ring_1_of_int_int @ ( minus_minus_int @ W2 @ Z2 ) )
      = ( minus_minus_int @ ( ring_1_of_int_int @ W2 ) @ ( ring_1_of_int_int @ Z2 ) ) ) ).

% of_int_diff
thf(fact_4819_division__segment__numeral,axiom,
    ! [K: num] :
      ( ( euclid3398187327856392827nt_nat @ ( numeral_numeral_nat @ K ) )
      = one_one_nat ) ).

% division_segment_numeral
thf(fact_4820_division__segment__numeral,axiom,
    ! [K: num] :
      ( ( euclid3395696857347342551nt_int @ ( numeral_numeral_int @ K ) )
      = one_one_int ) ).

% division_segment_numeral
thf(fact_4821_division__segment__numeral,axiom,
    ! [K: num] :
      ( ( euclid6289375185220004616nteger @ ( numera6620942414471956472nteger @ K ) )
      = one_one_Code_integer ) ).

% division_segment_numeral
thf(fact_4822_division__segment__numeral,axiom,
    ! [K: num] :
      ( ( euclid5112970336976722131atural @ ( numera5444537566228673987atural @ K ) )
      = one_one_Code_natural ) ).

% division_segment_numeral
thf(fact_4823_of__int__of__nat__eq,axiom,
    ! [N: nat] :
      ( ( ring_1_of_int_int @ ( semiri1314217659103216013at_int @ N ) )
      = ( semiri1314217659103216013at_int @ N ) ) ).

% of_int_of_nat_eq
thf(fact_4824_of__int__of__nat__eq,axiom,
    ! [N: nat] :
      ( ( ring_18347121197199848620nteger @ ( semiri1314217659103216013at_int @ N ) )
      = ( semiri4939895301339042750nteger @ N ) ) ).

% of_int_of_nat_eq
thf(fact_4825_of__int__of__nat__eq,axiom,
    ! [N: nat] :
      ( ( ring_1_of_int_rat @ ( semiri1314217659103216013at_int @ N ) )
      = ( semiri681578069525770553at_rat @ N ) ) ).

% of_int_of_nat_eq
thf(fact_4826_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_4827_of__int__abs,axiom,
    ! [X: int] :
      ( ( ring_18347121197199848620nteger @ ( abs_abs_int @ X ) )
      = ( abs_abs_Code_integer @ ( ring_18347121197199848620nteger @ X ) ) ) ).

% of_int_abs
thf(fact_4828_of__int__abs,axiom,
    ! [X: int] :
      ( ( ring_1_of_int_rat @ ( abs_abs_int @ X ) )
      = ( abs_abs_rat @ ( ring_1_of_int_rat @ X ) ) ) ).

% of_int_abs
thf(fact_4829_of__int__abs,axiom,
    ! [X: int] :
      ( ( ring_1_of_int_int @ ( abs_abs_int @ X ) )
      = ( abs_abs_int @ ( ring_1_of_int_int @ X ) ) ) ).

% of_int_abs
thf(fact_4830_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W2: nat] :
      ( ( ( ring_18347121197199848620nteger @ X )
        = ( power_8256067586552552935nteger @ ( ring_18347121197199848620nteger @ B ) @ W2 ) )
      = ( X
        = ( power_power_int @ B @ W2 ) ) ) ).

% of_int_power_eq_of_int_cancel_iff
thf(fact_4831_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W2: nat] :
      ( ( ( ring_1_of_int_rat @ X )
        = ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W2 ) )
      = ( X
        = ( power_power_int @ B @ W2 ) ) ) ).

% of_int_power_eq_of_int_cancel_iff
thf(fact_4832_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W2: nat] :
      ( ( ( ring_1_of_int_int @ X )
        = ( power_power_int @ ( ring_1_of_int_int @ B ) @ W2 ) )
      = ( X
        = ( power_power_int @ B @ W2 ) ) ) ).

% of_int_power_eq_of_int_cancel_iff
thf(fact_4833_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W2: nat,X: int] :
      ( ( ( power_8256067586552552935nteger @ ( ring_18347121197199848620nteger @ B ) @ W2 )
        = ( ring_18347121197199848620nteger @ X ) )
      = ( ( power_power_int @ B @ W2 )
        = X ) ) ).

% of_int_eq_of_int_power_cancel_iff
thf(fact_4834_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W2: nat,X: int] :
      ( ( ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W2 )
        = ( ring_1_of_int_rat @ X ) )
      = ( ( power_power_int @ B @ W2 )
        = X ) ) ).

% of_int_eq_of_int_power_cancel_iff
thf(fact_4835_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W2: nat,X: int] :
      ( ( ( power_power_int @ ( ring_1_of_int_int @ B ) @ W2 )
        = ( ring_1_of_int_int @ X ) )
      = ( ( power_power_int @ B @ W2 )
        = X ) ) ).

% of_int_eq_of_int_power_cancel_iff
thf(fact_4836_of__int__power,axiom,
    ! [Z2: int,N: nat] :
      ( ( ring_18347121197199848620nteger @ ( power_power_int @ Z2 @ N ) )
      = ( power_8256067586552552935nteger @ ( ring_18347121197199848620nteger @ Z2 ) @ N ) ) ).

% of_int_power
thf(fact_4837_of__int__power,axiom,
    ! [Z2: int,N: nat] :
      ( ( ring_1_of_int_rat @ ( power_power_int @ Z2 @ N ) )
      = ( power_power_rat @ ( ring_1_of_int_rat @ Z2 ) @ N ) ) ).

% of_int_power
thf(fact_4838_of__int__power,axiom,
    ! [Z2: int,N: nat] :
      ( ( ring_1_of_int_int @ ( power_power_int @ Z2 @ N ) )
      = ( power_power_int @ ( ring_1_of_int_int @ Z2 ) @ N ) ) ).

% of_int_power
thf(fact_4839_frac__of__int,axiom,
    ! [Z2: int] :
      ( ( archimedean_frac_rat @ ( ring_1_of_int_rat @ Z2 ) )
      = zero_zero_rat ) ).

% frac_of_int
thf(fact_4840_nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y3: int,X: num,N: nat] :
      ( ( ( nat2 @ Y3 )
        = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) )
      = ( Y3
        = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).

% nat_eq_numeral_power_cancel_iff
thf(fact_4841_numeral__power__eq__nat__cancel__iff,axiom,
    ! [X: num,N: nat,Y3: int] :
      ( ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
        = ( nat2 @ Y3 ) )
      = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
        = Y3 ) ) ).

% numeral_power_eq_nat_cancel_iff
thf(fact_4842_of__int__of__bool,axiom,
    ! [P: $o] :
      ( ( ring_18347121197199848620nteger @ ( zero_n2684676970156552555ol_int @ P ) )
      = ( zero_n356916108424825756nteger @ P ) ) ).

% of_int_of_bool
thf(fact_4843_of__int__of__bool,axiom,
    ! [P: $o] :
      ( ( ring_1_of_int_rat @ ( zero_n2684676970156552555ol_int @ P ) )
      = ( zero_n2052037380579107095ol_rat @ P ) ) ).

% of_int_of_bool
thf(fact_4844_of__int__of__bool,axiom,
    ! [P: $o] :
      ( ( ring_1_of_int_int @ ( zero_n2684676970156552555ol_int @ P ) )
      = ( zero_n2684676970156552555ol_int @ P ) ) ).

% of_int_of_bool
thf(fact_4845_le__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B: rat,W2: num] :
      ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W2 ) ) )
      = ( ord_less_eq_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W2 ) ) @ B ) ) ).

% le_divide_eq_numeral1(1)
thf(fact_4846_divide__le__eq__numeral1_I1_J,axiom,
    ! [B: rat,W2: num,A: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W2 ) ) @ A )
      = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W2 ) ) ) ) ).

% divide_le_eq_numeral1(1)
thf(fact_4847_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B: rat,W2: num,A: rat] :
      ( ( ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W2 ) )
        = A )
      = ( ( ( ( numeral_numeral_rat @ W2 )
           != zero_zero_rat )
         => ( B
            = ( times_times_rat @ A @ ( numeral_numeral_rat @ W2 ) ) ) )
        & ( ( ( numeral_numeral_rat @ W2 )
            = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% divide_eq_eq_numeral1(1)
thf(fact_4848_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B: rat,W2: num] :
      ( ( A
        = ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W2 ) ) )
      = ( ( ( ( numeral_numeral_rat @ W2 )
           != zero_zero_rat )
         => ( ( times_times_rat @ A @ ( numeral_numeral_rat @ W2 ) )
            = B ) )
        & ( ( ( numeral_numeral_rat @ W2 )
            = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral1(1)
thf(fact_4849_less__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B: rat,W2: num] :
      ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W2 ) ) )
      = ( ord_less_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W2 ) ) @ B ) ) ).

% less_divide_eq_numeral1(1)
thf(fact_4850_divide__less__eq__numeral1_I1_J,axiom,
    ! [B: rat,W2: num,A: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W2 ) ) @ A )
      = ( ord_less_rat @ B @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W2 ) ) ) ) ).

% divide_less_eq_numeral1(1)
thf(fact_4851_of__int__le__numeral__iff,axiom,
    ! [Z2: int,N: num] :
      ( ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ Z2 ) @ ( numera6620942414471956472nteger @ N ) )
      = ( ord_less_eq_int @ Z2 @ ( numeral_numeral_int @ N ) ) ) ).

% of_int_le_numeral_iff
thf(fact_4852_of__int__le__numeral__iff,axiom,
    ! [Z2: int,N: num] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z2 ) @ ( numeral_numeral_rat @ N ) )
      = ( ord_less_eq_int @ Z2 @ ( numeral_numeral_int @ N ) ) ) ).

% of_int_le_numeral_iff
thf(fact_4853_of__int__le__numeral__iff,axiom,
    ! [Z2: int,N: num] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z2 ) @ ( numeral_numeral_int @ N ) )
      = ( ord_less_eq_int @ Z2 @ ( numeral_numeral_int @ N ) ) ) ).

% of_int_le_numeral_iff
thf(fact_4854_of__int__numeral__le__iff,axiom,
    ! [N: num,Z2: int] :
      ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ N ) @ ( ring_18347121197199848620nteger @ Z2 ) )
      = ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z2 ) ) ).

% of_int_numeral_le_iff
thf(fact_4855_of__int__numeral__le__iff,axiom,
    ! [N: num,Z2: int] :
      ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ N ) @ ( ring_1_of_int_rat @ Z2 ) )
      = ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z2 ) ) ).

% of_int_numeral_le_iff
thf(fact_4856_of__int__numeral__le__iff,axiom,
    ! [N: num,Z2: int] :
      ( ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ ( ring_1_of_int_int @ Z2 ) )
      = ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z2 ) ) ).

% of_int_numeral_le_iff
thf(fact_4857_of__int__numeral__less__iff,axiom,
    ! [N: num,Z2: int] :
      ( ( ord_less_int @ ( numeral_numeral_int @ N ) @ ( ring_1_of_int_int @ Z2 ) )
      = ( ord_less_int @ ( numeral_numeral_int @ N ) @ Z2 ) ) ).

% of_int_numeral_less_iff
thf(fact_4858_of__int__numeral__less__iff,axiom,
    ! [N: num,Z2: int] :
      ( ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ N ) @ ( ring_18347121197199848620nteger @ Z2 ) )
      = ( ord_less_int @ ( numeral_numeral_int @ N ) @ Z2 ) ) ).

% of_int_numeral_less_iff
thf(fact_4859_of__int__numeral__less__iff,axiom,
    ! [N: num,Z2: int] :
      ( ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ ( ring_1_of_int_rat @ Z2 ) )
      = ( ord_less_int @ ( numeral_numeral_int @ N ) @ Z2 ) ) ).

% of_int_numeral_less_iff
thf(fact_4860_of__int__less__numeral__iff,axiom,
    ! [Z2: int,N: num] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ Z2 ) @ ( numeral_numeral_int @ N ) )
      = ( ord_less_int @ Z2 @ ( numeral_numeral_int @ N ) ) ) ).

% of_int_less_numeral_iff
thf(fact_4861_of__int__less__numeral__iff,axiom,
    ! [Z2: int,N: num] :
      ( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ Z2 ) @ ( numera6620942414471956472nteger @ N ) )
      = ( ord_less_int @ Z2 @ ( numeral_numeral_int @ N ) ) ) ).

% of_int_less_numeral_iff
thf(fact_4862_of__int__less__numeral__iff,axiom,
    ! [Z2: int,N: num] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z2 ) @ ( numeral_numeral_rat @ N ) )
      = ( ord_less_int @ Z2 @ ( numeral_numeral_int @ N ) ) ) ).

% of_int_less_numeral_iff
thf(fact_4863_ceiling__le__numeral,axiom,
    ! [X: rat,V: num] :
      ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ ( numeral_numeral_int @ V ) )
      = ( ord_less_eq_rat @ X @ ( numeral_numeral_rat @ V ) ) ) ).

% ceiling_le_numeral
thf(fact_4864_numeral__less__ceiling,axiom,
    ! [V: num,X: rat] :
      ( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim2889992004027027881ng_rat @ X ) )
      = ( ord_less_rat @ ( numeral_numeral_rat @ V ) @ X ) ) ).

% numeral_less_ceiling
thf(fact_4865_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y3: nat,X: num,N: nat] :
      ( ( ( semiri1314217659103216013at_int @ Y3 )
        = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) )
      = ( Y3
        = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_4866_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y3: nat,X: num,N: nat] :
      ( ( ( semiri4939895301339042750nteger @ Y3 )
        = ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ X ) @ N ) )
      = ( Y3
        = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_4867_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y3: nat,X: num,N: nat] :
      ( ( ( semiri1316708129612266289at_nat @ Y3 )
        = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) )
      = ( Y3
        = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_4868_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y3: nat,X: num,N: nat] :
      ( ( ( semiri681578069525770553at_rat @ Y3 )
        = ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) )
      = ( Y3
        = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_4869_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y3: nat,X: num,N: nat] :
      ( ( ( semiri3763490453095760265atural @ Y3 )
        = ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ X ) @ N ) )
      = ( Y3
        = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_4870_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X: num,N: nat,Y3: nat] :
      ( ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
        = ( semiri1314217659103216013at_int @ Y3 ) )
      = ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
        = Y3 ) ) ).

% numeral_power_eq_of_nat_cancel_iff
thf(fact_4871_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X: num,N: nat,Y3: nat] :
      ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ X ) @ N )
        = ( semiri4939895301339042750nteger @ Y3 ) )
      = ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
        = Y3 ) ) ).

% numeral_power_eq_of_nat_cancel_iff
thf(fact_4872_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X: num,N: nat,Y3: nat] :
      ( ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
        = ( semiri1316708129612266289at_nat @ Y3 ) )
      = ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
        = Y3 ) ) ).

% numeral_power_eq_of_nat_cancel_iff
thf(fact_4873_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X: num,N: nat,Y3: nat] :
      ( ( ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N )
        = ( semiri681578069525770553at_rat @ Y3 ) )
      = ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
        = Y3 ) ) ).

% numeral_power_eq_of_nat_cancel_iff
thf(fact_4874_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X: num,N: nat,Y3: nat] :
      ( ( ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ X ) @ N )
        = ( semiri3763490453095760265atural @ Y3 ) )
      = ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
        = Y3 ) ) ).

% numeral_power_eq_of_nat_cancel_iff
thf(fact_4875_ceiling__add__numeral,axiom,
    ! [X: rat,V: num] :
      ( ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X @ ( numeral_numeral_rat @ V ) ) )
      = ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X ) @ ( numeral_numeral_int @ V ) ) ) ).

% ceiling_add_numeral
thf(fact_4876_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_4877_ceiling__diff__numeral,axiom,
    ! [X: rat,V: num] :
      ( ( archim2889992004027027881ng_rat @ ( minus_minus_rat @ X @ ( numeral_numeral_rat @ V ) ) )
      = ( minus_minus_int @ ( archim2889992004027027881ng_rat @ X ) @ ( numeral_numeral_int @ V ) ) ) ).

% ceiling_diff_numeral
thf(fact_4878_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_4879_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,Y3: int] :
      ( ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
        = ( ring_1_of_int_int @ Y3 ) )
      = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
        = Y3 ) ) ).

% numeral_power_eq_of_int_cancel_iff
thf(fact_4880_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,Y3: int] :
      ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ X ) @ N )
        = ( ring_18347121197199848620nteger @ Y3 ) )
      = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
        = Y3 ) ) ).

% numeral_power_eq_of_int_cancel_iff
thf(fact_4881_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,Y3: int] :
      ( ( ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N )
        = ( ring_1_of_int_rat @ Y3 ) )
      = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
        = Y3 ) ) ).

% numeral_power_eq_of_int_cancel_iff
thf(fact_4882_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y3: int,X: num,N: nat] :
      ( ( ( ring_1_of_int_int @ Y3 )
        = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) )
      = ( Y3
        = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).

% of_int_eq_numeral_power_cancel_iff
thf(fact_4883_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y3: int,X: num,N: nat] :
      ( ( ( ring_18347121197199848620nteger @ Y3 )
        = ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ X ) @ N ) )
      = ( Y3
        = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).

% of_int_eq_numeral_power_cancel_iff
thf(fact_4884_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y3: int,X: num,N: nat] :
      ( ( ( ring_1_of_int_rat @ Y3 )
        = ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) )
      = ( Y3
        = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).

% of_int_eq_numeral_power_cancel_iff
thf(fact_4885_ceiling__add__of__int,axiom,
    ! [X: rat,Z2: int] :
      ( ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X @ ( ring_1_of_int_rat @ Z2 ) ) )
      = ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X ) @ Z2 ) ) ).

% ceiling_add_of_int
thf(fact_4886_ceiling__diff__of__int,axiom,
    ! [X: rat,Z2: int] :
      ( ( archim2889992004027027881ng_rat @ ( minus_minus_rat @ X @ ( ring_1_of_int_rat @ Z2 ) ) )
      = ( minus_minus_int @ ( archim2889992004027027881ng_rat @ X ) @ Z2 ) ) ).

% ceiling_diff_of_int
thf(fact_4887_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_4888_le__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B: rat,W2: num] :
      ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) )
      = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ) ).

% le_divide_eq_numeral1(2)
thf(fact_4889_divide__le__eq__numeral1_I2_J,axiom,
    ! [B: rat,W2: num,A: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) @ A )
      = ( ord_less_eq_rat @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) @ B ) ) ).

% divide_le_eq_numeral1(2)
thf(fact_4890_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B: rat,W2: num,A: rat] :
      ( ( ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) )
        = A )
      = ( ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
           != zero_zero_rat )
         => ( B
            = ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) )
        & ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
            = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% divide_eq_eq_numeral1(2)
thf(fact_4891_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B: rat,W2: num] :
      ( ( A
        = ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) )
      = ( ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
           != zero_zero_rat )
         => ( ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) )
            = B ) )
        & ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
            = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral1(2)
thf(fact_4892_less__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B: rat,W2: num] :
      ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) )
      = ( ord_less_rat @ B @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ) ).

% less_divide_eq_numeral1(2)
thf(fact_4893_divide__less__eq__numeral1_I2_J,axiom,
    ! [B: rat,W2: num,A: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) @ A )
      = ( ord_less_rat @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) @ B ) ) ).

% divide_less_eq_numeral1(2)
thf(fact_4894_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_4895_dbl__dec__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu7757733837767384882nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
      = ( uminus1351360451143612070nteger @ ( neg_nu5831290666863070958nteger @ ( numera6620942414471956472nteger @ K ) ) ) ) ).

% dbl_dec_simps(1)
thf(fact_4896_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_4897_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_4898_dbl__inc__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu5831290666863070958nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
      = ( uminus1351360451143612070nteger @ ( neg_nu7757733837767384882nteger @ ( numera6620942414471956472nteger @ K ) ) ) ) ).

% dbl_inc_simps(1)
thf(fact_4899_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_4900_of__int__le__0__iff,axiom,
    ! [Z2: int] :
      ( ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ Z2 ) @ zero_z3403309356797280102nteger )
      = ( ord_less_eq_int @ Z2 @ zero_zero_int ) ) ).

% of_int_le_0_iff
thf(fact_4901_of__int__le__0__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z2 ) @ zero_zero_rat )
      = ( ord_less_eq_int @ Z2 @ zero_zero_int ) ) ).

% of_int_le_0_iff
thf(fact_4902_of__int__le__0__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z2 ) @ zero_zero_int )
      = ( ord_less_eq_int @ Z2 @ zero_zero_int ) ) ).

% of_int_le_0_iff
thf(fact_4903_of__int__0__le__iff,axiom,
    ! [Z2: int] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( ring_18347121197199848620nteger @ Z2 ) )
      = ( ord_less_eq_int @ zero_zero_int @ Z2 ) ) ).

% of_int_0_le_iff
thf(fact_4904_of__int__0__le__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z2 ) )
      = ( ord_less_eq_int @ zero_zero_int @ Z2 ) ) ).

% of_int_0_le_iff
thf(fact_4905_of__int__0__le__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z2 ) )
      = ( ord_less_eq_int @ zero_zero_int @ Z2 ) ) ).

% of_int_0_le_iff
thf(fact_4906_of__int__less__0__iff,axiom,
    ! [Z2: int] :
      ( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ Z2 ) @ zero_z3403309356797280102nteger )
      = ( ord_less_int @ Z2 @ zero_zero_int ) ) ).

% of_int_less_0_iff
thf(fact_4907_of__int__less__0__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z2 ) @ zero_zero_rat )
      = ( ord_less_int @ Z2 @ zero_zero_int ) ) ).

% of_int_less_0_iff
thf(fact_4908_of__int__less__0__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ Z2 ) @ zero_zero_int )
      = ( ord_less_int @ Z2 @ zero_zero_int ) ) ).

% of_int_less_0_iff
thf(fact_4909_of__int__0__less__iff,axiom,
    ! [Z2: int] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( ring_18347121197199848620nteger @ Z2 ) )
      = ( ord_less_int @ zero_zero_int @ Z2 ) ) ).

% of_int_0_less_iff
thf(fact_4910_of__int__0__less__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z2 ) )
      = ( ord_less_int @ zero_zero_int @ Z2 ) ) ).

% of_int_0_less_iff
thf(fact_4911_of__int__0__less__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z2 ) )
      = ( ord_less_int @ zero_zero_int @ Z2 ) ) ).

% of_int_0_less_iff
thf(fact_4912_of__int__1__le__iff,axiom,
    ! [Z2: int] :
      ( ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( ring_18347121197199848620nteger @ Z2 ) )
      = ( ord_less_eq_int @ one_one_int @ Z2 ) ) ).

% of_int_1_le_iff
thf(fact_4913_of__int__1__le__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_eq_rat @ one_one_rat @ ( ring_1_of_int_rat @ Z2 ) )
      = ( ord_less_eq_int @ one_one_int @ Z2 ) ) ).

% of_int_1_le_iff
thf(fact_4914_of__int__1__le__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_eq_int @ one_one_int @ ( ring_1_of_int_int @ Z2 ) )
      = ( ord_less_eq_int @ one_one_int @ Z2 ) ) ).

% of_int_1_le_iff
thf(fact_4915_of__int__le__1__iff,axiom,
    ! [Z2: int] :
      ( ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ Z2 ) @ one_one_Code_integer )
      = ( ord_less_eq_int @ Z2 @ one_one_int ) ) ).

% of_int_le_1_iff
thf(fact_4916_of__int__le__1__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z2 ) @ one_one_rat )
      = ( ord_less_eq_int @ Z2 @ one_one_int ) ) ).

% of_int_le_1_iff
thf(fact_4917_of__int__le__1__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z2 ) @ one_one_int )
      = ( ord_less_eq_int @ Z2 @ one_one_int ) ) ).

% of_int_le_1_iff
thf(fact_4918_of__int__1__less__iff,axiom,
    ! [Z2: int] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( ring_18347121197199848620nteger @ Z2 ) )
      = ( ord_less_int @ one_one_int @ Z2 ) ) ).

% of_int_1_less_iff
thf(fact_4919_of__int__1__less__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_rat @ one_one_rat @ ( ring_1_of_int_rat @ Z2 ) )
      = ( ord_less_int @ one_one_int @ Z2 ) ) ).

% of_int_1_less_iff
thf(fact_4920_of__int__1__less__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_int @ one_one_int @ ( ring_1_of_int_int @ Z2 ) )
      = ( ord_less_int @ one_one_int @ Z2 ) ) ).

% of_int_1_less_iff
thf(fact_4921_of__int__less__1__iff,axiom,
    ! [Z2: int] :
      ( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ Z2 ) @ one_one_Code_integer )
      = ( ord_less_int @ Z2 @ one_one_int ) ) ).

% of_int_less_1_iff
thf(fact_4922_of__int__less__1__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z2 ) @ one_one_rat )
      = ( ord_less_int @ Z2 @ one_one_int ) ) ).

% of_int_less_1_iff
thf(fact_4923_of__int__less__1__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ Z2 ) @ one_one_int )
      = ( ord_less_int @ Z2 @ one_one_int ) ) ).

% of_int_less_1_iff
thf(fact_4924_of__nat__nat,axiom,
    ! [Z2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
     => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z2 ) )
        = ( ring_1_of_int_int @ Z2 ) ) ) ).

% of_nat_nat
thf(fact_4925_of__nat__nat,axiom,
    ! [Z2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
     => ( ( semiri4939895301339042750nteger @ ( nat2 @ Z2 ) )
        = ( ring_18347121197199848620nteger @ Z2 ) ) ) ).

% of_nat_nat
thf(fact_4926_of__nat__nat,axiom,
    ! [Z2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
     => ( ( semiri681578069525770553at_rat @ ( nat2 @ Z2 ) )
        = ( ring_1_of_int_rat @ Z2 ) ) ) ).

% of_nat_nat
thf(fact_4927_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W2: nat,X: int] :
      ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ ( ring_18347121197199848620nteger @ B ) @ W2 ) @ ( ring_18347121197199848620nteger @ X ) )
      = ( ord_less_eq_int @ ( power_power_int @ B @ W2 ) @ X ) ) ).

% of_int_le_of_int_power_cancel_iff
thf(fact_4928_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W2: nat,X: int] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W2 ) @ ( ring_1_of_int_rat @ X ) )
      = ( ord_less_eq_int @ ( power_power_int @ B @ W2 ) @ X ) ) ).

% of_int_le_of_int_power_cancel_iff
thf(fact_4929_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W2: nat,X: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W2 ) @ ( ring_1_of_int_int @ X ) )
      = ( ord_less_eq_int @ ( power_power_int @ B @ W2 ) @ X ) ) ).

% of_int_le_of_int_power_cancel_iff
thf(fact_4930_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W2: nat] :
      ( ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ X ) @ ( power_8256067586552552935nteger @ ( ring_18347121197199848620nteger @ B ) @ W2 ) )
      = ( ord_less_eq_int @ X @ ( power_power_int @ B @ W2 ) ) ) ).

% of_int_power_le_of_int_cancel_iff
thf(fact_4931_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W2: nat] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ X ) @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W2 ) )
      = ( ord_less_eq_int @ X @ ( power_power_int @ B @ W2 ) ) ) ).

% of_int_power_le_of_int_cancel_iff
thf(fact_4932_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W2: nat] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ X ) @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W2 ) )
      = ( ord_less_eq_int @ X @ ( power_power_int @ B @ W2 ) ) ) ).

% of_int_power_le_of_int_cancel_iff
thf(fact_4933_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W2: nat] :
      ( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ X ) @ ( power_8256067586552552935nteger @ ( ring_18347121197199848620nteger @ B ) @ W2 ) )
      = ( ord_less_int @ X @ ( power_power_int @ B @ W2 ) ) ) ).

% of_int_power_less_of_int_cancel_iff
thf(fact_4934_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W2: nat] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ X ) @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W2 ) )
      = ( ord_less_int @ X @ ( power_power_int @ B @ W2 ) ) ) ).

% of_int_power_less_of_int_cancel_iff
thf(fact_4935_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W2: nat] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ X ) @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W2 ) )
      = ( ord_less_int @ X @ ( power_power_int @ B @ W2 ) ) ) ).

% of_int_power_less_of_int_cancel_iff
thf(fact_4936_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W2: nat,X: int] :
      ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ ( ring_18347121197199848620nteger @ B ) @ W2 ) @ ( ring_18347121197199848620nteger @ X ) )
      = ( ord_less_int @ ( power_power_int @ B @ W2 ) @ X ) ) ).

% of_int_less_of_int_power_cancel_iff
thf(fact_4937_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W2: nat,X: int] :
      ( ( ord_less_rat @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W2 ) @ ( ring_1_of_int_rat @ X ) )
      = ( ord_less_int @ ( power_power_int @ B @ W2 ) @ X ) ) ).

% of_int_less_of_int_power_cancel_iff
thf(fact_4938_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W2: nat,X: int] :
      ( ( ord_less_int @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W2 ) @ ( ring_1_of_int_int @ X ) )
      = ( ord_less_int @ ( power_power_int @ B @ W2 ) @ X ) ) ).

% of_int_less_of_int_power_cancel_iff
thf(fact_4939_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_4940_nat__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( ( ord_less_nat @ ( nat2 @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).

% nat_less_numeral_power_cancel_iff
thf(fact_4941_numeral__power__less__nat__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) @ ( nat2 @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).

% numeral_power_less_nat_cancel_iff
thf(fact_4942_numeral__power__le__nat__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) @ ( nat2 @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).

% numeral_power_le_nat_cancel_iff
thf(fact_4943_nat__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( ( ord_less_eq_nat @ ( nat2 @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).

% nat_le_numeral_power_cancel_iff
thf(fact_4944_ceiling__less__numeral,axiom,
    ! [X: rat,V: num] :
      ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ ( numeral_numeral_int @ V ) )
      = ( ord_less_eq_rat @ X @ ( minus_minus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) ) ) ).

% ceiling_less_numeral
thf(fact_4945_numeral__le__ceiling,axiom,
    ! [V: num,X: rat] :
      ( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim2889992004027027881ng_rat @ X ) )
      = ( ord_less_rat @ ( minus_minus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) @ X ) ) ).

% numeral_le_ceiling
thf(fact_4946_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I2: num,N: nat,X: nat] :
      ( ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ I2 ) @ N ) @ ( semiri1314217659103216013at_int @ X ) )
      = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) @ X ) ) ).

% numeral_power_less_of_nat_cancel_iff
thf(fact_4947_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I2: num,N: nat,X: nat] :
      ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ I2 ) @ N ) @ ( semiri4939895301339042750nteger @ X ) )
      = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) @ X ) ) ).

% numeral_power_less_of_nat_cancel_iff
thf(fact_4948_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I2: num,N: nat,X: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) @ ( semiri1316708129612266289at_nat @ X ) )
      = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) @ X ) ) ).

% numeral_power_less_of_nat_cancel_iff
thf(fact_4949_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I2: num,N: nat,X: nat] :
      ( ( ord_less_rat @ ( power_power_rat @ ( numeral_numeral_rat @ I2 ) @ N ) @ ( semiri681578069525770553at_rat @ X ) )
      = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) @ X ) ) ).

% numeral_power_less_of_nat_cancel_iff
thf(fact_4950_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I2: num,N: nat,X: nat] :
      ( ( ord_le5570908160329646204atural @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ I2 ) @ N ) @ ( semiri3763490453095760265atural @ X ) )
      = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) @ X ) ) ).

% numeral_power_less_of_nat_cancel_iff
thf(fact_4951_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I2: num,N: nat] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( numeral_numeral_int @ I2 ) @ N ) )
      = ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) ) ) ).

% of_nat_less_numeral_power_cancel_iff
thf(fact_4952_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I2: num,N: nat] :
      ( ( ord_le6747313008572928689nteger @ ( semiri4939895301339042750nteger @ X ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ I2 ) @ N ) )
      = ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) ) ) ).

% of_nat_less_numeral_power_cancel_iff
thf(fact_4953_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I2: num,N: nat] :
      ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) )
      = ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) ) ) ).

% of_nat_less_numeral_power_cancel_iff
thf(fact_4954_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I2: num,N: nat] :
      ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( numeral_numeral_rat @ I2 ) @ N ) )
      = ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) ) ) ).

% of_nat_less_numeral_power_cancel_iff
thf(fact_4955_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I2: num,N: nat] :
      ( ( ord_le5570908160329646204atural @ ( semiri3763490453095760265atural @ X ) @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ I2 ) @ N ) )
      = ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) ) ) ).

% of_nat_less_numeral_power_cancel_iff
thf(fact_4956_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I2: num,N: nat] :
      ( ( ord_le3102999989581377725nteger @ ( semiri4939895301339042750nteger @ X ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ I2 ) @ N ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) ) ) ).

% of_nat_le_numeral_power_cancel_iff
thf(fact_4957_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I2: num,N: nat] :
      ( ( ord_le1926595141338095240atural @ ( semiri3763490453095760265atural @ X ) @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ I2 ) @ N ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) ) ) ).

% of_nat_le_numeral_power_cancel_iff
thf(fact_4958_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I2: num,N: nat] :
      ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( numeral_numeral_rat @ I2 ) @ N ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) ) ) ).

% of_nat_le_numeral_power_cancel_iff
thf(fact_4959_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I2: num,N: nat] :
      ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) ) ) ).

% of_nat_le_numeral_power_cancel_iff
thf(fact_4960_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I2: num,N: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( numeral_numeral_int @ I2 ) @ N ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) ) ) ).

% of_nat_le_numeral_power_cancel_iff
thf(fact_4961_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I2: num,N: nat,X: nat] :
      ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ I2 ) @ N ) @ ( semiri4939895301339042750nteger @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) @ X ) ) ).

% numeral_power_le_of_nat_cancel_iff
thf(fact_4962_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I2: num,N: nat,X: nat] :
      ( ( ord_le1926595141338095240atural @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ I2 ) @ N ) @ ( semiri3763490453095760265atural @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) @ X ) ) ).

% numeral_power_le_of_nat_cancel_iff
thf(fact_4963_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I2: num,N: nat,X: nat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ ( numeral_numeral_rat @ I2 ) @ N ) @ ( semiri681578069525770553at_rat @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) @ X ) ) ).

% numeral_power_le_of_nat_cancel_iff
thf(fact_4964_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I2: num,N: nat,X: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) @ ( semiri1316708129612266289at_nat @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) @ X ) ) ).

% numeral_power_le_of_nat_cancel_iff
thf(fact_4965_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I2: num,N: nat,X: nat] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ I2 ) @ N ) @ ( semiri1314217659103216013at_int @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) @ X ) ) ).

% numeral_power_le_of_nat_cancel_iff
thf(fact_4966_ceiling__le__neg__numeral,axiom,
    ! [X: rat,V: num] :
      ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
      = ( ord_less_eq_rat @ X @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) ) ).

% ceiling_le_neg_numeral
thf(fact_4967_neg__numeral__less__ceiling,axiom,
    ! [V: num,X: rat] :
      ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim2889992004027027881ng_rat @ X ) )
      = ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ X ) ) ).

% neg_numeral_less_ceiling
thf(fact_4968_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ A ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ X ) @ N ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).

% of_int_le_numeral_power_cancel_iff
thf(fact_4969_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).

% of_int_le_numeral_power_cancel_iff
thf(fact_4970_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).

% of_int_le_numeral_power_cancel_iff
thf(fact_4971_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ X ) @ N ) @ ( ring_18347121197199848620nteger @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).

% numeral_power_le_of_int_cancel_iff
thf(fact_4972_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) @ ( ring_1_of_int_rat @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).

% numeral_power_le_of_int_cancel_iff
thf(fact_4973_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ ( ring_1_of_int_int @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).

% numeral_power_le_of_int_cancel_iff
thf(fact_4974_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ ( ring_1_of_int_int @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).

% numeral_power_less_of_int_cancel_iff
thf(fact_4975_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ X ) @ N ) @ ( ring_18347121197199848620nteger @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).

% numeral_power_less_of_int_cancel_iff
thf(fact_4976_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( ( ord_less_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) @ ( ring_1_of_int_rat @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).

% numeral_power_less_of_int_cancel_iff
thf(fact_4977_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).

% of_int_less_numeral_power_cancel_iff
thf(fact_4978_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ A ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ X ) @ N ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).

% of_int_less_numeral_power_cancel_iff
thf(fact_4979_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).

% of_int_less_numeral_power_cancel_iff
thf(fact_4980_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y3: int,X: num,N: nat] :
      ( ( ( ring_1_of_int_int @ Y3 )
        = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) )
      = ( Y3
        = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_4981_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y3: int,X: num,N: nat] :
      ( ( ( ring_18347121197199848620nteger @ Y3 )
        = ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N ) )
      = ( Y3
        = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_4982_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y3: int,X: num,N: nat] :
      ( ( ( ring_1_of_int_rat @ Y3 )
        = ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N ) )
      = ( Y3
        = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_4983_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,Y3: int] :
      ( ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N )
        = ( ring_1_of_int_int @ Y3 ) )
      = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N )
        = Y3 ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_4984_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,Y3: int] :
      ( ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N )
        = ( ring_18347121197199848620nteger @ Y3 ) )
      = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N )
        = Y3 ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_4985_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,Y3: int] :
      ( ( ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N )
        = ( ring_1_of_int_rat @ Y3 ) )
      = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N )
        = Y3 ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_4986_ceiling__less__neg__numeral,axiom,
    ! [X: rat,V: num] :
      ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
      = ( ord_less_eq_rat @ X @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) ) ) ).

% ceiling_less_neg_numeral
thf(fact_4987_neg__numeral__le__ceiling,axiom,
    ! [V: num,X: rat] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim2889992004027027881ng_rat @ X ) )
      = ( ord_less_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) @ X ) ) ).

% neg_numeral_le_ceiling
thf(fact_4988_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N ) @ ( ring_18347121197199848620nteger @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).

% neg_numeral_power_le_of_int_cancel_iff
thf(fact_4989_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N ) @ ( ring_1_of_int_rat @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).

% neg_numeral_power_le_of_int_cancel_iff
thf(fact_4990_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ ( ring_1_of_int_int @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).

% neg_numeral_power_le_of_int_cancel_iff
thf(fact_4991_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ A ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).

% of_int_le_neg_numeral_power_cancel_iff
thf(fact_4992_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).

% of_int_le_neg_numeral_power_cancel_iff
thf(fact_4993_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).

% of_int_le_neg_numeral_power_cancel_iff
thf(fact_4994_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ ( ring_1_of_int_int @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).

% neg_numeral_power_less_of_int_cancel_iff
thf(fact_4995_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N ) @ ( ring_18347121197199848620nteger @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).

% neg_numeral_power_less_of_int_cancel_iff
thf(fact_4996_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( ( ord_less_rat @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N ) @ ( ring_1_of_int_rat @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).

% neg_numeral_power_less_of_int_cancel_iff
thf(fact_4997_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).

% of_int_less_neg_numeral_power_cancel_iff
thf(fact_4998_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ A ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).

% of_int_less_neg_numeral_power_cancel_iff
thf(fact_4999_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).

% of_int_less_neg_numeral_power_cancel_iff
thf(fact_5000_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_5001_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_5002_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_5003_nat__numeral__as__int,axiom,
    ( numeral_numeral_nat
    = ( ^ [I4: num] : ( nat2 @ ( numeral_numeral_int @ I4 ) ) ) ) ).

% nat_numeral_as_int
thf(fact_5004_int__ops_I3_J,axiom,
    ! [N: num] :
      ( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N ) )
      = ( numeral_numeral_int @ N ) ) ).

% int_ops(3)
thf(fact_5005_ex__le__of__int,axiom,
    ! [X: rat] :
    ? [Z3: int] : ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ Z3 ) ) ).

% ex_le_of_int
thf(fact_5006_ex__of__int__less,axiom,
    ! [X: rat] :
    ? [Z3: int] : ( ord_less_rat @ ( ring_1_of_int_rat @ Z3 ) @ X ) ).

% ex_of_int_less
thf(fact_5007_ex__less__of__int,axiom,
    ! [X: rat] :
    ? [Z3: int] : ( ord_less_rat @ X @ ( ring_1_of_int_rat @ Z3 ) ) ).

% ex_less_of_int
thf(fact_5008_mult__of__int__commute,axiom,
    ! [X: int,Y3: code_integer] :
      ( ( times_3573771949741848930nteger @ ( ring_18347121197199848620nteger @ X ) @ Y3 )
      = ( times_3573771949741848930nteger @ Y3 @ ( ring_18347121197199848620nteger @ X ) ) ) ).

% mult_of_int_commute
thf(fact_5009_mult__of__int__commute,axiom,
    ! [X: int,Y3: rat] :
      ( ( times_times_rat @ ( ring_1_of_int_rat @ X ) @ Y3 )
      = ( times_times_rat @ Y3 @ ( ring_1_of_int_rat @ X ) ) ) ).

% mult_of_int_commute
thf(fact_5010_mult__of__int__commute,axiom,
    ! [X: int,Y3: int] :
      ( ( times_times_int @ ( ring_1_of_int_int @ X ) @ Y3 )
      = ( times_times_int @ Y3 @ ( ring_1_of_int_int @ X ) ) ) ).

% mult_of_int_commute
thf(fact_5011_zero__neq__numeral,axiom,
    ! [N: num] :
      ( zero_zero_int
     != ( numeral_numeral_int @ N ) ) ).

% zero_neq_numeral
thf(fact_5012_zero__neq__numeral,axiom,
    ! [N: num] :
      ( zero_zero_nat
     != ( numeral_numeral_nat @ N ) ) ).

% zero_neq_numeral
thf(fact_5013_zero__neq__numeral,axiom,
    ! [N: num] :
      ( zero_z3403309356797280102nteger
     != ( numera6620942414471956472nteger @ N ) ) ).

% zero_neq_numeral
thf(fact_5014_zero__neq__numeral,axiom,
    ! [N: num] :
      ( zero_zero_rat
     != ( numeral_numeral_rat @ N ) ) ).

% zero_neq_numeral
thf(fact_5015_zero__neq__numeral,axiom,
    ! [N: num] :
      ( zero_z2226904508553997617atural
     != ( numera5444537566228673987atural @ N ) ) ).

% zero_neq_numeral
thf(fact_5016_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_5017_numeral__neq__neg__numeral,axiom,
    ! [M: num,N: num] :
      ( ( numera6620942414471956472nteger @ M )
     != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).

% numeral_neq_neg_numeral
thf(fact_5018_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_5019_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_5020_neg__numeral__neq__numeral,axiom,
    ! [M: num,N: num] :
      ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) )
     != ( numera6620942414471956472nteger @ N ) ) ).

% neg_numeral_neq_numeral
thf(fact_5021_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_5022_div__mult2__numeral__eq,axiom,
    ! [A: nat,K: num,L: num] :
      ( ( divide_divide_nat @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ L ) )
      = ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( times_times_num @ K @ L ) ) ) ) ).

% div_mult2_numeral_eq
thf(fact_5023_div__mult2__numeral__eq,axiom,
    ! [A: int,K: num,L: num] :
      ( ( divide_divide_int @ ( divide_divide_int @ A @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ L ) )
      = ( divide_divide_int @ A @ ( numeral_numeral_int @ ( times_times_num @ K @ L ) ) ) ) ).

% div_mult2_numeral_eq
thf(fact_5024_div__mult2__numeral__eq,axiom,
    ! [A: code_integer,K: num,L: num] :
      ( ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ K ) ) @ ( numera6620942414471956472nteger @ L ) )
      = ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( times_times_num @ K @ L ) ) ) ) ).

% div_mult2_numeral_eq
thf(fact_5025_div__mult2__numeral__eq,axiom,
    ! [A: code_natural,K: num,L: num] :
      ( ( divide5121882707175180666atural @ ( divide5121882707175180666atural @ A @ ( numera5444537566228673987atural @ K ) ) @ ( numera5444537566228673987atural @ L ) )
      = ( divide5121882707175180666atural @ A @ ( numera5444537566228673987atural @ ( times_times_num @ K @ L ) ) ) ) ).

% div_mult2_numeral_eq
thf(fact_5026_pochhammer__of__int,axiom,
    ! [X: int,N: nat] :
      ( ( comm_s8582702949713902594nteger @ ( ring_18347121197199848620nteger @ X ) @ N )
      = ( ring_18347121197199848620nteger @ ( comm_s4660882817536571857er_int @ X @ N ) ) ) ).

% pochhammer_of_int
thf(fact_5027_pochhammer__of__int,axiom,
    ! [X: int,N: nat] :
      ( ( comm_s4028243227959126397er_rat @ ( ring_1_of_int_rat @ X ) @ N )
      = ( ring_1_of_int_rat @ ( comm_s4660882817536571857er_int @ X @ N ) ) ) ).

% pochhammer_of_int
thf(fact_5028_pochhammer__of__int,axiom,
    ! [X: int,N: nat] :
      ( ( comm_s4660882817536571857er_int @ ( ring_1_of_int_int @ X ) @ N )
      = ( ring_1_of_int_int @ ( comm_s4660882817536571857er_int @ X @ N ) ) ) ).

% pochhammer_of_int
thf(fact_5029_Ints__of__int,axiom,
    ! [Z2: int] : ( member_Code_integer @ ( ring_18347121197199848620nteger @ Z2 ) @ ring_11222124179247155820nteger ) ).

% Ints_of_int
thf(fact_5030_Ints__of__int,axiom,
    ! [Z2: int] : ( member_rat @ ( ring_1_of_int_rat @ Z2 ) @ ring_1_Ints_rat ) ).

% Ints_of_int
thf(fact_5031_Ints__of__int,axiom,
    ! [Z2: int] : ( member_int @ ( ring_1_of_int_int @ Z2 ) @ ring_1_Ints_int ) ).

% Ints_of_int
thf(fact_5032_Ints__induct,axiom,
    ! [Q4: code_integer,P: code_integer > $o] :
      ( ( member_Code_integer @ Q4 @ ring_11222124179247155820nteger )
     => ( ! [Z3: int] : ( P @ ( ring_18347121197199848620nteger @ Z3 ) )
       => ( P @ Q4 ) ) ) ).

% Ints_induct
thf(fact_5033_Ints__induct,axiom,
    ! [Q4: rat,P: rat > $o] :
      ( ( member_rat @ Q4 @ ring_1_Ints_rat )
     => ( ! [Z3: int] : ( P @ ( ring_1_of_int_rat @ Z3 ) )
       => ( P @ Q4 ) ) ) ).

% Ints_induct
thf(fact_5034_Ints__induct,axiom,
    ! [Q4: int,P: int > $o] :
      ( ( member_int @ Q4 @ ring_1_Ints_int )
     => ( ! [Z3: int] : ( P @ ( ring_1_of_int_int @ Z3 ) )
       => ( P @ Q4 ) ) ) ).

% Ints_induct
thf(fact_5035_Ints__cases,axiom,
    ! [Q4: code_integer] :
      ( ( member_Code_integer @ Q4 @ ring_11222124179247155820nteger )
     => ~ ! [Z3: int] :
            ( Q4
           != ( ring_18347121197199848620nteger @ Z3 ) ) ) ).

% Ints_cases
thf(fact_5036_Ints__cases,axiom,
    ! [Q4: rat] :
      ( ( member_rat @ Q4 @ ring_1_Ints_rat )
     => ~ ! [Z3: int] :
            ( Q4
           != ( ring_1_of_int_rat @ Z3 ) ) ) ).

% Ints_cases
thf(fact_5037_Ints__cases,axiom,
    ! [Q4: int] :
      ( ( member_int @ Q4 @ ring_1_Ints_int )
     => ~ ! [Z3: int] :
            ( Q4
           != ( ring_1_of_int_int @ Z3 ) ) ) ).

% Ints_cases
thf(fact_5038_Ints__numeral,axiom,
    ! [N: num] : ( member_int @ ( numeral_numeral_int @ N ) @ ring_1_Ints_int ) ).

% Ints_numeral
thf(fact_5039_Ints__numeral,axiom,
    ! [N: num] : ( member_Code_integer @ ( numera6620942414471956472nteger @ N ) @ ring_11222124179247155820nteger ) ).

% Ints_numeral
thf(fact_5040_Ints__numeral,axiom,
    ! [N: num] : ( member_rat @ ( numeral_numeral_rat @ N ) @ ring_1_Ints_rat ) ).

% Ints_numeral
thf(fact_5041_not__numeral__le__zero,axiom,
    ! [N: num] :
      ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ N ) @ zero_z3403309356797280102nteger ) ).

% not_numeral_le_zero
thf(fact_5042_not__numeral__le__zero,axiom,
    ! [N: num] :
      ~ ( ord_le1926595141338095240atural @ ( numera5444537566228673987atural @ N ) @ zero_z2226904508553997617atural ) ).

% not_numeral_le_zero
thf(fact_5043_not__numeral__le__zero,axiom,
    ! [N: num] :
      ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ N ) @ zero_zero_rat ) ).

% not_numeral_le_zero
thf(fact_5044_not__numeral__le__zero,axiom,
    ! [N: num] :
      ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).

% not_numeral_le_zero
thf(fact_5045_not__numeral__le__zero,axiom,
    ! [N: num] :
      ~ ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).

% not_numeral_le_zero
thf(fact_5046_zero__le__numeral,axiom,
    ! [N: num] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ N ) ) ).

% zero_le_numeral
thf(fact_5047_zero__le__numeral,axiom,
    ! [N: num] : ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ ( numera5444537566228673987atural @ N ) ) ).

% zero_le_numeral
thf(fact_5048_zero__le__numeral,axiom,
    ! [N: num] : ( ord_less_eq_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N ) ) ).

% zero_le_numeral
thf(fact_5049_zero__le__numeral,axiom,
    ! [N: num] : ( ord_less_eq_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).

% zero_le_numeral
thf(fact_5050_zero__le__numeral,axiom,
    ! [N: num] : ( ord_less_eq_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).

% zero_le_numeral
thf(fact_5051_not__numeral__less__zero,axiom,
    ! [N: num] :
      ~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).

% not_numeral_less_zero
thf(fact_5052_not__numeral__less__zero,axiom,
    ! [N: num] :
      ~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).

% not_numeral_less_zero
thf(fact_5053_not__numeral__less__zero,axiom,
    ! [N: num] :
      ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ N ) @ zero_z3403309356797280102nteger ) ).

% not_numeral_less_zero
thf(fact_5054_not__numeral__less__zero,axiom,
    ! [N: num] :
      ~ ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ zero_zero_rat ) ).

% not_numeral_less_zero
thf(fact_5055_not__numeral__less__zero,axiom,
    ! [N: num] :
      ~ ( ord_le5570908160329646204atural @ ( numera5444537566228673987atural @ N ) @ zero_z2226904508553997617atural ) ).

% not_numeral_less_zero
thf(fact_5056_zero__less__numeral,axiom,
    ! [N: num] : ( ord_less_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).

% zero_less_numeral
thf(fact_5057_zero__less__numeral,axiom,
    ! [N: num] : ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).

% zero_less_numeral
thf(fact_5058_zero__less__numeral,axiom,
    ! [N: num] : ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ N ) ) ).

% zero_less_numeral
thf(fact_5059_zero__less__numeral,axiom,
    ! [N: num] : ( ord_less_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N ) ) ).

% zero_less_numeral
thf(fact_5060_zero__less__numeral,axiom,
    ! [N: num] : ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ ( numera5444537566228673987atural @ N ) ) ).

% zero_less_numeral
thf(fact_5061_le__of__int__ceiling,axiom,
    ! [X: rat] : ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X ) ) ) ).

% le_of_int_ceiling
thf(fact_5062_one__le__numeral,axiom,
    ! [N: num] : ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ N ) ) ).

% one_le_numeral
thf(fact_5063_one__le__numeral,axiom,
    ! [N: num] : ( ord_le1926595141338095240atural @ one_one_Code_natural @ ( numera5444537566228673987atural @ N ) ) ).

% one_le_numeral
thf(fact_5064_one__le__numeral,axiom,
    ! [N: num] : ( ord_less_eq_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) ) ).

% one_le_numeral
thf(fact_5065_one__le__numeral,axiom,
    ! [N: num] : ( ord_less_eq_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) ) ).

% one_le_numeral
thf(fact_5066_one__le__numeral,axiom,
    ! [N: num] : ( ord_less_eq_int @ one_one_int @ ( numeral_numeral_int @ N ) ) ).

% one_le_numeral
thf(fact_5067_not__numeral__less__one,axiom,
    ! [N: num] :
      ~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ one_one_int ) ).

% not_numeral_less_one
thf(fact_5068_not__numeral__less__one,axiom,
    ! [N: num] :
      ~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat ) ).

% not_numeral_less_one
thf(fact_5069_not__numeral__less__one,axiom,
    ! [N: num] :
      ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ N ) @ one_one_Code_integer ) ).

% not_numeral_less_one
thf(fact_5070_not__numeral__less__one,axiom,
    ! [N: num] :
      ~ ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ one_one_rat ) ).

% not_numeral_less_one
thf(fact_5071_not__numeral__less__one,axiom,
    ! [N: num] :
      ~ ( ord_le5570908160329646204atural @ ( numera5444537566228673987atural @ N ) @ one_one_Code_natural ) ).

% not_numeral_less_one
thf(fact_5072_not__numeral__le__neg__numeral,axiom,
    ! [M: num,N: num] :
      ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).

% not_numeral_le_neg_numeral
thf(fact_5073_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_5074_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_5075_neg__numeral__le__numeral,axiom,
    ! [M: num,N: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) ) ).

% neg_numeral_le_numeral
thf(fact_5076_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_5077_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_5078_zero__neq__neg__numeral,axiom,
    ! [N: num] :
      ( zero_zero_int
     != ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).

% zero_neq_neg_numeral
thf(fact_5079_zero__neq__neg__numeral,axiom,
    ! [N: num] :
      ( zero_z3403309356797280102nteger
     != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).

% zero_neq_neg_numeral
thf(fact_5080_zero__neq__neg__numeral,axiom,
    ! [N: num] :
      ( zero_zero_rat
     != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).

% zero_neq_neg_numeral
thf(fact_5081_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_5082_not__numeral__less__neg__numeral,axiom,
    ! [M: num,N: num] :
      ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).

% not_numeral_less_neg_numeral
thf(fact_5083_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_5084_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_5085_neg__numeral__less__numeral,axiom,
    ! [M: num,N: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) ) ).

% neg_numeral_less_numeral
thf(fact_5086_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_5087_one__plus__numeral__commute,axiom,
    ! [X: num] :
      ( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ X ) )
      = ( plus_plus_int @ ( numeral_numeral_int @ X ) @ one_one_int ) ) ).

% one_plus_numeral_commute
thf(fact_5088_one__plus__numeral__commute,axiom,
    ! [X: num] :
      ( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ X ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ X ) @ one_one_nat ) ) ).

% one_plus_numeral_commute
thf(fact_5089_one__plus__numeral__commute,axiom,
    ! [X: num] :
      ( ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ X ) )
      = ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ X ) @ one_one_Code_integer ) ) ).

% one_plus_numeral_commute
thf(fact_5090_one__plus__numeral__commute,axiom,
    ! [X: num] :
      ( ( plus_plus_rat @ one_one_rat @ ( numeral_numeral_rat @ X ) )
      = ( plus_plus_rat @ ( numeral_numeral_rat @ X ) @ one_one_rat ) ) ).

% one_plus_numeral_commute
thf(fact_5091_one__plus__numeral__commute,axiom,
    ! [X: num] :
      ( ( plus_p4538020629002901425atural @ one_one_Code_natural @ ( numera5444537566228673987atural @ X ) )
      = ( plus_p4538020629002901425atural @ ( numera5444537566228673987atural @ X ) @ one_one_Code_natural ) ) ).

% one_plus_numeral_commute
thf(fact_5092_numeral__times__minus__swap,axiom,
    ! [W2: num,X: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ W2 ) @ ( uminus_uminus_int @ X ) )
      = ( times_times_int @ X @ ( uminus_uminus_int @ ( numeral_numeral_int @ W2 ) ) ) ) ).

% numeral_times_minus_swap
thf(fact_5093_numeral__times__minus__swap,axiom,
    ! [W2: num,X: code_integer] :
      ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ W2 ) @ ( uminus1351360451143612070nteger @ X ) )
      = ( times_3573771949741848930nteger @ X @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W2 ) ) ) ) ).

% numeral_times_minus_swap
thf(fact_5094_numeral__times__minus__swap,axiom,
    ! [W2: num,X: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ ( uminus_uminus_rat @ X ) )
      = ( times_times_rat @ X @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ).

% numeral_times_minus_swap
thf(fact_5095_numeral__neq__neg__one,axiom,
    ! [N: num] :
      ( ( numeral_numeral_int @ N )
     != ( uminus_uminus_int @ one_one_int ) ) ).

% numeral_neq_neg_one
thf(fact_5096_numeral__neq__neg__one,axiom,
    ! [N: num] :
      ( ( numera6620942414471956472nteger @ N )
     != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% numeral_neq_neg_one
thf(fact_5097_numeral__neq__neg__one,axiom,
    ! [N: num] :
      ( ( numeral_numeral_rat @ N )
     != ( uminus_uminus_rat @ one_one_rat ) ) ).

% numeral_neq_neg_one
thf(fact_5098_one__neq__neg__numeral,axiom,
    ! [N: num] :
      ( one_one_int
     != ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).

% one_neq_neg_numeral
thf(fact_5099_one__neq__neg__numeral,axiom,
    ! [N: num] :
      ( one_one_Code_integer
     != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).

% one_neq_neg_numeral
thf(fact_5100_one__neq__neg__numeral,axiom,
    ! [N: num] :
      ( one_one_rat
     != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).

% one_neq_neg_numeral
thf(fact_5101_ceiling__le__iff,axiom,
    ! [X: rat,Z2: int] :
      ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ Z2 )
      = ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ Z2 ) ) ) ).

% ceiling_le_iff
thf(fact_5102_less__ceiling__iff,axiom,
    ! [Z2: int,X: rat] :
      ( ( ord_less_int @ Z2 @ ( archim2889992004027027881ng_rat @ X ) )
      = ( ord_less_rat @ ( ring_1_of_int_rat @ Z2 ) @ X ) ) ).

% less_ceiling_iff
thf(fact_5103_neg__numeral__le__zero,axiom,
    ! [N: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) @ zero_z3403309356797280102nteger ) ).

% neg_numeral_le_zero
thf(fact_5104_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_5105_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_5106_not__zero__le__neg__numeral,axiom,
    ! [N: num] :
      ~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).

% not_zero_le_neg_numeral
thf(fact_5107_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_5108_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_5109_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_5110_neg__numeral__less__zero,axiom,
    ! [N: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) @ zero_z3403309356797280102nteger ) ).

% neg_numeral_less_zero
thf(fact_5111_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_5112_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_5113_not__zero__less__neg__numeral,axiom,
    ! [N: num] :
      ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).

% not_zero_less_neg_numeral
thf(fact_5114_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_5115_not__one__le__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).

% not_one_le_neg_numeral
thf(fact_5116_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_5117_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_5118_not__numeral__le__neg__one,axiom,
    ! [M: num] :
      ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% not_numeral_le_neg_one
thf(fact_5119_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_5120_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_5121_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_5122_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_5123_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_5124_neg__one__le__numeral,axiom,
    ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ M ) ) ).

% neg_one_le_numeral
thf(fact_5125_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_5126_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_5127_neg__numeral__le__one,axiom,
    ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer ) ).

% neg_numeral_le_one
thf(fact_5128_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_5129_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_5130_divide__eq__eq__numeral_I1_J,axiom,
    ! [B: rat,C: rat,W2: num] :
      ( ( ( divide_divide_rat @ B @ C )
        = ( numeral_numeral_rat @ W2 ) )
      = ( ( ( C != zero_zero_rat )
         => ( B
            = ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) ) )
        & ( ( C = zero_zero_rat )
         => ( ( numeral_numeral_rat @ W2 )
            = zero_zero_rat ) ) ) ) ).

% divide_eq_eq_numeral(1)
thf(fact_5131_eq__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B: rat,C: rat] :
      ( ( ( numeral_numeral_rat @ W2 )
        = ( divide_divide_rat @ B @ C ) )
      = ( ( ( C != zero_zero_rat )
         => ( ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C )
            = B ) )
        & ( ( C = zero_zero_rat )
         => ( ( numeral_numeral_rat @ W2 )
            = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral(1)
thf(fact_5132_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_5133_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_5134_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_5135_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_5136_not__one__less__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).

% not_one_less_neg_numeral
thf(fact_5137_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_5138_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_5139_not__numeral__less__neg__one,axiom,
    ! [M: num] :
      ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% not_numeral_less_neg_one
thf(fact_5140_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_5141_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_5142_neg__one__less__numeral,axiom,
    ! [M: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ M ) ) ).

% neg_one_less_numeral
thf(fact_5143_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_5144_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_5145_neg__numeral__less__one,axiom,
    ! [M: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer ) ).

% neg_numeral_less_one
thf(fact_5146_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_5147_of__int__divide__in__Ints,axiom,
    ! [B: int,A: int] :
      ( ( dvd_dvd_int @ B @ A )
     => ( member_int @ ( divide_divide_int @ ( ring_1_of_int_int @ A ) @ ( ring_1_of_int_int @ B ) ) @ ring_1_Ints_int ) ) ).

% of_int_divide_in_Ints
thf(fact_5148_of__int__divide__in__Ints,axiom,
    ! [B: int,A: int] :
      ( ( dvd_dvd_int @ B @ A )
     => ( member_Code_integer @ ( divide6298287555418463151nteger @ ( ring_18347121197199848620nteger @ A ) @ ( ring_18347121197199848620nteger @ B ) ) @ ring_11222124179247155820nteger ) ) ).

% of_int_divide_in_Ints
thf(fact_5149_of__int__divide__in__Ints,axiom,
    ! [B: int,A: int] :
      ( ( dvd_dvd_int @ B @ A )
     => ( member_rat @ ( divide_divide_rat @ ( ring_1_of_int_rat @ A ) @ ( ring_1_of_int_rat @ B ) ) @ ring_1_Ints_rat ) ) ).

% of_int_divide_in_Ints
thf(fact_5150_of__int__nonneg,axiom,
    ! [Z2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
     => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( ring_18347121197199848620nteger @ Z2 ) ) ) ).

% of_int_nonneg
thf(fact_5151_of__int__nonneg,axiom,
    ! [Z2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z2 ) ) ) ).

% of_int_nonneg
thf(fact_5152_of__int__nonneg,axiom,
    ! [Z2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
     => ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z2 ) ) ) ).

% of_int_nonneg
thf(fact_5153_of__int__pos,axiom,
    ! [Z2: int] :
      ( ( ord_less_int @ zero_zero_int @ Z2 )
     => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( ring_18347121197199848620nteger @ Z2 ) ) ) ).

% of_int_pos
thf(fact_5154_of__int__pos,axiom,
    ! [Z2: int] :
      ( ( ord_less_int @ zero_zero_int @ Z2 )
     => ( ord_less_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z2 ) ) ) ).

% of_int_pos
thf(fact_5155_of__int__pos,axiom,
    ! [Z2: int] :
      ( ( ord_less_int @ zero_zero_int @ Z2 )
     => ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z2 ) ) ) ).

% of_int_pos
thf(fact_5156_of__int__leD,axiom,
    ! [N: int,X: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( ring_18347121197199848620nteger @ N ) ) @ X )
     => ( ( N = zero_zero_int )
        | ( ord_le3102999989581377725nteger @ one_one_Code_integer @ X ) ) ) ).

% of_int_leD
thf(fact_5157_of__int__leD,axiom,
    ! [N: int,X: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ ( ring_1_of_int_rat @ N ) ) @ X )
     => ( ( N = zero_zero_int )
        | ( ord_less_eq_rat @ one_one_rat @ X ) ) ) ).

% of_int_leD
thf(fact_5158_of__int__leD,axiom,
    ! [N: int,X: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N ) ) @ X )
     => ( ( N = zero_zero_int )
        | ( ord_less_eq_int @ one_one_int @ X ) ) ) ).

% of_int_leD
thf(fact_5159_of__int__lessD,axiom,
    ! [N: int,X: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( ring_18347121197199848620nteger @ N ) ) @ X )
     => ( ( N = zero_zero_int )
        | ( ord_le6747313008572928689nteger @ one_one_Code_integer @ X ) ) ) ).

% of_int_lessD
thf(fact_5160_of__int__lessD,axiom,
    ! [N: int,X: rat] :
      ( ( ord_less_rat @ ( abs_abs_rat @ ( ring_1_of_int_rat @ N ) ) @ X )
     => ( ( N = zero_zero_int )
        | ( ord_less_rat @ one_one_rat @ X ) ) ) ).

% of_int_lessD
thf(fact_5161_of__int__lessD,axiom,
    ! [N: int,X: int] :
      ( ( ord_less_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N ) ) @ X )
     => ( ( N = zero_zero_int )
        | ( ord_less_int @ one_one_int @ X ) ) ) ).

% of_int_lessD
thf(fact_5162_floor__exists1,axiom,
    ! [X: rat] :
    ? [X3: int] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ X3 ) @ X )
      & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ X3 @ one_one_int ) ) )
      & ! [Y5: int] :
          ( ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Y5 ) @ X )
            & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ Y5 @ one_one_int ) ) ) )
         => ( Y5 = X3 ) ) ) ).

% floor_exists1
thf(fact_5163_floor__exists,axiom,
    ! [X: rat] :
    ? [Z3: int] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z3 ) @ X )
      & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ Z3 @ one_one_int ) ) ) ) ).

% floor_exists
thf(fact_5164_of__nat__less__of__int__iff,axiom,
    ! [N: nat,X: int] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( ring_1_of_int_int @ X ) )
      = ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X ) ) ).

% of_nat_less_of_int_iff
thf(fact_5165_of__nat__less__of__int__iff,axiom,
    ! [N: nat,X: int] :
      ( ( ord_le6747313008572928689nteger @ ( semiri4939895301339042750nteger @ N ) @ ( ring_18347121197199848620nteger @ X ) )
      = ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X ) ) ).

% of_nat_less_of_int_iff
thf(fact_5166_of__nat__less__of__int__iff,axiom,
    ! [N: nat,X: int] :
      ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ N ) @ ( ring_1_of_int_rat @ X ) )
      = ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X ) ) ).

% of_nat_less_of_int_iff
thf(fact_5167_divide__less__eq__numeral_I1_J,axiom,
    ! [B: rat,C: rat,W2: num] :
      ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ ( numeral_numeral_rat @ W2 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) @ B ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ zero_zero_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ) ) ) ).

% divide_less_eq_numeral(1)
thf(fact_5168_less__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B: rat,C: rat] :
      ( ( ord_less_rat @ ( numeral_numeral_rat @ W2 ) @ ( divide_divide_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) @ B ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ ( numeral_numeral_rat @ W2 ) @ zero_zero_rat ) ) ) ) ) ) ).

% less_divide_eq_numeral(1)
thf(fact_5169_divide__eq__eq__numeral_I2_J,axiom,
    ! [B: rat,C: rat,W2: num] :
      ( ( ( divide_divide_rat @ B @ C )
        = ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) )
      = ( ( ( C != zero_zero_rat )
         => ( B
            = ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) ) )
        & ( ( C = zero_zero_rat )
         => ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
            = zero_zero_rat ) ) ) ) ).

% divide_eq_eq_numeral(2)
thf(fact_5170_eq__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B: rat,C: rat] :
      ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
        = ( divide_divide_rat @ B @ C ) )
      = ( ( ( C != zero_zero_rat )
         => ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C )
            = B ) )
        & ( ( C = zero_zero_rat )
         => ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
            = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral(2)
thf(fact_5171_round__diff__minimal,axiom,
    ! [Z2: rat,M: int] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ Z2 @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ Z2 ) ) ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ Z2 @ ( ring_1_of_int_rat @ M ) ) ) ) ).

% round_diff_minimal
thf(fact_5172_ceiling__correct,axiom,
    ! [X: rat] :
      ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X ) ) @ one_one_rat ) @ X )
      & ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X ) ) ) ) ).

% ceiling_correct
thf(fact_5173_ceiling__unique,axiom,
    ! [Z2: int,X: rat] :
      ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z2 ) @ one_one_rat ) @ X )
     => ( ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ Z2 ) )
       => ( ( archim2889992004027027881ng_rat @ X )
          = Z2 ) ) ) ).

% ceiling_unique
thf(fact_5174_ceiling__eq__iff,axiom,
    ! [X: rat,A: int] :
      ( ( ( archim2889992004027027881ng_rat @ X )
        = A )
      = ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ A ) @ one_one_rat ) @ X )
        & ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ A ) ) ) ) ).

% ceiling_eq_iff
thf(fact_5175_ceiling__split,axiom,
    ! [P: int > $o,T: rat] :
      ( ( P @ ( archim2889992004027027881ng_rat @ T ) )
      = ( ! [I4: int] :
            ( ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ I4 ) @ one_one_rat ) @ T )
              & ( ord_less_eq_rat @ T @ ( ring_1_of_int_rat @ I4 ) ) )
           => ( P @ I4 ) ) ) ) ).

% ceiling_split
thf(fact_5176_ceiling__less__iff,axiom,
    ! [X: rat,Z2: int] :
      ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ Z2 )
      = ( ord_less_eq_rat @ X @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z2 ) @ one_one_rat ) ) ) ).

% ceiling_less_iff
thf(fact_5177_le__ceiling__iff,axiom,
    ! [Z2: int,X: rat] :
      ( ( ord_less_eq_int @ Z2 @ ( archim2889992004027027881ng_rat @ X ) )
      = ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z2 ) @ one_one_rat ) @ X ) ) ).

% le_ceiling_iff
thf(fact_5178_divide__le__eq__numeral_I1_J,axiom,
    ! [B: rat,C: rat,W2: num] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ ( numeral_numeral_rat @ W2 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ 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 @ W2 ) @ C ) @ B ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ) ) ) ).

% divide_le_eq_numeral(1)
thf(fact_5179_le__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ W2 ) @ ( divide_divide_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) @ B ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( numeral_numeral_rat @ W2 ) @ zero_zero_rat ) ) ) ) ) ) ).

% le_divide_eq_numeral(1)
thf(fact_5180_divide__less__eq__numeral_I2_J,axiom,
    ! [B: rat,C: rat,W2: num] :
      ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ 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 @ W2 ) ) @ C ) @ B ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ) ) ) ) ).

% divide_less_eq_numeral(2)
thf(fact_5181_less__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B: rat,C: rat] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ ( divide_divide_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) @ B ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ zero_zero_rat ) ) ) ) ) ) ).

% less_divide_eq_numeral(2)
thf(fact_5182_distrib__left__NO__MATCH,axiom,
    ! [X: nat,Y3: nat,A: code_integer,B: code_integer,C: code_integer] :
      ( ( nO_MAT8694959213317031834nteger @ ( divide_divide_nat @ X @ Y3 ) @ A )
     => ( ( times_3573771949741848930nteger @ A @ ( plus_p5714425477246183910nteger @ B @ C ) )
        = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ B ) @ ( times_3573771949741848930nteger @ A @ C ) ) ) ) ).

% distrib_left_NO_MATCH
thf(fact_5183_distrib__left__NO__MATCH,axiom,
    ! [X: nat,Y3: nat,A: rat,B: rat,C: rat] :
      ( ( nO_MATCH_nat_rat @ ( divide_divide_nat @ X @ Y3 ) @ A )
     => ( ( times_times_rat @ A @ ( plus_plus_rat @ B @ C ) )
        = ( plus_plus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).

% distrib_left_NO_MATCH
thf(fact_5184_distrib__left__NO__MATCH,axiom,
    ! [X: nat,Y3: nat,A: nat,B: nat,C: nat] :
      ( ( nO_MATCH_nat_nat @ ( divide_divide_nat @ X @ Y3 ) @ A )
     => ( ( times_times_nat @ A @ ( plus_plus_nat @ B @ C ) )
        = ( plus_plus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ) ).

% distrib_left_NO_MATCH
thf(fact_5185_distrib__left__NO__MATCH,axiom,
    ! [X: nat,Y3: nat,A: int,B: int,C: int] :
      ( ( nO_MATCH_nat_int @ ( divide_divide_nat @ X @ Y3 ) @ A )
     => ( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
        = ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ) ).

% distrib_left_NO_MATCH
thf(fact_5186_distrib__left__NO__MATCH,axiom,
    ! [X: int,Y3: int,A: code_integer,B: code_integer,C: code_integer] :
      ( ( nO_MAT9066612773553876470nteger @ ( divide_divide_int @ X @ Y3 ) @ A )
     => ( ( times_3573771949741848930nteger @ A @ ( plus_p5714425477246183910nteger @ B @ C ) )
        = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ B ) @ ( times_3573771949741848930nteger @ A @ C ) ) ) ) ).

% distrib_left_NO_MATCH
thf(fact_5187_distrib__left__NO__MATCH,axiom,
    ! [X: int,Y3: int,A: rat,B: rat,C: rat] :
      ( ( nO_MATCH_int_rat @ ( divide_divide_int @ X @ Y3 ) @ A )
     => ( ( times_times_rat @ A @ ( plus_plus_rat @ B @ C ) )
        = ( plus_plus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).

% distrib_left_NO_MATCH
thf(fact_5188_distrib__left__NO__MATCH,axiom,
    ! [X: int,Y3: int,A: nat,B: nat,C: nat] :
      ( ( nO_MATCH_int_nat @ ( divide_divide_int @ X @ Y3 ) @ A )
     => ( ( times_times_nat @ A @ ( plus_plus_nat @ B @ C ) )
        = ( plus_plus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ) ).

% distrib_left_NO_MATCH
thf(fact_5189_distrib__left__NO__MATCH,axiom,
    ! [X: int,Y3: int,A: int,B: int,C: int] :
      ( ( nO_MATCH_int_int @ ( divide_divide_int @ X @ Y3 ) @ A )
     => ( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
        = ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ) ).

% distrib_left_NO_MATCH
thf(fact_5190_distrib__left__NO__MATCH,axiom,
    ! [X: code_integer,Y3: code_integer,A: code_integer,B: code_integer,C: code_integer] :
      ( ( nO_MAT8252062027627875367nteger @ ( divide6298287555418463151nteger @ X @ Y3 ) @ A )
     => ( ( times_3573771949741848930nteger @ A @ ( plus_p5714425477246183910nteger @ B @ C ) )
        = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ B ) @ ( times_3573771949741848930nteger @ A @ C ) ) ) ) ).

% distrib_left_NO_MATCH
thf(fact_5191_distrib__left__NO__MATCH,axiom,
    ! [X: code_integer,Y3: code_integer,A: rat,B: rat,C: rat] :
      ( ( nO_MAT7795273704451493282er_rat @ ( divide6298287555418463151nteger @ X @ Y3 ) @ A )
     => ( ( times_times_rat @ A @ ( plus_plus_rat @ B @ C ) )
        = ( plus_plus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).

% distrib_left_NO_MATCH
thf(fact_5192_distrib__right__NO__MATCH,axiom,
    ! [X: nat,Y3: nat,C: code_integer,A: code_integer,B: code_integer] :
      ( ( nO_MAT8694959213317031834nteger @ ( divide_divide_nat @ X @ Y3 ) @ C )
     => ( ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
        = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) ) ) ) ).

% distrib_right_NO_MATCH
thf(fact_5193_distrib__right__NO__MATCH,axiom,
    ! [X: nat,Y3: nat,C: rat,A: rat,B: rat] :
      ( ( nO_MATCH_nat_rat @ ( divide_divide_nat @ X @ Y3 ) @ C )
     => ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C )
        = ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).

% distrib_right_NO_MATCH
thf(fact_5194_distrib__right__NO__MATCH,axiom,
    ! [X: nat,Y3: nat,C: nat,A: nat,B: nat] :
      ( ( nO_MATCH_nat_nat @ ( divide_divide_nat @ X @ Y3 ) @ C )
     => ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
        = ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).

% distrib_right_NO_MATCH
thf(fact_5195_distrib__right__NO__MATCH,axiom,
    ! [X: nat,Y3: nat,C: int,A: int,B: int] :
      ( ( nO_MATCH_nat_int @ ( divide_divide_nat @ X @ Y3 ) @ C )
     => ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
        = ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).

% distrib_right_NO_MATCH
thf(fact_5196_distrib__right__NO__MATCH,axiom,
    ! [X: int,Y3: int,C: code_integer,A: code_integer,B: code_integer] :
      ( ( nO_MAT9066612773553876470nteger @ ( divide_divide_int @ X @ Y3 ) @ C )
     => ( ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
        = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) ) ) ) ).

% distrib_right_NO_MATCH
thf(fact_5197_distrib__right__NO__MATCH,axiom,
    ! [X: int,Y3: int,C: rat,A: rat,B: rat] :
      ( ( nO_MATCH_int_rat @ ( divide_divide_int @ X @ Y3 ) @ C )
     => ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C )
        = ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).

% distrib_right_NO_MATCH
thf(fact_5198_distrib__right__NO__MATCH,axiom,
    ! [X: int,Y3: int,C: nat,A: nat,B: nat] :
      ( ( nO_MATCH_int_nat @ ( divide_divide_int @ X @ Y3 ) @ C )
     => ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
        = ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).

% distrib_right_NO_MATCH
thf(fact_5199_distrib__right__NO__MATCH,axiom,
    ! [X: int,Y3: int,C: int,A: int,B: int] :
      ( ( nO_MATCH_int_int @ ( divide_divide_int @ X @ Y3 ) @ C )
     => ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
        = ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).

% distrib_right_NO_MATCH
thf(fact_5200_distrib__right__NO__MATCH,axiom,
    ! [X: code_integer,Y3: code_integer,C: code_integer,A: code_integer,B: code_integer] :
      ( ( nO_MAT8252062027627875367nteger @ ( divide6298287555418463151nteger @ X @ Y3 ) @ C )
     => ( ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
        = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) ) ) ) ).

% distrib_right_NO_MATCH
thf(fact_5201_distrib__right__NO__MATCH,axiom,
    ! [X: code_integer,Y3: code_integer,C: rat,A: rat,B: rat] :
      ( ( nO_MAT7795273704451493282er_rat @ ( divide6298287555418463151nteger @ X @ Y3 ) @ C )
     => ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C )
        = ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).

% distrib_right_NO_MATCH
thf(fact_5202_left__diff__distrib__NO__MATCH,axiom,
    ! [X: nat,Y3: nat,C: code_integer,A: code_integer,B: code_integer] :
      ( ( nO_MAT8694959213317031834nteger @ ( divide_divide_nat @ X @ Y3 ) @ C )
     => ( ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C )
        = ( minus_8373710615458151222nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) ) ) ) ).

% left_diff_distrib_NO_MATCH
thf(fact_5203_left__diff__distrib__NO__MATCH,axiom,
    ! [X: nat,Y3: nat,C: rat,A: rat,B: rat] :
      ( ( nO_MATCH_nat_rat @ ( divide_divide_nat @ X @ Y3 ) @ C )
     => ( ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ C )
        = ( minus_minus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).

% left_diff_distrib_NO_MATCH
thf(fact_5204_left__diff__distrib__NO__MATCH,axiom,
    ! [X: nat,Y3: nat,C: int,A: int,B: int] :
      ( ( nO_MATCH_nat_int @ ( divide_divide_nat @ X @ Y3 ) @ C )
     => ( ( times_times_int @ ( minus_minus_int @ A @ B ) @ C )
        = ( minus_minus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).

% left_diff_distrib_NO_MATCH
thf(fact_5205_left__diff__distrib__NO__MATCH,axiom,
    ! [X: int,Y3: int,C: code_integer,A: code_integer,B: code_integer] :
      ( ( nO_MAT9066612773553876470nteger @ ( divide_divide_int @ X @ Y3 ) @ C )
     => ( ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C )
        = ( minus_8373710615458151222nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) ) ) ) ).

% left_diff_distrib_NO_MATCH
thf(fact_5206_left__diff__distrib__NO__MATCH,axiom,
    ! [X: int,Y3: int,C: rat,A: rat,B: rat] :
      ( ( nO_MATCH_int_rat @ ( divide_divide_int @ X @ Y3 ) @ C )
     => ( ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ C )
        = ( minus_minus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).

% left_diff_distrib_NO_MATCH
thf(fact_5207_left__diff__distrib__NO__MATCH,axiom,
    ! [X: int,Y3: int,C: int,A: int,B: int] :
      ( ( nO_MATCH_int_int @ ( divide_divide_int @ X @ Y3 ) @ C )
     => ( ( times_times_int @ ( minus_minus_int @ A @ B ) @ C )
        = ( minus_minus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).

% left_diff_distrib_NO_MATCH
thf(fact_5208_left__diff__distrib__NO__MATCH,axiom,
    ! [X: code_integer,Y3: code_integer,C: code_integer,A: code_integer,B: code_integer] :
      ( ( nO_MAT8252062027627875367nteger @ ( divide6298287555418463151nteger @ X @ Y3 ) @ C )
     => ( ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C )
        = ( minus_8373710615458151222nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) ) ) ) ).

% left_diff_distrib_NO_MATCH
thf(fact_5209_left__diff__distrib__NO__MATCH,axiom,
    ! [X: code_integer,Y3: code_integer,C: rat,A: rat,B: rat] :
      ( ( nO_MAT7795273704451493282er_rat @ ( divide6298287555418463151nteger @ X @ Y3 ) @ C )
     => ( ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ C )
        = ( minus_minus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).

% left_diff_distrib_NO_MATCH
thf(fact_5210_left__diff__distrib__NO__MATCH,axiom,
    ! [X: code_integer,Y3: code_integer,C: int,A: int,B: int] :
      ( ( nO_MAT8427913294028938742er_int @ ( divide6298287555418463151nteger @ X @ Y3 ) @ C )
     => ( ( times_times_int @ ( minus_minus_int @ A @ B ) @ C )
        = ( minus_minus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).

% left_diff_distrib_NO_MATCH
thf(fact_5211_left__diff__distrib__NO__MATCH,axiom,
    ! [X: rat,Y3: rat,C: code_integer,A: code_integer,B: code_integer] :
      ( ( nO_MAT243659986023775650nteger @ ( divide_divide_rat @ X @ Y3 ) @ C )
     => ( ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C )
        = ( minus_8373710615458151222nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) ) ) ) ).

% left_diff_distrib_NO_MATCH
thf(fact_5212_right__diff__distrib__NO__MATCH,axiom,
    ! [X: nat,Y3: nat,A: code_integer,B: code_integer,C: code_integer] :
      ( ( nO_MAT8694959213317031834nteger @ ( divide_divide_nat @ X @ Y3 ) @ A )
     => ( ( times_3573771949741848930nteger @ A @ ( minus_8373710615458151222nteger @ B @ C ) )
        = ( minus_8373710615458151222nteger @ ( times_3573771949741848930nteger @ A @ B ) @ ( times_3573771949741848930nteger @ A @ C ) ) ) ) ).

% right_diff_distrib_NO_MATCH
thf(fact_5213_right__diff__distrib__NO__MATCH,axiom,
    ! [X: nat,Y3: nat,A: rat,B: rat,C: rat] :
      ( ( nO_MATCH_nat_rat @ ( divide_divide_nat @ X @ Y3 ) @ A )
     => ( ( times_times_rat @ A @ ( minus_minus_rat @ B @ C ) )
        = ( minus_minus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).

% right_diff_distrib_NO_MATCH
thf(fact_5214_right__diff__distrib__NO__MATCH,axiom,
    ! [X: nat,Y3: nat,A: int,B: int,C: int] :
      ( ( nO_MATCH_nat_int @ ( divide_divide_nat @ X @ Y3 ) @ A )
     => ( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
        = ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ) ).

% right_diff_distrib_NO_MATCH
thf(fact_5215_right__diff__distrib__NO__MATCH,axiom,
    ! [X: int,Y3: int,A: code_integer,B: code_integer,C: code_integer] :
      ( ( nO_MAT9066612773553876470nteger @ ( divide_divide_int @ X @ Y3 ) @ A )
     => ( ( times_3573771949741848930nteger @ A @ ( minus_8373710615458151222nteger @ B @ C ) )
        = ( minus_8373710615458151222nteger @ ( times_3573771949741848930nteger @ A @ B ) @ ( times_3573771949741848930nteger @ A @ C ) ) ) ) ).

% right_diff_distrib_NO_MATCH
thf(fact_5216_right__diff__distrib__NO__MATCH,axiom,
    ! [X: int,Y3: int,A: rat,B: rat,C: rat] :
      ( ( nO_MATCH_int_rat @ ( divide_divide_int @ X @ Y3 ) @ A )
     => ( ( times_times_rat @ A @ ( minus_minus_rat @ B @ C ) )
        = ( minus_minus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).

% right_diff_distrib_NO_MATCH
thf(fact_5217_right__diff__distrib__NO__MATCH,axiom,
    ! [X: int,Y3: int,A: int,B: int,C: int] :
      ( ( nO_MATCH_int_int @ ( divide_divide_int @ X @ Y3 ) @ A )
     => ( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
        = ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ) ).

% right_diff_distrib_NO_MATCH
thf(fact_5218_right__diff__distrib__NO__MATCH,axiom,
    ! [X: code_integer,Y3: code_integer,A: code_integer,B: code_integer,C: code_integer] :
      ( ( nO_MAT8252062027627875367nteger @ ( divide6298287555418463151nteger @ X @ Y3 ) @ A )
     => ( ( times_3573771949741848930nteger @ A @ ( minus_8373710615458151222nteger @ B @ C ) )
        = ( minus_8373710615458151222nteger @ ( times_3573771949741848930nteger @ A @ B ) @ ( times_3573771949741848930nteger @ A @ C ) ) ) ) ).

% right_diff_distrib_NO_MATCH
thf(fact_5219_right__diff__distrib__NO__MATCH,axiom,
    ! [X: code_integer,Y3: code_integer,A: rat,B: rat,C: rat] :
      ( ( nO_MAT7795273704451493282er_rat @ ( divide6298287555418463151nteger @ X @ Y3 ) @ A )
     => ( ( times_times_rat @ A @ ( minus_minus_rat @ B @ C ) )
        = ( minus_minus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).

% right_diff_distrib_NO_MATCH
thf(fact_5220_right__diff__distrib__NO__MATCH,axiom,
    ! [X: code_integer,Y3: code_integer,A: int,B: int,C: int] :
      ( ( nO_MAT8427913294028938742er_int @ ( divide6298287555418463151nteger @ X @ Y3 ) @ A )
     => ( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
        = ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ) ).

% right_diff_distrib_NO_MATCH
thf(fact_5221_right__diff__distrib__NO__MATCH,axiom,
    ! [X: rat,Y3: rat,A: code_integer,B: code_integer,C: code_integer] :
      ( ( nO_MAT243659986023775650nteger @ ( divide_divide_rat @ X @ Y3 ) @ A )
     => ( ( times_3573771949741848930nteger @ A @ ( minus_8373710615458151222nteger @ B @ C ) )
        = ( minus_8373710615458151222nteger @ ( times_3573771949741848930nteger @ A @ B ) @ ( times_3573771949741848930nteger @ A @ C ) ) ) ) ).

% right_diff_distrib_NO_MATCH
thf(fact_5222_ceiling__divide__upper,axiom,
    ! [Q4: rat,P5: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Q4 )
     => ( ord_less_eq_rat @ P5 @ ( times_times_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ ( divide_divide_rat @ P5 @ Q4 ) ) ) @ Q4 ) ) ) ).

% ceiling_divide_upper
thf(fact_5223_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_5224_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_5225_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_5226_floor__le__neg__numeral,axiom,
    ! [X: rat,V: num] :
      ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
      = ( ord_less_rat @ X @ ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) ) ) ).

% floor_le_neg_numeral
thf(fact_5227_neg__numeral__less__floor,axiom,
    ! [V: num,X: rat] :
      ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim3151403230148437115or_rat @ X ) )
      = ( ord_less_eq_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) @ X ) ) ).

% neg_numeral_less_floor
thf(fact_5228_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_5229_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_5230_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_5231_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_5232_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_5233_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_5234_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_5235_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_5236_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_5237_floor__of__int,axiom,
    ! [Z2: int] :
      ( ( archim3151403230148437115or_rat @ ( ring_1_of_int_rat @ Z2 ) )
      = Z2 ) ).

% floor_of_int
thf(fact_5238_floor__zero,axiom,
    ( ( archim3151403230148437115or_rat @ zero_zero_rat )
    = zero_zero_int ) ).

% floor_zero
thf(fact_5239_floor__numeral,axiom,
    ! [V: num] :
      ( ( archim3151403230148437115or_rat @ ( numeral_numeral_rat @ V ) )
      = ( numeral_numeral_int @ V ) ) ).

% floor_numeral
thf(fact_5240_floor__one,axiom,
    ( ( archim3151403230148437115or_rat @ one_one_rat )
    = one_one_int ) ).

% floor_one
thf(fact_5241_floor__of__nat,axiom,
    ! [N: nat] :
      ( ( archim3151403230148437115or_rat @ ( semiri681578069525770553at_rat @ N ) )
      = ( semiri1314217659103216013at_int @ N ) ) ).

% floor_of_nat
thf(fact_5242_floor__uminus__of__int,axiom,
    ! [Z2: int] :
      ( ( archim3151403230148437115or_rat @ ( uminus_uminus_rat @ ( ring_1_of_int_rat @ Z2 ) ) )
      = ( uminus_uminus_int @ Z2 ) ) ).

% floor_uminus_of_int
thf(fact_5243_floor__diff__of__int,axiom,
    ! [X: rat,Z2: int] :
      ( ( archim3151403230148437115or_rat @ ( minus_minus_rat @ X @ ( ring_1_of_int_rat @ Z2 ) ) )
      = ( minus_minus_int @ ( archim3151403230148437115or_rat @ X ) @ Z2 ) ) ).

% floor_diff_of_int
thf(fact_5244_floor__add2,axiom,
    ! [X: rat,Y3: rat] :
      ( ( ( member_rat @ X @ ring_1_Ints_rat )
        | ( member_rat @ Y3 @ ring_1_Ints_rat ) )
     => ( ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X @ Y3 ) )
        = ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim3151403230148437115or_rat @ Y3 ) ) ) ) ).

% floor_add2
thf(fact_5245_zero__le__floor,axiom,
    ! [X: rat] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( archim3151403230148437115or_rat @ X ) )
      = ( ord_less_eq_rat @ zero_zero_rat @ X ) ) ).

% zero_le_floor
thf(fact_5246_floor__less__zero,axiom,
    ! [X: rat] :
      ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X ) @ zero_zero_int )
      = ( ord_less_rat @ X @ zero_zero_rat ) ) ).

% floor_less_zero
thf(fact_5247_numeral__le__floor,axiom,
    ! [V: num,X: rat] :
      ( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim3151403230148437115or_rat @ X ) )
      = ( ord_less_eq_rat @ ( numeral_numeral_rat @ V ) @ X ) ) ).

% numeral_le_floor
thf(fact_5248_zero__less__floor,axiom,
    ! [X: rat] :
      ( ( ord_less_int @ zero_zero_int @ ( archim3151403230148437115or_rat @ X ) )
      = ( ord_less_eq_rat @ one_one_rat @ X ) ) ).

% zero_less_floor
thf(fact_5249_floor__le__zero,axiom,
    ! [X: rat] :
      ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ zero_zero_int )
      = ( ord_less_rat @ X @ one_one_rat ) ) ).

% floor_le_zero
thf(fact_5250_floor__less__numeral,axiom,
    ! [X: rat,V: num] :
      ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X ) @ ( numeral_numeral_int @ V ) )
      = ( ord_less_rat @ X @ ( numeral_numeral_rat @ V ) ) ) ).

% floor_less_numeral
thf(fact_5251_one__le__floor,axiom,
    ! [X: rat] :
      ( ( ord_less_eq_int @ one_one_int @ ( archim3151403230148437115or_rat @ X ) )
      = ( ord_less_eq_rat @ one_one_rat @ X ) ) ).

% one_le_floor
thf(fact_5252_floor__less__one,axiom,
    ! [X: rat] :
      ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X ) @ one_one_int )
      = ( ord_less_rat @ X @ one_one_rat ) ) ).

% floor_less_one
thf(fact_5253_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_5254_floor__diff__numeral,axiom,
    ! [X: rat,V: num] :
      ( ( archim3151403230148437115or_rat @ ( minus_minus_rat @ X @ ( numeral_numeral_rat @ V ) ) )
      = ( minus_minus_int @ ( archim3151403230148437115or_rat @ X ) @ ( numeral_numeral_int @ V ) ) ) ).

% floor_diff_numeral
thf(fact_5255_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_5256_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_5257_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_5258_floor__diff__one,axiom,
    ! [X: rat] :
      ( ( archim3151403230148437115or_rat @ ( minus_minus_rat @ X @ one_one_rat ) )
      = ( minus_minus_int @ ( archim3151403230148437115or_rat @ X ) @ one_one_int ) ) ).

% floor_diff_one
thf(fact_5259_numeral__less__floor,axiom,
    ! [V: num,X: rat] :
      ( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim3151403230148437115or_rat @ X ) )
      = ( ord_less_eq_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) @ X ) ) ).

% numeral_less_floor
thf(fact_5260_floor__le__numeral,axiom,
    ! [X: rat,V: num] :
      ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ ( numeral_numeral_int @ V ) )
      = ( ord_less_rat @ X @ ( plus_plus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) ) ) ).

% floor_le_numeral
thf(fact_5261_neg__numeral__le__floor,axiom,
    ! [V: num,X: rat] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim3151403230148437115or_rat @ X ) )
      = ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ X ) ) ).

% neg_numeral_le_floor
thf(fact_5262_floor__less__neg__numeral,axiom,
    ! [X: rat,V: num] :
      ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
      = ( ord_less_rat @ X @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) ) ).

% floor_less_neg_numeral
thf(fact_5263_add__inc,axiom,
    ! [X: num,Y3: num] :
      ( ( plus_plus_num @ X @ ( inc @ Y3 ) )
      = ( inc @ ( plus_plus_num @ X @ Y3 ) ) ) ).

% add_inc
thf(fact_5264_mult__inc,axiom,
    ! [X: num,Y3: num] :
      ( ( times_times_num @ X @ ( inc @ Y3 ) )
      = ( plus_plus_num @ ( times_times_num @ X @ Y3 ) @ X ) ) ).

% mult_inc
thf(fact_5265_floor__mono,axiom,
    ! [X: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ X @ Y3 )
     => ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim3151403230148437115or_rat @ Y3 ) ) ) ).

% floor_mono
thf(fact_5266_floor__less__cancel,axiom,
    ! [X: rat,Y3: rat] :
      ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim3151403230148437115or_rat @ Y3 ) )
     => ( ord_less_rat @ X @ Y3 ) ) ).

% floor_less_cancel
thf(fact_5267_of__int__floor__le,axiom,
    ! [X: rat] : ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X ) ) @ X ) ).

% of_int_floor_le
thf(fact_5268_floor__le__ceiling,axiom,
    ! [X: rat] : ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim2889992004027027881ng_rat @ X ) ) ).

% floor_le_ceiling
thf(fact_5269_floor__le__round,axiom,
    ! [X: rat] : ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim7778729529865785530nd_rat @ X ) ) ).

% floor_le_round
thf(fact_5270_le__floor__iff,axiom,
    ! [Z2: int,X: rat] :
      ( ( ord_less_eq_int @ Z2 @ ( archim3151403230148437115or_rat @ X ) )
      = ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z2 ) @ X ) ) ).

% le_floor_iff
thf(fact_5271_floor__less__iff,axiom,
    ! [X: rat,Z2: int] :
      ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X ) @ Z2 )
      = ( ord_less_rat @ X @ ( ring_1_of_int_rat @ Z2 ) ) ) ).

% floor_less_iff
thf(fact_5272_le__floor__add,axiom,
    ! [X: rat,Y3: rat] : ( ord_less_eq_int @ ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim3151403230148437115or_rat @ Y3 ) ) @ ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X @ Y3 ) ) ) ).

% le_floor_add
thf(fact_5273_int__add__floor,axiom,
    ! [Z2: int,X: rat] :
      ( ( plus_plus_int @ Z2 @ ( archim3151403230148437115or_rat @ X ) )
      = ( archim3151403230148437115or_rat @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z2 ) @ X ) ) ) ).

% int_add_floor
thf(fact_5274_floor__add__int,axiom,
    ! [X: rat,Z2: int] :
      ( ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ Z2 )
      = ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X @ ( ring_1_of_int_rat @ Z2 ) ) ) ) ).

% floor_add_int
thf(fact_5275_floor__divide__of__int__eq,axiom,
    ! [K: int,L: int] :
      ( ( archim3151403230148437115or_rat @ ( divide_divide_rat @ ( ring_1_of_int_rat @ K ) @ ( ring_1_of_int_rat @ L ) ) )
      = ( divide_divide_int @ K @ L ) ) ).

% floor_divide_of_int_eq
thf(fact_5276_ceiling__def,axiom,
    ( archim2889992004027027881ng_rat
    = ( ^ [X2: rat] : ( uminus_uminus_int @ ( archim3151403230148437115or_rat @ ( uminus_uminus_rat @ X2 ) ) ) ) ) ).

% ceiling_def
thf(fact_5277_floor__minus,axiom,
    ! [X: rat] :
      ( ( archim3151403230148437115or_rat @ ( uminus_uminus_rat @ X ) )
      = ( uminus_uminus_int @ ( archim2889992004027027881ng_rat @ X ) ) ) ).

% floor_minus
thf(fact_5278_ceiling__minus,axiom,
    ! [X: rat] :
      ( ( archim2889992004027027881ng_rat @ ( uminus_uminus_rat @ X ) )
      = ( uminus_uminus_int @ ( archim3151403230148437115or_rat @ X ) ) ) ).

% ceiling_minus
thf(fact_5279_frac__def,axiom,
    ( archimedean_frac_rat
    = ( ^ [X2: rat] : ( minus_minus_rat @ X2 @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X2 ) ) ) ) ) ).

% frac_def
thf(fact_5280_of__nat__floor,axiom,
    ! [R: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ R )
     => ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ ( nat2 @ ( archim3151403230148437115or_rat @ R ) ) ) @ R ) ) ).

% of_nat_floor
thf(fact_5281_one__add__floor,axiom,
    ! [X: rat] :
      ( ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ one_one_int )
      = ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X @ one_one_rat ) ) ) ).

% one_add_floor
thf(fact_5282_numeral__inc,axiom,
    ! [X: num] :
      ( ( numeral_numeral_int @ ( inc @ X ) )
      = ( plus_plus_int @ ( numeral_numeral_int @ X ) @ one_one_int ) ) ).

% numeral_inc
thf(fact_5283_numeral__inc,axiom,
    ! [X: num] :
      ( ( numeral_numeral_nat @ ( inc @ X ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ X ) @ one_one_nat ) ) ).

% numeral_inc
thf(fact_5284_numeral__inc,axiom,
    ! [X: num] :
      ( ( numera6620942414471956472nteger @ ( inc @ X ) )
      = ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ X ) @ one_one_Code_integer ) ) ).

% numeral_inc
thf(fact_5285_numeral__inc,axiom,
    ! [X: num] :
      ( ( numeral_numeral_rat @ ( inc @ X ) )
      = ( plus_plus_rat @ ( numeral_numeral_rat @ X ) @ one_one_rat ) ) ).

% numeral_inc
thf(fact_5286_numeral__inc,axiom,
    ! [X: num] :
      ( ( numera5444537566228673987atural @ ( inc @ X ) )
      = ( plus_p4538020629002901425atural @ ( numera5444537566228673987atural @ X ) @ one_one_Code_natural ) ) ).

% numeral_inc
thf(fact_5287_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_5288_ceiling__diff__floor__le__1,axiom,
    ! [X: rat] : ( ord_less_eq_int @ ( minus_minus_int @ ( archim2889992004027027881ng_rat @ X ) @ ( archim3151403230148437115or_rat @ X ) ) @ one_one_int ) ).

% ceiling_diff_floor_le_1
thf(fact_5289_ceiling__altdef,axiom,
    ( archim2889992004027027881ng_rat
    = ( ^ [X2: rat] :
          ( if_int
          @ ( X2
            = ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X2 ) ) )
          @ ( archim3151403230148437115or_rat @ X2 )
          @ ( plus_plus_int @ ( archim3151403230148437115or_rat @ X2 ) @ one_one_int ) ) ) ) ).

% ceiling_altdef
thf(fact_5290_floor__split,axiom,
    ! [P: int > $o,T: rat] :
      ( ( P @ ( archim3151403230148437115or_rat @ T ) )
      = ( ! [I4: int] :
            ( ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ I4 ) @ T )
              & ( ord_less_rat @ T @ ( plus_plus_rat @ ( ring_1_of_int_rat @ I4 ) @ one_one_rat ) ) )
           => ( P @ I4 ) ) ) ) ).

% floor_split
thf(fact_5291_floor__eq__iff,axiom,
    ! [X: rat,A: int] :
      ( ( ( archim3151403230148437115or_rat @ X )
        = A )
      = ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A ) @ X )
        & ( ord_less_rat @ X @ ( plus_plus_rat @ ( ring_1_of_int_rat @ A ) @ one_one_rat ) ) ) ) ).

% floor_eq_iff
thf(fact_5292_floor__unique,axiom,
    ! [Z2: int,X: rat] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z2 ) @ X )
     => ( ( ord_less_rat @ X @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z2 ) @ one_one_rat ) )
       => ( ( archim3151403230148437115or_rat @ X )
          = Z2 ) ) ) ).

% floor_unique
thf(fact_5293_le__mult__floor,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
       => ( ord_less_eq_int @ ( times_times_int @ ( archim3151403230148437115or_rat @ A ) @ ( archim3151403230148437115or_rat @ B ) ) @ ( archim3151403230148437115or_rat @ ( times_times_rat @ A @ B ) ) ) ) ) ).

% le_mult_floor
thf(fact_5294_less__floor__iff,axiom,
    ! [Z2: int,X: rat] :
      ( ( ord_less_int @ Z2 @ ( archim3151403230148437115or_rat @ X ) )
      = ( ord_less_eq_rat @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z2 ) @ one_one_rat ) @ X ) ) ).

% less_floor_iff
thf(fact_5295_floor__le__iff,axiom,
    ! [X: rat,Z2: int] :
      ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ Z2 )
      = ( ord_less_rat @ X @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z2 ) @ one_one_rat ) ) ) ).

% floor_le_iff
thf(fact_5296_floor__correct,axiom,
    ! [X: rat] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X ) ) @ X )
      & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ one_one_int ) ) ) ) ).

% floor_correct
thf(fact_5297_floor__divide__lower,axiom,
    ! [Q4: rat,P5: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Q4 )
     => ( ord_less_eq_rat @ ( times_times_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ ( divide_divide_rat @ P5 @ Q4 ) ) ) @ Q4 ) @ P5 ) ) ).

% floor_divide_lower
thf(fact_5298_le__mult__floor__Ints,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( member_rat @ A @ ring_1_Ints_rat )
       => ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ ( times_times_int @ ( archim3151403230148437115or_rat @ A ) @ ( archim3151403230148437115or_rat @ B ) ) ) @ ( ring_18347121197199848620nteger @ ( archim3151403230148437115or_rat @ ( times_times_rat @ A @ B ) ) ) ) ) ) ).

% le_mult_floor_Ints
thf(fact_5299_le__mult__floor__Ints,axiom,
    ! [A: rat,B: 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 @ B ) ) ) @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ ( times_times_rat @ A @ B ) ) ) ) ) ) ).

% le_mult_floor_Ints
thf(fact_5300_le__mult__floor__Ints,axiom,
    ! [A: rat,B: 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 @ B ) ) ) @ ( ring_1_of_int_int @ ( archim3151403230148437115or_rat @ ( times_times_rat @ A @ B ) ) ) ) ) ) ).

% le_mult_floor_Ints
thf(fact_5301_floor__add,axiom,
    ! [X: rat,Y3: rat] :
      ( ( ( ord_less_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y3 ) ) @ one_one_rat )
       => ( ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X @ Y3 ) )
          = ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim3151403230148437115or_rat @ Y3 ) ) ) )
      & ( ~ ( ord_less_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y3 ) ) @ one_one_rat )
       => ( ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X @ Y3 ) )
          = ( plus_plus_int @ ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim3151403230148437115or_rat @ Y3 ) ) @ one_one_int ) ) ) ) ).

% floor_add
thf(fact_5302_floor__divide__upper,axiom,
    ! [Q4: rat,P5: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Q4 )
     => ( ord_less_rat @ P5 @ ( times_times_rat @ ( plus_plus_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ ( divide_divide_rat @ P5 @ Q4 ) ) ) @ one_one_rat ) @ Q4 ) ) ) ).

% floor_divide_upper
thf(fact_5303_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_5304_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_5305_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_5306_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_5307_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_5308_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_5309_cpmi,axiom,
    ! [D3: int,P: int > $o,P2: int > $o,B5: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D3 )
     => ( ? [Z4: int] :
          ! [X3: int] :
            ( ( ord_less_int @ X3 @ Z4 )
           => ( ( P @ X3 )
              = ( P2 @ X3 ) ) )
       => ( ! [X3: int] :
              ( ! [Xa: int] :
                  ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
                 => ! [Xb: int] :
                      ( ( member_int @ Xb @ B5 )
                     => ( X3
                       != ( plus_plus_int @ Xb @ Xa ) ) ) )
             => ( ( P @ X3 )
               => ( P @ ( minus_minus_int @ X3 @ D3 ) ) ) )
         => ( ! [X3: int,K2: int] :
                ( ( P2 @ X3 )
                = ( P2 @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D3 ) ) ) )
           => ( ( ? [X7: int] : ( P @ X7 ) )
              = ( ? [X2: int] :
                    ( ( member_int @ X2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
                    & ( P2 @ X2 ) )
                | ? [X2: int] :
                    ( ( member_int @ X2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
                    & ? [Y6: int] :
                        ( ( member_int @ Y6 @ B5 )
                        & ( P @ ( plus_plus_int @ Y6 @ X2 ) ) ) ) ) ) ) ) ) ) ).

% cpmi
thf(fact_5310_cppi,axiom,
    ! [D3: int,P: int > $o,P2: int > $o,A3: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D3 )
     => ( ? [Z4: int] :
          ! [X3: int] :
            ( ( ord_less_int @ Z4 @ X3 )
           => ( ( P @ X3 )
              = ( P2 @ X3 ) ) )
       => ( ! [X3: int] :
              ( ! [Xa: int] :
                  ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
                 => ! [Xb: int] :
                      ( ( member_int @ Xb @ A3 )
                     => ( X3
                       != ( minus_minus_int @ Xb @ Xa ) ) ) )
             => ( ( P @ X3 )
               => ( P @ ( plus_plus_int @ X3 @ D3 ) ) ) )
         => ( ! [X3: int,K2: int] :
                ( ( P2 @ X3 )
                = ( P2 @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D3 ) ) ) )
           => ( ( ? [X7: int] : ( P @ X7 ) )
              = ( ? [X2: int] :
                    ( ( member_int @ X2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
                    & ( P2 @ X2 ) )
                | ? [X2: int] :
                    ( ( member_int @ X2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
                    & ? [Y6: int] :
                        ( ( member_int @ Y6 @ A3 )
                        & ( P @ ( minus_minus_int @ Y6 @ X2 ) ) ) ) ) ) ) ) ) ) ).

% cppi
thf(fact_5311_aset_I8_J,axiom,
    ! [D3: int,A3: set_int,T: int] :
      ( ( ord_less_int @ zero_zero_int @ D3 )
     => ! [X4: int] :
          ( ! [Xa2: int] :
              ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
             => ! [Xb2: int] :
                  ( ( member_int @ Xb2 @ A3 )
                 => ( X4
                   != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
         => ( ( ord_less_eq_int @ T @ X4 )
           => ( ord_less_eq_int @ T @ ( plus_plus_int @ X4 @ D3 ) ) ) ) ) ).

% aset(8)
thf(fact_5312_aset_I6_J,axiom,
    ! [D3: int,T: int,A3: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D3 )
     => ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A3 )
       => ! [X4: int] :
            ( ! [Xa2: int] :
                ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ A3 )
                   => ( X4
                     != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
           => ( ( ord_less_eq_int @ X4 @ T )
             => ( ord_less_eq_int @ ( plus_plus_int @ X4 @ D3 ) @ T ) ) ) ) ) ).

% aset(6)
thf(fact_5313_bset_I8_J,axiom,
    ! [D3: int,T: int,B5: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D3 )
     => ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B5 )
       => ! [X4: int] :
            ( ! [Xa2: int] :
                ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ B5 )
                   => ( X4
                     != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
           => ( ( ord_less_eq_int @ T @ X4 )
             => ( ord_less_eq_int @ T @ ( minus_minus_int @ X4 @ D3 ) ) ) ) ) ) ).

% bset(8)
thf(fact_5314_atLeastatMost__subset__iff,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat,D: set_nat] :
      ( ( ord_le6893508408891458716et_nat @ ( set_or4548717258645045905et_nat @ A @ B ) @ ( set_or4548717258645045905et_nat @ C @ D ) )
      = ( ~ ( ord_less_eq_set_nat @ A @ B )
        | ( ( ord_less_eq_set_nat @ C @ A )
          & ( ord_less_eq_set_nat @ B @ D ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_5315_atLeastatMost__subset__iff,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_eq_set_rat @ ( set_or633870826150836451st_rat @ A @ B ) @ ( set_or633870826150836451st_rat @ C @ D ) )
      = ( ~ ( ord_less_eq_rat @ A @ B )
        | ( ( ord_less_eq_rat @ C @ A )
          & ( ord_less_eq_rat @ B @ D ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_5316_atLeastatMost__subset__iff,axiom,
    ! [A: num,B: num,C: num,D: num] :
      ( ( ord_less_eq_set_num @ ( set_or7049704709247886629st_num @ A @ B ) @ ( set_or7049704709247886629st_num @ C @ D ) )
      = ( ~ ( ord_less_eq_num @ A @ B )
        | ( ( ord_less_eq_num @ C @ A )
          & ( ord_less_eq_num @ B @ D ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_5317_atLeastatMost__subset__iff,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_eq_set_int @ ( set_or1266510415728281911st_int @ A @ B ) @ ( set_or1266510415728281911st_int @ C @ D ) )
      = ( ~ ( ord_less_eq_int @ A @ B )
        | ( ( ord_less_eq_int @ C @ A )
          & ( ord_less_eq_int @ B @ D ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_5318_atLeastatMost__subset__iff,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ A @ B ) @ ( set_or1269000886237332187st_nat @ C @ D ) )
      = ( ~ ( ord_less_eq_nat @ A @ B )
        | ( ( ord_less_eq_nat @ C @ A )
          & ( ord_less_eq_nat @ B @ D ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_5319_atLeastAtMost__iff,axiom,
    ! [I2: set_Pr1261947904930325089at_nat,L: set_Pr1261947904930325089at_nat,U: set_Pr1261947904930325089at_nat] :
      ( ( member2643936169264416010at_nat @ I2 @ ( set_or2784603332384507286at_nat @ L @ U ) )
      = ( ( ord_le3146513528884898305at_nat @ L @ I2 )
        & ( ord_le3146513528884898305at_nat @ I2 @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_5320_atLeastAtMost__iff,axiom,
    ! [I2: set_nat,L: set_nat,U: set_nat] :
      ( ( member_set_nat @ I2 @ ( set_or4548717258645045905et_nat @ L @ U ) )
      = ( ( ord_less_eq_set_nat @ L @ I2 )
        & ( ord_less_eq_set_nat @ I2 @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_5321_atLeastAtMost__iff,axiom,
    ! [I2: rat,L: rat,U: rat] :
      ( ( member_rat @ I2 @ ( set_or633870826150836451st_rat @ L @ U ) )
      = ( ( ord_less_eq_rat @ L @ I2 )
        & ( ord_less_eq_rat @ I2 @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_5322_atLeastAtMost__iff,axiom,
    ! [I2: num,L: num,U: num] :
      ( ( member_num @ I2 @ ( set_or7049704709247886629st_num @ L @ U ) )
      = ( ( ord_less_eq_num @ L @ I2 )
        & ( ord_less_eq_num @ I2 @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_5323_atLeastAtMost__iff,axiom,
    ! [I2: int,L: int,U: int] :
      ( ( member_int @ I2 @ ( set_or1266510415728281911st_int @ L @ U ) )
      = ( ( ord_less_eq_int @ L @ I2 )
        & ( ord_less_eq_int @ I2 @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_5324_atLeastAtMost__iff,axiom,
    ! [I2: nat,L: nat,U: nat] :
      ( ( member_nat @ I2 @ ( set_or1269000886237332187st_nat @ L @ U ) )
      = ( ( ord_less_eq_nat @ L @ I2 )
        & ( ord_less_eq_nat @ I2 @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_5325_Icc__eq__Icc,axiom,
    ! [L: set_nat,H2: set_nat,L4: set_nat,H: set_nat] :
      ( ( ( set_or4548717258645045905et_nat @ L @ H2 )
        = ( set_or4548717258645045905et_nat @ L4 @ H ) )
      = ( ( ( L = L4 )
          & ( H2 = H ) )
        | ( ~ ( ord_less_eq_set_nat @ L @ H2 )
          & ~ ( ord_less_eq_set_nat @ L4 @ H ) ) ) ) ).

% Icc_eq_Icc
thf(fact_5326_Icc__eq__Icc,axiom,
    ! [L: rat,H2: rat,L4: rat,H: rat] :
      ( ( ( set_or633870826150836451st_rat @ L @ H2 )
        = ( set_or633870826150836451st_rat @ L4 @ H ) )
      = ( ( ( L = L4 )
          & ( H2 = H ) )
        | ( ~ ( ord_less_eq_rat @ L @ H2 )
          & ~ ( ord_less_eq_rat @ L4 @ H ) ) ) ) ).

% Icc_eq_Icc
thf(fact_5327_Icc__eq__Icc,axiom,
    ! [L: num,H2: num,L4: num,H: num] :
      ( ( ( set_or7049704709247886629st_num @ L @ H2 )
        = ( set_or7049704709247886629st_num @ L4 @ H ) )
      = ( ( ( L = L4 )
          & ( H2 = H ) )
        | ( ~ ( ord_less_eq_num @ L @ H2 )
          & ~ ( ord_less_eq_num @ L4 @ H ) ) ) ) ).

% Icc_eq_Icc
thf(fact_5328_Icc__eq__Icc,axiom,
    ! [L: int,H2: int,L4: int,H: int] :
      ( ( ( set_or1266510415728281911st_int @ L @ H2 )
        = ( set_or1266510415728281911st_int @ L4 @ H ) )
      = ( ( ( L = L4 )
          & ( H2 = H ) )
        | ( ~ ( ord_less_eq_int @ L @ H2 )
          & ~ ( ord_less_eq_int @ L4 @ H ) ) ) ) ).

% Icc_eq_Icc
thf(fact_5329_Icc__eq__Icc,axiom,
    ! [L: nat,H2: nat,L4: nat,H: nat] :
      ( ( ( set_or1269000886237332187st_nat @ L @ H2 )
        = ( set_or1269000886237332187st_nat @ L4 @ H ) )
      = ( ( ( L = L4 )
          & ( H2 = H ) )
        | ( ~ ( ord_less_eq_nat @ L @ H2 )
          & ~ ( ord_less_eq_nat @ L4 @ H ) ) ) ) ).

% Icc_eq_Icc
thf(fact_5330_semiring__norm_I12_J,axiom,
    ! [N: num] :
      ( ( times_times_num @ one @ N )
      = N ) ).

% semiring_norm(12)
thf(fact_5331_semiring__norm_I11_J,axiom,
    ! [M: num] :
      ( ( times_times_num @ M @ one )
      = M ) ).

% semiring_norm(11)
thf(fact_5332_semiring__norm_I68_J,axiom,
    ! [N: num] : ( ord_less_eq_num @ one @ N ) ).

% semiring_norm(68)
thf(fact_5333_semiring__norm_I75_J,axiom,
    ! [M: num] :
      ~ ( ord_less_num @ M @ one ) ).

% semiring_norm(75)
thf(fact_5334_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_int
        = ( numeral_numeral_int @ N ) )
      = ( one = N ) ) ).

% one_eq_numeral_iff
thf(fact_5335_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_nat
        = ( numeral_numeral_nat @ N ) )
      = ( one = N ) ) ).

% one_eq_numeral_iff
thf(fact_5336_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_Code_integer
        = ( numera6620942414471956472nteger @ N ) )
      = ( one = N ) ) ).

% one_eq_numeral_iff
thf(fact_5337_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_rat
        = ( numeral_numeral_rat @ N ) )
      = ( one = N ) ) ).

% one_eq_numeral_iff
thf(fact_5338_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_Code_natural
        = ( numera5444537566228673987atural @ N ) )
      = ( one = N ) ) ).

% one_eq_numeral_iff
thf(fact_5339_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( ( numeral_numeral_int @ N )
        = one_one_int )
      = ( N = one ) ) ).

% numeral_eq_one_iff
thf(fact_5340_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( ( numeral_numeral_nat @ N )
        = one_one_nat )
      = ( N = one ) ) ).

% numeral_eq_one_iff
thf(fact_5341_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( ( numera6620942414471956472nteger @ N )
        = one_one_Code_integer )
      = ( N = one ) ) ).

% numeral_eq_one_iff
thf(fact_5342_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( ( numeral_numeral_rat @ N )
        = one_one_rat )
      = ( N = one ) ) ).

% numeral_eq_one_iff
thf(fact_5343_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( ( numera5444537566228673987atural @ N )
        = one_one_Code_natural )
      = ( N = one ) ) ).

% numeral_eq_one_iff
thf(fact_5344_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_5345_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_5346_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_5347_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_5348_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_5349_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_5350_Suc__numeral,axiom,
    ! [N: num] :
      ( ( suc @ ( numeral_numeral_nat @ N ) )
      = ( numeral_numeral_nat @ ( plus_plus_num @ N @ one ) ) ) ).

% Suc_numeral
thf(fact_5351_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_5352_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_5353_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_5354_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_5355_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_5356_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_5357_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_5358_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_5359_one__plus__numeral,axiom,
    ! [N: num] :
      ( ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ N ) )
      = ( numera6620942414471956472nteger @ ( plus_plus_num @ one @ N ) ) ) ).

% one_plus_numeral
thf(fact_5360_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_5361_one__plus__numeral,axiom,
    ! [N: num] :
      ( ( plus_p4538020629002901425atural @ one_one_Code_natural @ ( numera5444537566228673987atural @ N ) )
      = ( numera5444537566228673987atural @ ( plus_plus_num @ one @ N ) ) ) ).

% one_plus_numeral
thf(fact_5362_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_5363_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_5364_numeral__plus__one,axiom,
    ! [N: num] :
      ( ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ N ) @ one_one_Code_integer )
      = ( numera6620942414471956472nteger @ ( plus_plus_num @ N @ one ) ) ) ).

% numeral_plus_one
thf(fact_5365_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_5366_numeral__plus__one,axiom,
    ! [N: num] :
      ( ( plus_p4538020629002901425atural @ ( numera5444537566228673987atural @ N ) @ one_one_Code_natural )
      = ( numera5444537566228673987atural @ ( plus_plus_num @ N @ one ) ) ) ).

% numeral_plus_one
thf(fact_5367_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ N ) @ one_one_Code_integer )
      = ( ord_less_eq_num @ N @ one ) ) ).

% numeral_le_one_iff
thf(fact_5368_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( ( ord_le1926595141338095240atural @ ( numera5444537566228673987atural @ N ) @ one_one_Code_natural )
      = ( ord_less_eq_num @ N @ one ) ) ).

% numeral_le_one_iff
thf(fact_5369_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_5370_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_5371_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_5372_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_5373_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_5374_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ N ) )
      = ( ord_less_num @ one @ N ) ) ).

% one_less_numeral_iff
thf(fact_5375_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_5376_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( ( ord_le5570908160329646204atural @ one_one_Code_natural @ ( numera5444537566228673987atural @ N ) )
      = ( ord_less_num @ one @ N ) ) ).

% one_less_numeral_iff
thf(fact_5377_add__One__commute,axiom,
    ! [N: num] :
      ( ( plus_plus_num @ one @ N )
      = ( plus_plus_num @ N @ one ) ) ).

% add_One_commute
thf(fact_5378_num__induct,axiom,
    ! [P: num > $o,X: num] :
      ( ( P @ one )
     => ( ! [X3: num] :
            ( ( P @ X3 )
           => ( P @ ( inc @ X3 ) ) )
       => ( P @ X ) ) ) ).

% num_induct
thf(fact_5379_le__num__One__iff,axiom,
    ! [X: num] :
      ( ( ord_less_eq_num @ X @ one )
      = ( X = one ) ) ).

% le_num_One_iff
thf(fact_5380_atLeastatMost__psubset__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
      ( ( ord_le1307284697595431911nteger @ ( set_or189985376899183464nteger @ A @ B ) @ ( set_or189985376899183464nteger @ C @ D ) )
      = ( ( ~ ( ord_le3102999989581377725nteger @ A @ B )
          | ( ( ord_le3102999989581377725nteger @ C @ A )
            & ( ord_le3102999989581377725nteger @ B @ D )
            & ( ( ord_le6747313008572928689nteger @ C @ A )
              | ( ord_le6747313008572928689nteger @ B @ D ) ) ) )
        & ( ord_le3102999989581377725nteger @ C @ D ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_5381_atLeastatMost__psubset__iff,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat,D: set_nat] :
      ( ( ord_less_set_set_nat @ ( set_or4548717258645045905et_nat @ A @ B ) @ ( set_or4548717258645045905et_nat @ C @ D ) )
      = ( ( ~ ( ord_less_eq_set_nat @ A @ B )
          | ( ( ord_less_eq_set_nat @ C @ A )
            & ( ord_less_eq_set_nat @ B @ D )
            & ( ( ord_less_set_nat @ C @ A )
              | ( ord_less_set_nat @ B @ D ) ) ) )
        & ( ord_less_eq_set_nat @ C @ D ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_5382_atLeastatMost__psubset__iff,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_set_rat @ ( set_or633870826150836451st_rat @ A @ B ) @ ( set_or633870826150836451st_rat @ C @ D ) )
      = ( ( ~ ( ord_less_eq_rat @ A @ B )
          | ( ( ord_less_eq_rat @ C @ A )
            & ( ord_less_eq_rat @ B @ D )
            & ( ( ord_less_rat @ C @ A )
              | ( ord_less_rat @ B @ D ) ) ) )
        & ( ord_less_eq_rat @ C @ D ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_5383_atLeastatMost__psubset__iff,axiom,
    ! [A: num,B: num,C: num,D: num] :
      ( ( ord_less_set_num @ ( set_or7049704709247886629st_num @ A @ B ) @ ( set_or7049704709247886629st_num @ C @ D ) )
      = ( ( ~ ( ord_less_eq_num @ A @ B )
          | ( ( ord_less_eq_num @ C @ A )
            & ( ord_less_eq_num @ B @ D )
            & ( ( ord_less_num @ C @ A )
              | ( ord_less_num @ B @ D ) ) ) )
        & ( ord_less_eq_num @ C @ D ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_5384_atLeastatMost__psubset__iff,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_set_int @ ( set_or1266510415728281911st_int @ A @ B ) @ ( set_or1266510415728281911st_int @ C @ D ) )
      = ( ( ~ ( ord_less_eq_int @ A @ B )
          | ( ( ord_less_eq_int @ C @ A )
            & ( ord_less_eq_int @ B @ D )
            & ( ( ord_less_int @ C @ A )
              | ( ord_less_int @ B @ D ) ) ) )
        & ( ord_less_eq_int @ C @ D ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_5385_atLeastatMost__psubset__iff,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_set_nat @ ( set_or1269000886237332187st_nat @ A @ B ) @ ( set_or1269000886237332187st_nat @ C @ D ) )
      = ( ( ~ ( ord_less_eq_nat @ A @ B )
          | ( ( ord_less_eq_nat @ C @ A )
            & ( ord_less_eq_nat @ B @ D )
            & ( ( ord_less_nat @ C @ A )
              | ( ord_less_nat @ B @ D ) ) ) )
        & ( ord_less_eq_nat @ C @ D ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_5386_mult__numeral__1,axiom,
    ! [A: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ one ) @ A )
      = A ) ).

% mult_numeral_1
thf(fact_5387_mult__numeral__1,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ one ) @ A )
      = A ) ).

% mult_numeral_1
thf(fact_5388_mult__numeral__1,axiom,
    ! [A: code_integer] :
      ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ one ) @ A )
      = A ) ).

% mult_numeral_1
thf(fact_5389_mult__numeral__1,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ one ) @ A )
      = A ) ).

% mult_numeral_1
thf(fact_5390_mult__numeral__1,axiom,
    ! [A: code_natural] :
      ( ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ one ) @ A )
      = A ) ).

% mult_numeral_1
thf(fact_5391_mult__numeral__1__right,axiom,
    ! [A: int] :
      ( ( times_times_int @ A @ ( numeral_numeral_int @ one ) )
      = A ) ).

% mult_numeral_1_right
thf(fact_5392_mult__numeral__1__right,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ ( numeral_numeral_nat @ one ) )
      = A ) ).

% mult_numeral_1_right
thf(fact_5393_mult__numeral__1__right,axiom,
    ! [A: code_integer] :
      ( ( times_3573771949741848930nteger @ A @ ( numera6620942414471956472nteger @ one ) )
      = A ) ).

% mult_numeral_1_right
thf(fact_5394_mult__numeral__1__right,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ A @ ( numeral_numeral_rat @ one ) )
      = A ) ).

% mult_numeral_1_right
thf(fact_5395_mult__numeral__1__right,axiom,
    ! [A: code_natural] :
      ( ( times_2397367101498566445atural @ A @ ( numera5444537566228673987atural @ one ) )
      = A ) ).

% mult_numeral_1_right
thf(fact_5396_numeral__One,axiom,
    ( ( numeral_numeral_int @ one )
    = one_one_int ) ).

% numeral_One
thf(fact_5397_numeral__One,axiom,
    ( ( numeral_numeral_nat @ one )
    = one_one_nat ) ).

% numeral_One
thf(fact_5398_numeral__One,axiom,
    ( ( numera6620942414471956472nteger @ one )
    = one_one_Code_integer ) ).

% numeral_One
thf(fact_5399_numeral__One,axiom,
    ( ( numeral_numeral_rat @ one )
    = one_one_rat ) ).

% numeral_One
thf(fact_5400_numeral__One,axiom,
    ( ( numera5444537566228673987atural @ one )
    = one_one_Code_natural ) ).

% numeral_One
thf(fact_5401_divide__numeral__1,axiom,
    ! [A: rat] :
      ( ( divide_divide_rat @ A @ ( numeral_numeral_rat @ one ) )
      = A ) ).

% divide_numeral_1
thf(fact_5402_numerals_I1_J,axiom,
    ( ( numeral_numeral_nat @ one )
    = one_one_nat ) ).

% numerals(1)
thf(fact_5403_add__One,axiom,
    ! [X: num] :
      ( ( plus_plus_num @ X @ one )
      = ( inc @ X ) ) ).

% add_One
thf(fact_5404_bset_I1_J,axiom,
    ! [D3: int,B5: set_int,P: int > $o,Q: int > $o] :
      ( ! [X3: int] :
          ( ! [Xa: int] :
              ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
             => ! [Xb: int] :
                  ( ( member_int @ Xb @ B5 )
                 => ( X3
                   != ( plus_plus_int @ Xb @ Xa ) ) ) )
         => ( ( P @ X3 )
           => ( P @ ( minus_minus_int @ X3 @ D3 ) ) ) )
     => ( ! [X3: int] :
            ( ! [Xa: int] :
                ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
               => ! [Xb: int] :
                    ( ( member_int @ Xb @ B5 )
                   => ( X3
                     != ( plus_plus_int @ Xb @ Xa ) ) ) )
           => ( ( Q @ X3 )
             => ( Q @ ( minus_minus_int @ X3 @ D3 ) ) ) )
       => ! [X4: int] :
            ( ! [Xa2: int] :
                ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ B5 )
                   => ( X4
                     != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
           => ( ( ( P @ X4 )
                & ( Q @ X4 ) )
             => ( ( P @ ( minus_minus_int @ X4 @ D3 ) )
                & ( Q @ ( minus_minus_int @ X4 @ D3 ) ) ) ) ) ) ) ).

% bset(1)
thf(fact_5405_bset_I2_J,axiom,
    ! [D3: int,B5: set_int,P: int > $o,Q: int > $o] :
      ( ! [X3: int] :
          ( ! [Xa: int] :
              ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
             => ! [Xb: int] :
                  ( ( member_int @ Xb @ B5 )
                 => ( X3
                   != ( plus_plus_int @ Xb @ Xa ) ) ) )
         => ( ( P @ X3 )
           => ( P @ ( minus_minus_int @ X3 @ D3 ) ) ) )
     => ( ! [X3: int] :
            ( ! [Xa: int] :
                ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
               => ! [Xb: int] :
                    ( ( member_int @ Xb @ B5 )
                   => ( X3
                     != ( plus_plus_int @ Xb @ Xa ) ) ) )
           => ( ( Q @ X3 )
             => ( Q @ ( minus_minus_int @ X3 @ D3 ) ) ) )
       => ! [X4: int] :
            ( ! [Xa2: int] :
                ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ B5 )
                   => ( X4
                     != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
           => ( ( ( P @ X4 )
                | ( Q @ X4 ) )
             => ( ( P @ ( minus_minus_int @ X4 @ D3 ) )
                | ( Q @ ( minus_minus_int @ X4 @ D3 ) ) ) ) ) ) ) ).

% bset(2)
thf(fact_5406_aset_I1_J,axiom,
    ! [D3: int,A3: set_int,P: int > $o,Q: int > $o] :
      ( ! [X3: int] :
          ( ! [Xa: int] :
              ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
             => ! [Xb: int] :
                  ( ( member_int @ Xb @ A3 )
                 => ( X3
                   != ( minus_minus_int @ Xb @ Xa ) ) ) )
         => ( ( P @ X3 )
           => ( P @ ( plus_plus_int @ X3 @ D3 ) ) ) )
     => ( ! [X3: int] :
            ( ! [Xa: int] :
                ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
               => ! [Xb: int] :
                    ( ( member_int @ Xb @ A3 )
                   => ( X3
                     != ( minus_minus_int @ Xb @ Xa ) ) ) )
           => ( ( Q @ X3 )
             => ( Q @ ( plus_plus_int @ X3 @ D3 ) ) ) )
       => ! [X4: int] :
            ( ! [Xa2: int] :
                ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ A3 )
                   => ( X4
                     != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
           => ( ( ( P @ X4 )
                & ( Q @ X4 ) )
             => ( ( P @ ( plus_plus_int @ X4 @ D3 ) )
                & ( Q @ ( plus_plus_int @ X4 @ D3 ) ) ) ) ) ) ) ).

% aset(1)
thf(fact_5407_aset_I2_J,axiom,
    ! [D3: int,A3: set_int,P: int > $o,Q: int > $o] :
      ( ! [X3: int] :
          ( ! [Xa: int] :
              ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
             => ! [Xb: int] :
                  ( ( member_int @ Xb @ A3 )
                 => ( X3
                   != ( minus_minus_int @ Xb @ Xa ) ) ) )
         => ( ( P @ X3 )
           => ( P @ ( plus_plus_int @ X3 @ D3 ) ) ) )
     => ( ! [X3: int] :
            ( ! [Xa: int] :
                ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
               => ! [Xb: int] :
                    ( ( member_int @ Xb @ A3 )
                   => ( X3
                     != ( minus_minus_int @ Xb @ Xa ) ) ) )
           => ( ( Q @ X3 )
             => ( Q @ ( plus_plus_int @ X3 @ D3 ) ) ) )
       => ! [X4: int] :
            ( ! [Xa2: int] :
                ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ A3 )
                   => ( X4
                     != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
           => ( ( ( P @ X4 )
                | ( Q @ X4 ) )
             => ( ( P @ ( plus_plus_int @ X4 @ D3 ) )
                | ( Q @ ( plus_plus_int @ X4 @ D3 ) ) ) ) ) ) ) ).

% aset(2)
thf(fact_5408_mult__1s__ring__1_I1_J,axiom,
    ! [B: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ one ) ) @ B )
      = ( uminus_uminus_int @ B ) ) ).

% mult_1s_ring_1(1)
thf(fact_5409_mult__1s__ring__1_I1_J,axiom,
    ! [B: code_integer] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) ) @ B )
      = ( uminus1351360451143612070nteger @ B ) ) ).

% mult_1s_ring_1(1)
thf(fact_5410_mult__1s__ring__1_I1_J,axiom,
    ! [B: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) ) @ B )
      = ( uminus_uminus_rat @ B ) ) ).

% mult_1s_ring_1(1)
thf(fact_5411_mult__1s__ring__1_I2_J,axiom,
    ! [B: int] :
      ( ( times_times_int @ B @ ( uminus_uminus_int @ ( numeral_numeral_int @ one ) ) )
      = ( uminus_uminus_int @ B ) ) ).

% mult_1s_ring_1(2)
thf(fact_5412_mult__1s__ring__1_I2_J,axiom,
    ! [B: code_integer] :
      ( ( times_3573771949741848930nteger @ B @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) ) )
      = ( uminus1351360451143612070nteger @ B ) ) ).

% mult_1s_ring_1(2)
thf(fact_5413_mult__1s__ring__1_I2_J,axiom,
    ! [B: rat] :
      ( ( times_times_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) ) )
      = ( uminus_uminus_rat @ B ) ) ).

% mult_1s_ring_1(2)
thf(fact_5414_uminus__numeral__One,axiom,
    ( ( uminus_uminus_int @ ( numeral_numeral_int @ one ) )
    = ( uminus_uminus_int @ one_one_int ) ) ).

% uminus_numeral_One
thf(fact_5415_uminus__numeral__One,axiom,
    ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) )
    = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% uminus_numeral_One
thf(fact_5416_uminus__numeral__One,axiom,
    ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) )
    = ( uminus_uminus_rat @ one_one_rat ) ) ).

% uminus_numeral_One
thf(fact_5417_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_5418_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_5419_cong__exp__iff__simps_I1_J,axiom,
    ! [N: num] :
      ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ one ) )
      = zero_z3403309356797280102nteger ) ).

% cong_exp_iff_simps(1)
thf(fact_5420_numeral__1__eq__Suc__0,axiom,
    ( ( numeral_numeral_nat @ one )
    = ( suc @ zero_zero_nat ) ) ).

% numeral_1_eq_Suc_0
thf(fact_5421_bset_I9_J,axiom,
    ! [D: int,D3: int,B5: set_int,T: int] :
      ( ( dvd_dvd_int @ D @ D3 )
     => ! [X4: int] :
          ( ! [Xa2: int] :
              ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
             => ! [Xb2: int] :
                  ( ( member_int @ Xb2 @ B5 )
                 => ( X4
                   != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
         => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X4 @ T ) )
           => ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X4 @ D3 ) @ T ) ) ) ) ) ).

% bset(9)
thf(fact_5422_bset_I10_J,axiom,
    ! [D: int,D3: int,B5: set_int,T: int] :
      ( ( dvd_dvd_int @ D @ D3 )
     => ! [X4: int] :
          ( ! [Xa2: int] :
              ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
             => ! [Xb2: int] :
                  ( ( member_int @ Xb2 @ B5 )
                 => ( X4
                   != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
         => ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X4 @ T ) )
           => ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X4 @ D3 ) @ T ) ) ) ) ) ).

% bset(10)
thf(fact_5423_aset_I9_J,axiom,
    ! [D: int,D3: int,A3: set_int,T: int] :
      ( ( dvd_dvd_int @ D @ D3 )
     => ! [X4: int] :
          ( ! [Xa2: int] :
              ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
             => ! [Xb2: int] :
                  ( ( member_int @ Xb2 @ A3 )
                 => ( X4
                   != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
         => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X4 @ T ) )
           => ( dvd_dvd_int @ D @ ( plus_plus_int @ ( plus_plus_int @ X4 @ D3 ) @ T ) ) ) ) ) ).

% aset(9)
thf(fact_5424_aset_I10_J,axiom,
    ! [D: int,D3: int,A3: set_int,T: int] :
      ( ( dvd_dvd_int @ D @ D3 )
     => ! [X4: int] :
          ( ! [Xa2: int] :
              ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
             => ! [Xb2: int] :
                  ( ( member_int @ Xb2 @ A3 )
                 => ( X4
                   != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
         => ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X4 @ T ) )
           => ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( plus_plus_int @ X4 @ D3 ) @ T ) ) ) ) ) ).

% aset(10)
thf(fact_5425_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_5426_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 ) ) ) )
       => ( ( ? [X7: int] : ( P @ X7 ) )
          = ( ? [X2: int] :
                ( ( member_int @ X2 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
                & ( P @ X2 ) ) ) ) ) ) ).

% periodic_finite_ex
thf(fact_5427_aset_I7_J,axiom,
    ! [D3: int,A3: set_int,T: int] :
      ( ( ord_less_int @ zero_zero_int @ D3 )
     => ! [X4: int] :
          ( ! [Xa2: int] :
              ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
             => ! [Xb2: int] :
                  ( ( member_int @ Xb2 @ A3 )
                 => ( X4
                   != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
         => ( ( ord_less_int @ T @ X4 )
           => ( ord_less_int @ T @ ( plus_plus_int @ X4 @ D3 ) ) ) ) ) ).

% aset(7)
thf(fact_5428_aset_I5_J,axiom,
    ! [D3: int,T: int,A3: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D3 )
     => ( ( member_int @ T @ A3 )
       => ! [X4: int] :
            ( ! [Xa2: int] :
                ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ A3 )
                   => ( X4
                     != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
           => ( ( ord_less_int @ X4 @ T )
             => ( ord_less_int @ ( plus_plus_int @ X4 @ D3 ) @ T ) ) ) ) ) ).

% aset(5)
thf(fact_5429_aset_I4_J,axiom,
    ! [D3: int,T: int,A3: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D3 )
     => ( ( member_int @ T @ A3 )
       => ! [X4: int] :
            ( ! [Xa2: int] :
                ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ A3 )
                   => ( X4
                     != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
           => ( ( X4 != T )
             => ( ( plus_plus_int @ X4 @ D3 )
               != T ) ) ) ) ) ).

% aset(4)
thf(fact_5430_aset_I3_J,axiom,
    ! [D3: int,T: int,A3: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D3 )
     => ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A3 )
       => ! [X4: int] :
            ( ! [Xa2: int] :
                ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ A3 )
                   => ( X4
                     != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
           => ( ( X4 = T )
             => ( ( plus_plus_int @ X4 @ D3 )
                = T ) ) ) ) ) ).

% aset(3)
thf(fact_5431_bset_I7_J,axiom,
    ! [D3: int,T: int,B5: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D3 )
     => ( ( member_int @ T @ B5 )
       => ! [X4: int] :
            ( ! [Xa2: int] :
                ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ B5 )
                   => ( X4
                     != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
           => ( ( ord_less_int @ T @ X4 )
             => ( ord_less_int @ T @ ( minus_minus_int @ X4 @ D3 ) ) ) ) ) ) ).

% bset(7)
thf(fact_5432_bset_I5_J,axiom,
    ! [D3: int,B5: set_int,T: int] :
      ( ( ord_less_int @ zero_zero_int @ D3 )
     => ! [X4: int] :
          ( ! [Xa2: int] :
              ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
             => ! [Xb2: int] :
                  ( ( member_int @ Xb2 @ B5 )
                 => ( X4
                   != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
         => ( ( ord_less_int @ X4 @ T )
           => ( ord_less_int @ ( minus_minus_int @ X4 @ D3 ) @ T ) ) ) ) ).

% bset(5)
thf(fact_5433_bset_I4_J,axiom,
    ! [D3: int,T: int,B5: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D3 )
     => ( ( member_int @ T @ B5 )
       => ! [X4: int] :
            ( ! [Xa2: int] :
                ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ B5 )
                   => ( X4
                     != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
           => ( ( X4 != T )
             => ( ( minus_minus_int @ X4 @ D3 )
               != T ) ) ) ) ) ).

% bset(4)
thf(fact_5434_bset_I3_J,axiom,
    ! [D3: int,T: int,B5: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D3 )
     => ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B5 )
       => ! [X4: int] :
            ( ! [Xa2: int] :
                ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ B5 )
                   => ( X4
                     != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
           => ( ( X4 = T )
             => ( ( minus_minus_int @ X4 @ D3 )
                = T ) ) ) ) ) ).

% bset(3)
thf(fact_5435_bset_I6_J,axiom,
    ! [D3: int,B5: set_int,T: int] :
      ( ( ord_less_int @ zero_zero_int @ D3 )
     => ! [X4: int] :
          ( ! [Xa2: int] :
              ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
             => ! [Xb2: int] :
                  ( ( member_int @ Xb2 @ B5 )
                 => ( X4
                   != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
         => ( ( ord_less_eq_int @ X4 @ T )
           => ( ord_less_eq_int @ ( minus_minus_int @ X4 @ D3 ) @ T ) ) ) ) ).

% bset(6)
thf(fact_5436_pochhammer__double,axiom,
    ! [Z2: rat,N: nat] :
      ( ( comm_s4028243227959126397er_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ Z2 ) @ ( 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 @ Z2 @ N ) ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z2 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ N ) ) ) ).

% pochhammer_double
thf(fact_5437_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_5438_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_5439_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_5440_even__succ__mod__exp,axiom,
    ! [A: code_natural,N: nat] :
      ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ one_one_Code_natural @ A ) @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) )
          = ( plus_p4538020629002901425atural @ one_one_Code_natural @ ( modulo8411746178871703098atural @ A @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).

% even_succ_mod_exp
thf(fact_5441_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_5442_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_5443_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_5444_even__succ__div__exp,axiom,
    ! [A: code_natural,N: nat] :
      ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( divide5121882707175180666atural @ ( plus_p4538020629002901425atural @ one_one_Code_natural @ A ) @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) )
          = ( divide5121882707175180666atural @ A @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).

% even_succ_div_exp
thf(fact_5445_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_5446_round__unique,axiom,
    ! [X: rat,Y3: int] :
      ( ( ord_less_rat @ ( minus_minus_rat @ X @ ( 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 @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) )
       => ( ( archim7778729529865785530nd_rat @ X )
          = Y3 ) ) ) ).

% round_unique
thf(fact_5447_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_5448_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_5449_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_5450_even__mult__exp__div__exp__iff,axiom,
    ! [A: code_natural,M: nat,N: nat] :
      ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( divide5121882707175180666atural @ ( times_2397367101498566445atural @ A @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ M ) ) @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) ) )
      = ( ( ord_less_nat @ N @ M )
        | ( ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N )
          = zero_z2226904508553997617atural )
        | ( ( ord_less_eq_nat @ M @ N )
          & ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( divide5121882707175180666atural @ A @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).

% even_mult_exp_div_exp_iff
thf(fact_5451_divmod__digit__1_I1_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
       => ( ( ord_le3102999989581377725nteger @ B @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) )
         => ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) @ one_one_Code_integer )
            = ( divide6298287555418463151nteger @ A @ B ) ) ) ) ) ).

% divmod_digit_1(1)
thf(fact_5452_divmod__digit__1_I1_J,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ B )
       => ( ( ord_less_eq_nat @ B @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) )
         => ( ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) @ one_one_nat )
            = ( divide_divide_nat @ A @ B ) ) ) ) ) ).

% divmod_digit_1(1)
thf(fact_5453_divmod__digit__1_I1_J,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ( ord_less_eq_int @ B @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) )
         => ( ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) @ one_one_int )
            = ( divide_divide_int @ A @ B ) ) ) ) ) ).

% divmod_digit_1(1)
thf(fact_5454_verit__eq__simplify_I8_J,axiom,
    ! [X22: num,Y2: num] :
      ( ( ( bit0 @ X22 )
        = ( bit0 @ Y2 ) )
      = ( X22 = Y2 ) ) ).

% verit_eq_simplify(8)
thf(fact_5455_semiring__norm_I87_J,axiom,
    ! [M: num,N: num] :
      ( ( ( bit0 @ M )
        = ( bit0 @ N ) )
      = ( M = N ) ) ).

% semiring_norm(87)
thf(fact_5456_semiring__norm_I85_J,axiom,
    ! [M: num] :
      ( ( bit0 @ M )
     != one ) ).

% semiring_norm(85)
thf(fact_5457_semiring__norm_I83_J,axiom,
    ! [N: num] :
      ( one
     != ( bit0 @ N ) ) ).

% semiring_norm(83)
thf(fact_5458_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_5459_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_5460_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_5461_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_5462_semiring__norm_I2_J,axiom,
    ( ( plus_plus_num @ one @ one )
    = ( bit0 @ one ) ) ).

% semiring_norm(2)
thf(fact_5463_num__double,axiom,
    ! [N: num] :
      ( ( times_times_num @ ( bit0 @ one ) @ N )
      = ( bit0 @ N ) ) ).

% num_double
thf(fact_5464_semiring__norm_I69_J,axiom,
    ! [M: num] :
      ~ ( ord_less_eq_num @ ( bit0 @ M ) @ one ) ).

% semiring_norm(69)
thf(fact_5465_semiring__norm_I76_J,axiom,
    ! [N: num] : ( ord_less_num @ one @ ( bit0 @ N ) ) ).

% semiring_norm(76)
thf(fact_5466_one__add__one,axiom,
    ( ( plus_plus_int @ one_one_int @ one_one_int )
    = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_5467_one__add__one,axiom,
    ( ( plus_plus_nat @ one_one_nat @ one_one_nat )
    = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_5468_one__add__one,axiom,
    ( ( plus_p5714425477246183910nteger @ one_one_Code_integer @ one_one_Code_integer )
    = ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_5469_one__add__one,axiom,
    ( ( plus_plus_rat @ one_one_rat @ one_one_rat )
    = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_5470_one__add__one,axiom,
    ( ( plus_p4538020629002901425atural @ one_one_Code_natural @ one_one_Code_natural )
    = ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_5471_even__mult__iff,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( times_times_int @ A @ B ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
        | ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ).

% even_mult_iff
thf(fact_5472_even__mult__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ A @ B ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
        | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ).

% even_mult_iff
thf(fact_5473_even__mult__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( times_3573771949741848930nteger @ A @ B ) )
      = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
        | ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) ).

% even_mult_iff
thf(fact_5474_even__mult__iff,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( times_2397367101498566445atural @ A @ B ) )
      = ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A )
        | ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ B ) ) ) ).

% even_mult_iff
thf(fact_5475_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_5476_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_5477_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_5478_bits__one__mod__two__eq__one,axiom,
    ( ( modulo8411746178871703098atural @ one_one_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
    = one_one_Code_natural ) ).

% bits_one_mod_two_eq_one
thf(fact_5479_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_5480_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_5481_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_5482_one__mod__two__eq__one,axiom,
    ( ( modulo8411746178871703098atural @ one_one_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
    = one_one_Code_natural ) ).

% one_mod_two_eq_one
thf(fact_5483_zero__eq__power2,axiom,
    ! [A: code_integer] :
      ( ( ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = zero_z3403309356797280102nteger )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% zero_eq_power2
thf(fact_5484_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_5485_zero__eq__power2,axiom,
    ! [A: code_natural] :
      ( ( ( power_7079662738309270450atural @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = zero_z2226904508553997617atural )
      = ( A = zero_z2226904508553997617atural ) ) ).

% zero_eq_power2
thf(fact_5486_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_5487_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_5488_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_5489_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_5490_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_5491_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_5492_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_5493_Suc__1,axiom,
    ( ( suc @ one_one_nat )
    = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% Suc_1
thf(fact_5494_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_5495_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_5496_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_5497_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_5498_fact__2,axiom,
    ( ( semiri1408675320244567234ct_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% fact_2
thf(fact_5499_fact__2,axiom,
    ( ( semiri2447717529341329178atural @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ).

% fact_2
thf(fact_5500_fact__2,axiom,
    ( ( semiri773545260158071498ct_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).

% fact_2
thf(fact_5501_fact__2,axiom,
    ( ( semiri3624122377584611663nteger @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ).

% fact_2
thf(fact_5502_fact__2,axiom,
    ( ( semiri1406184849735516958ct_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).

% fact_2
thf(fact_5503_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_5504_zmod__numeral__Bit0,axiom,
    ! [V: num,W2: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W2 ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W2 ) ) ) ) ).

% zmod_numeral_Bit0
thf(fact_5505_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_5506_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_5507_one__div__two__eq__zero,axiom,
    ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
    = zero_z3403309356797280102nteger ) ).

% one_div_two_eq_zero
thf(fact_5508_one__div__two__eq__zero,axiom,
    ( ( divide5121882707175180666atural @ one_one_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
    = zero_z2226904508553997617atural ) ).

% one_div_two_eq_zero
thf(fact_5509_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_5510_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_5511_bits__1__div__2,axiom,
    ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
    = zero_z3403309356797280102nteger ) ).

% bits_1_div_2
thf(fact_5512_bits__1__div__2,axiom,
    ( ( divide5121882707175180666atural @ one_one_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
    = zero_z2226904508553997617atural ) ).

% bits_1_div_2
thf(fact_5513_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_5514_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_5515_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_5516_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_5517_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_5518_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_5519_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_5520_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_5521_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_5522_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_5523_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_5524_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_5525_even__plus__one__iff,axiom,
    ! [A: code_natural] :
      ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( plus_p4538020629002901425atural @ A @ one_one_Code_natural ) )
      = ( ~ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A ) ) ) ).

% even_plus_one_iff
thf(fact_5526_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_5527_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_5528_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_5529_not__mod__2__eq__0__eq__1,axiom,
    ! [A: code_natural] :
      ( ( ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
       != zero_z2226904508553997617atural )
      = ( ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
        = one_one_Code_natural ) ) ).

% not_mod_2_eq_0_eq_1
thf(fact_5530_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_5531_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_5532_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_5533_not__mod__2__eq__1__eq__0,axiom,
    ! [A: code_natural] :
      ( ( ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
       != one_one_Code_natural )
      = ( ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
        = zero_z2226904508553997617atural ) ) ).

% not_mod_2_eq_1_eq_0
thf(fact_5534_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_5535_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_5536_power2__less__eq__zero__iff,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_z3403309356797280102nteger )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% power2_less_eq_zero_iff
thf(fact_5537_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_5538_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_5539_power2__eq__iff__nonneg,axiom,
    ! [X: code_integer,Y3: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ Y3 )
       => ( ( ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = ( power_8256067586552552935nteger @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          = ( X = Y3 ) ) ) ) ).

% power2_eq_iff_nonneg
thf(fact_5540_power2__eq__iff__nonneg,axiom,
    ! [X: code_natural,Y3: code_natural] :
      ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ X )
     => ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ Y3 )
       => ( ( ( power_7079662738309270450atural @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = ( power_7079662738309270450atural @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          = ( X = Y3 ) ) ) ) ).

% power2_eq_iff_nonneg
thf(fact_5541_power2__eq__iff__nonneg,axiom,
    ! [X: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
       => ( ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          = ( X = Y3 ) ) ) ) ).

% power2_eq_iff_nonneg
thf(fact_5542_power2__eq__iff__nonneg,axiom,
    ! [X: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ X )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ Y3 )
       => ( ( ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = ( power_power_nat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          = ( X = Y3 ) ) ) ) ).

% power2_eq_iff_nonneg
thf(fact_5543_power2__eq__iff__nonneg,axiom,
    ! [X: int,Y3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
       => ( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          = ( X = Y3 ) ) ) ) ).

% power2_eq_iff_nonneg
thf(fact_5544_zero__less__power2,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( A != zero_z3403309356797280102nteger ) ) ).

% zero_less_power2
thf(fact_5545_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_5546_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_5547_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_5548_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_5549_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_5550_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_5551_of__bool__half__eq__0,axiom,
    ! [B: $o] :
      ( ( divide6298287555418463151nteger @ ( zero_n356916108424825756nteger @ B ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
      = zero_z3403309356797280102nteger ) ).

% of_bool_half_eq_0
thf(fact_5552_of__bool__half__eq__0,axiom,
    ! [B: $o] :
      ( ( divide5121882707175180666atural @ ( zero_n8403883297036319079atural @ B ) @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
      = zero_z2226904508553997617atural ) ).

% of_bool_half_eq_0
thf(fact_5553_of__bool__half__eq__0,axiom,
    ! [B: $o] :
      ( ( divide_divide_nat @ ( zero_n2687167440665602831ol_nat @ B ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = zero_zero_nat ) ).

% of_bool_half_eq_0
thf(fact_5554_of__bool__half__eq__0,axiom,
    ! [B: $o] :
      ( ( divide_divide_int @ ( zero_n2684676970156552555ol_int @ B ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
      = zero_zero_int ) ).

% of_bool_half_eq_0
thf(fact_5555_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_5556_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_5557_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_5558_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_5559_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_5560_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_5561_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_5562_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_5563_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_5564_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_5565_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_5566_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_5567_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_5568_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_5569_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_5570_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_5571_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_5572_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_5573_even__succ__div__2,axiom,
    ! [A: code_natural] :
      ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A )
     => ( ( divide5121882707175180666atural @ ( plus_p4538020629002901425atural @ one_one_Code_natural @ A ) @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
        = ( divide5121882707175180666atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ) ).

% even_succ_div_2
thf(fact_5574_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_5575_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_5576_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_5577_even__succ__div__two,axiom,
    ! [A: code_natural] :
      ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A )
     => ( ( divide5121882707175180666atural @ ( plus_p4538020629002901425atural @ A @ one_one_Code_natural ) @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
        = ( divide5121882707175180666atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ) ).

% even_succ_div_two
thf(fact_5578_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_5579_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_5580_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_5581_odd__succ__div__two,axiom,
    ! [A: code_natural] :
      ( ~ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A )
     => ( ( divide5121882707175180666atural @ ( plus_p4538020629002901425atural @ A @ one_one_Code_natural ) @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
        = ( plus_p4538020629002901425atural @ ( divide5121882707175180666atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) @ one_one_Code_natural ) ) ) ).

% odd_succ_div_two
thf(fact_5582_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_5583_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_5584_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_5585_even__power,axiom,
    ! [A: code_natural,N: nat] :
      ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( power_7079662738309270450atural @ A @ N ) )
      = ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A )
        & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% even_power
thf(fact_5586_zero__le__power__eq__numeral,axiom,
    ! [A: code_integer,W2: num] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ W2 ) ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
          & ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ) ) ).

% zero_le_power_eq_numeral
thf(fact_5587_zero__le__power__eq__numeral,axiom,
    ! [A: rat,W2: num] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W2 ) ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
          & ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ).

% zero_le_power_eq_numeral
thf(fact_5588_zero__le__power__eq__numeral,axiom,
    ! [A: int,W2: num] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W2 ) ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
          & ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ) ).

% zero_le_power_eq_numeral
thf(fact_5589_power__less__zero__eq__numeral,axiom,
    ! [A: code_integer,W2: num] :
      ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ W2 ) ) @ zero_z3403309356797280102nteger )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
        & ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ) ).

% power_less_zero_eq_numeral
thf(fact_5590_power__less__zero__eq__numeral,axiom,
    ! [A: rat,W2: num] :
      ( ( ord_less_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W2 ) ) @ zero_zero_rat )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
        & ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).

% power_less_zero_eq_numeral
thf(fact_5591_power__less__zero__eq__numeral,axiom,
    ! [A: int,W2: num] :
      ( ( ord_less_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W2 ) ) @ zero_zero_int )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
        & ( ord_less_int @ A @ zero_zero_int ) ) ) ).

% power_less_zero_eq_numeral
thf(fact_5592_power__less__zero__eq,axiom,
    ! [A: code_integer,N: nat] :
      ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A @ N ) @ zero_z3403309356797280102nteger )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
        & ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ) ).

% power_less_zero_eq
thf(fact_5593_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_5594_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_5595_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_5596_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_5597_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_5598_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_5599_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_5600_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_5601_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_5602_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_5603_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_5604_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_5605_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_5606_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_5607_even__of__nat,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( semiri3763490453095760265atural @ N ) )
      = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).

% even_of_nat
thf(fact_5608_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_5609_one__less__floor,axiom,
    ! [X: rat] :
      ( ( ord_less_int @ one_one_int @ ( archim3151403230148437115or_rat @ X ) )
      = ( ord_less_eq_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X ) ) ).

% one_less_floor
thf(fact_5610_floor__le__one,axiom,
    ! [X: rat] :
      ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ one_one_int )
      = ( ord_less_rat @ X @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).

% floor_le_one
thf(fact_5611_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_5612_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_5613_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_5614_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_5615_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_5616_odd__two__times__div__two__succ,axiom,
    ! [A: code_natural] :
      ( ~ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A )
     => ( ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( divide5121882707175180666atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) @ one_one_Code_natural )
        = A ) ) ).

% odd_two_times_div_two_succ
thf(fact_5617_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_5618_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_5619_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_5620_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( minus_7197305767214868737atural @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) @ one_one_Code_natural ) )
      = ( N = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
thf(fact_5621_zero__less__power__eq__numeral,axiom,
    ! [A: code_integer,W2: num] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ W2 ) ) )
      = ( ( ( numeral_numeral_nat @ W2 )
          = zero_zero_nat )
        | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
          & ( A != zero_z3403309356797280102nteger ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
          & ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ) ) ).

% zero_less_power_eq_numeral
thf(fact_5622_zero__less__power__eq__numeral,axiom,
    ! [A: rat,W2: num] :
      ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W2 ) ) )
      = ( ( ( numeral_numeral_nat @ W2 )
          = zero_zero_nat )
        | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
          & ( A != zero_zero_rat ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
          & ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ).

% zero_less_power_eq_numeral
thf(fact_5623_zero__less__power__eq__numeral,axiom,
    ! [A: int,W2: num] :
      ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W2 ) ) )
      = ( ( ( numeral_numeral_nat @ W2 )
          = zero_zero_nat )
        | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
          & ( A != zero_zero_int ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
          & ( ord_less_int @ zero_zero_int @ A ) ) ) ) ).

% zero_less_power_eq_numeral
thf(fact_5624_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_5625_bits__1__div__exp,axiom,
    ! [N: nat] :
      ( ( divide5121882707175180666atural @ one_one_Code_natural @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) )
      = ( zero_n8403883297036319079atural @ ( N = zero_zero_nat ) ) ) ).

% bits_1_div_exp
thf(fact_5626_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_5627_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_5628_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_5629_one__div__2__pow__eq,axiom,
    ! [N: nat] :
      ( ( divide5121882707175180666atural @ one_one_Code_natural @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) )
      = ( zero_n8403883297036319079atural @ ( N = zero_zero_nat ) ) ) ).

% one_div_2_pow_eq
thf(fact_5630_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_5631_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_5632_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_5633_power__le__zero__eq__numeral,axiom,
    ! [A: code_integer,W2: num] :
      ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ W2 ) ) @ zero_z3403309356797280102nteger )
      = ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W2 ) )
        & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
            & ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) )
          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
            & ( A = zero_z3403309356797280102nteger ) ) ) ) ) ).

% power_le_zero_eq_numeral
thf(fact_5634_power__le__zero__eq__numeral,axiom,
    ! [A: rat,W2: num] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W2 ) ) @ zero_zero_rat )
      = ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W2 ) )
        & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
            & ( ord_less_eq_rat @ A @ zero_zero_rat ) )
          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
            & ( A = zero_zero_rat ) ) ) ) ) ).

% power_le_zero_eq_numeral
thf(fact_5635_power__le__zero__eq__numeral,axiom,
    ! [A: int,W2: num] :
      ( ( ord_less_eq_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W2 ) ) @ zero_zero_int )
      = ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W2 ) )
        & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
            & ( ord_less_eq_int @ A @ zero_zero_int ) )
          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
            & ( A = zero_zero_int ) ) ) ) ) ).

% power_le_zero_eq_numeral
thf(fact_5636_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_5637_one__mod__2__pow__eq,axiom,
    ! [N: nat] :
      ( ( modulo8411746178871703098atural @ one_one_Code_natural @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) )
      = ( zero_n8403883297036319079atural @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% one_mod_2_pow_eq
thf(fact_5638_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_5639_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_5640_verit__eq__simplify_I10_J,axiom,
    ! [X22: num] :
      ( one
     != ( bit0 @ X22 ) ) ).

% verit_eq_simplify(10)
thf(fact_5641_ex__nat__less,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ? [M2: nat] :
            ( ( ord_less_eq_nat @ M2 @ N )
            & ( P @ M2 ) ) )
      = ( ? [X2: nat] :
            ( ( member_nat @ X2 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
            & ( P @ X2 ) ) ) ) ).

% ex_nat_less
thf(fact_5642_all__nat__less,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ! [M2: nat] :
            ( ( ord_less_eq_nat @ M2 @ N )
           => ( P @ M2 ) ) )
      = ( ! [X2: nat] :
            ( ( member_nat @ X2 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
           => ( P @ X2 ) ) ) ) ).

% all_nat_less
thf(fact_5643_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_5644_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_5645_numeral__Bit0,axiom,
    ! [N: num] :
      ( ( numera6620942414471956472nteger @ ( bit0 @ N ) )
      = ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ N ) ) ) ).

% numeral_Bit0
thf(fact_5646_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_5647_numeral__Bit0,axiom,
    ! [N: num] :
      ( ( numera5444537566228673987atural @ ( bit0 @ N ) )
      = ( plus_p4538020629002901425atural @ ( numera5444537566228673987atural @ N ) @ ( numera5444537566228673987atural @ N ) ) ) ).

% numeral_Bit0
thf(fact_5648_inc_Osimps_I1_J,axiom,
    ( ( inc @ one )
    = ( bit0 @ one ) ) ).

% inc.simps(1)
thf(fact_5649_cong__exp__iff__simps_I2_J,axiom,
    ! [N: num,Q4: num] :
      ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q4 ) ) )
        = zero_zero_nat )
      = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q4 ) )
        = zero_zero_nat ) ) ).

% cong_exp_iff_simps(2)
thf(fact_5650_cong__exp__iff__simps_I2_J,axiom,
    ! [N: num,Q4: num] :
      ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q4 ) ) )
        = zero_zero_int )
      = ( ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q4 ) )
        = zero_zero_int ) ) ).

% cong_exp_iff_simps(2)
thf(fact_5651_cong__exp__iff__simps_I2_J,axiom,
    ! [N: num,Q4: num] :
      ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q4 ) ) )
        = zero_z3403309356797280102nteger )
      = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q4 ) )
        = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(2)
thf(fact_5652_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_5653_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_5654_numeral__Bit0__div__2,axiom,
    ! [N: num] :
      ( ( divide6298287555418463151nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
      = ( numera6620942414471956472nteger @ N ) ) ).

% numeral_Bit0_div_2
thf(fact_5655_numeral__Bit0__div__2,axiom,
    ! [N: num] :
      ( ( divide5121882707175180666atural @ ( numera5444537566228673987atural @ ( bit0 @ N ) ) @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
      = ( numera5444537566228673987atural @ N ) ) ).

% numeral_Bit0_div_2
thf(fact_5656_power__minus__Bit0,axiom,
    ! [X: int,K: num] :
      ( ( power_power_int @ ( uminus_uminus_int @ X ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
      = ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).

% power_minus_Bit0
thf(fact_5657_power__minus__Bit0,axiom,
    ! [X: code_integer,K: num] :
      ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ X ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
      = ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).

% power_minus_Bit0
thf(fact_5658_power__minus__Bit0,axiom,
    ! [X: rat,K: num] :
      ( ( power_power_rat @ ( uminus_uminus_rat @ X ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
      = ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).

% power_minus_Bit0
thf(fact_5659_mult__2,axiom,
    ! [Z2: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Z2 )
      = ( plus_plus_int @ Z2 @ Z2 ) ) ).

% mult_2
thf(fact_5660_mult__2,axiom,
    ! [Z2: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Z2 )
      = ( plus_plus_nat @ Z2 @ Z2 ) ) ).

% mult_2
thf(fact_5661_mult__2,axiom,
    ! [Z2: code_integer] :
      ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Z2 )
      = ( plus_p5714425477246183910nteger @ Z2 @ Z2 ) ) ).

% mult_2
thf(fact_5662_mult__2,axiom,
    ! [Z2: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ Z2 )
      = ( plus_plus_rat @ Z2 @ Z2 ) ) ).

% mult_2
thf(fact_5663_mult__2,axiom,
    ! [Z2: code_natural] :
      ( ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ Z2 )
      = ( plus_p4538020629002901425atural @ Z2 @ Z2 ) ) ).

% mult_2
thf(fact_5664_mult__2__right,axiom,
    ! [Z2: int] :
      ( ( times_times_int @ Z2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
      = ( plus_plus_int @ Z2 @ Z2 ) ) ).

% mult_2_right
thf(fact_5665_mult__2__right,axiom,
    ! [Z2: nat] :
      ( ( times_times_nat @ Z2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_nat @ Z2 @ Z2 ) ) ).

% mult_2_right
thf(fact_5666_mult__2__right,axiom,
    ! [Z2: code_integer] :
      ( ( times_3573771949741848930nteger @ Z2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
      = ( plus_p5714425477246183910nteger @ Z2 @ Z2 ) ) ).

% mult_2_right
thf(fact_5667_mult__2__right,axiom,
    ! [Z2: rat] :
      ( ( times_times_rat @ Z2 @ ( numeral_numeral_rat @ ( bit0 @ one ) ) )
      = ( plus_plus_rat @ Z2 @ Z2 ) ) ).

% mult_2_right
thf(fact_5668_mult__2__right,axiom,
    ! [Z2: code_natural] :
      ( ( times_2397367101498566445atural @ Z2 @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
      = ( plus_p4538020629002901425atural @ Z2 @ Z2 ) ) ).

% mult_2_right
thf(fact_5669_left__add__twice,axiom,
    ! [A: int,B: int] :
      ( ( plus_plus_int @ A @ ( plus_plus_int @ A @ B ) )
      = ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) @ B ) ) ).

% left_add_twice
thf(fact_5670_left__add__twice,axiom,
    ! [A: nat,B: nat] :
      ( ( plus_plus_nat @ A @ ( plus_plus_nat @ A @ B ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ B ) ) ).

% left_add_twice
thf(fact_5671_left__add__twice,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger @ A @ ( plus_p5714425477246183910nteger @ A @ B ) )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) @ B ) ) ).

% left_add_twice
thf(fact_5672_left__add__twice,axiom,
    ! [A: rat,B: rat] :
      ( ( plus_plus_rat @ A @ ( plus_plus_rat @ A @ B ) )
      = ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ A ) @ B ) ) ).

% left_add_twice
thf(fact_5673_left__add__twice,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( plus_p4538020629002901425atural @ A @ ( plus_p4538020629002901425atural @ A @ B ) )
      = ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A ) @ B ) ) ).

% left_add_twice
thf(fact_5674_even__zero,axiom,
    dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ zero_zero_int ).

% even_zero
thf(fact_5675_even__zero,axiom,
    dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ zero_zero_nat ).

% even_zero
thf(fact_5676_even__zero,axiom,
    dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ zero_z3403309356797280102nteger ).

% even_zero
thf(fact_5677_even__zero,axiom,
    dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ zero_z2226904508553997617atural ).

% even_zero
thf(fact_5678_evenE,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ~ ! [B4: int] :
            ( A
           != ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B4 ) ) ) ).

% evenE
thf(fact_5679_evenE,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ~ ! [B4: nat] :
            ( A
           != ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B4 ) ) ) ).

% evenE
thf(fact_5680_evenE,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ~ ! [B4: code_integer] :
            ( A
           != ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B4 ) ) ) ).

% evenE
thf(fact_5681_evenE,axiom,
    ! [A: code_natural] :
      ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A )
     => ~ ! [B4: code_natural] :
            ( A
           != ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ B4 ) ) ) ).

% evenE
thf(fact_5682_odd__one,axiom,
    ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ one_one_int ) ).

% odd_one
thf(fact_5683_odd__one,axiom,
    ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ one_one_nat ) ).

% odd_one
thf(fact_5684_odd__one,axiom,
    ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ one_one_Code_integer ) ).

% odd_one
thf(fact_5685_odd__one,axiom,
    ~ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ one_one_Code_natural ) ).

% odd_one
thf(fact_5686_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_5687_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_5688_zero__power2,axiom,
    ( ( power_8256067586552552935nteger @ zero_z3403309356797280102nteger @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_z3403309356797280102nteger ) ).

% zero_power2
thf(fact_5689_zero__power2,axiom,
    ( ( power_power_rat @ zero_zero_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_rat ) ).

% zero_power2
thf(fact_5690_zero__power2,axiom,
    ( ( power_7079662738309270450atural @ zero_z2226904508553997617atural @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_z2226904508553997617atural ) ).

% zero_power2
thf(fact_5691_zero__power2,axiom,
    ( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_int ) ).

% zero_power2
thf(fact_5692_zero__power2,axiom,
    ( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_nat ) ).

% zero_power2
thf(fact_5693_power2__eq__square,axiom,
    ! [A: code_integer] :
      ( ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( times_3573771949741848930nteger @ A @ A ) ) ).

% power2_eq_square
thf(fact_5694_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_5695_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_5696_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_5697_power4__eq__xxxx,axiom,
    ! [X: code_integer] :
      ( ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
      = ( times_3573771949741848930nteger @ ( times_3573771949741848930nteger @ ( times_3573771949741848930nteger @ X @ X ) @ X ) @ X ) ) ).

% power4_eq_xxxx
thf(fact_5698_power4__eq__xxxx,axiom,
    ! [X: rat] :
      ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
      = ( times_times_rat @ ( times_times_rat @ ( times_times_rat @ X @ X ) @ X ) @ X ) ) ).

% power4_eq_xxxx
thf(fact_5699_power4__eq__xxxx,axiom,
    ! [X: nat] :
      ( ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
      = ( times_times_nat @ ( times_times_nat @ ( times_times_nat @ X @ X ) @ X ) @ X ) ) ).

% power4_eq_xxxx
thf(fact_5700_power4__eq__xxxx,axiom,
    ! [X: int] :
      ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
      = ( times_times_int @ ( times_times_int @ ( times_times_int @ X @ X ) @ X ) @ X ) ) ).

% power4_eq_xxxx
thf(fact_5701_numeral__2__eq__2,axiom,
    ( ( numeral_numeral_nat @ ( bit0 @ one ) )
    = ( suc @ ( suc @ zero_zero_nat ) ) ) ).

% numeral_2_eq_2
thf(fact_5702_one__power2,axiom,
    ( ( power_8256067586552552935nteger @ one_one_Code_integer @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_Code_integer ) ).

% one_power2
thf(fact_5703_one__power2,axiom,
    ( ( power_power_rat @ one_one_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_rat ) ).

% one_power2
thf(fact_5704_one__power2,axiom,
    ( ( power_7079662738309270450atural @ one_one_Code_natural @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_Code_natural ) ).

% one_power2
thf(fact_5705_one__power2,axiom,
    ( ( power_power_int @ one_one_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_int ) ).

% one_power2
thf(fact_5706_one__power2,axiom,
    ( ( power_power_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_nat ) ).

% one_power2
thf(fact_5707_power2__eq__iff,axiom,
    ! [X: int,Y3: int] :
      ( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( ( X = Y3 )
        | ( X
          = ( uminus_uminus_int @ Y3 ) ) ) ) ).

% power2_eq_iff
thf(fact_5708_power2__eq__iff,axiom,
    ! [X: code_integer,Y3: code_integer] :
      ( ( ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_8256067586552552935nteger @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( ( X = Y3 )
        | ( X
          = ( uminus1351360451143612070nteger @ Y3 ) ) ) ) ).

% power2_eq_iff
thf(fact_5709_power2__eq__iff,axiom,
    ! [X: rat,Y3: rat] :
      ( ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( ( X = Y3 )
        | ( X
          = ( uminus_uminus_rat @ Y3 ) ) ) ) ).

% power2_eq_iff
thf(fact_5710_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_5711_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_5712_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_5713_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_5714_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_5715_less__exp,axiom,
    ! [N: nat] : ( ord_less_nat @ N @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).

% less_exp
thf(fact_5716_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_5717_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_5718_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_5719_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_5720_div2__even__ext__nat,axiom,
    ! [X: nat,Y3: nat] :
      ( ( ( divide_divide_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( divide_divide_nat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X )
          = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Y3 ) )
       => ( X = Y3 ) ) ) ).

% div2_even_ext_nat
thf(fact_5721_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_5722_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_5723_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_5724_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_5725_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_5726_even__two__times__div__two,axiom,
    ! [A: code_natural] :
      ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A )
     => ( ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( divide5121882707175180666atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) )
        = A ) ) ).

% even_two_times_div_two
thf(fact_5727_zero__le__power2,axiom,
    ! [A: code_integer] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% zero_le_power2
thf(fact_5728_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_5729_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_5730_power2__eq__imp__eq,axiom,
    ! [X: code_integer,Y3: code_integer] :
      ( ( ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_8256067586552552935nteger @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X )
       => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ Y3 )
         => ( X = Y3 ) ) ) ) ).

% power2_eq_imp_eq
thf(fact_5731_power2__eq__imp__eq,axiom,
    ! [X: code_natural,Y3: code_natural] :
      ( ( ( power_7079662738309270450atural @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_7079662738309270450atural @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ X )
       => ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ Y3 )
         => ( X = Y3 ) ) ) ) ).

% power2_eq_imp_eq
thf(fact_5732_power2__eq__imp__eq,axiom,
    ! [X: rat,Y3: rat] :
      ( ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ X )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
         => ( X = Y3 ) ) ) ) ).

% power2_eq_imp_eq
thf(fact_5733_power2__eq__imp__eq,axiom,
    ! [X: nat,Y3: nat] :
      ( ( ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_nat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ X )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ Y3 )
         => ( X = Y3 ) ) ) ) ).

% power2_eq_imp_eq
thf(fact_5734_power2__eq__imp__eq,axiom,
    ! [X: int,Y3: int] :
      ( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ X )
       => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
         => ( X = Y3 ) ) ) ) ).

% power2_eq_imp_eq
thf(fact_5735_power2__le__imp__le,axiom,
    ! [X: code_integer,Y3: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ Y3 )
       => ( ord_le3102999989581377725nteger @ X @ Y3 ) ) ) ).

% power2_le_imp_le
thf(fact_5736_power2__le__imp__le,axiom,
    ! [X: code_natural,Y3: code_natural] :
      ( ( ord_le1926595141338095240atural @ ( power_7079662738309270450atural @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_7079662738309270450atural @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ Y3 )
       => ( ord_le1926595141338095240atural @ X @ Y3 ) ) ) ).

% power2_le_imp_le
thf(fact_5737_power2__le__imp__le,axiom,
    ! [X: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ X @ ( 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 @ X @ Y3 ) ) ) ).

% power2_le_imp_le
thf(fact_5738_power2__le__imp__le,axiom,
    ! [X: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ X @ ( 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 @ X @ Y3 ) ) ) ).

% power2_le_imp_le
thf(fact_5739_power2__le__imp__le,axiom,
    ! [X: int,Y3: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ X @ ( 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 @ X @ Y3 ) ) ) ).

% power2_le_imp_le
thf(fact_5740_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_5741_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_5742_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_5743_even__iff__mod__2__eq__zero,axiom,
    ! [A: code_natural] :
      ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A )
      = ( ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
        = zero_z2226904508553997617atural ) ) ).

% even_iff_mod_2_eq_zero
thf(fact_5744_power2__less__0,axiom,
    ! [A: code_integer] :
      ~ ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_z3403309356797280102nteger ) ).

% power2_less_0
thf(fact_5745_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_5746_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_5747_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_5748_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_5749_exp__add__not__zero__imp__left,axiom,
    ! [M: nat,N: nat] :
      ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
       != zero_z3403309356797280102nteger )
     => ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M )
       != zero_z3403309356797280102nteger ) ) ).

% exp_add_not_zero_imp_left
thf(fact_5750_exp__add__not__zero__imp__left,axiom,
    ! [M: nat,N: nat] :
      ( ( ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
       != zero_z2226904508553997617atural )
     => ( ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ M )
       != zero_z2226904508553997617atural ) ) ).

% exp_add_not_zero_imp_left
thf(fact_5751_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_5752_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_5753_exp__add__not__zero__imp__right,axiom,
    ! [M: nat,N: nat] :
      ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
       != zero_z3403309356797280102nteger )
     => ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N )
       != zero_z3403309356797280102nteger ) ) ).

% exp_add_not_zero_imp_right
thf(fact_5754_exp__add__not__zero__imp__right,axiom,
    ! [M: nat,N: nat] :
      ( ( ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
       != zero_z2226904508553997617atural )
     => ( ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N )
       != zero_z2226904508553997617atural ) ) ).

% exp_add_not_zero_imp_right
thf(fact_5755_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_5756_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_5757_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_5758_odd__iff__mod__2__eq__one,axiom,
    ! [A: code_natural] :
      ( ( ~ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A ) )
      = ( ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
        = one_one_Code_natural ) ) ).

% odd_iff_mod_2_eq_one
thf(fact_5759_sum__power2__eq__zero__iff,axiom,
    ! [X: code_integer,Y3: code_integer] :
      ( ( ( plus_p5714425477246183910nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = zero_z3403309356797280102nteger )
      = ( ( X = zero_z3403309356797280102nteger )
        & ( Y3 = zero_z3403309356797280102nteger ) ) ) ).

% sum_power2_eq_zero_iff
thf(fact_5760_sum__power2__eq__zero__iff,axiom,
    ! [X: rat,Y3: rat] :
      ( ( ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = zero_zero_rat )
      = ( ( X = zero_zero_rat )
        & ( Y3 = zero_zero_rat ) ) ) ).

% sum_power2_eq_zero_iff
thf(fact_5761_sum__power2__eq__zero__iff,axiom,
    ! [X: int,Y3: int] :
      ( ( ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = zero_zero_int )
      = ( ( X = zero_zero_int )
        & ( Y3 = zero_zero_int ) ) ) ).

% sum_power2_eq_zero_iff
thf(fact_5762_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_5763_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_5764_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [N: nat,M: nat] :
      ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N )
       != zero_z3403309356797280102nteger )
     => ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) )
       != zero_z3403309356797280102nteger ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
thf(fact_5765_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [N: nat,M: nat] :
      ( ( ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N )
       != zero_z2226904508553997617atural )
     => ( ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) )
       != zero_z2226904508553997617atural ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
thf(fact_5766_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_5767_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_5768_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_5769_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_5770_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_5771_abs__le__square__iff,axiom,
    ! [X: code_integer,Y3: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X ) @ ( abs_abs_Code_integer @ Y3 ) )
      = ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_le_square_iff
thf(fact_5772_abs__le__square__iff,axiom,
    ! [X: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ X ) @ ( abs_abs_rat @ Y3 ) )
      = ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_le_square_iff
thf(fact_5773_abs__le__square__iff,axiom,
    ! [X: int,Y3: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ X ) @ ( abs_abs_int @ Y3 ) )
      = ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_le_square_iff
thf(fact_5774_power__mono__odd,axiom,
    ! [N: nat,A: rat,B: rat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( ord_less_eq_rat @ A @ B )
       => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ) ).

% power_mono_odd
thf(fact_5775_power__mono__odd,axiom,
    ! [N: nat,A: int,B: int] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( ord_less_eq_int @ A @ B )
       => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).

% power_mono_odd
thf(fact_5776_abs__square__eq__1,axiom,
    ! [X: code_integer] :
      ( ( ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_Code_integer )
      = ( ( abs_abs_Code_integer @ X )
        = one_one_Code_integer ) ) ).

% abs_square_eq_1
thf(fact_5777_abs__square__eq__1,axiom,
    ! [X: rat] :
      ( ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_rat )
      = ( ( abs_abs_rat @ X )
        = one_one_rat ) ) ).

% abs_square_eq_1
thf(fact_5778_abs__square__eq__1,axiom,
    ! [X: int] :
      ( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_int )
      = ( ( abs_abs_int @ X )
        = one_one_int ) ) ).

% abs_square_eq_1
thf(fact_5779_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_5780_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_5781_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_5782_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_5783_nat__2,axiom,
    ( ( nat2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
    = ( suc @ ( suc @ zero_zero_nat ) ) ) ).

% nat_2
thf(fact_5784_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_5785_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_5786_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_5787_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_5788_divmod__digit__0_I2_J,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B )
     => ( ( ord_less_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ B )
       => ( ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) )
          = ( modulo_modulo_nat @ A @ B ) ) ) ) ).

% divmod_digit_0(2)
thf(fact_5789_divmod__digit__0_I2_J,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ B )
       => ( ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) )
          = ( modulo_modulo_int @ A @ B ) ) ) ) ).

% divmod_digit_0(2)
thf(fact_5790_divmod__digit__0_I2_J,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
     => ( ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ B )
       => ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) )
          = ( modulo364778990260209775nteger @ A @ B ) ) ) ) ).

% divmod_digit_0(2)
thf(fact_5791_oddE,axiom,
    ! [A: int] :
      ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ~ ! [B4: int] :
            ( A
           != ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B4 ) @ one_one_int ) ) ) ).

% oddE
thf(fact_5792_oddE,axiom,
    ! [A: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ~ ! [B4: nat] :
            ( A
           != ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B4 ) @ one_one_nat ) ) ) ).

% oddE
thf(fact_5793_oddE,axiom,
    ! [A: code_integer] :
      ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ~ ! [B4: code_integer] :
            ( A
           != ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B4 ) @ one_one_Code_integer ) ) ) ).

% oddE
thf(fact_5794_oddE,axiom,
    ! [A: code_natural] :
      ( ~ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A )
     => ~ ! [B4: code_natural] :
            ( A
           != ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ B4 ) @ one_one_Code_natural ) ) ) ).

% oddE
thf(fact_5795_power2__less__imp__less,axiom,
    ! [X: code_natural,Y3: code_natural] :
      ( ( ord_le5570908160329646204atural @ ( power_7079662738309270450atural @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_7079662738309270450atural @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ Y3 )
       => ( ord_le5570908160329646204atural @ X @ Y3 ) ) ) ).

% power2_less_imp_less
thf(fact_5796_power2__less__imp__less,axiom,
    ! [X: code_integer,Y3: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ Y3 )
       => ( ord_le6747313008572928689nteger @ X @ Y3 ) ) ) ).

% power2_less_imp_less
thf(fact_5797_power2__less__imp__less,axiom,
    ! [X: rat,Y3: rat] :
      ( ( ord_less_rat @ ( power_power_rat @ X @ ( 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 @ X @ Y3 ) ) ) ).

% power2_less_imp_less
thf(fact_5798_power2__less__imp__less,axiom,
    ! [X: nat,Y3: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ X @ ( 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 @ X @ Y3 ) ) ) ).

% power2_less_imp_less
thf(fact_5799_power2__less__imp__less,axiom,
    ! [X: int,Y3: int] :
      ( ( ord_less_int @ ( power_power_int @ X @ ( 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 @ X @ Y3 ) ) ) ).

% power2_less_imp_less
thf(fact_5800_sum__power2__le__zero__iff,axiom,
    ! [X: code_integer,Y3: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_z3403309356797280102nteger )
      = ( ( X = zero_z3403309356797280102nteger )
        & ( Y3 = zero_z3403309356797280102nteger ) ) ) ).

% sum_power2_le_zero_iff
thf(fact_5801_sum__power2__le__zero__iff,axiom,
    ! [X: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_rat )
      = ( ( X = zero_zero_rat )
        & ( Y3 = zero_zero_rat ) ) ) ).

% sum_power2_le_zero_iff
thf(fact_5802_sum__power2__le__zero__iff,axiom,
    ! [X: int,Y3: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_int )
      = ( ( X = zero_zero_int )
        & ( Y3 = zero_zero_int ) ) ) ).

% sum_power2_le_zero_iff
thf(fact_5803_sum__power2__ge__zero,axiom,
    ! [X: code_integer,Y3: code_integer] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sum_power2_ge_zero
thf(fact_5804_sum__power2__ge__zero,axiom,
    ! [X: rat,Y3: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sum_power2_ge_zero
thf(fact_5805_sum__power2__ge__zero,axiom,
    ! [X: int,Y3: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sum_power2_ge_zero
thf(fact_5806_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_5807_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_5808_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_5809_mod2__eq__if,axiom,
    ! [A: code_natural] :
      ( ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A )
       => ( ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
          = zero_z2226904508553997617atural ) )
      & ( ~ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A )
       => ( ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
          = one_one_Code_natural ) ) ) ).

% mod2_eq_if
thf(fact_5810_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_5811_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_5812_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_5813_parity__cases,axiom,
    ! [A: code_natural] :
      ( ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A )
       => ( ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
         != zero_z2226904508553997617atural ) )
     => ~ ( ~ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A )
         => ( ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
           != one_one_Code_natural ) ) ) ).

% parity_cases
thf(fact_5814_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_5815_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_5816_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_5817_bits__stable__imp__add__self,axiom,
    ! [A: code_natural] :
      ( ( ( divide5121882707175180666atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
        = A )
     => ( ( plus_p4538020629002901425atural @ A @ ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) )
        = zero_z2226904508553997617atural ) ) ).

% bits_stable_imp_add_self
thf(fact_5818_not__sum__power2__lt__zero,axiom,
    ! [X: code_integer,Y3: code_integer] :
      ~ ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_z3403309356797280102nteger ) ).

% not_sum_power2_lt_zero
thf(fact_5819_not__sum__power2__lt__zero,axiom,
    ! [X: rat,Y3: rat] :
      ~ ( ord_less_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_rat ) ).

% not_sum_power2_lt_zero
thf(fact_5820_not__sum__power2__lt__zero,axiom,
    ! [X: int,Y3: int] :
      ~ ( ord_less_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_int ) ).

% not_sum_power2_lt_zero
thf(fact_5821_sum__power2__gt__zero__iff,axiom,
    ! [X: code_integer,Y3: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
      = ( ( X != zero_z3403309356797280102nteger )
        | ( Y3 != zero_z3403309356797280102nteger ) ) ) ).

% sum_power2_gt_zero_iff
thf(fact_5822_sum__power2__gt__zero__iff,axiom,
    ! [X: rat,Y3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
      = ( ( X != zero_zero_rat )
        | ( Y3 != zero_zero_rat ) ) ) ).

% sum_power2_gt_zero_iff
thf(fact_5823_sum__power2__gt__zero__iff,axiom,
    ! [X: int,Y3: int] :
      ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
      = ( ( X != zero_zero_int )
        | ( Y3 != zero_zero_int ) ) ) ).

% sum_power2_gt_zero_iff
thf(fact_5824_power2__sum,axiom,
    ! [X: int,Y3: int] :
      ( ( power_power_int @ ( plus_plus_int @ X @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_int @ ( plus_plus_int @ ( power_power_int @ X @ ( 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 ) ) @ X ) @ Y3 ) ) ) ).

% power2_sum
thf(fact_5825_power2__sum,axiom,
    ! [X: nat,Y3: nat] :
      ( ( power_power_nat @ ( plus_plus_nat @ X @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_nat @ ( plus_plus_nat @ ( power_power_nat @ X @ ( 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 ) ) @ X ) @ Y3 ) ) ) ).

% power2_sum
thf(fact_5826_power2__sum,axiom,
    ! [X: code_integer,Y3: code_integer] :
      ( ( power_8256067586552552935nteger @ ( plus_p5714425477246183910nteger @ X @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_3573771949741848930nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ X ) @ Y3 ) ) ) ).

% power2_sum
thf(fact_5827_power2__sum,axiom,
    ! [X: rat,Y3: rat] :
      ( ( power_power_rat @ ( plus_plus_rat @ X @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( 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 ) ) @ X ) @ Y3 ) ) ) ).

% power2_sum
thf(fact_5828_power2__sum,axiom,
    ! [X: code_natural,Y3: code_natural] :
      ( ( power_7079662738309270450atural @ ( plus_p4538020629002901425atural @ X @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_p4538020629002901425atural @ ( plus_p4538020629002901425atural @ ( power_7079662738309270450atural @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_7079662738309270450atural @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_2397367101498566445atural @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ X ) @ Y3 ) ) ) ).

% power2_sum
thf(fact_5829_square__le__1,axiom,
    ! [X: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ X )
     => ( ( ord_le3102999989581377725nteger @ X @ one_one_Code_integer )
       => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer ) ) ) ).

% square_le_1
thf(fact_5830_square__le__1,axiom,
    ! [X: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ X )
     => ( ( ord_less_eq_rat @ X @ one_one_rat )
       => ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat ) ) ) ).

% square_le_1
thf(fact_5831_square__le__1,axiom,
    ! [X: int] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ X )
     => ( ( ord_less_eq_int @ X @ one_one_int )
       => ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int ) ) ) ).

% square_le_1
thf(fact_5832_zero__le__even__power_H,axiom,
    ! [A: code_integer,N: nat] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% zero_le_even_power'
thf(fact_5833_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_5834_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_5835_power2__le__iff__abs__le,axiom,
    ! [Y3: code_integer,X: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ Y3 )
     => ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X ) @ Y3 ) ) ) ).

% power2_le_iff_abs_le
thf(fact_5836_power2__le__iff__abs__le,axiom,
    ! [Y3: rat,X: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
     => ( ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = ( ord_less_eq_rat @ ( abs_abs_rat @ X ) @ Y3 ) ) ) ).

% power2_le_iff_abs_le
thf(fact_5837_power2__le__iff__abs__le,axiom,
    ! [Y3: int,X: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
     => ( ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = ( ord_less_eq_int @ ( abs_abs_int @ X ) @ Y3 ) ) ) ).

% power2_le_iff_abs_le
thf(fact_5838_zero__le__power__eq,axiom,
    ! [A: code_integer,N: nat] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ A @ N ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
          & ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ) ) ).

% zero_le_power_eq
thf(fact_5839_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_5840_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_5841_zero__le__odd__power,axiom,
    ! [N: nat,A: code_integer] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ A @ N ) )
        = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ) ).

% zero_le_odd_power
thf(fact_5842_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_5843_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_5844_zero__le__even__power,axiom,
    ! [N: nat,A: code_integer] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ A @ N ) ) ) ).

% zero_le_even_power
thf(fact_5845_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_5846_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_5847_abs__square__le__1,axiom,
    ! [X: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer )
      = ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X ) @ one_one_Code_integer ) ) ).

% abs_square_le_1
thf(fact_5848_abs__square__le__1,axiom,
    ! [X: rat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat )
      = ( ord_less_eq_rat @ ( abs_abs_rat @ X ) @ one_one_rat ) ) ).

% abs_square_le_1
thf(fact_5849_abs__square__le__1,axiom,
    ! [X: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int )
      = ( ord_less_eq_int @ ( abs_abs_int @ X ) @ one_one_int ) ) ).

% abs_square_le_1
thf(fact_5850_abs__square__less__1,axiom,
    ! [X: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer )
      = ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ X ) @ one_one_Code_integer ) ) ).

% abs_square_less_1
thf(fact_5851_abs__square__less__1,axiom,
    ! [X: rat] :
      ( ( ord_less_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat )
      = ( ord_less_rat @ ( abs_abs_rat @ X ) @ one_one_rat ) ) ).

% abs_square_less_1
thf(fact_5852_abs__square__less__1,axiom,
    ! [X: int] :
      ( ( ord_less_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int )
      = ( ord_less_int @ ( abs_abs_int @ X ) @ one_one_int ) ) ).

% abs_square_less_1
thf(fact_5853_bits__induct,axiom,
    ! [P: code_integer > $o,A: code_integer] :
      ( ! [A5: code_integer] :
          ( ( ( divide6298287555418463151nteger @ A5 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
            = A5 )
         => ( P @ A5 ) )
     => ( ! [A5: code_integer,B4: $o] :
            ( ( P @ A5 )
           => ( ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ B4 ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A5 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
                = A5 )
             => ( P @ ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ B4 ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A5 ) ) ) ) )
       => ( P @ A ) ) ) ).

% bits_induct
thf(fact_5854_bits__induct,axiom,
    ! [P: code_natural > $o,A: code_natural] :
      ( ! [A5: code_natural] :
          ( ( ( divide5121882707175180666atural @ A5 @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
            = A5 )
         => ( P @ A5 ) )
     => ( ! [A5: code_natural,B4: $o] :
            ( ( P @ A5 )
           => ( ( ( divide5121882707175180666atural @ ( plus_p4538020629002901425atural @ ( zero_n8403883297036319079atural @ B4 ) @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A5 ) ) @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
                = A5 )
             => ( P @ ( plus_p4538020629002901425atural @ ( zero_n8403883297036319079atural @ B4 ) @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A5 ) ) ) ) )
       => ( P @ A ) ) ) ).

% bits_induct
thf(fact_5855_bits__induct,axiom,
    ! [P: nat > $o,A: nat] :
      ( ! [A5: nat] :
          ( ( ( divide_divide_nat @ A5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = A5 )
         => ( P @ A5 ) )
     => ( ! [A5: nat,B4: $o] :
            ( ( P @ A5 )
           => ( ( ( divide_divide_nat @ ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ B4 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A5 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
                = A5 )
             => ( P @ ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ B4 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A5 ) ) ) ) )
       => ( P @ A ) ) ) ).

% bits_induct
thf(fact_5856_bits__induct,axiom,
    ! [P: int > $o,A: int] :
      ( ! [A5: int] :
          ( ( ( divide_divide_int @ A5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
            = A5 )
         => ( P @ A5 ) )
     => ( ! [A5: int,B4: $o] :
            ( ( P @ A5 )
           => ( ( ( divide_divide_int @ ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ B4 ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A5 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
                = A5 )
             => ( P @ ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ B4 ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A5 ) ) ) ) )
       => ( P @ A ) ) ) ).

% bits_induct
thf(fact_5857_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_5858_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_5859_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_5860_power__odd__eq,axiom,
    ! [A: code_integer,N: nat] :
      ( ( power_8256067586552552935nteger @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ ( power_8256067586552552935nteger @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% power_odd_eq
thf(fact_5861_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_5862_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_5863_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_5864_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_5865_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_5866_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_5867_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_5868_power__mono__even,axiom,
    ! [N: nat,A: code_integer,B: code_integer] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) )
       => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A @ N ) @ ( power_8256067586552552935nteger @ B @ N ) ) ) ) ).

% power_mono_even
thf(fact_5869_power__mono__even,axiom,
    ! [N: nat,A: rat,B: rat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) )
       => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ) ).

% power_mono_even
thf(fact_5870_power__mono__even,axiom,
    ! [N: nat,A: int,B: int] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) )
       => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).

% power_mono_even
thf(fact_5871_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_5872_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_5873_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_5874_triangle__def,axiom,
    ( nat_triangle
    = ( ^ [N3: nat] : ( divide_divide_nat @ ( times_times_nat @ N3 @ ( suc @ N3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% triangle_def
thf(fact_5875_divmod__digit__0_I1_J,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B )
     => ( ( ord_less_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ B )
       => ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) )
          = ( divide_divide_nat @ A @ B ) ) ) ) ).

% divmod_digit_0(1)
thf(fact_5876_divmod__digit__0_I1_J,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ B )
       => ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) )
          = ( divide_divide_int @ A @ B ) ) ) ) ).

% divmod_digit_0(1)
thf(fact_5877_divmod__digit__0_I1_J,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
     => ( ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ B )
       => ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) )
          = ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).

% divmod_digit_0(1)
thf(fact_5878_power2__diff,axiom,
    ! [X: int,Y3: int] :
      ( ( power_power_int @ ( minus_minus_int @ X @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_int @ ( plus_plus_int @ ( power_power_int @ X @ ( 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 ) ) @ X ) @ Y3 ) ) ) ).

% power2_diff
thf(fact_5879_power2__diff,axiom,
    ! [X: code_integer,Y3: code_integer] :
      ( ( power_8256067586552552935nteger @ ( minus_8373710615458151222nteger @ X @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_3573771949741848930nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ X ) @ Y3 ) ) ) ).

% power2_diff
thf(fact_5880_power2__diff,axiom,
    ! [X: rat,Y3: rat] :
      ( ( power_power_rat @ ( minus_minus_rat @ X @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( 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 ) ) @ X ) @ Y3 ) ) ) ).

% power2_diff
thf(fact_5881_odd__0__le__power__imp__0__le,axiom,
    ! [A: code_integer,N: nat] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
     => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).

% odd_0_le_power_imp_0_le
thf(fact_5882_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_5883_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_5884_odd__power__less__zero,axiom,
    ! [A: code_integer,N: nat] :
      ( ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger )
     => ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ zero_z3403309356797280102nteger ) ) ).

% odd_power_less_zero
thf(fact_5885_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_5886_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_5887_zero__less__power__eq,axiom,
    ! [A: code_integer,N: nat] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ A @ N ) )
      = ( ( N = zero_zero_nat )
        | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
          & ( A != zero_z3403309356797280102nteger ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
          & ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ) ) ).

% zero_less_power_eq
thf(fact_5888_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_5889_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_5890_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_5891_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_5892_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_5893_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,N: nat,A: code_natural] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ A @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ M ) ) @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) )
        = ( times_2397367101498566445atural @ ( modulo8411746178871703098atural @ A @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ M ) ) ) ) ).

% mult_exp_mod_exp_eq
thf(fact_5894_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_5895_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_5896_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_5897_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_5898_exp__mod__exp,axiom,
    ! [M: nat,N: nat] :
      ( ( modulo8411746178871703098atural @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ M ) @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) )
      = ( times_2397367101498566445atural @ ( zero_n8403883297036319079atural @ ( ord_less_nat @ M @ N ) ) @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ M ) ) ) ).

% exp_mod_exp
thf(fact_5899_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_5900_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_5901_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_5902_ex__power__ivl1,axiom,
    ! [B: nat,K: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
     => ( ( ord_less_eq_nat @ one_one_nat @ K )
       => ? [N4: nat] :
            ( ( ord_less_eq_nat @ ( power_power_nat @ B @ N4 ) @ K )
            & ( ord_less_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N4 @ one_one_nat ) ) ) ) ) ) ).

% ex_power_ivl1
thf(fact_5903_ex__power__ivl2,axiom,
    ! [B: nat,K: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
     => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
       => ? [N4: nat] :
            ( ( ord_less_nat @ ( power_power_nat @ B @ N4 ) @ K )
            & ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N4 @ one_one_nat ) ) ) ) ) ) ).

% ex_power_ivl2
thf(fact_5904_round__def,axiom,
    ( archim7778729529865785530nd_rat
    = ( ^ [X2: rat] : ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X2 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% round_def
thf(fact_5905_mod__double__modulus,axiom,
    ! [M: code_integer,X: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ M )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X )
       => ( ( ( modulo364778990260209775nteger @ X @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
            = ( modulo364778990260209775nteger @ X @ M ) )
          | ( ( modulo364778990260209775nteger @ X @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
            = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ X @ M ) @ M ) ) ) ) ) ).

% mod_double_modulus
thf(fact_5906_mod__double__modulus,axiom,
    ! [M: nat,X: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ X )
       => ( ( ( modulo_modulo_nat @ X @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
            = ( modulo_modulo_nat @ X @ M ) )
          | ( ( modulo_modulo_nat @ X @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
            = ( plus_plus_nat @ ( modulo_modulo_nat @ X @ M ) @ M ) ) ) ) ) ).

% mod_double_modulus
thf(fact_5907_mod__double__modulus,axiom,
    ! [M: int,X: int] :
      ( ( ord_less_int @ zero_zero_int @ M )
     => ( ( ord_less_eq_int @ zero_zero_int @ X )
       => ( ( ( modulo_modulo_int @ X @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
            = ( modulo_modulo_int @ X @ M ) )
          | ( ( modulo_modulo_int @ X @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
            = ( plus_plus_int @ ( modulo_modulo_int @ X @ M ) @ M ) ) ) ) ) ).

% mod_double_modulus
thf(fact_5908_divmod__digit__1_I2_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
       => ( ( ord_le3102999989581377725nteger @ B @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) )
         => ( ( minus_8373710615458151222nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ B )
            = ( modulo364778990260209775nteger @ A @ B ) ) ) ) ) ).

% divmod_digit_1(2)
thf(fact_5909_divmod__digit__1_I2_J,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ B )
       => ( ( ord_less_eq_nat @ B @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) )
         => ( ( minus_minus_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ B )
            = ( modulo_modulo_nat @ A @ B ) ) ) ) ) ).

% divmod_digit_1(2)
thf(fact_5910_divmod__digit__1_I2_J,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ( ord_less_eq_int @ B @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) )
         => ( ( minus_minus_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ B )
            = ( modulo_modulo_int @ A @ B ) ) ) ) ) ).

% divmod_digit_1(2)
thf(fact_5911_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_5912_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_5913_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_5914_even__mask__div__iff_H,axiom,
    ! [M: nat,N: nat] :
      ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( divide5121882707175180666atural @ ( minus_7197305767214868737atural @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ M ) @ one_one_Code_natural ) @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% even_mask_div_iff'
thf(fact_5915_power__le__zero__eq,axiom,
    ! [A: code_integer,N: nat] :
      ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A @ N ) @ zero_z3403309356797280102nteger )
      = ( ( ord_less_nat @ zero_zero_nat @ N )
        & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
            & ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) )
          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
            & ( A = zero_z3403309356797280102nteger ) ) ) ) ) ).

% power_le_zero_eq
thf(fact_5916_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_5917_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_5918_of__int__round__le,axiom,
    ! [X: rat] : ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X ) ) @ ( plus_plus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).

% of_int_round_le
thf(fact_5919_of__int__round__ge,axiom,
    ! [X: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X ) ) ) ).

% of_int_round_ge
thf(fact_5920_of__int__round__gt,axiom,
    ! [X: rat] : ( ord_less_rat @ ( minus_minus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X ) ) ) ).

% of_int_round_gt
thf(fact_5921_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_5922_neg__zdiv__mult__2,axiom,
    ! [A: int,B: 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 ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
        = ( divide_divide_int @ ( plus_plus_int @ B @ one_one_int ) @ A ) ) ) ).

% neg_zdiv_mult_2
thf(fact_5923_pos__zdiv__mult__2,axiom,
    ! [A: int,B: 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 ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
        = ( divide_divide_int @ B @ A ) ) ) ).

% pos_zdiv_mult_2
thf(fact_5924_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_5925_pos__zmod__mult__2,axiom,
    ! [A: int,B: 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 ) ) @ B ) ) @ ( 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 @ B @ A ) ) ) ) ) ).

% pos_zmod_mult_2
thf(fact_5926_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_5927_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_5928_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_5929_even__mask__div__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( divide5121882707175180666atural @ ( minus_7197305767214868737atural @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ M ) @ one_one_Code_natural ) @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) ) )
      = ( ( ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N )
          = zero_z2226904508553997617atural )
        | ( ord_less_eq_nat @ M @ N ) ) ) ).

% even_mask_div_iff
thf(fact_5930_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_5931_exp__div__exp__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( divide5121882707175180666atural @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ M ) @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) )
      = ( times_2397367101498566445atural
        @ ( zero_n8403883297036319079atural
          @ ( ( ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ M )
             != zero_z2226904508553997617atural )
            & ( ord_less_eq_nat @ N @ M ) ) )
        @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) ) ) ) ).

% exp_div_exp_eq
thf(fact_5932_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_5933_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_5934_of__int__round__abs__le,axiom,
    ! [X: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X ) ) @ X ) ) @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).

% of_int_round_abs_le
thf(fact_5935_round__unique_H,axiom,
    ! [X: rat,N: int] :
      ( ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X @ ( ring_1_of_int_rat @ N ) ) ) @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
     => ( ( archim7778729529865785530nd_rat @ X )
        = N ) ) ).

% round_unique'
thf(fact_5936_neg__zmod__mult__2,axiom,
    ! [A: int,B: 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 ) ) @ B ) ) @ ( 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 @ B @ one_one_int ) @ A ) ) @ one_one_int ) ) ) ).

% neg_zmod_mult_2
thf(fact_5937_round__altdef,axiom,
    ( archim7778729529865785530nd_rat
    = ( ^ [X2: rat] : ( if_int @ ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( archimedean_frac_rat @ X2 ) ) @ ( archim2889992004027027881ng_rat @ X2 ) @ ( archim3151403230148437115or_rat @ X2 ) ) ) ) ).

% round_altdef
thf(fact_5938_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_5939_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_5940_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_5941_flip__bit__0,axiom,
    ! [A: code_natural] :
      ( ( bit_se168947363167071951atural @ zero_zero_nat @ A )
      = ( plus_p4538020629002901425atural @ ( zero_n8403883297036319079atural @ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A ) ) @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( divide5121882707175180666atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ) ) ).

% flip_bit_0
thf(fact_5942_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_5943_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_5944_set__bit__0,axiom,
    ! [A: code_integer] :
      ( ( bit_se2793503036327961859nteger @ zero_zero_nat @ A )
      = ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ).

% set_bit_0
thf(fact_5945_set__bit__0,axiom,
    ! [A: code_natural] :
      ( ( bit_se1617098188084679374atural @ zero_zero_nat @ A )
      = ( plus_p4538020629002901425atural @ one_one_Code_natural @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( divide5121882707175180666atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ) ) ).

% set_bit_0
thf(fact_5946_psubsetI,axiom,
    ! [A3: set_nat,B5: set_nat] :
      ( ( ord_less_eq_set_nat @ A3 @ B5 )
     => ( ( A3 != B5 )
       => ( ord_less_set_nat @ A3 @ B5 ) ) ) ).

% psubsetI
thf(fact_5947_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_5948_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_5949_unset__bit__0,axiom,
    ! [A: code_integer] :
      ( ( bit_se8260200283734997820nteger @ zero_zero_nat @ A )
      = ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).

% unset_bit_0
thf(fact_5950_unset__bit__0,axiom,
    ! [A: code_natural] :
      ( ( bit_se7083795435491715335atural @ zero_zero_nat @ A )
      = ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( divide5121882707175180666atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ) ).

% unset_bit_0
thf(fact_5951_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_5952_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_5953_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_5954_unset__bit__Suc,axiom,
    ! [N: nat,A: code_natural] :
      ( ( bit_se7083795435491715335atural @ ( suc @ N ) @ A )
      = ( plus_p4538020629002901425atural @ ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( bit_se7083795435491715335atural @ N @ ( divide5121882707175180666atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ) ) ) ).

% unset_bit_Suc
thf(fact_5955_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_5956_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_5957_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_5958_flip__bit__Suc,axiom,
    ! [N: nat,A: code_natural] :
      ( ( bit_se168947363167071951atural @ ( suc @ N ) @ A )
      = ( plus_p4538020629002901425atural @ ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( bit_se168947363167071951atural @ N @ ( divide5121882707175180666atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ) ) ) ).

% flip_bit_Suc
thf(fact_5959_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_5960_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_5961_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_5962_set__bit__Suc,axiom,
    ! [N: nat,A: code_natural] :
      ( ( bit_se1617098188084679374atural @ ( suc @ N ) @ A )
      = ( plus_p4538020629002901425atural @ ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( bit_se1617098188084679374atural @ N @ ( divide5121882707175180666atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ) ) ) ).

% set_bit_Suc
thf(fact_5963_Compl__subset__Compl__iff,axiom,
    ! [A3: set_nat,B5: set_nat] :
      ( ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ A3 ) @ ( uminus5710092332889474511et_nat @ B5 ) )
      = ( ord_less_eq_set_nat @ B5 @ A3 ) ) ).

% Compl_subset_Compl_iff
thf(fact_5964_Compl__anti__mono,axiom,
    ! [A3: set_nat,B5: set_nat] :
      ( ( ord_less_eq_set_nat @ A3 @ B5 )
     => ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ B5 ) @ ( uminus5710092332889474511et_nat @ A3 ) ) ) ).

% Compl_anti_mono
thf(fact_5965_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_5966_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_5967_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_5968_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_5969_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_5970_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_5971_unset__bit__nat__def,axiom,
    ( bit_se4205575877204974255it_nat
    = ( ^ [M2: nat,N3: nat] : ( nat2 @ ( bit_se4203085406695923979it_int @ M2 @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).

% unset_bit_nat_def
thf(fact_5972_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_5973_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_5974_psubset__imp__ex__mem,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( ord_le7866589430770878221at_nat @ A3 @ B5 )
     => ? [B4: product_prod_nat_nat] : ( member8440522571783428010at_nat @ B4 @ ( minus_1356011639430497352at_nat @ B5 @ A3 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_5975_psubset__imp__ex__mem,axiom,
    ! [A3: set_se7855581050983116737at_nat,B5: set_se7855581050983116737at_nat] :
      ( ( ord_le3519150613208228973at_nat @ A3 @ B5 )
     => ? [B4: set_Pr1261947904930325089at_nat] : ( member2643936169264416010at_nat @ B4 @ ( minus_4207664762107033000at_nat @ B5 @ A3 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_5976_psubset__imp__ex__mem,axiom,
    ! [A3: set_set_nat,B5: set_set_nat] :
      ( ( ord_less_set_set_nat @ A3 @ B5 )
     => ? [B4: set_nat] : ( member_set_nat @ B4 @ ( minus_2163939370556025621et_nat @ B5 @ A3 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_5977_psubset__imp__ex__mem,axiom,
    ! [A3: set_int,B5: set_int] :
      ( ( ord_less_set_int @ A3 @ B5 )
     => ? [B4: int] : ( member_int @ B4 @ ( minus_minus_set_int @ B5 @ A3 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_5978_psubset__imp__ex__mem,axiom,
    ! [A3: set_nat,B5: set_nat] :
      ( ( ord_less_set_nat @ A3 @ B5 )
     => ? [B4: nat] : ( member_nat @ B4 @ ( minus_minus_set_nat @ B5 @ A3 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_5979_psubsetD,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat,C: product_prod_nat_nat] :
      ( ( ord_le7866589430770878221at_nat @ A3 @ B5 )
     => ( ( member8440522571783428010at_nat @ C @ A3 )
       => ( member8440522571783428010at_nat @ C @ B5 ) ) ) ).

% psubsetD
thf(fact_5980_psubsetD,axiom,
    ! [A3: set_se7855581050983116737at_nat,B5: set_se7855581050983116737at_nat,C: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3519150613208228973at_nat @ A3 @ B5 )
     => ( ( member2643936169264416010at_nat @ C @ A3 )
       => ( member2643936169264416010at_nat @ C @ B5 ) ) ) ).

% psubsetD
thf(fact_5981_psubsetD,axiom,
    ! [A3: set_set_nat,B5: set_set_nat,C: set_nat] :
      ( ( ord_less_set_set_nat @ A3 @ B5 )
     => ( ( member_set_nat @ C @ A3 )
       => ( member_set_nat @ C @ B5 ) ) ) ).

% psubsetD
thf(fact_5982_psubsetD,axiom,
    ! [A3: set_nat,B5: set_nat,C: nat] :
      ( ( ord_less_set_nat @ A3 @ B5 )
     => ( ( member_nat @ C @ A3 )
       => ( member_nat @ C @ B5 ) ) ) ).

% psubsetD
thf(fact_5983_psubsetD,axiom,
    ! [A3: set_int,B5: set_int,C: int] :
      ( ( ord_less_set_int @ A3 @ B5 )
     => ( ( member_int @ C @ A3 )
       => ( member_int @ C @ B5 ) ) ) ).

% psubsetD
thf(fact_5984_psubsetE,axiom,
    ! [A3: set_nat,B5: set_nat] :
      ( ( ord_less_set_nat @ A3 @ B5 )
     => ~ ( ( ord_less_eq_set_nat @ A3 @ B5 )
         => ( ord_less_eq_set_nat @ B5 @ A3 ) ) ) ).

% psubsetE
thf(fact_5985_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_5986_psubset__imp__subset,axiom,
    ! [A3: set_nat,B5: set_nat] :
      ( ( ord_less_set_nat @ A3 @ B5 )
     => ( ord_less_eq_set_nat @ A3 @ B5 ) ) ).

% psubset_imp_subset
thf(fact_5987_psubset__subset__trans,axiom,
    ! [A3: set_nat,B5: set_nat,C5: set_nat] :
      ( ( ord_less_set_nat @ A3 @ B5 )
     => ( ( ord_less_eq_set_nat @ B5 @ C5 )
       => ( ord_less_set_nat @ A3 @ C5 ) ) ) ).

% psubset_subset_trans
thf(fact_5988_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_5989_subset__psubset__trans,axiom,
    ! [A3: set_nat,B5: set_nat,C5: set_nat] :
      ( ( ord_less_eq_set_nat @ A3 @ B5 )
     => ( ( ord_less_set_nat @ B5 @ C5 )
       => ( ord_less_set_nat @ A3 @ C5 ) ) ) ).

% subset_psubset_trans
thf(fact_5990_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_5991_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_5992_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_5993_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_5994_even__set__bit__iff,axiom,
    ! [M: nat,A: code_natural] :
      ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( bit_se1617098188084679374atural @ M @ A ) )
      = ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A )
        & ( M != zero_zero_nat ) ) ) ).

% even_set_bit_iff
thf(fact_5995_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_5996_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_5997_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_5998_even__flip__bit__iff,axiom,
    ! [M: nat,A: code_natural] :
      ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( bit_se168947363167071951atural @ M @ A ) )
      = ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A )
       != ( M = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
thf(fact_5999_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_6000_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_6001_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_6002_even__unset__bit__iff,axiom,
    ! [M: nat,A: code_natural] :
      ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( bit_se7083795435491715335atural @ M @ A ) )
      = ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A )
        | ( M = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
thf(fact_6003_signed__take__bit__rec,axiom,
    ( bit_ri6519982836138164636nteger
    = ( ^ [N3: nat,A2: code_integer] : ( if_Code_integer @ ( N3 = zero_zero_nat ) @ ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( divide6298287555418463151nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% signed_take_bit_rec
thf(fact_6004_signed__take__bit__rec,axiom,
    ( bit_ri631733984087533419it_int
    = ( ^ [N3: nat,A2: int] : ( if_int @ ( N3 = zero_zero_nat ) @ ( uminus_uminus_int @ ( modulo_modulo_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( plus_plus_int @ ( modulo_modulo_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( divide_divide_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% signed_take_bit_rec
thf(fact_6005_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_6006_dbl__simps_I4_J,axiom,
    ( ( neg_nu8804712462038260780nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% dbl_simps(4)
thf(fact_6007_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_6008_dbl__simps_I3_J,axiom,
    ( ( neg_numeral_dbl_int @ one_one_int )
    = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).

% dbl_simps(3)
thf(fact_6009_dbl__simps_I3_J,axiom,
    ( ( neg_nu8804712462038260780nteger @ one_one_Code_integer )
    = ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ).

% dbl_simps(3)
thf(fact_6010_dbl__simps_I3_J,axiom,
    ( ( neg_numeral_dbl_rat @ one_one_rat )
    = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).

% dbl_simps(3)
thf(fact_6011_concat__bit__Suc,axiom,
    ! [N: nat,K: int,L: int] :
      ( ( bit_concat_bit @ ( suc @ N ) @ K @ L )
      = ( 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 ) ) ) @ L ) ) ) ) ).

% concat_bit_Suc
thf(fact_6012_set__decode__Suc,axiom,
    ! [N: nat,X: nat] :
      ( ( member_nat @ ( suc @ N ) @ ( nat_set_decode @ X ) )
      = ( member_nat @ N @ ( nat_set_decode @ ( divide_divide_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% set_decode_Suc
thf(fact_6013_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_6014_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_6015_Compl__iff,axiom,
    ! [C: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ C @ ( uminus6524753893492686040at_nat @ A3 ) )
      = ( ~ ( member8440522571783428010at_nat @ C @ A3 ) ) ) ).

% Compl_iff
thf(fact_6016_Compl__iff,axiom,
    ! [C: set_Pr1261947904930325089at_nat,A3: set_se7855581050983116737at_nat] :
      ( ( member2643936169264416010at_nat @ C @ ( uminus8018730102153342520at_nat @ A3 ) )
      = ( ~ ( member2643936169264416010at_nat @ C @ A3 ) ) ) ).

% Compl_iff
thf(fact_6017_Compl__iff,axiom,
    ! [C: set_nat,A3: set_set_nat] :
      ( ( member_set_nat @ C @ ( uminus613421341184616069et_nat @ A3 ) )
      = ( ~ ( member_set_nat @ C @ A3 ) ) ) ).

% Compl_iff
thf(fact_6018_Compl__iff,axiom,
    ! [C: nat,A3: set_nat] :
      ( ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A3 ) )
      = ( ~ ( member_nat @ C @ A3 ) ) ) ).

% Compl_iff
thf(fact_6019_Compl__iff,axiom,
    ! [C: int,A3: set_int] :
      ( ( member_int @ C @ ( uminus1532241313380277803et_int @ A3 ) )
      = ( ~ ( member_int @ C @ A3 ) ) ) ).

% Compl_iff
thf(fact_6020_ComplI,axiom,
    ! [C: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ~ ( member8440522571783428010at_nat @ C @ A3 )
     => ( member8440522571783428010at_nat @ C @ ( uminus6524753893492686040at_nat @ A3 ) ) ) ).

% ComplI
thf(fact_6021_ComplI,axiom,
    ! [C: set_Pr1261947904930325089at_nat,A3: set_se7855581050983116737at_nat] :
      ( ~ ( member2643936169264416010at_nat @ C @ A3 )
     => ( member2643936169264416010at_nat @ C @ ( uminus8018730102153342520at_nat @ A3 ) ) ) ).

% ComplI
thf(fact_6022_ComplI,axiom,
    ! [C: set_nat,A3: set_set_nat] :
      ( ~ ( member_set_nat @ C @ A3 )
     => ( member_set_nat @ C @ ( uminus613421341184616069et_nat @ A3 ) ) ) ).

% ComplI
thf(fact_6023_ComplI,axiom,
    ! [C: nat,A3: set_nat] :
      ( ~ ( member_nat @ C @ A3 )
     => ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A3 ) ) ) ).

% ComplI
thf(fact_6024_ComplI,axiom,
    ! [C: int,A3: set_int] :
      ( ~ ( member_int @ C @ A3 )
     => ( member_int @ C @ ( uminus1532241313380277803et_int @ A3 ) ) ) ).

% ComplI
thf(fact_6025_signed__take__bit__of__0,axiom,
    ! [N: nat] :
      ( ( bit_ri6519982836138164636nteger @ N @ zero_z3403309356797280102nteger )
      = zero_z3403309356797280102nteger ) ).

% signed_take_bit_of_0
thf(fact_6026_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_6027_concat__bit__0,axiom,
    ! [K: int,L: int] :
      ( ( bit_concat_bit @ zero_zero_nat @ K @ L )
      = L ) ).

% concat_bit_0
thf(fact_6028_dbl__simps_I2_J,axiom,
    ( ( neg_numeral_dbl_int @ zero_zero_int )
    = zero_zero_int ) ).

% dbl_simps(2)
thf(fact_6029_dbl__simps_I2_J,axiom,
    ( ( neg_nu8804712462038260780nteger @ zero_z3403309356797280102nteger )
    = zero_z3403309356797280102nteger ) ).

% dbl_simps(2)
thf(fact_6030_dbl__simps_I2_J,axiom,
    ( ( neg_numeral_dbl_rat @ zero_zero_rat )
    = zero_zero_rat ) ).

% dbl_simps(2)
thf(fact_6031_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_6032_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_6033_signed__take__bit__Suc__1,axiom,
    ! [N: nat] :
      ( ( bit_ri6519982836138164636nteger @ ( suc @ N ) @ one_one_Code_integer )
      = one_one_Code_integer ) ).

% signed_take_bit_Suc_1
thf(fact_6034_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_6035_signed__take__bit__numeral__of__1,axiom,
    ! [K: num] :
      ( ( bit_ri6519982836138164636nteger @ ( numeral_numeral_nat @ K ) @ one_one_Code_integer )
      = one_one_Code_integer ) ).

% signed_take_bit_numeral_of_1
thf(fact_6036_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_6037_concat__bit__nonnegative__iff,axiom,
    ! [N: nat,K: int,L: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_concat_bit @ N @ K @ L ) )
      = ( ord_less_eq_int @ zero_zero_int @ L ) ) ).

% concat_bit_nonnegative_iff
thf(fact_6038_concat__bit__negative__iff,axiom,
    ! [N: nat,K: int,L: int] :
      ( ( ord_less_int @ ( bit_concat_bit @ N @ K @ L ) @ zero_zero_int )
      = ( ord_less_int @ L @ zero_zero_int ) ) ).

% concat_bit_negative_iff
thf(fact_6039_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_6040_dbl__simps_I5_J,axiom,
    ! [K: num] :
      ( ( neg_nu8804712462038260780nteger @ ( numera6620942414471956472nteger @ K ) )
      = ( numera6620942414471956472nteger @ ( bit0 @ K ) ) ) ).

% dbl_simps(5)
thf(fact_6041_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_6042_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_6043_dbl__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu8804712462038260780nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
      = ( uminus1351360451143612070nteger @ ( neg_nu8804712462038260780nteger @ ( numera6620942414471956472nteger @ K ) ) ) ) ).

% dbl_simps(1)
thf(fact_6044_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_6045_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_6046_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_6047_set__decode__0,axiom,
    ! [X: nat] :
      ( ( member_nat @ zero_zero_nat @ ( nat_set_decode @ X ) )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X ) ) ) ).

% set_decode_0
thf(fact_6048_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_6049_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_6050_ComplD,axiom,
    ! [C: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ C @ ( uminus6524753893492686040at_nat @ A3 ) )
     => ~ ( member8440522571783428010at_nat @ C @ A3 ) ) ).

% ComplD
thf(fact_6051_ComplD,axiom,
    ! [C: set_Pr1261947904930325089at_nat,A3: set_se7855581050983116737at_nat] :
      ( ( member2643936169264416010at_nat @ C @ ( uminus8018730102153342520at_nat @ A3 ) )
     => ~ ( member2643936169264416010at_nat @ C @ A3 ) ) ).

% ComplD
thf(fact_6052_ComplD,axiom,
    ! [C: set_nat,A3: set_set_nat] :
      ( ( member_set_nat @ C @ ( uminus613421341184616069et_nat @ A3 ) )
     => ~ ( member_set_nat @ C @ A3 ) ) ).

% ComplD
thf(fact_6053_ComplD,axiom,
    ! [C: nat,A3: set_nat] :
      ( ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A3 ) )
     => ~ ( member_nat @ C @ A3 ) ) ).

% ComplD
thf(fact_6054_ComplD,axiom,
    ! [C: int,A3: set_int] :
      ( ( member_int @ C @ ( uminus1532241313380277803et_int @ A3 ) )
     => ~ ( member_int @ C @ A3 ) ) ).

% ComplD
thf(fact_6055_signed__take__bit__mult,axiom,
    ! [N: nat,K: int,L: int] :
      ( ( bit_ri631733984087533419it_int @ N @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( bit_ri631733984087533419it_int @ N @ L ) ) )
      = ( bit_ri631733984087533419it_int @ N @ ( times_times_int @ K @ L ) ) ) ).

% signed_take_bit_mult
thf(fact_6056_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_6057_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_6058_dbl__def,axiom,
    ( neg_nu8804712462038260780nteger
    = ( ^ [X2: code_integer] : ( plus_p5714425477246183910nteger @ X2 @ X2 ) ) ) ).

% dbl_def
thf(fact_6059_dbl__def,axiom,
    ( neg_numeral_dbl_rat
    = ( ^ [X2: rat] : ( plus_plus_rat @ X2 @ X2 ) ) ) ).

% dbl_def
thf(fact_6060_dbl__def,axiom,
    ( neg_numeral_dbl_int
    = ( ^ [X2: int] : ( plus_plus_int @ X2 @ X2 ) ) ) ).

% dbl_def
thf(fact_6061_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_6062_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_6063_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_6064_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_6065_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_6066_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_6067_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_6068_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_6069_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_6070_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_6071_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_6072_fact__code,axiom,
    ( semiri1408675320244567234ct_nat
    = ( ^ [N3: nat] : ( semiri1316708129612266289at_nat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 @ one_one_nat ) ) ) ) ).

% fact_code
thf(fact_6073_fact__code,axiom,
    ( semiri2447717529341329178atural
    = ( ^ [N3: nat] : ( semiri3763490453095760265atural @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 @ one_one_nat ) ) ) ) ).

% fact_code
thf(fact_6074_fact__code,axiom,
    ( semiri773545260158071498ct_rat
    = ( ^ [N3: nat] : ( semiri681578069525770553at_rat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 @ one_one_nat ) ) ) ) ).

% fact_code
thf(fact_6075_fact__code,axiom,
    ( semiri3624122377584611663nteger
    = ( ^ [N3: nat] : ( semiri4939895301339042750nteger @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 @ one_one_nat ) ) ) ) ).

% fact_code
thf(fact_6076_fact__code,axiom,
    ( semiri1406184849735516958ct_int
    = ( ^ [N3: nat] : ( semiri1314217659103216013at_int @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 @ one_one_nat ) ) ) ) ).

% fact_code
thf(fact_6077_take__bit__rec,axiom,
    ( bit_se2923211474154528505it_int
    = ( ^ [N3: nat,A2: int] : ( if_int @ ( N3 = zero_zero_nat ) @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( divide_divide_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).

% take_bit_rec
thf(fact_6078_take__bit__rec,axiom,
    ( bit_se2925701944663578781it_nat
    = ( ^ [N3: nat,A2: nat] : ( if_nat @ ( N3 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( divide_divide_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% take_bit_rec
thf(fact_6079_take__bit__rec,axiom,
    ( bit_se1745604003318907178nteger
    = ( ^ [N3: nat,A2: code_integer] : ( if_Code_integer @ ( N3 = zero_zero_nat ) @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( bit_se1745604003318907178nteger @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( divide6298287555418463151nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( modulo364778990260209775nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% take_bit_rec
thf(fact_6080_take__bit__rec,axiom,
    ( bit_se569199155075624693atural
    = ( ^ [N3: nat,A2: code_natural] : ( if_Code_natural @ ( N3 = zero_zero_nat ) @ zero_z2226904508553997617atural @ ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ ( bit_se569199155075624693atural @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( divide5121882707175180666atural @ A2 @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) @ ( modulo8411746178871703098atural @ A2 @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ) ) ) ).

% take_bit_rec
thf(fact_6081_num_Osize__gen_I2_J,axiom,
    ! [X22: num] :
      ( ( size_num @ ( bit0 @ X22 ) )
      = ( plus_plus_nat @ ( size_num @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).

% num.size_gen(2)
thf(fact_6082_and__int__unfold,axiom,
    ( bit_se725231765392027082nd_int
    = ( ^ [K3: int,L3: int] :
          ( if_int
          @ ( ( K3 = zero_zero_int )
            | ( L3 = zero_zero_int ) )
          @ zero_zero_int
          @ ( if_int
            @ ( K3
              = ( uminus_uminus_int @ one_one_int ) )
            @ L3
            @ ( if_int
              @ ( L3
                = ( 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 @ L3 @ ( 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 @ L3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).

% and_int_unfold
thf(fact_6083_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_6084_semiring__norm_I90_J,axiom,
    ! [M: num,N: num] :
      ( ( ( bit1 @ M )
        = ( bit1 @ N ) )
      = ( M = N ) ) ).

% semiring_norm(90)
thf(fact_6085_verit__eq__simplify_I9_J,axiom,
    ! [X32: num,Y32: num] :
      ( ( ( bit1 @ X32 )
        = ( bit1 @ Y32 ) )
      = ( X32 = Y32 ) ) ).

% verit_eq_simplify(9)
thf(fact_6086_take__bit__of__0,axiom,
    ! [N: nat] :
      ( ( bit_se2923211474154528505it_int @ N @ zero_zero_int )
      = zero_zero_int ) ).

% take_bit_of_0
thf(fact_6087_take__bit__of__0,axiom,
    ! [N: nat] :
      ( ( bit_se2925701944663578781it_nat @ N @ zero_zero_nat )
      = zero_zero_nat ) ).

% take_bit_of_0
thf(fact_6088_take__bit__of__0,axiom,
    ! [N: nat] :
      ( ( bit_se1745604003318907178nteger @ N @ zero_z3403309356797280102nteger )
      = zero_z3403309356797280102nteger ) ).

% take_bit_of_0
thf(fact_6089_take__bit__of__0,axiom,
    ! [N: nat] :
      ( ( bit_se569199155075624693atural @ N @ zero_z2226904508553997617atural )
      = zero_z2226904508553997617atural ) ).

% take_bit_of_0
thf(fact_6090_and__zero__eq,axiom,
    ! [A: int] :
      ( ( bit_se725231765392027082nd_int @ A @ zero_zero_int )
      = zero_zero_int ) ).

% and_zero_eq
thf(fact_6091_and__zero__eq,axiom,
    ! [A: nat] :
      ( ( bit_se727722235901077358nd_nat @ A @ zero_zero_nat )
      = zero_zero_nat ) ).

% and_zero_eq
thf(fact_6092_and__zero__eq,axiom,
    ! [A: code_integer] :
      ( ( bit_se3949692690581998587nteger @ A @ zero_z3403309356797280102nteger )
      = zero_z3403309356797280102nteger ) ).

% and_zero_eq
thf(fact_6093_and__zero__eq,axiom,
    ! [A: code_natural] :
      ( ( bit_se2773287842338716102atural @ A @ zero_z2226904508553997617atural )
      = zero_z2226904508553997617atural ) ).

% and_zero_eq
thf(fact_6094_zero__and__eq,axiom,
    ! [A: int] :
      ( ( bit_se725231765392027082nd_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% zero_and_eq
thf(fact_6095_zero__and__eq,axiom,
    ! [A: nat] :
      ( ( bit_se727722235901077358nd_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% zero_and_eq
thf(fact_6096_zero__and__eq,axiom,
    ! [A: code_integer] :
      ( ( bit_se3949692690581998587nteger @ zero_z3403309356797280102nteger @ A )
      = zero_z3403309356797280102nteger ) ).

% zero_and_eq
thf(fact_6097_zero__and__eq,axiom,
    ! [A: code_natural] :
      ( ( bit_se2773287842338716102atural @ zero_z2226904508553997617atural @ A )
      = zero_z2226904508553997617atural ) ).

% zero_and_eq
thf(fact_6098_bit_Oconj__zero__left,axiom,
    ! [X: int] :
      ( ( bit_se725231765392027082nd_int @ zero_zero_int @ X )
      = zero_zero_int ) ).

% bit.conj_zero_left
thf(fact_6099_bit_Oconj__zero__left,axiom,
    ! [X: code_integer] :
      ( ( bit_se3949692690581998587nteger @ zero_z3403309356797280102nteger @ X )
      = zero_z3403309356797280102nteger ) ).

% bit.conj_zero_left
thf(fact_6100_bit_Oconj__zero__right,axiom,
    ! [X: int] :
      ( ( bit_se725231765392027082nd_int @ X @ zero_zero_int )
      = zero_zero_int ) ).

% bit.conj_zero_right
thf(fact_6101_bit_Oconj__zero__right,axiom,
    ! [X: code_integer] :
      ( ( bit_se3949692690581998587nteger @ X @ zero_z3403309356797280102nteger )
      = zero_z3403309356797280102nteger ) ).

% bit.conj_zero_right
thf(fact_6102_semiring__norm_I88_J,axiom,
    ! [M: num,N: num] :
      ( ( bit0 @ M )
     != ( bit1 @ N ) ) ).

% semiring_norm(88)
thf(fact_6103_semiring__norm_I89_J,axiom,
    ! [M: num,N: num] :
      ( ( bit1 @ M )
     != ( bit0 @ N ) ) ).

% semiring_norm(89)
thf(fact_6104_semiring__norm_I84_J,axiom,
    ! [N: num] :
      ( one
     != ( bit1 @ N ) ) ).

% semiring_norm(84)
thf(fact_6105_semiring__norm_I86_J,axiom,
    ! [M: num] :
      ( ( bit1 @ M )
     != one ) ).

% semiring_norm(86)
thf(fact_6106_bit_Oxor__self,axiom,
    ! [X: int] :
      ( ( bit_se6526347334894502574or_int @ X @ X )
      = zero_zero_int ) ).

% bit.xor_self
thf(fact_6107_bit_Oxor__self,axiom,
    ! [X: code_integer] :
      ( ( bit_se3222712562003087583nteger @ X @ X )
      = zero_z3403309356797280102nteger ) ).

% bit.xor_self
thf(fact_6108_xor__self__eq,axiom,
    ! [A: nat] :
      ( ( bit_se6528837805403552850or_nat @ A @ A )
      = zero_zero_nat ) ).

% xor_self_eq
thf(fact_6109_xor__self__eq,axiom,
    ! [A: int] :
      ( ( bit_se6526347334894502574or_int @ A @ A )
      = zero_zero_int ) ).

% xor_self_eq
thf(fact_6110_xor__self__eq,axiom,
    ! [A: code_integer] :
      ( ( bit_se3222712562003087583nteger @ A @ A )
      = zero_z3403309356797280102nteger ) ).

% xor_self_eq
thf(fact_6111_xor__self__eq,axiom,
    ! [A: code_natural] :
      ( ( bit_se2046307713759805098atural @ A @ A )
      = zero_z2226904508553997617atural ) ).

% xor_self_eq
thf(fact_6112_xor_Oleft__neutral,axiom,
    ! [A: nat] :
      ( ( bit_se6528837805403552850or_nat @ zero_zero_nat @ A )
      = A ) ).

% xor.left_neutral
thf(fact_6113_xor_Oleft__neutral,axiom,
    ! [A: int] :
      ( ( bit_se6526347334894502574or_int @ zero_zero_int @ A )
      = A ) ).

% xor.left_neutral
thf(fact_6114_xor_Oleft__neutral,axiom,
    ! [A: code_integer] :
      ( ( bit_se3222712562003087583nteger @ zero_z3403309356797280102nteger @ A )
      = A ) ).

% xor.left_neutral
thf(fact_6115_xor_Oleft__neutral,axiom,
    ! [A: code_natural] :
      ( ( bit_se2046307713759805098atural @ zero_z2226904508553997617atural @ A )
      = A ) ).

% xor.left_neutral
thf(fact_6116_xor_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( bit_se6528837805403552850or_nat @ A @ zero_zero_nat )
      = A ) ).

% xor.right_neutral
thf(fact_6117_xor_Oright__neutral,axiom,
    ! [A: int] :
      ( ( bit_se6526347334894502574or_int @ A @ zero_zero_int )
      = A ) ).

% xor.right_neutral
thf(fact_6118_xor_Oright__neutral,axiom,
    ! [A: code_integer] :
      ( ( bit_se3222712562003087583nteger @ A @ zero_z3403309356797280102nteger )
      = A ) ).

% xor.right_neutral
thf(fact_6119_xor_Oright__neutral,axiom,
    ! [A: code_natural] :
      ( ( bit_se2046307713759805098atural @ A @ zero_z2226904508553997617atural )
      = A ) ).

% xor.right_neutral
thf(fact_6120_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_6121_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_6122_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_6123_take__bit__0,axiom,
    ! [A: int] :
      ( ( bit_se2923211474154528505it_int @ zero_zero_nat @ A )
      = zero_zero_int ) ).

% take_bit_0
thf(fact_6124_take__bit__0,axiom,
    ! [A: nat] :
      ( ( bit_se2925701944663578781it_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% take_bit_0
thf(fact_6125_take__bit__0,axiom,
    ! [A: code_integer] :
      ( ( bit_se1745604003318907178nteger @ zero_zero_nat @ A )
      = zero_z3403309356797280102nteger ) ).

% take_bit_0
thf(fact_6126_take__bit__0,axiom,
    ! [A: code_natural] :
      ( ( bit_se569199155075624693atural @ zero_zero_nat @ A )
      = zero_z2226904508553997617atural ) ).

% take_bit_0
thf(fact_6127_take__bit__Suc__1,axiom,
    ! [N: nat] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ one_one_int )
      = one_one_int ) ).

% take_bit_Suc_1
thf(fact_6128_take__bit__Suc__1,axiom,
    ! [N: nat] :
      ( ( bit_se2925701944663578781it_nat @ ( suc @ N ) @ one_one_nat )
      = one_one_nat ) ).

% take_bit_Suc_1
thf(fact_6129_take__bit__Suc__1,axiom,
    ! [N: nat] :
      ( ( bit_se1745604003318907178nteger @ ( suc @ N ) @ one_one_Code_integer )
      = one_one_Code_integer ) ).

% take_bit_Suc_1
thf(fact_6130_take__bit__Suc__1,axiom,
    ! [N: nat] :
      ( ( bit_se569199155075624693atural @ ( suc @ N ) @ one_one_Code_natural )
      = one_one_Code_natural ) ).

% take_bit_Suc_1
thf(fact_6131_bit_Oconj__one__right,axiom,
    ! [X: int] :
      ( ( bit_se725231765392027082nd_int @ X @ ( uminus_uminus_int @ one_one_int ) )
      = X ) ).

% bit.conj_one_right
thf(fact_6132_bit_Oconj__one__right,axiom,
    ! [X: code_integer] :
      ( ( bit_se3949692690581998587nteger @ X @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = X ) ).

% bit.conj_one_right
thf(fact_6133_and_Oright__neutral,axiom,
    ! [A: int] :
      ( ( bit_se725231765392027082nd_int @ A @ ( uminus_uminus_int @ one_one_int ) )
      = A ) ).

% and.right_neutral
thf(fact_6134_and_Oright__neutral,axiom,
    ! [A: code_integer] :
      ( ( bit_se3949692690581998587nteger @ A @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = A ) ).

% and.right_neutral
thf(fact_6135_and_Oleft__neutral,axiom,
    ! [A: int] :
      ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ one_one_int ) @ A )
      = A ) ).

% and.left_neutral
thf(fact_6136_and_Oleft__neutral,axiom,
    ! [A: code_integer] :
      ( ( bit_se3949692690581998587nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ A )
      = A ) ).

% and.left_neutral
thf(fact_6137_take__bit__numeral__1,axiom,
    ! [L: num] :
      ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ one_one_int )
      = one_one_int ) ).

% take_bit_numeral_1
thf(fact_6138_take__bit__numeral__1,axiom,
    ! [L: num] :
      ( ( bit_se2925701944663578781it_nat @ ( numeral_numeral_nat @ L ) @ one_one_nat )
      = one_one_nat ) ).

% take_bit_numeral_1
thf(fact_6139_take__bit__numeral__1,axiom,
    ! [L: num] :
      ( ( bit_se1745604003318907178nteger @ ( numeral_numeral_nat @ L ) @ one_one_Code_integer )
      = one_one_Code_integer ) ).

% take_bit_numeral_1
thf(fact_6140_take__bit__numeral__1,axiom,
    ! [L: num] :
      ( ( bit_se569199155075624693atural @ ( numeral_numeral_nat @ L ) @ one_one_Code_natural )
      = one_one_Code_natural ) ).

% take_bit_numeral_1
thf(fact_6141_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_6142_of__nat__nat__take__bit__eq,axiom,
    ! [N: nat,K: int] :
      ( ( semiri4939895301339042750nteger @ ( nat2 @ ( bit_se2923211474154528505it_int @ N @ K ) ) )
      = ( ring_18347121197199848620nteger @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).

% of_nat_nat_take_bit_eq
thf(fact_6143_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_6144_and__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se725231765392027082nd_int @ K @ L ) )
      = ( ( ord_less_eq_int @ zero_zero_int @ K )
        | ( ord_less_eq_int @ zero_zero_int @ L ) ) ) ).

% and_nonnegative_int_iff
thf(fact_6145_and__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_int @ ( bit_se725231765392027082nd_int @ K @ L ) @ zero_zero_int )
      = ( ( ord_less_int @ K @ zero_zero_int )
        & ( ord_less_int @ L @ zero_zero_int ) ) ) ).

% and_negative_int_iff
thf(fact_6146_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_6147_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_6148_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_6149_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_6150_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_6151_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_6152_semiring__norm_I77_J,axiom,
    ! [N: num] : ( ord_less_num @ one @ ( bit1 @ N ) ) ).

% semiring_norm(77)
thf(fact_6153_semiring__norm_I70_J,axiom,
    ! [M: num] :
      ~ ( ord_less_eq_num @ ( bit1 @ M ) @ one ) ).

% semiring_norm(70)
thf(fact_6154_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_6155_dbl__inc__simps_I5_J,axiom,
    ! [K: num] :
      ( ( neg_nu5831290666863070958nteger @ ( numera6620942414471956472nteger @ K ) )
      = ( numera6620942414471956472nteger @ ( bit1 @ K ) ) ) ).

% dbl_inc_simps(5)
thf(fact_6156_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_6157_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_6158_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_6159_take__bit__of__1__eq__0__iff,axiom,
    ! [N: nat] :
      ( ( ( bit_se1745604003318907178nteger @ N @ one_one_Code_integer )
        = zero_z3403309356797280102nteger )
      = ( N = zero_zero_nat ) ) ).

% take_bit_of_1_eq_0_iff
thf(fact_6160_take__bit__of__1__eq__0__iff,axiom,
    ! [N: nat] :
      ( ( ( bit_se569199155075624693atural @ N @ one_one_Code_natural )
        = zero_z2226904508553997617atural )
      = ( N = zero_zero_nat ) ) ).

% take_bit_of_1_eq_0_iff
thf(fact_6161_and__numerals_I8_J,axiom,
    ! [X: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ one_one_int )
      = one_one_int ) ).

% and_numerals(8)
thf(fact_6162_and__numerals_I8_J,axiom,
    ! [X: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ one_one_nat )
      = one_one_nat ) ).

% and_numerals(8)
thf(fact_6163_and__numerals_I8_J,axiom,
    ! [X: num] :
      ( ( bit_se3949692690581998587nteger @ ( numera6620942414471956472nteger @ ( bit1 @ X ) ) @ one_one_Code_integer )
      = one_one_Code_integer ) ).

% and_numerals(8)
thf(fact_6164_and__numerals_I8_J,axiom,
    ! [X: num] :
      ( ( bit_se2773287842338716102atural @ ( numera5444537566228673987atural @ ( bit1 @ X ) ) @ one_one_Code_natural )
      = one_one_Code_natural ) ).

% and_numerals(8)
thf(fact_6165_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_6166_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_6167_and__numerals_I2_J,axiom,
    ! [Y3: num] :
      ( ( bit_se3949692690581998587nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit1 @ Y3 ) ) )
      = one_one_Code_integer ) ).

% and_numerals(2)
thf(fact_6168_and__numerals_I2_J,axiom,
    ! [Y3: num] :
      ( ( bit_se2773287842338716102atural @ one_one_Code_natural @ ( numera5444537566228673987atural @ ( bit1 @ Y3 ) ) )
      = one_one_Code_natural ) ).

% and_numerals(2)
thf(fact_6169_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_6170_semiring__norm_I8_J,axiom,
    ! [M: num] :
      ( ( plus_plus_num @ ( bit1 @ M ) @ one )
      = ( bit0 @ ( plus_plus_num @ M @ one ) ) ) ).

% semiring_norm(8)
thf(fact_6171_semiring__norm_I5_J,axiom,
    ! [M: num] :
      ( ( plus_plus_num @ ( bit0 @ M ) @ one )
      = ( bit1 @ M ) ) ).

% semiring_norm(5)
thf(fact_6172_semiring__norm_I4_J,axiom,
    ! [N: num] :
      ( ( plus_plus_num @ one @ ( bit1 @ N ) )
      = ( bit0 @ ( plus_plus_num @ N @ one ) ) ) ).

% semiring_norm(4)
thf(fact_6173_semiring__norm_I3_J,axiom,
    ! [N: num] :
      ( ( plus_plus_num @ one @ ( bit0 @ N ) )
      = ( bit1 @ N ) ) ).

% semiring_norm(3)
thf(fact_6174_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_6175_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_6176_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_6177_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_6178_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_6179_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_6180_and__numerals_I1_J,axiom,
    ! [Y3: num] :
      ( ( bit_se3949692690581998587nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ Y3 ) ) )
      = zero_z3403309356797280102nteger ) ).

% and_numerals(1)
thf(fact_6181_and__numerals_I1_J,axiom,
    ! [Y3: num] :
      ( ( bit_se2773287842338716102atural @ one_one_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ Y3 ) ) )
      = zero_z2226904508553997617atural ) ).

% and_numerals(1)
thf(fact_6182_and__numerals_I5_J,axiom,
    ! [X: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ one_one_int )
      = zero_zero_int ) ).

% and_numerals(5)
thf(fact_6183_and__numerals_I5_J,axiom,
    ! [X: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ one_one_nat )
      = zero_zero_nat ) ).

% and_numerals(5)
thf(fact_6184_and__numerals_I5_J,axiom,
    ! [X: num] :
      ( ( bit_se3949692690581998587nteger @ ( numera6620942414471956472nteger @ ( bit0 @ X ) ) @ one_one_Code_integer )
      = zero_z3403309356797280102nteger ) ).

% and_numerals(5)
thf(fact_6185_and__numerals_I5_J,axiom,
    ! [X: num] :
      ( ( bit_se2773287842338716102atural @ ( numera5444537566228673987atural @ ( bit0 @ X ) ) @ one_one_Code_natural )
      = zero_z2226904508553997617atural ) ).

% and_numerals(5)
thf(fact_6186_and__numerals_I3_J,axiom,
    ! [X: num,Y3: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ ( numeral_numeral_int @ ( bit0 @ Y3 ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y3 ) ) ) ) ).

% and_numerals(3)
thf(fact_6187_and__numerals_I3_J,axiom,
    ! [X: num,Y3: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) )
      = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y3 ) ) ) ) ).

% and_numerals(3)
thf(fact_6188_and__numerals_I3_J,axiom,
    ! [X: num,Y3: num] :
      ( ( bit_se3949692690581998587nteger @ ( numera6620942414471956472nteger @ ( bit0 @ X ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Y3 ) ) )
      = ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se3949692690581998587nteger @ ( numera6620942414471956472nteger @ X ) @ ( numera6620942414471956472nteger @ Y3 ) ) ) ) ).

% and_numerals(3)
thf(fact_6189_and__numerals_I3_J,axiom,
    ! [X: num,Y3: num] :
      ( ( bit_se2773287842338716102atural @ ( numera5444537566228673987atural @ ( bit0 @ X ) ) @ ( numera5444537566228673987atural @ ( bit0 @ Y3 ) ) )
      = ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( bit_se2773287842338716102atural @ ( numera5444537566228673987atural @ X ) @ ( numera5444537566228673987atural @ Y3 ) ) ) ) ).

% and_numerals(3)
thf(fact_6190_xor__numerals_I3_J,axiom,
    ! [X: num,Y3: num] :
      ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) )
      = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y3 ) ) ) ) ).

% xor_numerals(3)
thf(fact_6191_xor__numerals_I3_J,axiom,
    ! [X: num,Y3: num] :
      ( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ ( numeral_numeral_int @ ( bit0 @ Y3 ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y3 ) ) ) ) ).

% xor_numerals(3)
thf(fact_6192_xor__numerals_I3_J,axiom,
    ! [X: num,Y3: num] :
      ( ( bit_se3222712562003087583nteger @ ( numera6620942414471956472nteger @ ( bit0 @ X ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Y3 ) ) )
      = ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se3222712562003087583nteger @ ( numera6620942414471956472nteger @ X ) @ ( numera6620942414471956472nteger @ Y3 ) ) ) ) ).

% xor_numerals(3)
thf(fact_6193_xor__numerals_I3_J,axiom,
    ! [X: num,Y3: num] :
      ( ( bit_se2046307713759805098atural @ ( numera5444537566228673987atural @ ( bit0 @ X ) ) @ ( numera5444537566228673987atural @ ( bit0 @ Y3 ) ) )
      = ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( bit_se2046307713759805098atural @ ( numera5444537566228673987atural @ X ) @ ( numera5444537566228673987atural @ Y3 ) ) ) ) ).

% xor_numerals(3)
thf(fact_6194_xor__numerals_I8_J,axiom,
    ! [X: num] :
      ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ one_one_nat )
      = ( numeral_numeral_nat @ ( bit0 @ X ) ) ) ).

% xor_numerals(8)
thf(fact_6195_xor__numerals_I8_J,axiom,
    ! [X: num] :
      ( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ one_one_int )
      = ( numeral_numeral_int @ ( bit0 @ X ) ) ) ).

% xor_numerals(8)
thf(fact_6196_xor__numerals_I8_J,axiom,
    ! [X: num] :
      ( ( bit_se3222712562003087583nteger @ ( numera6620942414471956472nteger @ ( bit1 @ X ) ) @ one_one_Code_integer )
      = ( numera6620942414471956472nteger @ ( bit0 @ X ) ) ) ).

% xor_numerals(8)
thf(fact_6197_xor__numerals_I8_J,axiom,
    ! [X: num] :
      ( ( bit_se2046307713759805098atural @ ( numera5444537566228673987atural @ ( bit1 @ X ) ) @ one_one_Code_natural )
      = ( numera5444537566228673987atural @ ( bit0 @ X ) ) ) ).

% xor_numerals(8)
thf(fact_6198_xor__numerals_I5_J,axiom,
    ! [X: num] :
      ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ one_one_nat )
      = ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).

% xor_numerals(5)
thf(fact_6199_xor__numerals_I5_J,axiom,
    ! [X: num] :
      ( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ one_one_int )
      = ( numeral_numeral_int @ ( bit1 @ X ) ) ) ).

% xor_numerals(5)
thf(fact_6200_xor__numerals_I5_J,axiom,
    ! [X: num] :
      ( ( bit_se3222712562003087583nteger @ ( numera6620942414471956472nteger @ ( bit0 @ X ) ) @ one_one_Code_integer )
      = ( numera6620942414471956472nteger @ ( bit1 @ X ) ) ) ).

% xor_numerals(5)
thf(fact_6201_xor__numerals_I5_J,axiom,
    ! [X: num] :
      ( ( bit_se2046307713759805098atural @ ( numera5444537566228673987atural @ ( bit0 @ X ) ) @ one_one_Code_natural )
      = ( numera5444537566228673987atural @ ( bit1 @ X ) ) ) ).

% xor_numerals(5)
thf(fact_6202_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_6203_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_6204_xor__numerals_I2_J,axiom,
    ! [Y3: num] :
      ( ( bit_se3222712562003087583nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit1 @ Y3 ) ) )
      = ( numera6620942414471956472nteger @ ( bit0 @ Y3 ) ) ) ).

% xor_numerals(2)
thf(fact_6205_xor__numerals_I2_J,axiom,
    ! [Y3: num] :
      ( ( bit_se2046307713759805098atural @ one_one_Code_natural @ ( numera5444537566228673987atural @ ( bit1 @ Y3 ) ) )
      = ( numera5444537566228673987atural @ ( bit0 @ Y3 ) ) ) ).

% xor_numerals(2)
thf(fact_6206_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_6207_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_6208_xor__numerals_I1_J,axiom,
    ! [Y3: num] :
      ( ( bit_se3222712562003087583nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ Y3 ) ) )
      = ( numera6620942414471956472nteger @ ( bit1 @ Y3 ) ) ) ).

% xor_numerals(1)
thf(fact_6209_xor__numerals_I1_J,axiom,
    ! [Y3: num] :
      ( ( bit_se2046307713759805098atural @ one_one_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ Y3 ) ) )
      = ( numera5444537566228673987atural @ ( bit1 @ Y3 ) ) ) ).

% xor_numerals(1)
thf(fact_6210_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_6211_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_6212_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_6213_take__bit__of__1,axiom,
    ! [N: nat] :
      ( ( bit_se569199155075624693atural @ N @ one_one_Code_natural )
      = ( zero_n8403883297036319079atural @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% take_bit_of_1
thf(fact_6214_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_6215_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_6216_dbl__inc__simps_I3_J,axiom,
    ( ( neg_nu5851722552734809277nc_int @ one_one_int )
    = ( numeral_numeral_int @ ( bit1 @ one ) ) ) ).

% dbl_inc_simps(3)
thf(fact_6217_dbl__inc__simps_I3_J,axiom,
    ( ( neg_nu5831290666863070958nteger @ one_one_Code_integer )
    = ( numera6620942414471956472nteger @ ( bit1 @ one ) ) ) ).

% dbl_inc_simps(3)
thf(fact_6218_dbl__inc__simps_I3_J,axiom,
    ( ( neg_nu5219082963157363817nc_rat @ one_one_rat )
    = ( numeral_numeral_rat @ ( bit1 @ one ) ) ) ).

% dbl_inc_simps(3)
thf(fact_6219_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_6220_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_6221_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_6222_even__take__bit__eq,axiom,
    ! [N: nat,A: code_natural] :
      ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( bit_se569199155075624693atural @ N @ A ) )
      = ( ( N = zero_zero_nat )
        | ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A ) ) ) ).

% even_take_bit_eq
thf(fact_6223_and__numerals_I4_J,axiom,
    ! [X: num,Y3: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ ( numeral_numeral_int @ ( bit1 @ Y3 ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y3 ) ) ) ) ).

% and_numerals(4)
thf(fact_6224_and__numerals_I4_J,axiom,
    ! [X: num,Y3: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) )
      = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y3 ) ) ) ) ).

% and_numerals(4)
thf(fact_6225_and__numerals_I4_J,axiom,
    ! [X: num,Y3: num] :
      ( ( bit_se3949692690581998587nteger @ ( numera6620942414471956472nteger @ ( bit0 @ X ) ) @ ( numera6620942414471956472nteger @ ( bit1 @ Y3 ) ) )
      = ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se3949692690581998587nteger @ ( numera6620942414471956472nteger @ X ) @ ( numera6620942414471956472nteger @ Y3 ) ) ) ) ).

% and_numerals(4)
thf(fact_6226_and__numerals_I4_J,axiom,
    ! [X: num,Y3: num] :
      ( ( bit_se2773287842338716102atural @ ( numera5444537566228673987atural @ ( bit0 @ X ) ) @ ( numera5444537566228673987atural @ ( bit1 @ Y3 ) ) )
      = ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( bit_se2773287842338716102atural @ ( numera5444537566228673987atural @ X ) @ ( numera5444537566228673987atural @ Y3 ) ) ) ) ).

% and_numerals(4)
thf(fact_6227_and__numerals_I6_J,axiom,
    ! [X: num,Y3: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ ( numeral_numeral_int @ ( bit0 @ Y3 ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y3 ) ) ) ) ).

% and_numerals(6)
thf(fact_6228_and__numerals_I6_J,axiom,
    ! [X: num,Y3: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) )
      = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y3 ) ) ) ) ).

% and_numerals(6)
thf(fact_6229_and__numerals_I6_J,axiom,
    ! [X: num,Y3: num] :
      ( ( bit_se3949692690581998587nteger @ ( numera6620942414471956472nteger @ ( bit1 @ X ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Y3 ) ) )
      = ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se3949692690581998587nteger @ ( numera6620942414471956472nteger @ X ) @ ( numera6620942414471956472nteger @ Y3 ) ) ) ) ).

% and_numerals(6)
thf(fact_6230_and__numerals_I6_J,axiom,
    ! [X: num,Y3: num] :
      ( ( bit_se2773287842338716102atural @ ( numera5444537566228673987atural @ ( bit1 @ X ) ) @ ( numera5444537566228673987atural @ ( bit0 @ Y3 ) ) )
      = ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( bit_se2773287842338716102atural @ ( numera5444537566228673987atural @ X ) @ ( numera5444537566228673987atural @ Y3 ) ) ) ) ).

% and_numerals(6)
thf(fact_6231_xor__numerals_I7_J,axiom,
    ! [X: num,Y3: num] :
      ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) )
      = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y3 ) ) ) ) ).

% xor_numerals(7)
thf(fact_6232_xor__numerals_I7_J,axiom,
    ! [X: num,Y3: num] :
      ( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ ( numeral_numeral_int @ ( bit1 @ Y3 ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y3 ) ) ) ) ).

% xor_numerals(7)
thf(fact_6233_xor__numerals_I7_J,axiom,
    ! [X: num,Y3: num] :
      ( ( bit_se3222712562003087583nteger @ ( numera6620942414471956472nteger @ ( bit1 @ X ) ) @ ( numera6620942414471956472nteger @ ( bit1 @ Y3 ) ) )
      = ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se3222712562003087583nteger @ ( numera6620942414471956472nteger @ X ) @ ( numera6620942414471956472nteger @ Y3 ) ) ) ) ).

% xor_numerals(7)
thf(fact_6234_xor__numerals_I7_J,axiom,
    ! [X: num,Y3: num] :
      ( ( bit_se2046307713759805098atural @ ( numera5444537566228673987atural @ ( bit1 @ X ) ) @ ( numera5444537566228673987atural @ ( bit1 @ Y3 ) ) )
      = ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( bit_se2046307713759805098atural @ ( numera5444537566228673987atural @ X ) @ ( numera5444537566228673987atural @ Y3 ) ) ) ) ).

% xor_numerals(7)
thf(fact_6235_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_6236_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_6237_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_6238_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_6239_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_6240_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_6241_xor__nat__numerals_I4_J,axiom,
    ! [X: num] :
      ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( suc @ zero_zero_nat ) )
      = ( numeral_numeral_nat @ ( bit0 @ X ) ) ) ).

% xor_nat_numerals(4)
thf(fact_6242_xor__nat__numerals_I3_J,axiom,
    ! [X: num] :
      ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( suc @ zero_zero_nat ) )
      = ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).

% xor_nat_numerals(3)
thf(fact_6243_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_6244_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_6245_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_6246_dbl__dec__simps_I4_J,axiom,
    ( ( neg_nu7757733837767384882nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit1 @ one ) ) ) ) ).

% dbl_dec_simps(4)
thf(fact_6247_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_6248_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_6249_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_6250_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_6251_take__bit__Suc__0,axiom,
    ! [A: code_natural] :
      ( ( bit_se569199155075624693atural @ ( suc @ zero_zero_nat ) @ A )
      = ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ).

% take_bit_Suc_0
thf(fact_6252_and__numerals_I7_J,axiom,
    ! [X: num,Y3: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ ( 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 @ X ) @ ( numeral_numeral_int @ Y3 ) ) ) ) ) ).

% and_numerals(7)
thf(fact_6253_and__numerals_I7_J,axiom,
    ! [X: num,Y3: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( 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 @ X ) @ ( numeral_numeral_nat @ Y3 ) ) ) ) ) ).

% and_numerals(7)
thf(fact_6254_and__numerals_I7_J,axiom,
    ! [X: num,Y3: num] :
      ( ( bit_se3949692690581998587nteger @ ( numera6620942414471956472nteger @ ( bit1 @ X ) ) @ ( numera6620942414471956472nteger @ ( bit1 @ Y3 ) ) )
      = ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se3949692690581998587nteger @ ( numera6620942414471956472nteger @ X ) @ ( numera6620942414471956472nteger @ Y3 ) ) ) ) ) ).

% and_numerals(7)
thf(fact_6255_and__numerals_I7_J,axiom,
    ! [X: num,Y3: num] :
      ( ( bit_se2773287842338716102atural @ ( numera5444537566228673987atural @ ( bit1 @ X ) ) @ ( numera5444537566228673987atural @ ( bit1 @ Y3 ) ) )
      = ( plus_p4538020629002901425atural @ one_one_Code_natural @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( bit_se2773287842338716102atural @ ( numera5444537566228673987atural @ X ) @ ( numera5444537566228673987atural @ Y3 ) ) ) ) ) ).

% and_numerals(7)
thf(fact_6256_xor__numerals_I6_J,axiom,
    ! [X: num,Y3: num] :
      ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( 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 @ X ) @ ( numeral_numeral_nat @ Y3 ) ) ) ) ) ).

% xor_numerals(6)
thf(fact_6257_xor__numerals_I6_J,axiom,
    ! [X: num,Y3: num] :
      ( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ ( 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 @ X ) @ ( numeral_numeral_int @ Y3 ) ) ) ) ) ).

% xor_numerals(6)
thf(fact_6258_xor__numerals_I6_J,axiom,
    ! [X: num,Y3: num] :
      ( ( bit_se3222712562003087583nteger @ ( numera6620942414471956472nteger @ ( bit1 @ X ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Y3 ) ) )
      = ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se3222712562003087583nteger @ ( numera6620942414471956472nteger @ X ) @ ( numera6620942414471956472nteger @ Y3 ) ) ) ) ) ).

% xor_numerals(6)
thf(fact_6259_xor__numerals_I6_J,axiom,
    ! [X: num,Y3: num] :
      ( ( bit_se2046307713759805098atural @ ( numera5444537566228673987atural @ ( bit1 @ X ) ) @ ( numera5444537566228673987atural @ ( bit0 @ Y3 ) ) )
      = ( plus_p4538020629002901425atural @ one_one_Code_natural @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( bit_se2046307713759805098atural @ ( numera5444537566228673987atural @ X ) @ ( numera5444537566228673987atural @ Y3 ) ) ) ) ) ).

% xor_numerals(6)
thf(fact_6260_xor__numerals_I4_J,axiom,
    ! [X: num,Y3: num] :
      ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( 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 @ X ) @ ( numeral_numeral_nat @ Y3 ) ) ) ) ) ).

% xor_numerals(4)
thf(fact_6261_xor__numerals_I4_J,axiom,
    ! [X: num,Y3: num] :
      ( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ ( 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 @ X ) @ ( numeral_numeral_int @ Y3 ) ) ) ) ) ).

% xor_numerals(4)
thf(fact_6262_xor__numerals_I4_J,axiom,
    ! [X: num,Y3: num] :
      ( ( bit_se3222712562003087583nteger @ ( numera6620942414471956472nteger @ ( bit0 @ X ) ) @ ( numera6620942414471956472nteger @ ( bit1 @ Y3 ) ) )
      = ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se3222712562003087583nteger @ ( numera6620942414471956472nteger @ X ) @ ( numera6620942414471956472nteger @ Y3 ) ) ) ) ) ).

% xor_numerals(4)
thf(fact_6263_xor__numerals_I4_J,axiom,
    ! [X: num,Y3: num] :
      ( ( bit_se2046307713759805098atural @ ( numera5444537566228673987atural @ ( bit0 @ X ) ) @ ( numera5444537566228673987atural @ ( bit1 @ Y3 ) ) )
      = ( plus_p4538020629002901425atural @ one_one_Code_natural @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( bit_se2046307713759805098atural @ ( numera5444537566228673987atural @ X ) @ ( numera5444537566228673987atural @ Y3 ) ) ) ) ) ).

% xor_numerals(4)
thf(fact_6264_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_6265_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_6266_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_6267_take__bit__of__exp,axiom,
    ! [M: nat,N: nat] :
      ( ( bit_se569199155075624693atural @ M @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) )
      = ( times_2397367101498566445atural @ ( zero_n8403883297036319079atural @ ( ord_less_nat @ N @ M ) ) @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) ) ) ).

% take_bit_of_exp
thf(fact_6268_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_6269_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_6270_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_6271_take__bit__of__2,axiom,
    ! [N: nat] :
      ( ( bit_se569199155075624693atural @ N @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
      = ( times_2397367101498566445atural @ ( zero_n8403883297036319079atural @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ).

% take_bit_of_2
thf(fact_6272_zmod__numeral__Bit1,axiom,
    ! [V: num,W2: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W2 ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W2 ) ) ) @ one_one_int ) ) ).

% zmod_numeral_Bit1
thf(fact_6273_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_6274_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_6275_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_6276_take__bit__of__nat,axiom,
    ! [N: nat,M: nat] :
      ( ( bit_se1745604003318907178nteger @ N @ ( semiri4939895301339042750nteger @ M ) )
      = ( semiri4939895301339042750nteger @ ( bit_se2925701944663578781it_nat @ N @ M ) ) ) ).

% take_bit_of_nat
thf(fact_6277_take__bit__of__nat,axiom,
    ! [N: nat,M: nat] :
      ( ( bit_se569199155075624693atural @ N @ ( semiri3763490453095760265atural @ M ) )
      = ( semiri3763490453095760265atural @ ( bit_se2925701944663578781it_nat @ N @ M ) ) ) ).

% take_bit_of_nat
thf(fact_6278_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_6279_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_6280_of__nat__xor__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri4939895301339042750nteger @ ( bit_se6528837805403552850or_nat @ M @ N ) )
      = ( bit_se3222712562003087583nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) ) ) ).

% of_nat_xor_eq
thf(fact_6281_of__nat__xor__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri3763490453095760265atural @ ( bit_se6528837805403552850or_nat @ M @ N ) )
      = ( bit_se2046307713759805098atural @ ( semiri3763490453095760265atural @ M ) @ ( semiri3763490453095760265atural @ N ) ) ) ).

% of_nat_xor_eq
thf(fact_6282_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_6283_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_6284_of__nat__and__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri4939895301339042750nteger @ ( bit_se727722235901077358nd_nat @ M @ N ) )
      = ( bit_se3949692690581998587nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) ) ) ).

% of_nat_and_eq
thf(fact_6285_of__nat__and__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri3763490453095760265atural @ ( bit_se727722235901077358nd_nat @ M @ N ) )
      = ( bit_se2773287842338716102atural @ ( semiri3763490453095760265atural @ M ) @ ( semiri3763490453095760265atural @ N ) ) ) ).

% of_nat_and_eq
thf(fact_6286_take__bit__tightened,axiom,
    ! [N: nat,A: int,B: int,M: nat] :
      ( ( ( bit_se2923211474154528505it_int @ N @ A )
        = ( bit_se2923211474154528505it_int @ N @ B ) )
     => ( ( ord_less_eq_nat @ M @ N )
       => ( ( bit_se2923211474154528505it_int @ M @ A )
          = ( bit_se2923211474154528505it_int @ M @ B ) ) ) ) ).

% take_bit_tightened
thf(fact_6287_take__bit__tightened,axiom,
    ! [N: nat,A: nat,B: nat,M: nat] :
      ( ( ( bit_se2925701944663578781it_nat @ N @ A )
        = ( bit_se2925701944663578781it_nat @ N @ B ) )
     => ( ( ord_less_eq_nat @ M @ N )
       => ( ( bit_se2925701944663578781it_nat @ M @ A )
          = ( bit_se2925701944663578781it_nat @ M @ B ) ) ) ) ).

% take_bit_tightened
thf(fact_6288_take__bit__tightened,axiom,
    ! [N: nat,A: code_integer,B: code_integer,M: nat] :
      ( ( ( bit_se1745604003318907178nteger @ N @ A )
        = ( bit_se1745604003318907178nteger @ N @ B ) )
     => ( ( ord_less_eq_nat @ M @ N )
       => ( ( bit_se1745604003318907178nteger @ M @ A )
          = ( bit_se1745604003318907178nteger @ M @ B ) ) ) ) ).

% take_bit_tightened
thf(fact_6289_take__bit__tightened,axiom,
    ! [N: nat,A: code_natural,B: code_natural,M: nat] :
      ( ( ( bit_se569199155075624693atural @ N @ A )
        = ( bit_se569199155075624693atural @ N @ B ) )
     => ( ( ord_less_eq_nat @ M @ N )
       => ( ( bit_se569199155075624693atural @ M @ A )
          = ( bit_se569199155075624693atural @ M @ B ) ) ) ) ).

% take_bit_tightened
thf(fact_6290_take__bit__tightened__less__eq__nat,axiom,
    ! [M: nat,N: nat,Q4: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_nat @ ( bit_se2925701944663578781it_nat @ M @ Q4 ) @ ( bit_se2925701944663578781it_nat @ N @ Q4 ) ) ) ).

% take_bit_tightened_less_eq_nat
thf(fact_6291_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_6292_verit__eq__simplify_I14_J,axiom,
    ! [X22: num,X32: num] :
      ( ( bit0 @ X22 )
     != ( bit1 @ X32 ) ) ).

% verit_eq_simplify(14)
thf(fact_6293_verit__eq__simplify_I12_J,axiom,
    ! [X32: num] :
      ( one
     != ( bit1 @ X32 ) ) ).

% verit_eq_simplify(12)
thf(fact_6294_take__bit__mult,axiom,
    ! [N: nat,K: int,L: int] :
      ( ( bit_se2923211474154528505it_int @ N @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ ( bit_se2923211474154528505it_int @ N @ L ) ) )
      = ( bit_se2923211474154528505it_int @ N @ ( times_times_int @ K @ L ) ) ) ).

% take_bit_mult
thf(fact_6295_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_6296_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_6297_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_6298_and__eq__minus__1__iff,axiom,
    ! [A: int,B: int] :
      ( ( ( bit_se725231765392027082nd_int @ A @ B )
        = ( uminus_uminus_int @ one_one_int ) )
      = ( ( A
          = ( uminus_uminus_int @ one_one_int ) )
        & ( B
          = ( uminus_uminus_int @ one_one_int ) ) ) ) ).

% and_eq_minus_1_iff
thf(fact_6299_and__eq__minus__1__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( bit_se3949692690581998587nteger @ A @ B )
        = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( ( A
          = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
        & ( B
          = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).

% and_eq_minus_1_iff
thf(fact_6300_num_Oexhaust,axiom,
    ! [Y3: num] :
      ( ( Y3 != one )
     => ( ! [X23: num] :
            ( Y3
           != ( bit0 @ X23 ) )
       => ~ ! [X33: num] :
              ( Y3
             != ( bit1 @ X33 ) ) ) ) ).

% num.exhaust
thf(fact_6301_AND__upper2_H,axiom,
    ! [Y3: int,Z2: int,X: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
     => ( ( ord_less_eq_int @ Y3 @ Z2 )
       => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X @ Y3 ) @ Z2 ) ) ) ).

% AND_upper2'
thf(fact_6302_AND__upper1_H,axiom,
    ! [Y3: int,Z2: int,Ya: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
     => ( ( ord_less_eq_int @ Y3 @ Z2 )
       => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ Y3 @ Ya ) @ Z2 ) ) ) ).

% AND_upper1'
thf(fact_6303_AND__upper2,axiom,
    ! [Y3: int,X: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
     => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X @ Y3 ) @ Y3 ) ) ).

% AND_upper2
thf(fact_6304_AND__upper1,axiom,
    ! [X: int,Y3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X @ Y3 ) @ X ) ) ).

% AND_upper1
thf(fact_6305_AND__lower,axiom,
    ! [X: int,Y3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ord_less_eq_int @ zero_zero_int @ ( bit_se725231765392027082nd_int @ X @ Y3 ) ) ) ).

% AND_lower
thf(fact_6306_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_6307_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_6308_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_6309_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_6310_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_6311_signed__take__bit__eq__iff__take__bit__eq,axiom,
    ! [N: nat,A: code_integer,B: code_integer] :
      ( ( ( bit_ri6519982836138164636nteger @ N @ A )
        = ( bit_ri6519982836138164636nteger @ N @ B ) )
      = ( ( bit_se1745604003318907178nteger @ ( suc @ N ) @ A )
        = ( bit_se1745604003318907178nteger @ ( suc @ N ) @ B ) ) ) ).

% signed_take_bit_eq_iff_take_bit_eq
thf(fact_6312_signed__take__bit__eq__iff__take__bit__eq,axiom,
    ! [N: nat,A: int,B: int] :
      ( ( ( bit_ri631733984087533419it_int @ N @ A )
        = ( bit_ri631733984087533419it_int @ N @ B ) )
      = ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ A )
        = ( bit_se2923211474154528505it_int @ ( suc @ N ) @ B ) ) ) ).

% signed_take_bit_eq_iff_take_bit_eq
thf(fact_6313_signed__take__bit__take__bit,axiom,
    ! [M: nat,N: nat,A: code_integer] :
      ( ( bit_ri6519982836138164636nteger @ M @ ( bit_se1745604003318907178nteger @ N @ A ) )
      = ( if_Cod4779417660136461971nteger @ ( ord_less_eq_nat @ N @ M ) @ ( bit_se1745604003318907178nteger @ N ) @ ( bit_ri6519982836138164636nteger @ M ) @ A ) ) ).

% signed_take_bit_take_bit
thf(fact_6314_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_6315_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_6316_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_6317_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_6318_take__bit__unset__bit__eq,axiom,
    ! [N: nat,M: nat,A: code_integer] :
      ( ( ( ord_less_eq_nat @ N @ M )
       => ( ( bit_se1745604003318907178nteger @ N @ ( bit_se8260200283734997820nteger @ M @ A ) )
          = ( bit_se1745604003318907178nteger @ N @ A ) ) )
      & ( ~ ( ord_less_eq_nat @ N @ M )
       => ( ( bit_se1745604003318907178nteger @ N @ ( bit_se8260200283734997820nteger @ M @ A ) )
          = ( bit_se8260200283734997820nteger @ M @ ( bit_se1745604003318907178nteger @ N @ A ) ) ) ) ) ).

% take_bit_unset_bit_eq
thf(fact_6319_take__bit__unset__bit__eq,axiom,
    ! [N: nat,M: nat,A: code_natural] :
      ( ( ( ord_less_eq_nat @ N @ M )
       => ( ( bit_se569199155075624693atural @ N @ ( bit_se7083795435491715335atural @ M @ A ) )
          = ( bit_se569199155075624693atural @ N @ A ) ) )
      & ( ~ ( ord_less_eq_nat @ N @ M )
       => ( ( bit_se569199155075624693atural @ N @ ( bit_se7083795435491715335atural @ M @ A ) )
          = ( bit_se7083795435491715335atural @ M @ ( bit_se569199155075624693atural @ N @ A ) ) ) ) ) ).

% take_bit_unset_bit_eq
thf(fact_6320_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_6321_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_6322_take__bit__set__bit__eq,axiom,
    ! [N: nat,M: nat,A: code_integer] :
      ( ( ( ord_less_eq_nat @ N @ M )
       => ( ( bit_se1745604003318907178nteger @ N @ ( bit_se2793503036327961859nteger @ M @ A ) )
          = ( bit_se1745604003318907178nteger @ N @ A ) ) )
      & ( ~ ( ord_less_eq_nat @ N @ M )
       => ( ( bit_se1745604003318907178nteger @ N @ ( bit_se2793503036327961859nteger @ M @ A ) )
          = ( bit_se2793503036327961859nteger @ M @ ( bit_se1745604003318907178nteger @ N @ A ) ) ) ) ) ).

% take_bit_set_bit_eq
thf(fact_6323_take__bit__set__bit__eq,axiom,
    ! [N: nat,M: nat,A: code_natural] :
      ( ( ( ord_less_eq_nat @ N @ M )
       => ( ( bit_se569199155075624693atural @ N @ ( bit_se1617098188084679374atural @ M @ A ) )
          = ( bit_se569199155075624693atural @ N @ A ) ) )
      & ( ~ ( ord_less_eq_nat @ N @ M )
       => ( ( bit_se569199155075624693atural @ N @ ( bit_se1617098188084679374atural @ M @ A ) )
          = ( bit_se1617098188084679374atural @ M @ ( bit_se569199155075624693atural @ N @ A ) ) ) ) ) ).

% take_bit_set_bit_eq
thf(fact_6324_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_6325_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_6326_take__bit__flip__bit__eq,axiom,
    ! [N: nat,M: nat,A: code_integer] :
      ( ( ( ord_less_eq_nat @ N @ M )
       => ( ( bit_se1745604003318907178nteger @ N @ ( bit_se1345352211410354436nteger @ M @ A ) )
          = ( bit_se1745604003318907178nteger @ N @ A ) ) )
      & ( ~ ( ord_less_eq_nat @ N @ M )
       => ( ( bit_se1745604003318907178nteger @ N @ ( bit_se1345352211410354436nteger @ M @ A ) )
          = ( bit_se1345352211410354436nteger @ M @ ( bit_se1745604003318907178nteger @ N @ A ) ) ) ) ) ).

% take_bit_flip_bit_eq
thf(fact_6327_take__bit__flip__bit__eq,axiom,
    ! [N: nat,M: nat,A: code_natural] :
      ( ( ( ord_less_eq_nat @ N @ M )
       => ( ( bit_se569199155075624693atural @ N @ ( bit_se168947363167071951atural @ M @ A ) )
          = ( bit_se569199155075624693atural @ N @ A ) ) )
      & ( ~ ( ord_less_eq_nat @ N @ M )
       => ( ( bit_se569199155075624693atural @ N @ ( bit_se168947363167071951atural @ M @ A ) )
          = ( bit_se168947363167071951atural @ M @ ( bit_se569199155075624693atural @ N @ A ) ) ) ) ) ).

% take_bit_flip_bit_eq
thf(fact_6328_inc_Osimps_I3_J,axiom,
    ! [X: num] :
      ( ( inc @ ( bit1 @ X ) )
      = ( bit0 @ ( inc @ X ) ) ) ).

% inc.simps(3)
thf(fact_6329_inc_Osimps_I2_J,axiom,
    ! [X: num] :
      ( ( inc @ ( bit0 @ X ) )
      = ( bit1 @ X ) ) ).

% inc.simps(2)
thf(fact_6330_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_6331_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_6332_numeral__Bit1,axiom,
    ! [N: num] :
      ( ( numera6620942414471956472nteger @ ( bit1 @ N ) )
      = ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ N ) ) @ one_one_Code_integer ) ) ).

% numeral_Bit1
thf(fact_6333_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_6334_numeral__Bit1,axiom,
    ! [N: num] :
      ( ( numera5444537566228673987atural @ ( bit1 @ N ) )
      = ( plus_p4538020629002901425atural @ ( plus_p4538020629002901425atural @ ( numera5444537566228673987atural @ N ) @ ( numera5444537566228673987atural @ N ) ) @ one_one_Code_natural ) ) ).

% numeral_Bit1
thf(fact_6335_take__bit__signed__take__bit,axiom,
    ! [M: nat,N: nat,A: code_integer] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
     => ( ( bit_se1745604003318907178nteger @ M @ ( bit_ri6519982836138164636nteger @ N @ A ) )
        = ( bit_se1745604003318907178nteger @ M @ A ) ) ) ).

% take_bit_signed_take_bit
thf(fact_6336_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_6337_and__less__eq,axiom,
    ! [L: int,K: int] :
      ( ( ord_less_int @ L @ zero_zero_int )
     => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ K @ L ) @ K ) ) ).

% and_less_eq
thf(fact_6338_AND__upper1_H_H,axiom,
    ! [Y3: int,Z2: int,Ya: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
     => ( ( ord_less_int @ Y3 @ Z2 )
       => ( ord_less_int @ ( bit_se725231765392027082nd_int @ Y3 @ Ya ) @ Z2 ) ) ) ).

% AND_upper1''
thf(fact_6339_AND__upper2_H_H,axiom,
    ! [Y3: int,Z2: int,X: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
     => ( ( ord_less_int @ Y3 @ Z2 )
       => ( ord_less_int @ ( bit_se725231765392027082nd_int @ X @ Y3 ) @ Z2 ) ) ) ).

% AND_upper2''
thf(fact_6340_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_6341_power__minus__Bit1,axiom,
    ! [X: int,K: num] :
      ( ( power_power_int @ ( uminus_uminus_int @ X ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
      = ( uminus_uminus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).

% power_minus_Bit1
thf(fact_6342_power__minus__Bit1,axiom,
    ! [X: code_integer,K: num] :
      ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ X ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
      = ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).

% power_minus_Bit1
thf(fact_6343_power__minus__Bit1,axiom,
    ! [X: rat,K: num] :
      ( ( power_power_rat @ ( uminus_uminus_rat @ X ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
      = ( uminus_uminus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).

% power_minus_Bit1
thf(fact_6344_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_6345_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_6346_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_6347_take__bit__Suc__bit1,axiom,
    ! [N: nat,K: num] :
      ( ( bit_se1745604003318907178nteger @ ( suc @ N ) @ ( numera6620942414471956472nteger @ ( bit1 @ K ) ) )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( bit_se1745604003318907178nteger @ N @ ( numera6620942414471956472nteger @ K ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ one_one_Code_integer ) ) ).

% take_bit_Suc_bit1
thf(fact_6348_take__bit__Suc__bit1,axiom,
    ! [N: nat,K: num] :
      ( ( bit_se569199155075624693atural @ ( suc @ N ) @ ( numera5444537566228673987atural @ ( bit1 @ K ) ) )
      = ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ ( bit_se569199155075624693atural @ N @ ( numera5444537566228673987atural @ K ) ) @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) @ one_one_Code_natural ) ) ).

% take_bit_Suc_bit1
thf(fact_6349_cong__exp__iff__simps_I3_J,axiom,
    ! [N: num,Q4: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q4 ) ) )
     != zero_zero_nat ) ).

% cong_exp_iff_simps(3)
thf(fact_6350_cong__exp__iff__simps_I3_J,axiom,
    ! [N: num,Q4: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q4 ) ) )
     != zero_zero_int ) ).

% cong_exp_iff_simps(3)
thf(fact_6351_cong__exp__iff__simps_I3_J,axiom,
    ! [N: num,Q4: num] :
      ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q4 ) ) )
     != zero_z3403309356797280102nteger ) ).

% cong_exp_iff_simps(3)
thf(fact_6352_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_6353_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_6354_numeral__Bit1__div__2,axiom,
    ! [N: num] :
      ( ( divide6298287555418463151nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
      = ( numera6620942414471956472nteger @ N ) ) ).

% numeral_Bit1_div_2
thf(fact_6355_numeral__Bit1__div__2,axiom,
    ! [N: num] :
      ( ( divide5121882707175180666atural @ ( numera5444537566228673987atural @ ( bit1 @ N ) ) @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
      = ( numera5444537566228673987atural @ N ) ) ).

% numeral_Bit1_div_2
thf(fact_6356_power3__eq__cube,axiom,
    ! [A: code_integer] :
      ( ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
      = ( times_3573771949741848930nteger @ ( times_3573771949741848930nteger @ A @ A ) @ A ) ) ).

% power3_eq_cube
thf(fact_6357_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_6358_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_6359_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_6360_numeral__3__eq__3,axiom,
    ( ( numeral_numeral_nat @ ( bit1 @ one ) )
    = ( suc @ ( suc @ ( suc @ zero_zero_nat ) ) ) ) ).

% numeral_3_eq_3
thf(fact_6361_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_6362_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_6363_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_6364_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_6365_take__bit__Suc__bit0,axiom,
    ! [N: nat,K: num] :
      ( ( bit_se1745604003318907178nteger @ ( suc @ N ) @ ( numera6620942414471956472nteger @ ( bit0 @ K ) ) )
      = ( times_3573771949741848930nteger @ ( bit_se1745604003318907178nteger @ N @ ( numera6620942414471956472nteger @ K ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% take_bit_Suc_bit0
thf(fact_6366_take__bit__Suc__bit0,axiom,
    ! [N: nat,K: num] :
      ( ( bit_se569199155075624693atural @ ( suc @ N ) @ ( numera5444537566228673987atural @ ( bit0 @ K ) ) )
      = ( times_2397367101498566445atural @ ( bit_se569199155075624693atural @ N @ ( numera5444537566228673987atural @ K ) ) @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ).

% take_bit_Suc_bit0
thf(fact_6367_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_6368_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_6369_and__one__eq,axiom,
    ! [A: code_integer] :
      ( ( bit_se3949692690581998587nteger @ A @ one_one_Code_integer )
      = ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% and_one_eq
thf(fact_6370_and__one__eq,axiom,
    ! [A: code_natural] :
      ( ( bit_se2773287842338716102atural @ A @ one_one_Code_natural )
      = ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ).

% and_one_eq
thf(fact_6371_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_6372_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_6373_one__and__eq,axiom,
    ! [A: code_integer] :
      ( ( bit_se3949692690581998587nteger @ one_one_Code_integer @ A )
      = ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% one_and_eq
thf(fact_6374_one__and__eq,axiom,
    ! [A: code_natural] :
      ( ( bit_se2773287842338716102atural @ one_one_Code_natural @ A )
      = ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ).

% one_and_eq
thf(fact_6375_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_6376_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_6377_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_6378_cong__exp__iff__simps_I7_J,axiom,
    ! [Q4: num,N: num] :
      ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q4 ) ) )
        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q4 ) ) ) )
      = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q4 ) )
        = zero_zero_nat ) ) ).

% cong_exp_iff_simps(7)
thf(fact_6379_cong__exp__iff__simps_I7_J,axiom,
    ! [Q4: num,N: num] :
      ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q4 ) ) )
        = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q4 ) ) ) )
      = ( ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q4 ) )
        = zero_zero_int ) ) ).

% cong_exp_iff_simps(7)
thf(fact_6380_cong__exp__iff__simps_I7_J,axiom,
    ! [Q4: num,N: num] :
      ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q4 ) ) )
        = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q4 ) ) ) )
      = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q4 ) )
        = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(7)
thf(fact_6381_cong__exp__iff__simps_I11_J,axiom,
    ! [M: num,Q4: num] :
      ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q4 ) ) )
        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q4 ) ) ) )
      = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q4 ) )
        = zero_zero_nat ) ) ).

% cong_exp_iff_simps(11)
thf(fact_6382_cong__exp__iff__simps_I11_J,axiom,
    ! [M: num,Q4: num] :
      ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q4 ) ) )
        = ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q4 ) ) ) )
      = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q4 ) )
        = zero_zero_int ) ) ).

% cong_exp_iff_simps(11)
thf(fact_6383_cong__exp__iff__simps_I11_J,axiom,
    ! [M: num,Q4: num] :
      ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q4 ) ) )
        = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q4 ) ) ) )
      = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ Q4 ) )
        = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(11)
thf(fact_6384_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_6385_fold__atLeastAtMost__nat_Osimps,axiom,
    ( set_fo2584398358068434914at_nat
    = ( ^ [F2: nat > nat > nat,A2: nat,B2: nat,Acc: nat] : ( if_nat @ ( ord_less_nat @ B2 @ A2 ) @ Acc @ ( set_fo2584398358068434914at_nat @ F2 @ ( plus_plus_nat @ A2 @ one_one_nat ) @ B2 @ ( F2 @ A2 @ Acc ) ) ) ) ) ).

% fold_atLeastAtMost_nat.simps
thf(fact_6386_fold__atLeastAtMost__nat_Oelims,axiom,
    ! [X: nat > nat > nat,Xa3: nat,Xb3: nat,Xc: nat,Y3: nat] :
      ( ( ( set_fo2584398358068434914at_nat @ X @ Xa3 @ Xb3 @ Xc )
        = Y3 )
     => ( ( ( ord_less_nat @ Xb3 @ Xa3 )
         => ( Y3 = Xc ) )
        & ( ~ ( ord_less_nat @ Xb3 @ Xa3 )
         => ( Y3
            = ( set_fo2584398358068434914at_nat @ X @ ( plus_plus_nat @ Xa3 @ one_one_nat ) @ Xb3 @ ( X @ Xa3 @ Xc ) ) ) ) ) ) ).

% fold_atLeastAtMost_nat.elims
thf(fact_6387_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_6388_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_6389_num_Osize__gen_I1_J,axiom,
    ( ( size_num @ one )
    = zero_zero_nat ) ).

% num.size_gen(1)
thf(fact_6390_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_6391_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_6392_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_6393_take__bit__eq__0__iff,axiom,
    ! [N: nat,A: code_natural] :
      ( ( ( bit_se569199155075624693atural @ N @ A )
        = zero_z2226904508553997617atural )
      = ( dvd_dvd_Code_natural @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) @ A ) ) ).

% take_bit_eq_0_iff
thf(fact_6394_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_6395_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_6396_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_6397_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_6398_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_6399_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_6400_signed__take__bit__eq__take__bit__shift,axiom,
    ( bit_ri631733984087533419it_int
    = ( ^ [N3: nat,K3: int] : ( minus_minus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N3 ) @ ( plus_plus_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).

% signed_take_bit_eq_take_bit_shift
thf(fact_6401_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_6402_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_6403_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_6404_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_6405_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_6406_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_6407_take__bit__Suc,axiom,
    ! [N: nat,A: code_natural] :
      ( ( bit_se569199155075624693atural @ ( suc @ N ) @ A )
      = ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ ( bit_se569199155075624693atural @ N @ ( divide5121882707175180666atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) @ ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ) ).

% take_bit_Suc
thf(fact_6408_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_6409_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_6410_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_6411_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_6412_xor__nat__unfold,axiom,
    ( bit_se6528837805403552850or_nat
    = ( ^ [M2: nat,N3: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ N3 @ ( if_nat @ ( N3 = zero_zero_nat ) @ M2 @ ( plus_plus_nat @ ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( divide_divide_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% xor_nat_unfold
thf(fact_6413_and__int__rec,axiom,
    ( bit_se725231765392027082nd_int
    = ( ^ [K3: int,L3: int] :
          ( plus_plus_int
          @ ( zero_n2684676970156552555ol_int
            @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 )
              & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L3 ) ) )
          @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% and_int_rec
thf(fact_6414_xor__nat__rec,axiom,
    ( bit_se6528837805403552850or_nat
    = ( ^ [M2: nat,N3: nat] :
          ( plus_plus_nat
          @ ( zero_n2687167440665602831ol_nat
            @ ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
             != ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) )
          @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( divide_divide_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% xor_nat_rec
thf(fact_6415_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_6416_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_6417_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_6418_xor__one__eq,axiom,
    ! [A: code_natural] :
      ( ( bit_se2046307713759805098atural @ A @ one_one_Code_natural )
      = ( minus_7197305767214868737atural @ ( plus_p4538020629002901425atural @ A @ ( zero_n8403883297036319079atural @ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A ) ) )
        @ ( zero_n8403883297036319079atural
          @ ~ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A ) ) ) ) ).

% xor_one_eq
thf(fact_6419_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_6420_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_6421_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_6422_one__xor__eq,axiom,
    ! [A: code_natural] :
      ( ( bit_se2046307713759805098atural @ one_one_Code_natural @ A )
      = ( minus_7197305767214868737atural @ ( plus_p4538020629002901425atural @ A @ ( zero_n8403883297036319079atural @ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A ) ) )
        @ ( zero_n8403883297036319079atural
          @ ~ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A ) ) ) ) ).

% one_xor_eq
thf(fact_6423_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_6424_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_6425_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_6426_stable__imp__take__bit__eq,axiom,
    ! [A: code_natural,N: nat] :
      ( ( ( divide5121882707175180666atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
        = A )
     => ( ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A )
         => ( ( bit_se569199155075624693atural @ N @ A )
            = zero_z2226904508553997617atural ) )
        & ( ~ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A )
         => ( ( bit_se569199155075624693atural @ N @ A )
            = ( minus_7197305767214868737atural @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) @ one_one_Code_natural ) ) ) ) ) ).

% stable_imp_take_bit_eq
thf(fact_6427_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_6428_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_6429_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_6430_signed__take__bit__numeral__minus__bit1,axiom,
    ! [L: num,K: num] :
      ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L ) @ ( 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_6431_binomial__code,axiom,
    ( binomial
    = ( ^ [N3: nat,K3: nat] : ( if_nat @ ( ord_less_nat @ N3 @ K3 ) @ zero_zero_nat @ ( if_nat @ ( ord_less_nat @ N3 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K3 ) ) @ ( binomial @ N3 @ ( minus_minus_nat @ N3 @ K3 ) ) @ ( divide_divide_nat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( plus_plus_nat @ ( minus_minus_nat @ N3 @ K3 ) @ one_one_nat ) @ N3 @ one_one_nat ) @ ( semiri1408675320244567234ct_nat @ K3 ) ) ) ) ) ) ).

% binomial_code
thf(fact_6432_signed__take__bit__numeral__bit1,axiom,
    ! [L: num,K: num] :
      ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% signed_take_bit_numeral_bit1
thf(fact_6433_take__bit__numeral__minus__bit1,axiom,
    ! [L: num,K: num] :
      ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% take_bit_numeral_minus_bit1
thf(fact_6434_take__bit__numeral__bit1,axiom,
    ! [L: num,K: num] :
      ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% take_bit_numeral_bit1
thf(fact_6435_take__bit__numeral__bit1,axiom,
    ! [L: num,K: num] :
      ( ( bit_se2925701944663578781it_nat @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ ( pred_numeral @ L ) @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ).

% take_bit_numeral_bit1
thf(fact_6436_take__bit__numeral__bit1,axiom,
    ! [L: num,K: num] :
      ( ( bit_se1745604003318907178nteger @ ( numeral_numeral_nat @ L ) @ ( numera6620942414471956472nteger @ ( bit1 @ K ) ) )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( bit_se1745604003318907178nteger @ ( pred_numeral @ L ) @ ( numera6620942414471956472nteger @ K ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ one_one_Code_integer ) ) ).

% take_bit_numeral_bit1
thf(fact_6437_take__bit__numeral__bit1,axiom,
    ! [L: num,K: num] :
      ( ( bit_se569199155075624693atural @ ( numeral_numeral_nat @ L ) @ ( numera5444537566228673987atural @ ( bit1 @ K ) ) )
      = ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ ( bit_se569199155075624693atural @ ( pred_numeral @ L ) @ ( numera5444537566228673987atural @ K ) ) @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) @ one_one_Code_natural ) ) ).

% take_bit_numeral_bit1
thf(fact_6438_signed__take__bit__numeral__minus__bit0,axiom,
    ! [L: num,K: num] :
      ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
      = ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% signed_take_bit_numeral_minus_bit0
thf(fact_6439_xor__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se6526347334894502574or_int @ K @ L ) )
      = ( ( ord_less_eq_int @ zero_zero_int @ K )
        = ( ord_less_eq_int @ zero_zero_int @ L ) ) ) ).

% xor_nonnegative_int_iff
thf(fact_6440_xor__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_int @ ( bit_se6526347334894502574or_int @ K @ L ) @ zero_zero_int )
      = ( ( ord_less_int @ K @ zero_zero_int )
       != ( ord_less_int @ L @ zero_zero_int ) ) ) ).

% xor_negative_int_iff
thf(fact_6441_binomial__Suc__n,axiom,
    ! [N: nat] :
      ( ( binomial @ ( suc @ N ) @ N )
      = ( suc @ N ) ) ).

% binomial_Suc_n
thf(fact_6442_binomial__n__n,axiom,
    ! [N: nat] :
      ( ( binomial @ N @ N )
      = one_one_nat ) ).

% binomial_n_n
thf(fact_6443_binomial__1,axiom,
    ! [N: nat] :
      ( ( binomial @ N @ ( suc @ zero_zero_nat ) )
      = N ) ).

% binomial_1
thf(fact_6444_binomial__0__Suc,axiom,
    ! [K: nat] :
      ( ( binomial @ zero_zero_nat @ ( suc @ K ) )
      = zero_zero_nat ) ).

% binomial_0_Suc
thf(fact_6445_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_6446_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_6447_pred__numeral__simps_I1_J,axiom,
    ( ( pred_numeral @ one )
    = zero_zero_nat ) ).

% pred_numeral_simps(1)
thf(fact_6448_binomial__n__0,axiom,
    ! [N: nat] :
      ( ( binomial @ N @ zero_zero_nat )
      = one_one_nat ) ).

% binomial_n_0
thf(fact_6449_eq__numeral__Suc,axiom,
    ! [K: num,N: nat] :
      ( ( ( numeral_numeral_nat @ K )
        = ( suc @ N ) )
      = ( ( pred_numeral @ K )
        = N ) ) ).

% eq_numeral_Suc
thf(fact_6450_Suc__eq__numeral,axiom,
    ! [N: nat,K: num] :
      ( ( ( suc @ N )
        = ( numeral_numeral_nat @ K ) )
      = ( N
        = ( pred_numeral @ K ) ) ) ).

% Suc_eq_numeral
thf(fact_6451_pred__numeral__inc,axiom,
    ! [K: num] :
      ( ( pred_numeral @ ( inc @ K ) )
      = ( numeral_numeral_nat @ K ) ) ).

% pred_numeral_inc
thf(fact_6452_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_6453_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_6454_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_6455_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_6456_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_6457_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_6458_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_6459_pred__numeral__simps_I3_J,axiom,
    ! [K: num] :
      ( ( pred_numeral @ ( bit1 @ K ) )
      = ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ).

% pred_numeral_simps(3)
thf(fact_6460_and__nat__numerals_I3_J,axiom,
    ! [X: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( suc @ zero_zero_nat ) )
      = zero_zero_nat ) ).

% and_nat_numerals(3)
thf(fact_6461_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_6462_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_6463_and__nat__numerals_I4_J,axiom,
    ! [X: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( suc @ zero_zero_nat ) )
      = one_one_nat ) ).

% and_nat_numerals(4)
thf(fact_6464_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_6465_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_6466_signed__take__bit__numeral__bit0,axiom,
    ! [L: num,K: num] :
      ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
      = ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% signed_take_bit_numeral_bit0
thf(fact_6467_choose__one,axiom,
    ! [N: nat] :
      ( ( binomial @ N @ one_one_nat )
      = N ) ).

% choose_one
thf(fact_6468_binomial__eq__0,axiom,
    ! [N: nat,K: nat] :
      ( ( ord_less_nat @ N @ K )
     => ( ( binomial @ N @ K )
        = zero_zero_nat ) ) ).

% binomial_eq_0
thf(fact_6469_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_6470_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_6471_XOR__lower,axiom,
    ! [X: int,Y3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
       => ( ord_less_eq_int @ zero_zero_int @ ( bit_se6526347334894502574or_int @ X @ Y3 ) ) ) ) ).

% XOR_lower
thf(fact_6472_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_6473_choose__mult__lemma,axiom,
    ! [M: nat,R: nat,K: nat] :
      ( ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M @ R ) @ K ) @ ( plus_plus_nat @ M @ K ) ) @ ( binomial @ ( plus_plus_nat @ M @ K ) @ K ) )
      = ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M @ R ) @ K ) @ K ) @ ( binomial @ ( plus_plus_nat @ M @ R ) @ M ) ) ) ).

% choose_mult_lemma
thf(fact_6474_binomial__le__pow,axiom,
    ! [R: nat,N: nat] :
      ( ( ord_less_eq_nat @ R @ N )
     => ( ord_less_eq_nat @ ( binomial @ N @ R ) @ ( power_power_nat @ N @ R ) ) ) ).

% binomial_le_pow
thf(fact_6475_binomial__gbinomial,axiom,
    ! [N: nat,K: nat] :
      ( ( semiri681578069525770553at_rat @ ( binomial @ N @ K ) )
      = ( gbinomial_rat @ ( semiri681578069525770553at_rat @ N ) @ K ) ) ).

% binomial_gbinomial
thf(fact_6476_numeral__eq__Suc,axiom,
    ( numeral_numeral_nat
    = ( ^ [K3: num] : ( suc @ ( pred_numeral @ K3 ) ) ) ) ).

% numeral_eq_Suc
thf(fact_6477_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_6478_Suc__times__binomial__add,axiom,
    ! [A: nat,B: nat] :
      ( ( times_times_nat @ ( suc @ A ) @ ( binomial @ ( suc @ ( plus_plus_nat @ A @ B ) ) @ ( suc @ A ) ) )
      = ( times_times_nat @ ( suc @ B ) @ ( binomial @ ( suc @ ( plus_plus_nat @ A @ B ) ) @ A ) ) ) ).

% Suc_times_binomial_add
thf(fact_6479_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_6480_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_6481_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_6482_and__nat__def,axiom,
    ( bit_se727722235901077358nd_nat
    = ( ^ [M2: nat,N3: nat] : ( nat2 @ ( bit_se725231765392027082nd_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).

% and_nat_def
thf(fact_6483_xor__nat__def,axiom,
    ( bit_se6528837805403552850or_nat
    = ( ^ [M2: nat,N3: nat] : ( nat2 @ ( bit_se6526347334894502574or_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).

% xor_nat_def
thf(fact_6484_pred__numeral__def,axiom,
    ( pred_numeral
    = ( ^ [K3: num] : ( minus_minus_nat @ ( numeral_numeral_nat @ K3 ) @ one_one_nat ) ) ) ).

% pred_numeral_def
thf(fact_6485_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_6486_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_6487_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_6488_fact__numeral,axiom,
    ! [K: num] :
      ( ( semiri2447717529341329178atural @ ( numeral_numeral_nat @ K ) )
      = ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ K ) @ ( semiri2447717529341329178atural @ ( pred_numeral @ K ) ) ) ) ).

% fact_numeral
thf(fact_6489_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_6490_fact__numeral,axiom,
    ! [K: num] :
      ( ( semiri3624122377584611663nteger @ ( numeral_numeral_nat @ K ) )
      = ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ K ) @ ( semiri3624122377584611663nteger @ ( pred_numeral @ K ) ) ) ) ).

% fact_numeral
thf(fact_6491_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_6492_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_6493_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_6494_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_6495_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_6496_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_6497_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_6498_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_6499_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_6500_take__bit__numeral__bit0,axiom,
    ! [L: num,K: num] :
      ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
      = ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% take_bit_numeral_bit0
thf(fact_6501_take__bit__numeral__bit0,axiom,
    ! [L: num,K: num] :
      ( ( bit_se2925701944663578781it_nat @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
      = ( times_times_nat @ ( bit_se2925701944663578781it_nat @ ( pred_numeral @ L ) @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% take_bit_numeral_bit0
thf(fact_6502_take__bit__numeral__bit0,axiom,
    ! [L: num,K: num] :
      ( ( bit_se1745604003318907178nteger @ ( numeral_numeral_nat @ L ) @ ( numera6620942414471956472nteger @ ( bit0 @ K ) ) )
      = ( times_3573771949741848930nteger @ ( bit_se1745604003318907178nteger @ ( pred_numeral @ L ) @ ( numera6620942414471956472nteger @ K ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% take_bit_numeral_bit0
thf(fact_6503_take__bit__numeral__bit0,axiom,
    ! [L: num,K: num] :
      ( ( bit_se569199155075624693atural @ ( numeral_numeral_nat @ L ) @ ( numera5444537566228673987atural @ ( bit0 @ K ) ) )
      = ( times_2397367101498566445atural @ ( bit_se569199155075624693atural @ ( pred_numeral @ L ) @ ( numera5444537566228673987atural @ K ) ) @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ).

% take_bit_numeral_bit0
thf(fact_6504_take__bit__numeral__minus__bit0,axiom,
    ! [L: num,K: num] :
      ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
      = ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% take_bit_numeral_minus_bit0
thf(fact_6505_XOR__upper,axiom,
    ! [X: int,N: nat,Y3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_int @ X @ ( 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 @ X @ Y3 ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).

% XOR_upper
thf(fact_6506_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_6507_and__nat__unfold,axiom,
    ( bit_se727722235901077358nd_nat
    = ( ^ [M2: nat,N3: nat] :
          ( if_nat
          @ ( ( M2 = zero_zero_nat )
            | ( N3 = zero_zero_nat ) )
          @ zero_zero_nat
          @ ( plus_plus_nat @ ( times_times_nat @ ( modulo_modulo_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( divide_divide_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% and_nat_unfold
thf(fact_6508_and__nat__rec,axiom,
    ( bit_se727722235901077358nd_nat
    = ( ^ [M2: nat,N3: nat] :
          ( plus_plus_nat
          @ ( zero_n2687167440665602831ol_nat
            @ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 )
              & ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) )
          @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( divide_divide_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% and_nat_rec
thf(fact_6509_xor__int__rec,axiom,
    ( bit_se6526347334894502574or_int
    = ( ^ [K3: int,L3: int] :
          ( plus_plus_int
          @ ( zero_n2684676970156552555ol_int
            @ ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 ) )
             != ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L3 ) ) ) )
          @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% xor_int_rec
thf(fact_6510_xor__int__unfold,axiom,
    ( bit_se6526347334894502574or_int
    = ( ^ [K3: int,L3: int] :
          ( if_int
          @ ( K3
            = ( uminus_uminus_int @ one_one_int ) )
          @ ( bit_ri7919022796975470100ot_int @ L3 )
          @ ( if_int
            @ ( L3
              = ( uminus_uminus_int @ one_one_int ) )
            @ ( bit_ri7919022796975470100ot_int @ K3 )
            @ ( if_int @ ( K3 = zero_zero_int ) @ L3 @ ( if_int @ ( L3 = 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 @ L3 @ ( 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 @ L3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ).

% xor_int_unfold
thf(fact_6511_signed__take__bit__eq__take__bit__minus,axiom,
    ( bit_ri631733984087533419it_int
    = ( ^ [N3: nat,K3: int] : ( minus_minus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N3 ) @ K3 ) @ ( times_times_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N3 ) ) @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K3 @ N3 ) ) ) ) ) ) ).

% signed_take_bit_eq_take_bit_minus
thf(fact_6512_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_6513_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_6514_mask__numeral,axiom,
    ! [N: num] :
      ( ( bit_se2119862282449309892nteger @ ( numeral_numeral_nat @ N ) )
      = ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se2119862282449309892nteger @ ( pred_numeral @ N ) ) ) ) ) ).

% mask_numeral
thf(fact_6515_mask__numeral,axiom,
    ! [N: num] :
      ( ( bit_se943457434206027407atural @ ( numeral_numeral_nat @ N ) )
      = ( plus_p4538020629002901425atural @ one_one_Code_natural @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( bit_se943457434206027407atural @ ( pred_numeral @ N ) ) ) ) ) ).

% mask_numeral
thf(fact_6516_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_6517_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_6518_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_6519_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_6520_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_6521_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_6522_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_6523_push__bit__of__0,axiom,
    ! [N: nat] :
      ( ( bit_se545348938243370406it_int @ N @ zero_zero_int )
      = zero_zero_int ) ).

% push_bit_of_0
thf(fact_6524_push__bit__of__0,axiom,
    ! [N: nat] :
      ( ( bit_se547839408752420682it_nat @ N @ zero_zero_nat )
      = zero_zero_nat ) ).

% push_bit_of_0
thf(fact_6525_push__bit__of__0,axiom,
    ! [N: nat] :
      ( ( bit_se7788150548672797655nteger @ N @ zero_z3403309356797280102nteger )
      = zero_z3403309356797280102nteger ) ).

% push_bit_of_0
thf(fact_6526_push__bit__of__0,axiom,
    ! [N: nat] :
      ( ( bit_se6611745700429515170atural @ N @ zero_z2226904508553997617atural )
      = zero_z2226904508553997617atural ) ).

% push_bit_of_0
thf(fact_6527_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_6528_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_6529_push__bit__eq__0__iff,axiom,
    ! [N: nat,A: code_integer] :
      ( ( ( bit_se7788150548672797655nteger @ N @ A )
        = zero_z3403309356797280102nteger )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% push_bit_eq_0_iff
thf(fact_6530_push__bit__eq__0__iff,axiom,
    ! [N: nat,A: code_natural] :
      ( ( ( bit_se6611745700429515170atural @ N @ A )
        = zero_z2226904508553997617atural )
      = ( A = zero_z2226904508553997617atural ) ) ).

% push_bit_eq_0_iff
thf(fact_6531_concat__bit__of__zero__1,axiom,
    ! [N: nat,L: int] :
      ( ( bit_concat_bit @ N @ zero_zero_int @ L )
      = ( bit_se545348938243370406it_int @ N @ L ) ) ).

% concat_bit_of_zero_1
thf(fact_6532_mask__0,axiom,
    ( ( bit_se2000444600071755411sk_int @ zero_zero_nat )
    = zero_zero_int ) ).

% mask_0
thf(fact_6533_mask__0,axiom,
    ( ( bit_se2002935070580805687sk_nat @ zero_zero_nat )
    = zero_zero_nat ) ).

% mask_0
thf(fact_6534_mask__0,axiom,
    ( ( bit_se2119862282449309892nteger @ zero_zero_nat )
    = zero_z3403309356797280102nteger ) ).

% mask_0
thf(fact_6535_mask__0,axiom,
    ( ( bit_se943457434206027407atural @ zero_zero_nat )
    = zero_z2226904508553997617atural ) ).

% mask_0
thf(fact_6536_mask__eq__0__iff,axiom,
    ! [N: nat] :
      ( ( ( bit_se2000444600071755411sk_int @ N )
        = zero_zero_int )
      = ( N = zero_zero_nat ) ) ).

% mask_eq_0_iff
thf(fact_6537_mask__eq__0__iff,axiom,
    ! [N: nat] :
      ( ( ( bit_se2002935070580805687sk_nat @ N )
        = zero_zero_nat )
      = ( N = zero_zero_nat ) ) ).

% mask_eq_0_iff
thf(fact_6538_mask__eq__0__iff,axiom,
    ! [N: nat] :
      ( ( ( bit_se2119862282449309892nteger @ N )
        = zero_z3403309356797280102nteger )
      = ( N = zero_zero_nat ) ) ).

% mask_eq_0_iff
thf(fact_6539_mask__eq__0__iff,axiom,
    ! [N: nat] :
      ( ( ( bit_se943457434206027407atural @ N )
        = zero_z2226904508553997617atural )
      = ( N = zero_zero_nat ) ) ).

% mask_eq_0_iff
thf(fact_6540_bit_Oconj__cancel__right,axiom,
    ! [X: int] :
      ( ( bit_se725231765392027082nd_int @ X @ ( bit_ri7919022796975470100ot_int @ X ) )
      = zero_zero_int ) ).

% bit.conj_cancel_right
thf(fact_6541_bit_Oconj__cancel__right,axiom,
    ! [X: code_integer] :
      ( ( bit_se3949692690581998587nteger @ X @ ( bit_ri7632146776885996613nteger @ X ) )
      = zero_z3403309356797280102nteger ) ).

% bit.conj_cancel_right
thf(fact_6542_bit_Oconj__cancel__left,axiom,
    ! [X: int] :
      ( ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ X ) @ X )
      = zero_zero_int ) ).

% bit.conj_cancel_left
thf(fact_6543_bit_Oconj__cancel__left,axiom,
    ! [X: code_integer] :
      ( ( bit_se3949692690581998587nteger @ ( bit_ri7632146776885996613nteger @ X ) @ X )
      = zero_z3403309356797280102nteger ) ).

% bit.conj_cancel_left
thf(fact_6544_bit_Ocompl__zero,axiom,
    ( ( bit_ri7919022796975470100ot_int @ zero_zero_int )
    = ( uminus_uminus_int @ one_one_int ) ) ).

% bit.compl_zero
thf(fact_6545_bit_Ocompl__zero,axiom,
    ( ( bit_ri7632146776885996613nteger @ zero_z3403309356797280102nteger )
    = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% bit.compl_zero
thf(fact_6546_bit_Ocompl__one,axiom,
    ( ( bit_ri7919022796975470100ot_int @ ( uminus_uminus_int @ one_one_int ) )
    = zero_zero_int ) ).

% bit.compl_one
thf(fact_6547_bit_Ocompl__one,axiom,
    ( ( bit_ri7632146776885996613nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = zero_z3403309356797280102nteger ) ).

% bit.compl_one
thf(fact_6548_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_6549_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_6550_bit__numeral__Bit0__Suc__iff,axiom,
    ! [M: num,N: nat] :
      ( ( bit_se9216721137139052372nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) @ ( suc @ N ) )
      = ( bit_se9216721137139052372nteger @ ( numera6620942414471956472nteger @ M ) @ N ) ) ).

% bit_numeral_Bit0_Suc_iff
thf(fact_6551_bit__numeral__Bit0__Suc__iff,axiom,
    ! [M: num,N: nat] :
      ( ( bit_se8040316288895769887atural @ ( numera5444537566228673987atural @ ( bit0 @ M ) ) @ ( suc @ N ) )
      = ( bit_se8040316288895769887atural @ ( numera5444537566228673987atural @ M ) @ N ) ) ).

% bit_numeral_Bit0_Suc_iff
thf(fact_6552_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_6553_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_6554_push__bit__Suc__numeral,axiom,
    ! [N: nat,K: num] :
      ( ( bit_se7788150548672797655nteger @ ( suc @ N ) @ ( numera6620942414471956472nteger @ K ) )
      = ( bit_se7788150548672797655nteger @ N @ ( numera6620942414471956472nteger @ ( bit0 @ K ) ) ) ) ).

% push_bit_Suc_numeral
thf(fact_6555_push__bit__Suc__numeral,axiom,
    ! [N: nat,K: num] :
      ( ( bit_se6611745700429515170atural @ ( suc @ N ) @ ( numera5444537566228673987atural @ K ) )
      = ( bit_se6611745700429515170atural @ N @ ( numera5444537566228673987atural @ ( bit0 @ K ) ) ) ) ).

% push_bit_Suc_numeral
thf(fact_6556_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_6557_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_6558_bit__numeral__Bit1__Suc__iff,axiom,
    ! [M: num,N: nat] :
      ( ( bit_se9216721137139052372nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) @ ( suc @ N ) )
      = ( bit_se9216721137139052372nteger @ ( numera6620942414471956472nteger @ M ) @ N ) ) ).

% bit_numeral_Bit1_Suc_iff
thf(fact_6559_bit__numeral__Bit1__Suc__iff,axiom,
    ! [M: num,N: nat] :
      ( ( bit_se8040316288895769887atural @ ( numera5444537566228673987atural @ ( bit1 @ M ) ) @ ( suc @ N ) )
      = ( bit_se8040316288895769887atural @ ( numera5444537566228673987atural @ M ) @ N ) ) ).

% bit_numeral_Bit1_Suc_iff
thf(fact_6560_mask__Suc__0,axiom,
    ( ( bit_se2000444600071755411sk_int @ ( suc @ zero_zero_nat ) )
    = one_one_int ) ).

% mask_Suc_0
thf(fact_6561_mask__Suc__0,axiom,
    ( ( bit_se2002935070580805687sk_nat @ ( suc @ zero_zero_nat ) )
    = one_one_nat ) ).

% mask_Suc_0
thf(fact_6562_mask__Suc__0,axiom,
    ( ( bit_se2119862282449309892nteger @ ( suc @ zero_zero_nat ) )
    = one_one_Code_integer ) ).

% mask_Suc_0
thf(fact_6563_mask__Suc__0,axiom,
    ( ( bit_se943457434206027407atural @ ( suc @ zero_zero_nat ) )
    = one_one_Code_natural ) ).

% mask_Suc_0
thf(fact_6564_bit_Oxor__cancel__right,axiom,
    ! [X: int] :
      ( ( bit_se6526347334894502574or_int @ X @ ( bit_ri7919022796975470100ot_int @ X ) )
      = ( uminus_uminus_int @ one_one_int ) ) ).

% bit.xor_cancel_right
thf(fact_6565_bit_Oxor__cancel__right,axiom,
    ! [X: code_integer] :
      ( ( bit_se3222712562003087583nteger @ X @ ( bit_ri7632146776885996613nteger @ X ) )
      = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% bit.xor_cancel_right
thf(fact_6566_bit_Oxor__cancel__left,axiom,
    ! [X: int] :
      ( ( bit_se6526347334894502574or_int @ ( bit_ri7919022796975470100ot_int @ X ) @ X )
      = ( uminus_uminus_int @ one_one_int ) ) ).

% bit.xor_cancel_left
thf(fact_6567_bit_Oxor__cancel__left,axiom,
    ! [X: code_integer] :
      ( ( bit_se3222712562003087583nteger @ ( bit_ri7632146776885996613nteger @ X ) @ X )
      = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% bit.xor_cancel_left
thf(fact_6568_bit_Oxor__one__right,axiom,
    ! [X: int] :
      ( ( bit_se6526347334894502574or_int @ X @ ( uminus_uminus_int @ one_one_int ) )
      = ( bit_ri7919022796975470100ot_int @ X ) ) ).

% bit.xor_one_right
thf(fact_6569_bit_Oxor__one__right,axiom,
    ! [X: code_integer] :
      ( ( bit_se3222712562003087583nteger @ X @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( bit_ri7632146776885996613nteger @ X ) ) ).

% bit.xor_one_right
thf(fact_6570_bit_Oxor__one__left,axiom,
    ! [X: int] :
      ( ( bit_se6526347334894502574or_int @ ( uminus_uminus_int @ one_one_int ) @ X )
      = ( bit_ri7919022796975470100ot_int @ X ) ) ).

% bit.xor_one_left
thf(fact_6571_bit_Oxor__one__left,axiom,
    ! [X: code_integer] :
      ( ( bit_se3222712562003087583nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ X )
      = ( bit_ri7632146776885996613nteger @ X ) ) ).

% bit.xor_one_left
thf(fact_6572_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_6573_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_6574_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_6575_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_6576_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_6577_minus__not__numeral__eq,axiom,
    ! [N: num] :
      ( ( uminus1351360451143612070nteger @ ( bit_ri7632146776885996613nteger @ ( numera6620942414471956472nteger @ N ) ) )
      = ( numera6620942414471956472nteger @ ( inc @ N ) ) ) ).

% minus_not_numeral_eq
thf(fact_6578_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_6579_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_6580_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_6581_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_6582_bit__minus__numeral__Bit0__Suc__iff,axiom,
    ! [W2: num,N: nat] :
      ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ W2 ) ) ) @ ( suc @ N ) )
      = ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W2 ) ) @ N ) ) ).

% bit_minus_numeral_Bit0_Suc_iff
thf(fact_6583_bit__minus__numeral__Bit1__Suc__iff,axiom,
    ! [W2: num,N: nat] :
      ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ W2 ) ) ) @ ( suc @ N ) )
      = ( ~ ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W2 ) @ N ) ) ) ).

% bit_minus_numeral_Bit1_Suc_iff
thf(fact_6584_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_6585_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_6586_not__one__eq,axiom,
    ( ( bit_ri7919022796975470100ot_int @ one_one_int )
    = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% not_one_eq
thf(fact_6587_not__one__eq,axiom,
    ( ( bit_ri7632146776885996613nteger @ one_one_Code_integer )
    = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% not_one_eq
thf(fact_6588_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_6589_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_6590_bit__0,axiom,
    ! [A: code_integer] :
      ( ( bit_se9216721137139052372nteger @ A @ zero_zero_nat )
      = ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).

% bit_0
thf(fact_6591_bit__0,axiom,
    ! [A: code_natural] :
      ( ( bit_se8040316288895769887atural @ A @ zero_zero_nat )
      = ( ~ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A ) ) ) ).

% bit_0
thf(fact_6592_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_6593_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_6594_push__bit__Suc,axiom,
    ! [N: nat,A: code_integer] :
      ( ( bit_se7788150548672797655nteger @ ( suc @ N ) @ A )
      = ( bit_se7788150548672797655nteger @ N @ ( times_3573771949741848930nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).

% push_bit_Suc
thf(fact_6595_push__bit__Suc,axiom,
    ! [N: nat,A: code_natural] :
      ( ( bit_se6611745700429515170atural @ ( suc @ N ) @ A )
      = ( bit_se6611745700429515170atural @ N @ ( times_2397367101498566445atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ) ).

% push_bit_Suc
thf(fact_6596_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_6597_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_6598_push__bit__of__1,axiom,
    ! [N: nat] :
      ( ( bit_se7788150548672797655nteger @ N @ one_one_Code_integer )
      = ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ).

% push_bit_of_1
thf(fact_6599_push__bit__of__1,axiom,
    ! [N: nat] :
      ( ( bit_se6611745700429515170atural @ N @ one_one_Code_natural )
      = ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) ) ).

% push_bit_of_1
thf(fact_6600_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_6601_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_6602_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_6603_even__push__bit__iff,axiom,
    ! [N: nat,A: code_natural] :
      ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( bit_se6611745700429515170atural @ N @ A ) )
      = ( ( N != zero_zero_nat )
        | ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A ) ) ) ).

% even_push_bit_iff
thf(fact_6604_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_6605_push__bit__minus__numeral,axiom,
    ! [L: num,K: num] :
      ( ( bit_se545348938243370406it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
      = ( bit_se545348938243370406it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) ) ) ).

% push_bit_minus_numeral
thf(fact_6606_push__bit__minus__numeral,axiom,
    ! [L: num,K: num] :
      ( ( bit_se7788150548672797655nteger @ ( numeral_numeral_nat @ L ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
      = ( bit_se7788150548672797655nteger @ ( pred_numeral @ L ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ K ) ) ) ) ) ).

% push_bit_minus_numeral
thf(fact_6607_bit__minus__numeral__int_I1_J,axiom,
    ! [W2: num,N: num] :
      ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ W2 ) ) ) @ ( numeral_numeral_nat @ N ) )
      = ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W2 ) ) @ ( pred_numeral @ N ) ) ) ).

% bit_minus_numeral_int(1)
thf(fact_6608_bit__minus__numeral__int_I2_J,axiom,
    ! [W2: num,N: num] :
      ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ W2 ) ) ) @ ( numeral_numeral_nat @ N ) )
      = ( ~ ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W2 ) @ ( pred_numeral @ N ) ) ) ) ).

% bit_minus_numeral_int(2)
thf(fact_6609_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_6610_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_6611_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_6612_bit__mod__2__iff,axiom,
    ! [A: code_natural,N: nat] :
      ( ( bit_se8040316288895769887atural @ ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) @ N )
      = ( ( N = zero_zero_nat )
        & ~ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A ) ) ) ).

% bit_mod_2_iff
thf(fact_6613_of__nat__mask__eq,axiom,
    ! [N: nat] :
      ( ( semiri1314217659103216013at_int @ ( bit_se2002935070580805687sk_nat @ N ) )
      = ( bit_se2000444600071755411sk_int @ N ) ) ).

% of_nat_mask_eq
thf(fact_6614_of__nat__mask__eq,axiom,
    ! [N: nat] :
      ( ( semiri1316708129612266289at_nat @ ( bit_se2002935070580805687sk_nat @ N ) )
      = ( bit_se2002935070580805687sk_nat @ N ) ) ).

% of_nat_mask_eq
thf(fact_6615_of__nat__mask__eq,axiom,
    ! [N: nat] :
      ( ( semiri4939895301339042750nteger @ ( bit_se2002935070580805687sk_nat @ N ) )
      = ( bit_se2119862282449309892nteger @ N ) ) ).

% of_nat_mask_eq
thf(fact_6616_of__nat__mask__eq,axiom,
    ! [N: nat] :
      ( ( semiri3763490453095760265atural @ ( bit_se2002935070580805687sk_nat @ N ) )
      = ( bit_se943457434206027407atural @ N ) ) ).

% of_nat_mask_eq
thf(fact_6617_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_6618_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_6619_bit__of__nat__iff__bit,axiom,
    ! [M: nat,N: nat] :
      ( ( bit_se9216721137139052372nteger @ ( semiri4939895301339042750nteger @ M ) @ N )
      = ( bit_se1148574629649215175it_nat @ M @ N ) ) ).

% bit_of_nat_iff_bit
thf(fact_6620_bit__of__nat__iff__bit,axiom,
    ! [M: nat,N: nat] :
      ( ( bit_se8040316288895769887atural @ ( semiri3763490453095760265atural @ M ) @ N )
      = ( bit_se1148574629649215175it_nat @ M @ N ) ) ).

% bit_of_nat_iff_bit
thf(fact_6621_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_6622_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_6623_push__bit__of__nat,axiom,
    ! [N: nat,M: nat] :
      ( ( bit_se7788150548672797655nteger @ N @ ( semiri4939895301339042750nteger @ M ) )
      = ( semiri4939895301339042750nteger @ ( bit_se547839408752420682it_nat @ N @ M ) ) ) ).

% push_bit_of_nat
thf(fact_6624_push__bit__of__nat,axiom,
    ! [N: nat,M: nat] :
      ( ( bit_se6611745700429515170atural @ N @ ( semiri3763490453095760265atural @ M ) )
      = ( semiri3763490453095760265atural @ ( bit_se547839408752420682it_nat @ N @ M ) ) ) ).

% push_bit_of_nat
thf(fact_6625_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_6626_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_6627_of__nat__push__bit,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri4939895301339042750nteger @ ( bit_se547839408752420682it_nat @ M @ N ) )
      = ( bit_se7788150548672797655nteger @ M @ ( semiri4939895301339042750nteger @ N ) ) ) ).

% of_nat_push_bit
thf(fact_6628_of__nat__push__bit,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri3763490453095760265atural @ ( bit_se547839408752420682it_nat @ M @ N ) )
      = ( bit_se6611745700429515170atural @ M @ ( semiri3763490453095760265atural @ N ) ) ) ).

% of_nat_push_bit
thf(fact_6629_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_6630_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_6631_push__bit__minus,axiom,
    ! [N: nat,A: code_integer] :
      ( ( bit_se7788150548672797655nteger @ N @ ( uminus1351360451143612070nteger @ A ) )
      = ( uminus1351360451143612070nteger @ ( bit_se7788150548672797655nteger @ N @ A ) ) ) ).

% push_bit_minus
thf(fact_6632_less__eq__mask,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ N @ ( bit_se2002935070580805687sk_nat @ N ) ) ).

% less_eq_mask
thf(fact_6633_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_6634_bit__iff__and__push__bit__not__eq__0,axiom,
    ( bit_se1146084159140164899it_int
    = ( ^ [A2: int,N3: nat] :
          ( ( bit_se725231765392027082nd_int @ A2 @ ( bit_se545348938243370406it_int @ N3 @ one_one_int ) )
         != zero_zero_int ) ) ) ).

% bit_iff_and_push_bit_not_eq_0
thf(fact_6635_bit__iff__and__push__bit__not__eq__0,axiom,
    ( bit_se1148574629649215175it_nat
    = ( ^ [A2: nat,N3: nat] :
          ( ( bit_se727722235901077358nd_nat @ A2 @ ( bit_se547839408752420682it_nat @ N3 @ one_one_nat ) )
         != zero_zero_nat ) ) ) ).

% bit_iff_and_push_bit_not_eq_0
thf(fact_6636_bit__iff__and__push__bit__not__eq__0,axiom,
    ( bit_se9216721137139052372nteger
    = ( ^ [A2: code_integer,N3: nat] :
          ( ( bit_se3949692690581998587nteger @ A2 @ ( bit_se7788150548672797655nteger @ N3 @ one_one_Code_integer ) )
         != zero_z3403309356797280102nteger ) ) ) ).

% bit_iff_and_push_bit_not_eq_0
thf(fact_6637_bit__iff__and__push__bit__not__eq__0,axiom,
    ( bit_se8040316288895769887atural
    = ( ^ [A2: code_natural,N3: nat] :
          ( ( bit_se2773287842338716102atural @ A2 @ ( bit_se6611745700429515170atural @ N3 @ one_one_Code_natural ) )
         != zero_z2226904508553997617atural ) ) ) ).

% bit_iff_and_push_bit_not_eq_0
thf(fact_6638_unset__bit__eq__and__not,axiom,
    ( bit_se4203085406695923979it_int
    = ( ^ [N3: nat,A2: int] : ( bit_se725231765392027082nd_int @ A2 @ ( bit_ri7919022796975470100ot_int @ ( bit_se545348938243370406it_int @ N3 @ one_one_int ) ) ) ) ) ).

% unset_bit_eq_and_not
thf(fact_6639_unset__bit__eq__and__not,axiom,
    ( bit_se8260200283734997820nteger
    = ( ^ [N3: nat,A2: code_integer] : ( bit_se3949692690581998587nteger @ A2 @ ( bit_ri7632146776885996613nteger @ ( bit_se7788150548672797655nteger @ N3 @ one_one_Code_integer ) ) ) ) ) ).

% unset_bit_eq_and_not
thf(fact_6640_unset__bit__int__def,axiom,
    ( bit_se4203085406695923979it_int
    = ( ^ [N3: nat,K3: int] : ( bit_se725231765392027082nd_int @ K3 @ ( bit_ri7919022796975470100ot_int @ ( bit_se545348938243370406it_int @ N3 @ one_one_int ) ) ) ) ) ).

% unset_bit_int_def
thf(fact_6641_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_6642_take__bit__not__mask__eq__0,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( bit_se1745604003318907178nteger @ M @ ( bit_ri7632146776885996613nteger @ ( bit_se2119862282449309892nteger @ N ) ) )
        = zero_z3403309356797280102nteger ) ) ).

% take_bit_not_mask_eq_0
thf(fact_6643_not__bit__1__Suc,axiom,
    ! [N: nat] :
      ~ ( bit_se1146084159140164899it_int @ one_one_int @ ( suc @ N ) ) ).

% not_bit_1_Suc
thf(fact_6644_not__bit__1__Suc,axiom,
    ! [N: nat] :
      ~ ( bit_se1148574629649215175it_nat @ one_one_nat @ ( suc @ N ) ) ).

% not_bit_1_Suc
thf(fact_6645_not__bit__1__Suc,axiom,
    ! [N: nat] :
      ~ ( bit_se9216721137139052372nteger @ one_one_Code_integer @ ( suc @ N ) ) ).

% not_bit_1_Suc
thf(fact_6646_not__bit__1__Suc,axiom,
    ! [N: nat] :
      ~ ( bit_se8040316288895769887atural @ one_one_Code_natural @ ( suc @ N ) ) ).

% not_bit_1_Suc
thf(fact_6647_bit__1__iff,axiom,
    ! [N: nat] :
      ( ( bit_se1146084159140164899it_int @ one_one_int @ N )
      = ( N = zero_zero_nat ) ) ).

% bit_1_iff
thf(fact_6648_bit__1__iff,axiom,
    ! [N: nat] :
      ( ( bit_se1148574629649215175it_nat @ one_one_nat @ N )
      = ( N = zero_zero_nat ) ) ).

% bit_1_iff
thf(fact_6649_bit__1__iff,axiom,
    ! [N: nat] :
      ( ( bit_se9216721137139052372nteger @ one_one_Code_integer @ N )
      = ( N = zero_zero_nat ) ) ).

% bit_1_iff
thf(fact_6650_bit__1__iff,axiom,
    ! [N: nat] :
      ( ( bit_se8040316288895769887atural @ one_one_Code_natural @ N )
      = ( N = zero_zero_nat ) ) ).

% bit_1_iff
thf(fact_6651_bit__numeral__simps_I1_J,axiom,
    ! [N: num] :
      ~ ( bit_se1146084159140164899it_int @ one_one_int @ ( numeral_numeral_nat @ N ) ) ).

% bit_numeral_simps(1)
thf(fact_6652_bit__numeral__simps_I1_J,axiom,
    ! [N: num] :
      ~ ( bit_se1148574629649215175it_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) ) ).

% bit_numeral_simps(1)
thf(fact_6653_bit__numeral__simps_I1_J,axiom,
    ! [N: num] :
      ~ ( bit_se9216721137139052372nteger @ one_one_Code_integer @ ( numeral_numeral_nat @ N ) ) ).

% bit_numeral_simps(1)
thf(fact_6654_bit__numeral__simps_I1_J,axiom,
    ! [N: num] :
      ~ ( bit_se8040316288895769887atural @ one_one_Code_natural @ ( numeral_numeral_nat @ N ) ) ).

% bit_numeral_simps(1)
thf(fact_6655_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_6656_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_6657_bit__take__bit__iff,axiom,
    ! [M: nat,A: code_integer,N: nat] :
      ( ( bit_se9216721137139052372nteger @ ( bit_se1745604003318907178nteger @ M @ A ) @ N )
      = ( ( ord_less_nat @ N @ M )
        & ( bit_se9216721137139052372nteger @ A @ N ) ) ) ).

% bit_take_bit_iff
thf(fact_6658_bit__take__bit__iff,axiom,
    ! [M: nat,A: code_natural,N: nat] :
      ( ( bit_se8040316288895769887atural @ ( bit_se569199155075624693atural @ M @ A ) @ N )
      = ( ( ord_less_nat @ N @ M )
        & ( bit_se8040316288895769887atural @ A @ N ) ) ) ).

% bit_take_bit_iff
thf(fact_6659_bit__of__bool__iff,axiom,
    ! [B: $o,N: nat] :
      ( ( bit_se1146084159140164899it_int @ ( zero_n2684676970156552555ol_int @ B ) @ N )
      = ( B
        & ( N = zero_zero_nat ) ) ) ).

% bit_of_bool_iff
thf(fact_6660_bit__of__bool__iff,axiom,
    ! [B: $o,N: nat] :
      ( ( bit_se1148574629649215175it_nat @ ( zero_n2687167440665602831ol_nat @ B ) @ N )
      = ( B
        & ( N = zero_zero_nat ) ) ) ).

% bit_of_bool_iff
thf(fact_6661_bit__of__bool__iff,axiom,
    ! [B: $o,N: nat] :
      ( ( bit_se9216721137139052372nteger @ ( zero_n356916108424825756nteger @ B ) @ N )
      = ( B
        & ( N = zero_zero_nat ) ) ) ).

% bit_of_bool_iff
thf(fact_6662_bit__of__bool__iff,axiom,
    ! [B: $o,N: nat] :
      ( ( bit_se8040316288895769887atural @ ( zero_n8403883297036319079atural @ B ) @ N )
      = ( B
        & ( N = zero_zero_nat ) ) ) ).

% bit_of_bool_iff
thf(fact_6663_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_6664_bit__not__iff__eq,axiom,
    ! [A: code_integer,N: nat] :
      ( ( bit_se9216721137139052372nteger @ ( bit_ri7632146776885996613nteger @ A ) @ N )
      = ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N )
         != zero_z3403309356797280102nteger )
        & ~ ( bit_se9216721137139052372nteger @ A @ N ) ) ) ).

% bit_not_iff_eq
thf(fact_6665_mask__nonnegative__int,axiom,
    ! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( bit_se2000444600071755411sk_int @ N ) ) ).

% mask_nonnegative_int
thf(fact_6666_not__mask__negative__int,axiom,
    ! [N: nat] :
      ~ ( ord_less_int @ ( bit_se2000444600071755411sk_int @ N ) @ zero_zero_int ) ).

% not_mask_negative_int
thf(fact_6667_flip__bit__nat__def,axiom,
    ( bit_se2161824704523386999it_nat
    = ( ^ [M2: nat,N3: nat] : ( bit_se6528837805403552850or_nat @ N3 @ ( bit_se547839408752420682it_nat @ M2 @ one_one_nat ) ) ) ) ).

% flip_bit_nat_def
thf(fact_6668_flip__bit__int__def,axiom,
    ( bit_se2159334234014336723it_int
    = ( ^ [N3: nat,K3: int] : ( bit_se6526347334894502574or_int @ K3 @ ( bit_se545348938243370406it_int @ N3 @ one_one_int ) ) ) ) ).

% flip_bit_int_def
thf(fact_6669_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_6670_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_6671_minus__eq__not__plus__1,axiom,
    ( uminus_uminus_int
    = ( ^ [A2: int] : ( plus_plus_int @ ( bit_ri7919022796975470100ot_int @ A2 ) @ one_one_int ) ) ) ).

% minus_eq_not_plus_1
thf(fact_6672_minus__eq__not__plus__1,axiom,
    ( uminus1351360451143612070nteger
    = ( ^ [A2: code_integer] : ( plus_p5714425477246183910nteger @ ( bit_ri7632146776885996613nteger @ A2 ) @ one_one_Code_integer ) ) ) ).

% minus_eq_not_plus_1
thf(fact_6673_minus__eq__not__minus__1,axiom,
    ( uminus_uminus_int
    = ( ^ [A2: int] : ( bit_ri7919022796975470100ot_int @ ( minus_minus_int @ A2 @ one_one_int ) ) ) ) ).

% minus_eq_not_minus_1
thf(fact_6674_minus__eq__not__minus__1,axiom,
    ( uminus1351360451143612070nteger
    = ( ^ [A2: code_integer] : ( bit_ri7632146776885996613nteger @ ( minus_8373710615458151222nteger @ A2 @ one_one_Code_integer ) ) ) ) ).

% minus_eq_not_minus_1
thf(fact_6675_not__eq__complement,axiom,
    ( bit_ri7919022796975470100ot_int
    = ( ^ [A2: int] : ( minus_minus_int @ ( uminus_uminus_int @ A2 ) @ one_one_int ) ) ) ).

% not_eq_complement
thf(fact_6676_not__eq__complement,axiom,
    ( bit_ri7632146776885996613nteger
    = ( ^ [A2: code_integer] : ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ A2 ) @ one_one_Code_integer ) ) ) ).

% not_eq_complement
thf(fact_6677_not__int__def,axiom,
    ( bit_ri7919022796975470100ot_int
    = ( ^ [K3: int] : ( minus_minus_int @ ( uminus_uminus_int @ K3 ) @ one_one_int ) ) ) ).

% not_int_def
thf(fact_6678_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_6679_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_6680_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_6681_minus__numeral__inc__eq,axiom,
    ! [N: num] :
      ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( inc @ N ) ) )
      = ( bit_ri7632146776885996613nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).

% minus_numeral_inc_eq
thf(fact_6682_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_6683_flip__bit__eq__xor,axiom,
    ( bit_se2159334234014336723it_int
    = ( ^ [N3: nat,A2: int] : ( bit_se6526347334894502574or_int @ A2 @ ( bit_se545348938243370406it_int @ N3 @ one_one_int ) ) ) ) ).

% flip_bit_eq_xor
thf(fact_6684_flip__bit__eq__xor,axiom,
    ( bit_se2161824704523386999it_nat
    = ( ^ [N3: nat,A2: nat] : ( bit_se6528837805403552850or_nat @ A2 @ ( bit_se547839408752420682it_nat @ N3 @ one_one_nat ) ) ) ) ).

% flip_bit_eq_xor
thf(fact_6685_flip__bit__eq__xor,axiom,
    ( bit_se1345352211410354436nteger
    = ( ^ [N3: nat,A2: code_integer] : ( bit_se3222712562003087583nteger @ A2 @ ( bit_se7788150548672797655nteger @ N3 @ one_one_Code_integer ) ) ) ) ).

% flip_bit_eq_xor
thf(fact_6686_flip__bit__eq__xor,axiom,
    ( bit_se168947363167071951atural
    = ( ^ [N3: nat,A2: code_natural] : ( bit_se2046307713759805098atural @ A2 @ ( bit_se6611745700429515170atural @ N3 @ one_one_Code_natural ) ) ) ) ).

% flip_bit_eq_xor
thf(fact_6687_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_6688_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_6689_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_6690_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_6691_not__numeral__Bit0__eq,axiom,
    ! [N: num] :
      ( ( bit_ri7632146776885996613nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) )
      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) ) ) ).

% not_numeral_Bit0_eq
thf(fact_6692_bit__concat__bit__iff,axiom,
    ! [M: nat,K: int,L: int,N: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_concat_bit @ M @ K @ L ) @ N )
      = ( ( ( ord_less_nat @ N @ M )
          & ( bit_se1146084159140164899it_int @ K @ N ) )
        | ( ( ord_less_eq_nat @ M @ N )
          & ( bit_se1146084159140164899it_int @ L @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).

% bit_concat_bit_iff
thf(fact_6693_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_6694_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_6695_push__bit__double,axiom,
    ! [N: nat,A: code_integer] :
      ( ( bit_se7788150548672797655nteger @ N @ ( times_3573771949741848930nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) )
      = ( times_3573771949741848930nteger @ ( bit_se7788150548672797655nteger @ N @ A ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% push_bit_double
thf(fact_6696_push__bit__double,axiom,
    ! [N: nat,A: code_natural] :
      ( ( bit_se6611745700429515170atural @ N @ ( times_2397367101498566445atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) )
      = ( times_2397367101498566445atural @ ( bit_se6611745700429515170atural @ N @ A ) @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ).

% push_bit_double
thf(fact_6697_signed__take__bit__eq__concat__bit,axiom,
    ( bit_ri631733984087533419it_int
    = ( ^ [N3: nat,K3: int] : ( bit_concat_bit @ N3 @ K3 @ ( uminus_uminus_int @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K3 @ N3 ) ) ) ) ) ) ).

% signed_take_bit_eq_concat_bit
thf(fact_6698_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_6699_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_6700_exp__eq__0__imp__not__bit,axiom,
    ! [N: nat,A: code_integer] :
      ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N )
        = zero_z3403309356797280102nteger )
     => ~ ( bit_se9216721137139052372nteger @ A @ N ) ) ).

% exp_eq_0_imp_not_bit
thf(fact_6701_exp__eq__0__imp__not__bit,axiom,
    ! [N: nat,A: code_natural] :
      ( ( ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N )
        = zero_z2226904508553997617atural )
     => ~ ( bit_se8040316288895769887atural @ A @ N ) ) ).

% exp_eq_0_imp_not_bit
thf(fact_6702_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_6703_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_6704_bit__Suc,axiom,
    ! [A: code_integer,N: nat] :
      ( ( bit_se9216721137139052372nteger @ A @ ( suc @ N ) )
      = ( bit_se9216721137139052372nteger @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ N ) ) ).

% bit_Suc
thf(fact_6705_bit__Suc,axiom,
    ! [A: code_natural,N: nat] :
      ( ( bit_se8040316288895769887atural @ A @ ( suc @ N ) )
      = ( bit_se8040316288895769887atural @ ( divide5121882707175180666atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) @ N ) ) ).

% bit_Suc
thf(fact_6706_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_6707_int__bit__bound,axiom,
    ! [K: int] :
      ~ ! [N4: nat] :
          ( ! [M3: nat] :
              ( ( ord_less_eq_nat @ N4 @ M3 )
             => ( ( bit_se1146084159140164899it_int @ K @ M3 )
                = ( 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_6708_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_6709_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_6710_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_6711_push__bit__int__def,axiom,
    ( bit_se545348938243370406it_int
    = ( ^ [N3: nat,K3: int] : ( times_times_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).

% push_bit_int_def
thf(fact_6712_push__bit__nat__def,axiom,
    ( bit_se547839408752420682it_nat
    = ( ^ [N3: nat,M2: nat] : ( times_times_nat @ M2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).

% push_bit_nat_def
thf(fact_6713_push__bit__eq__mult,axiom,
    ( bit_se545348938243370406it_int
    = ( ^ [N3: nat,A2: int] : ( times_times_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).

% push_bit_eq_mult
thf(fact_6714_push__bit__eq__mult,axiom,
    ( bit_se547839408752420682it_nat
    = ( ^ [N3: nat,A2: nat] : ( times_times_nat @ A2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).

% push_bit_eq_mult
thf(fact_6715_push__bit__eq__mult,axiom,
    ( bit_se7788150548672797655nteger
    = ( ^ [N3: nat,A2: code_integer] : ( times_3573771949741848930nteger @ A2 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).

% push_bit_eq_mult
thf(fact_6716_push__bit__eq__mult,axiom,
    ( bit_se6611745700429515170atural
    = ( ^ [N3: nat,A2: code_natural] : ( times_2397367101498566445atural @ A2 @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).

% push_bit_eq_mult
thf(fact_6717_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_6718_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_6719_and__exp__eq__0__iff__not__bit,axiom,
    ! [A: code_integer,N: nat] :
      ( ( ( bit_se3949692690581998587nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
        = zero_z3403309356797280102nteger )
      = ( ~ ( bit_se9216721137139052372nteger @ A @ N ) ) ) ).

% and_exp_eq_0_iff_not_bit
thf(fact_6720_and__exp__eq__0__iff__not__bit,axiom,
    ! [A: code_natural,N: nat] :
      ( ( ( bit_se2773287842338716102atural @ A @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) )
        = zero_z2226904508553997617atural )
      = ( ~ ( bit_se8040316288895769887atural @ A @ N ) ) ) ).

% and_exp_eq_0_iff_not_bit
thf(fact_6721_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_6722_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_6723_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_6724_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_6725_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_6726_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_6727_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_6728_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_6729_semiring__bit__operations__class_Oeven__mask__iff,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( bit_se943457434206027407atural @ N ) )
      = ( N = zero_zero_nat ) ) ).

% semiring_bit_operations_class.even_mask_iff
thf(fact_6730_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_6731_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_6732_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_6733_even__bit__succ__iff,axiom,
    ! [A: code_natural,N: nat] :
      ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A )
     => ( ( bit_se8040316288895769887atural @ ( plus_p4538020629002901425atural @ one_one_Code_natural @ A ) @ N )
        = ( ( bit_se8040316288895769887atural @ A @ N )
          | ( N = zero_zero_nat ) ) ) ) ).

% even_bit_succ_iff
thf(fact_6734_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_6735_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_6736_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_6737_odd__bit__iff__bit__pred,axiom,
    ! [A: code_natural,N: nat] :
      ( ~ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A )
     => ( ( bit_se8040316288895769887atural @ A @ N )
        = ( ( bit_se8040316288895769887atural @ ( minus_7197305767214868737atural @ A @ one_one_Code_natural ) @ N )
          | ( N = zero_zero_nat ) ) ) ) ).

% odd_bit_iff_bit_pred
thf(fact_6738_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_6739_mask__int__def,axiom,
    ( bit_se2000444600071755411sk_int
    = ( ^ [N3: nat] : ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) @ one_one_int ) ) ) ).

% mask_int_def
thf(fact_6740_mask__nat__def,axiom,
    ( bit_se2002935070580805687sk_nat
    = ( ^ [N3: nat] : ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ one_one_nat ) ) ) ).

% mask_nat_def
thf(fact_6741_bit__sum__mult__2__cases,axiom,
    ! [A: int,B: 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 ) ) @ B ) ) @ 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 ) ) @ B ) @ N ) ) ) ) ) ).

% bit_sum_mult_2_cases
thf(fact_6742_bit__sum__mult__2__cases,axiom,
    ! [A: nat,B: 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 ) ) @ B ) ) @ 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 ) ) @ B ) @ N ) ) ) ) ) ).

% bit_sum_mult_2_cases
thf(fact_6743_bit__sum__mult__2__cases,axiom,
    ! [A: code_integer,B: code_integer,N: nat] :
      ( ! [J2: nat] :
          ~ ( bit_se9216721137139052372nteger @ A @ ( suc @ J2 ) )
     => ( ( bit_se9216721137139052372nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ N )
        = ( ( ( N = zero_zero_nat )
           => ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) )
          & ( ( N != zero_zero_nat )
           => ( bit_se9216721137139052372nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) @ N ) ) ) ) ) ).

% bit_sum_mult_2_cases
thf(fact_6744_bit__sum__mult__2__cases,axiom,
    ! [A: code_natural,B: code_natural,N: nat] :
      ( ! [J2: nat] :
          ~ ( bit_se8040316288895769887atural @ A @ ( suc @ J2 ) )
     => ( ( bit_se8040316288895769887atural @ ( plus_p4538020629002901425atural @ A @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ B ) ) @ N )
        = ( ( ( N = zero_zero_nat )
           => ~ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A ) )
          & ( ( N != zero_zero_nat )
           => ( bit_se8040316288895769887atural @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ B ) @ N ) ) ) ) ) ).

% bit_sum_mult_2_cases
thf(fact_6745_bit__rec,axiom,
    ( bit_se1146084159140164899it_int
    = ( ^ [A2: int,N3: nat] :
          ( ( ( N3 = zero_zero_nat )
           => ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 ) )
          & ( ( N3 != zero_zero_nat )
           => ( bit_se1146084159140164899it_int @ ( divide_divide_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( minus_minus_nat @ N3 @ one_one_nat ) ) ) ) ) ) ).

% bit_rec
thf(fact_6746_bit__rec,axiom,
    ( bit_se1148574629649215175it_nat
    = ( ^ [A2: nat,N3: nat] :
          ( ( ( N3 = zero_zero_nat )
           => ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 ) )
          & ( ( N3 != zero_zero_nat )
           => ( bit_se1148574629649215175it_nat @ ( divide_divide_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( minus_minus_nat @ N3 @ one_one_nat ) ) ) ) ) ) ).

% bit_rec
thf(fact_6747_bit__rec,axiom,
    ( bit_se9216721137139052372nteger
    = ( ^ [A2: code_integer,N3: nat] :
          ( ( ( N3 = zero_zero_nat )
           => ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A2 ) )
          & ( ( N3 != zero_zero_nat )
           => ( bit_se9216721137139052372nteger @ ( divide6298287555418463151nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( minus_minus_nat @ N3 @ one_one_nat ) ) ) ) ) ) ).

% bit_rec
thf(fact_6748_bit__rec,axiom,
    ( bit_se8040316288895769887atural
    = ( ^ [A2: code_natural,N3: nat] :
          ( ( ( N3 = zero_zero_nat )
           => ~ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A2 ) )
          & ( ( N3 != zero_zero_nat )
           => ( bit_se8040316288895769887atural @ ( divide5121882707175180666atural @ A2 @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) @ ( minus_minus_nat @ N3 @ one_one_nat ) ) ) ) ) ) ).

% bit_rec
thf(fact_6749_mask__eq__exp__minus__1,axiom,
    ( bit_se2000444600071755411sk_int
    = ( ^ [N3: nat] : ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) @ one_one_int ) ) ) ).

% mask_eq_exp_minus_1
thf(fact_6750_mask__eq__exp__minus__1,axiom,
    ( bit_se2002935070580805687sk_nat
    = ( ^ [N3: nat] : ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ one_one_nat ) ) ) ).

% mask_eq_exp_minus_1
thf(fact_6751_mask__eq__exp__minus__1,axiom,
    ( bit_se2119862282449309892nteger
    = ( ^ [N3: nat] : ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N3 ) @ one_one_Code_integer ) ) ) ).

% mask_eq_exp_minus_1
thf(fact_6752_mask__eq__exp__minus__1,axiom,
    ( bit_se943457434206027407atural
    = ( ^ [N3: nat] : ( minus_7197305767214868737atural @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N3 ) @ one_one_Code_natural ) ) ) ).

% mask_eq_exp_minus_1
thf(fact_6753_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_6754_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_6755_set__bit__eq,axiom,
    ( bit_se7879613467334960850it_int
    = ( ^ [N3: nat,K3: int] :
          ( plus_plus_int @ K3
          @ ( times_times_int
            @ ( zero_n2684676970156552555ol_int
              @ ~ ( bit_se1146084159140164899it_int @ K3 @ N3 ) )
            @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ) ).

% set_bit_eq
thf(fact_6756_unset__bit__eq,axiom,
    ( bit_se4203085406695923979it_int
    = ( ^ [N3: nat,K3: int] : ( minus_minus_int @ K3 @ ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K3 @ N3 ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ) ).

% unset_bit_eq
thf(fact_6757_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_6758_num_Osize_I5_J,axiom,
    ! [X22: num] :
      ( ( size_size_num @ ( bit0 @ X22 ) )
      = ( plus_plus_nat @ ( size_size_num @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).

% num.size(5)
thf(fact_6759_or__int__rec,axiom,
    ( bit_se1409905431419307370or_int
    = ( ^ [K3: int,L3: int] :
          ( plus_plus_int
          @ ( zero_n2684676970156552555ol_int
            @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 )
              | ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L3 ) ) )
          @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% or_int_rec
thf(fact_6760_drop__bit__numeral__minus__bit1,axiom,
    ! [L: num,K: num] :
      ( ( bit_se8568078237143864401it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
      = ( bit_se8568078237143864401it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) ) ).

% drop_bit_numeral_minus_bit1
thf(fact_6761_or__numerals_I7_J,axiom,
    ! [X: num,Y3: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ ( numeral_numeral_int @ ( bit1 @ Y3 ) ) )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y3 ) ) ) ) ) ).

% or_numerals(7)
thf(fact_6762_or__numerals_I7_J,axiom,
    ! [X: num,Y3: num] :
      ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) )
      = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y3 ) ) ) ) ) ).

% or_numerals(7)
thf(fact_6763_or__numerals_I7_J,axiom,
    ! [X: num,Y3: num] :
      ( ( bit_se1080825931792720795nteger @ ( numera6620942414471956472nteger @ ( bit1 @ X ) ) @ ( numera6620942414471956472nteger @ ( bit1 @ Y3 ) ) )
      = ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1080825931792720795nteger @ ( numera6620942414471956472nteger @ X ) @ ( numera6620942414471956472nteger @ Y3 ) ) ) ) ) ).

% or_numerals(7)
thf(fact_6764_or__numerals_I7_J,axiom,
    ! [X: num,Y3: num] :
      ( ( bit_se9127793120404214118atural @ ( numera5444537566228673987atural @ ( bit1 @ X ) ) @ ( numera5444537566228673987atural @ ( bit1 @ Y3 ) ) )
      = ( plus_p4538020629002901425atural @ one_one_Code_natural @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( bit_se9127793120404214118atural @ ( numera5444537566228673987atural @ X ) @ ( numera5444537566228673987atural @ Y3 ) ) ) ) ) ).

% or_numerals(7)
thf(fact_6765_or__numerals_I6_J,axiom,
    ! [X: num,Y3: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ ( numeral_numeral_int @ ( bit0 @ Y3 ) ) )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y3 ) ) ) ) ) ).

% or_numerals(6)
thf(fact_6766_or__numerals_I6_J,axiom,
    ! [X: num,Y3: num] :
      ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) )
      = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y3 ) ) ) ) ) ).

% or_numerals(6)
thf(fact_6767_or__numerals_I6_J,axiom,
    ! [X: num,Y3: num] :
      ( ( bit_se1080825931792720795nteger @ ( numera6620942414471956472nteger @ ( bit1 @ X ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Y3 ) ) )
      = ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1080825931792720795nteger @ ( numera6620942414471956472nteger @ X ) @ ( numera6620942414471956472nteger @ Y3 ) ) ) ) ) ).

% or_numerals(6)
thf(fact_6768_or__numerals_I6_J,axiom,
    ! [X: num,Y3: num] :
      ( ( bit_se9127793120404214118atural @ ( numera5444537566228673987atural @ ( bit1 @ X ) ) @ ( numera5444537566228673987atural @ ( bit0 @ Y3 ) ) )
      = ( plus_p4538020629002901425atural @ one_one_Code_natural @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( bit_se9127793120404214118atural @ ( numera5444537566228673987atural @ X ) @ ( numera5444537566228673987atural @ Y3 ) ) ) ) ) ).

% or_numerals(6)
thf(fact_6769_or_Oright__neutral,axiom,
    ! [A: int] :
      ( ( bit_se1409905431419307370or_int @ A @ zero_zero_int )
      = A ) ).

% or.right_neutral
thf(fact_6770_or_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( bit_se1412395901928357646or_nat @ A @ zero_zero_nat )
      = A ) ).

% or.right_neutral
thf(fact_6771_or_Oright__neutral,axiom,
    ! [A: code_integer] :
      ( ( bit_se1080825931792720795nteger @ A @ zero_z3403309356797280102nteger )
      = A ) ).

% or.right_neutral
thf(fact_6772_or_Oright__neutral,axiom,
    ! [A: code_natural] :
      ( ( bit_se9127793120404214118atural @ A @ zero_z2226904508553997617atural )
      = A ) ).

% or.right_neutral
thf(fact_6773_or_Oleft__neutral,axiom,
    ! [A: int] :
      ( ( bit_se1409905431419307370or_int @ zero_zero_int @ A )
      = A ) ).

% or.left_neutral
thf(fact_6774_or_Oleft__neutral,axiom,
    ! [A: nat] :
      ( ( bit_se1412395901928357646or_nat @ zero_zero_nat @ A )
      = A ) ).

% or.left_neutral
thf(fact_6775_or_Oleft__neutral,axiom,
    ! [A: code_integer] :
      ( ( bit_se1080825931792720795nteger @ zero_z3403309356797280102nteger @ A )
      = A ) ).

% or.left_neutral
thf(fact_6776_or_Oleft__neutral,axiom,
    ! [A: code_natural] :
      ( ( bit_se9127793120404214118atural @ zero_z2226904508553997617atural @ A )
      = A ) ).

% or.left_neutral
thf(fact_6777_drop__bit__of__0,axiom,
    ! [N: nat] :
      ( ( bit_se8568078237143864401it_int @ N @ zero_zero_int )
      = zero_zero_int ) ).

% drop_bit_of_0
thf(fact_6778_drop__bit__of__0,axiom,
    ! [N: nat] :
      ( ( bit_se8570568707652914677it_nat @ N @ zero_zero_nat )
      = zero_zero_nat ) ).

% drop_bit_of_0
thf(fact_6779_drop__bit__of__0,axiom,
    ! [N: nat] :
      ( ( bit_se3928097537394005634nteger @ N @ zero_z3403309356797280102nteger )
      = zero_z3403309356797280102nteger ) ).

% drop_bit_of_0
thf(fact_6780_drop__bit__of__0,axiom,
    ! [N: nat] :
      ( ( bit_se2751692689150723149atural @ N @ zero_z2226904508553997617atural )
      = zero_z2226904508553997617atural ) ).

% drop_bit_of_0
thf(fact_6781_bit_Odisj__one__left,axiom,
    ! [X: int] :
      ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ one_one_int ) @ X )
      = ( uminus_uminus_int @ one_one_int ) ) ).

% bit.disj_one_left
thf(fact_6782_bit_Odisj__one__left,axiom,
    ! [X: code_integer] :
      ( ( bit_se1080825931792720795nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ X )
      = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% bit.disj_one_left
thf(fact_6783_bit_Odisj__one__right,axiom,
    ! [X: int] :
      ( ( bit_se1409905431419307370or_int @ X @ ( uminus_uminus_int @ one_one_int ) )
      = ( uminus_uminus_int @ one_one_int ) ) ).

% bit.disj_one_right
thf(fact_6784_bit_Odisj__one__right,axiom,
    ! [X: code_integer] :
      ( ( bit_se1080825931792720795nteger @ X @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% bit.disj_one_right
thf(fact_6785_drop__bit__of__bool,axiom,
    ! [N: nat,B: $o] :
      ( ( bit_se8568078237143864401it_int @ N @ ( zero_n2684676970156552555ol_int @ B ) )
      = ( zero_n2684676970156552555ol_int
        @ ( ( N = zero_zero_nat )
          & B ) ) ) ).

% drop_bit_of_bool
thf(fact_6786_drop__bit__of__bool,axiom,
    ! [N: nat,B: $o] :
      ( ( bit_se8570568707652914677it_nat @ N @ ( zero_n2687167440665602831ol_nat @ B ) )
      = ( zero_n2687167440665602831ol_nat
        @ ( ( N = zero_zero_nat )
          & B ) ) ) ).

% drop_bit_of_bool
thf(fact_6787_drop__bit__of__bool,axiom,
    ! [N: nat,B: $o] :
      ( ( bit_se3928097537394005634nteger @ N @ ( zero_n356916108424825756nteger @ B ) )
      = ( zero_n356916108424825756nteger
        @ ( ( N = zero_zero_nat )
          & B ) ) ) ).

% drop_bit_of_bool
thf(fact_6788_drop__bit__of__bool,axiom,
    ! [N: nat,B: $o] :
      ( ( bit_se2751692689150723149atural @ N @ ( zero_n8403883297036319079atural @ B ) )
      = ( zero_n8403883297036319079atural
        @ ( ( N = zero_zero_nat )
          & B ) ) ) ).

% drop_bit_of_bool
thf(fact_6789_or__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se1409905431419307370or_int @ K @ L ) )
      = ( ( ord_less_eq_int @ zero_zero_int @ K )
        & ( ord_less_eq_int @ zero_zero_int @ L ) ) ) ).

% or_nonnegative_int_iff
thf(fact_6790_or__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_int @ ( bit_se1409905431419307370or_int @ K @ L ) @ zero_zero_int )
      = ( ( ord_less_int @ K @ zero_zero_int )
        | ( ord_less_int @ L @ zero_zero_int ) ) ) ).

% or_negative_int_iff
thf(fact_6791_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_6792_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_6793_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_6794_or__nat__numerals_I4_J,axiom,
    ! [X: num] :
      ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( suc @ zero_zero_nat ) )
      = ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).

% or_nat_numerals(4)
thf(fact_6795_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_6796_drop__bit__Suc__bit0,axiom,
    ! [N: nat,K: num] :
      ( ( bit_se8568078237143864401it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
      = ( bit_se8568078237143864401it_int @ N @ ( numeral_numeral_int @ K ) ) ) ).

% drop_bit_Suc_bit0
thf(fact_6797_drop__bit__Suc__bit0,axiom,
    ! [N: nat,K: num] :
      ( ( bit_se8570568707652914677it_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
      = ( bit_se8570568707652914677it_nat @ N @ ( numeral_numeral_nat @ K ) ) ) ).

% drop_bit_Suc_bit0
thf(fact_6798_drop__bit__Suc__bit0,axiom,
    ! [N: nat,K: num] :
      ( ( bit_se3928097537394005634nteger @ ( suc @ N ) @ ( numera6620942414471956472nteger @ ( bit0 @ K ) ) )
      = ( bit_se3928097537394005634nteger @ N @ ( numera6620942414471956472nteger @ K ) ) ) ).

% drop_bit_Suc_bit0
thf(fact_6799_drop__bit__Suc__bit0,axiom,
    ! [N: nat,K: num] :
      ( ( bit_se2751692689150723149atural @ ( suc @ N ) @ ( numera5444537566228673987atural @ ( bit0 @ K ) ) )
      = ( bit_se2751692689150723149atural @ N @ ( numera5444537566228673987atural @ K ) ) ) ).

% drop_bit_Suc_bit0
thf(fact_6800_or__numerals_I2_J,axiom,
    ! [Y3: num] :
      ( ( bit_se1409905431419307370or_int @ one_one_int @ ( numeral_numeral_int @ ( bit1 @ Y3 ) ) )
      = ( numeral_numeral_int @ ( bit1 @ Y3 ) ) ) ).

% or_numerals(2)
thf(fact_6801_or__numerals_I2_J,axiom,
    ! [Y3: num] :
      ( ( bit_se1412395901928357646or_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) )
      = ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) ) ).

% or_numerals(2)
thf(fact_6802_or__numerals_I2_J,axiom,
    ! [Y3: num] :
      ( ( bit_se1080825931792720795nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit1 @ Y3 ) ) )
      = ( numera6620942414471956472nteger @ ( bit1 @ Y3 ) ) ) ).

% or_numerals(2)
thf(fact_6803_or__numerals_I2_J,axiom,
    ! [Y3: num] :
      ( ( bit_se9127793120404214118atural @ one_one_Code_natural @ ( numera5444537566228673987atural @ ( bit1 @ Y3 ) ) )
      = ( numera5444537566228673987atural @ ( bit1 @ Y3 ) ) ) ).

% or_numerals(2)
thf(fact_6804_or__numerals_I8_J,axiom,
    ! [X: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ one_one_int )
      = ( numeral_numeral_int @ ( bit1 @ X ) ) ) ).

% or_numerals(8)
thf(fact_6805_or__numerals_I8_J,axiom,
    ! [X: num] :
      ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ one_one_nat )
      = ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).

% or_numerals(8)
thf(fact_6806_or__numerals_I8_J,axiom,
    ! [X: num] :
      ( ( bit_se1080825931792720795nteger @ ( numera6620942414471956472nteger @ ( bit1 @ X ) ) @ one_one_Code_integer )
      = ( numera6620942414471956472nteger @ ( bit1 @ X ) ) ) ).

% or_numerals(8)
thf(fact_6807_or__numerals_I8_J,axiom,
    ! [X: num] :
      ( ( bit_se9127793120404214118atural @ ( numera5444537566228673987atural @ ( bit1 @ X ) ) @ one_one_Code_natural )
      = ( numera5444537566228673987atural @ ( bit1 @ X ) ) ) ).

% or_numerals(8)
thf(fact_6808_bit_Odisj__cancel__left,axiom,
    ! [X: int] :
      ( ( bit_se1409905431419307370or_int @ ( bit_ri7919022796975470100ot_int @ X ) @ X )
      = ( uminus_uminus_int @ one_one_int ) ) ).

% bit.disj_cancel_left
thf(fact_6809_bit_Odisj__cancel__left,axiom,
    ! [X: code_integer] :
      ( ( bit_se1080825931792720795nteger @ ( bit_ri7632146776885996613nteger @ X ) @ X )
      = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% bit.disj_cancel_left
thf(fact_6810_bit_Odisj__cancel__right,axiom,
    ! [X: int] :
      ( ( bit_se1409905431419307370or_int @ X @ ( bit_ri7919022796975470100ot_int @ X ) )
      = ( uminus_uminus_int @ one_one_int ) ) ).

% bit.disj_cancel_right
thf(fact_6811_bit_Odisj__cancel__right,axiom,
    ! [X: code_integer] :
      ( ( bit_se1080825931792720795nteger @ X @ ( bit_ri7632146776885996613nteger @ X ) )
      = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% bit.disj_cancel_right
thf(fact_6812_drop__bit__Suc__bit1,axiom,
    ! [N: nat,K: num] :
      ( ( bit_se8568078237143864401it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
      = ( bit_se8568078237143864401it_int @ N @ ( numeral_numeral_int @ K ) ) ) ).

% drop_bit_Suc_bit1
thf(fact_6813_drop__bit__Suc__bit1,axiom,
    ! [N: nat,K: num] :
      ( ( bit_se8570568707652914677it_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
      = ( bit_se8570568707652914677it_nat @ N @ ( numeral_numeral_nat @ K ) ) ) ).

% drop_bit_Suc_bit1
thf(fact_6814_drop__bit__Suc__bit1,axiom,
    ! [N: nat,K: num] :
      ( ( bit_se3928097537394005634nteger @ ( suc @ N ) @ ( numera6620942414471956472nteger @ ( bit1 @ K ) ) )
      = ( bit_se3928097537394005634nteger @ N @ ( numera6620942414471956472nteger @ K ) ) ) ).

% drop_bit_Suc_bit1
thf(fact_6815_drop__bit__Suc__bit1,axiom,
    ! [N: nat,K: num] :
      ( ( bit_se2751692689150723149atural @ ( suc @ N ) @ ( numera5444537566228673987atural @ ( bit1 @ K ) ) )
      = ( bit_se2751692689150723149atural @ N @ ( numera5444537566228673987atural @ K ) ) ) ).

% drop_bit_Suc_bit1
thf(fact_6816_drop__bit__of__1,axiom,
    ! [N: nat] :
      ( ( bit_se8568078237143864401it_int @ N @ one_one_int )
      = ( zero_n2684676970156552555ol_int @ ( N = zero_zero_nat ) ) ) ).

% drop_bit_of_1
thf(fact_6817_drop__bit__of__1,axiom,
    ! [N: nat] :
      ( ( bit_se8570568707652914677it_nat @ N @ one_one_nat )
      = ( zero_n2687167440665602831ol_nat @ ( N = zero_zero_nat ) ) ) ).

% drop_bit_of_1
thf(fact_6818_drop__bit__of__1,axiom,
    ! [N: nat] :
      ( ( bit_se3928097537394005634nteger @ N @ one_one_Code_integer )
      = ( zero_n356916108424825756nteger @ ( N = zero_zero_nat ) ) ) ).

% drop_bit_of_1
thf(fact_6819_drop__bit__of__1,axiom,
    ! [N: nat] :
      ( ( bit_se2751692689150723149atural @ N @ one_one_Code_natural )
      = ( zero_n8403883297036319079atural @ ( N = zero_zero_nat ) ) ) ).

% drop_bit_of_1
thf(fact_6820_or__numerals_I3_J,axiom,
    ! [X: num,Y3: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ ( numeral_numeral_int @ ( bit0 @ Y3 ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y3 ) ) ) ) ).

% or_numerals(3)
thf(fact_6821_or__numerals_I3_J,axiom,
    ! [X: num,Y3: num] :
      ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) )
      = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y3 ) ) ) ) ).

% or_numerals(3)
thf(fact_6822_or__numerals_I3_J,axiom,
    ! [X: num,Y3: num] :
      ( ( bit_se1080825931792720795nteger @ ( numera6620942414471956472nteger @ ( bit0 @ X ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Y3 ) ) )
      = ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1080825931792720795nteger @ ( numera6620942414471956472nteger @ X ) @ ( numera6620942414471956472nteger @ Y3 ) ) ) ) ).

% or_numerals(3)
thf(fact_6823_or__numerals_I3_J,axiom,
    ! [X: num,Y3: num] :
      ( ( bit_se9127793120404214118atural @ ( numera5444537566228673987atural @ ( bit0 @ X ) ) @ ( numera5444537566228673987atural @ ( bit0 @ Y3 ) ) )
      = ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( bit_se9127793120404214118atural @ ( numera5444537566228673987atural @ X ) @ ( numera5444537566228673987atural @ Y3 ) ) ) ) ).

% or_numerals(3)
thf(fact_6824_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_6825_or__nat__numerals_I3_J,axiom,
    ! [X: num] :
      ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( suc @ zero_zero_nat ) )
      = ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).

% or_nat_numerals(3)
thf(fact_6826_or__numerals_I1_J,axiom,
    ! [Y3: num] :
      ( ( bit_se1409905431419307370or_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ Y3 ) ) )
      = ( numeral_numeral_int @ ( bit1 @ Y3 ) ) ) ).

% or_numerals(1)
thf(fact_6827_or__numerals_I1_J,axiom,
    ! [Y3: num] :
      ( ( bit_se1412395901928357646or_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) )
      = ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) ) ).

% or_numerals(1)
thf(fact_6828_or__numerals_I1_J,axiom,
    ! [Y3: num] :
      ( ( bit_se1080825931792720795nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ Y3 ) ) )
      = ( numera6620942414471956472nteger @ ( bit1 @ Y3 ) ) ) ).

% or_numerals(1)
thf(fact_6829_or__numerals_I1_J,axiom,
    ! [Y3: num] :
      ( ( bit_se9127793120404214118atural @ one_one_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ Y3 ) ) )
      = ( numera5444537566228673987atural @ ( bit1 @ Y3 ) ) ) ).

% or_numerals(1)
thf(fact_6830_or__numerals_I5_J,axiom,
    ! [X: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ one_one_int )
      = ( numeral_numeral_int @ ( bit1 @ X ) ) ) ).

% or_numerals(5)
thf(fact_6831_or__numerals_I5_J,axiom,
    ! [X: num] :
      ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ one_one_nat )
      = ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).

% or_numerals(5)
thf(fact_6832_or__numerals_I5_J,axiom,
    ! [X: num] :
      ( ( bit_se1080825931792720795nteger @ ( numera6620942414471956472nteger @ ( bit0 @ X ) ) @ one_one_Code_integer )
      = ( numera6620942414471956472nteger @ ( bit1 @ X ) ) ) ).

% or_numerals(5)
thf(fact_6833_or__numerals_I5_J,axiom,
    ! [X: num] :
      ( ( bit_se9127793120404214118atural @ ( numera5444537566228673987atural @ ( bit0 @ X ) ) @ one_one_Code_natural )
      = ( numera5444537566228673987atural @ ( bit1 @ X ) ) ) ).

% or_numerals(5)
thf(fact_6834_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_6835_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_6836_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_6837_drop__bit__numeral__minus__bit0,axiom,
    ! [L: num,K: num] :
      ( ( bit_se8568078237143864401it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
      = ( bit_se8568078237143864401it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) ) ).

% drop_bit_numeral_minus_bit0
thf(fact_6838_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_6839_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_6840_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_6841_or__numerals_I4_J,axiom,
    ! [X: num,Y3: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ ( numeral_numeral_int @ ( bit1 @ Y3 ) ) )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y3 ) ) ) ) ) ).

% or_numerals(4)
thf(fact_6842_or__numerals_I4_J,axiom,
    ! [X: num,Y3: num] :
      ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) )
      = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y3 ) ) ) ) ) ).

% or_numerals(4)
thf(fact_6843_or__numerals_I4_J,axiom,
    ! [X: num,Y3: num] :
      ( ( bit_se1080825931792720795nteger @ ( numera6620942414471956472nteger @ ( bit0 @ X ) ) @ ( numera6620942414471956472nteger @ ( bit1 @ Y3 ) ) )
      = ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1080825931792720795nteger @ ( numera6620942414471956472nteger @ X ) @ ( numera6620942414471956472nteger @ Y3 ) ) ) ) ) ).

% or_numerals(4)
thf(fact_6844_or__numerals_I4_J,axiom,
    ! [X: num,Y3: num] :
      ( ( bit_se9127793120404214118atural @ ( numera5444537566228673987atural @ ( bit0 @ X ) ) @ ( numera5444537566228673987atural @ ( bit1 @ Y3 ) ) )
      = ( plus_p4538020629002901425atural @ one_one_Code_natural @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( bit_se9127793120404214118atural @ ( numera5444537566228673987atural @ X ) @ ( numera5444537566228673987atural @ Y3 ) ) ) ) ) ).

% or_numerals(4)
thf(fact_6845_size__neq__size__imp__neq,axiom,
    ! [X: char,Y3: char] :
      ( ( ( size_size_char @ X )
       != ( size_size_char @ Y3 ) )
     => ( X != Y3 ) ) ).

% size_neq_size_imp_neq
thf(fact_6846_size__neq__size__imp__neq,axiom,
    ! [X: list_o,Y3: list_o] :
      ( ( ( size_size_list_o @ X )
       != ( size_size_list_o @ Y3 ) )
     => ( X != Y3 ) ) ).

% size_neq_size_imp_neq
thf(fact_6847_size__neq__size__imp__neq,axiom,
    ! [X: list_nat,Y3: list_nat] :
      ( ( ( size_size_list_nat @ X )
       != ( size_size_list_nat @ Y3 ) )
     => ( X != Y3 ) ) ).

% size_neq_size_imp_neq
thf(fact_6848_size__neq__size__imp__neq,axiom,
    ! [X: list_int,Y3: list_int] :
      ( ( ( size_size_list_int @ X )
       != ( size_size_list_int @ Y3 ) )
     => ( X != Y3 ) ) ).

% size_neq_size_imp_neq
thf(fact_6849_size__neq__size__imp__neq,axiom,
    ! [X: num,Y3: num] :
      ( ( ( size_size_num @ X )
       != ( size_size_num @ Y3 ) )
     => ( X != Y3 ) ) ).

% size_neq_size_imp_neq
thf(fact_6850_or__nat__def,axiom,
    ( bit_se1412395901928357646or_nat
    = ( ^ [M2: nat,N3: nat] : ( nat2 @ ( bit_se1409905431419307370or_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).

% or_nat_def
thf(fact_6851_of__nat__or__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri1314217659103216013at_int @ ( bit_se1412395901928357646or_nat @ M @ N ) )
      = ( bit_se1409905431419307370or_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).

% of_nat_or_eq
thf(fact_6852_of__nat__or__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri1316708129612266289at_nat @ ( bit_se1412395901928357646or_nat @ M @ N ) )
      = ( bit_se1412395901928357646or_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).

% of_nat_or_eq
thf(fact_6853_of__nat__or__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri4939895301339042750nteger @ ( bit_se1412395901928357646or_nat @ M @ N ) )
      = ( bit_se1080825931792720795nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) ) ) ).

% of_nat_or_eq
thf(fact_6854_of__nat__or__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri3763490453095760265atural @ ( bit_se1412395901928357646or_nat @ M @ N ) )
      = ( bit_se9127793120404214118atural @ ( semiri3763490453095760265atural @ M ) @ ( semiri3763490453095760265atural @ N ) ) ) ).

% of_nat_or_eq
thf(fact_6855_drop__bit__of__nat,axiom,
    ! [N: nat,M: nat] :
      ( ( bit_se8568078237143864401it_int @ N @ ( semiri1314217659103216013at_int @ M ) )
      = ( semiri1314217659103216013at_int @ ( bit_se8570568707652914677it_nat @ N @ M ) ) ) ).

% drop_bit_of_nat
thf(fact_6856_drop__bit__of__nat,axiom,
    ! [N: nat,M: nat] :
      ( ( bit_se8570568707652914677it_nat @ N @ ( semiri1316708129612266289at_nat @ M ) )
      = ( semiri1316708129612266289at_nat @ ( bit_se8570568707652914677it_nat @ N @ M ) ) ) ).

% drop_bit_of_nat
thf(fact_6857_drop__bit__of__nat,axiom,
    ! [N: nat,M: nat] :
      ( ( bit_se3928097537394005634nteger @ N @ ( semiri4939895301339042750nteger @ M ) )
      = ( semiri4939895301339042750nteger @ ( bit_se8570568707652914677it_nat @ N @ M ) ) ) ).

% drop_bit_of_nat
thf(fact_6858_drop__bit__of__nat,axiom,
    ! [N: nat,M: nat] :
      ( ( bit_se2751692689150723149atural @ N @ ( semiri3763490453095760265atural @ M ) )
      = ( semiri3763490453095760265atural @ ( bit_se8570568707652914677it_nat @ N @ M ) ) ) ).

% drop_bit_of_nat
thf(fact_6859_of__nat__drop__bit,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri1314217659103216013at_int @ ( bit_se8570568707652914677it_nat @ M @ N ) )
      = ( bit_se8568078237143864401it_int @ M @ ( semiri1314217659103216013at_int @ N ) ) ) ).

% of_nat_drop_bit
thf(fact_6860_of__nat__drop__bit,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri1316708129612266289at_nat @ ( bit_se8570568707652914677it_nat @ M @ N ) )
      = ( bit_se8570568707652914677it_nat @ M @ ( semiri1316708129612266289at_nat @ N ) ) ) ).

% of_nat_drop_bit
thf(fact_6861_of__nat__drop__bit,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri4939895301339042750nteger @ ( bit_se8570568707652914677it_nat @ M @ N ) )
      = ( bit_se3928097537394005634nteger @ M @ ( semiri4939895301339042750nteger @ N ) ) ) ).

% of_nat_drop_bit
thf(fact_6862_of__nat__drop__bit,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri3763490453095760265atural @ ( bit_se8570568707652914677it_nat @ M @ N ) )
      = ( bit_se2751692689150723149atural @ M @ ( semiri3763490453095760265atural @ N ) ) ) ).

% of_nat_drop_bit
thf(fact_6863_bit_Odisj__zero__right,axiom,
    ! [X: int] :
      ( ( bit_se1409905431419307370or_int @ X @ zero_zero_int )
      = X ) ).

% bit.disj_zero_right
thf(fact_6864_bit_Odisj__zero__right,axiom,
    ! [X: code_integer] :
      ( ( bit_se1080825931792720795nteger @ X @ zero_z3403309356797280102nteger )
      = X ) ).

% bit.disj_zero_right
thf(fact_6865_or__eq__0__iff,axiom,
    ! [A: int,B: int] :
      ( ( ( bit_se1409905431419307370or_int @ A @ B )
        = zero_zero_int )
      = ( ( A = zero_zero_int )
        & ( B = zero_zero_int ) ) ) ).

% or_eq_0_iff
thf(fact_6866_or__eq__0__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( ( bit_se1412395901928357646or_nat @ A @ B )
        = zero_zero_nat )
      = ( ( A = zero_zero_nat )
        & ( B = zero_zero_nat ) ) ) ).

% or_eq_0_iff
thf(fact_6867_or__eq__0__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( bit_se1080825931792720795nteger @ A @ B )
        = zero_z3403309356797280102nteger )
      = ( ( A = zero_z3403309356797280102nteger )
        & ( B = zero_z3403309356797280102nteger ) ) ) ).

% or_eq_0_iff
thf(fact_6868_or__eq__0__iff,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( ( bit_se9127793120404214118atural @ A @ B )
        = zero_z2226904508553997617atural )
      = ( ( A = zero_z2226904508553997617atural )
        & ( B = zero_z2226904508553997617atural ) ) ) ).

% or_eq_0_iff
thf(fact_6869_set__bit__nat__def,axiom,
    ( bit_se7882103937844011126it_nat
    = ( ^ [M2: nat,N3: nat] : ( bit_se1412395901928357646or_nat @ N3 @ ( bit_se547839408752420682it_nat @ M2 @ one_one_nat ) ) ) ) ).

% set_bit_nat_def
thf(fact_6870_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_6871_not__bit__Suc__0__Suc,axiom,
    ! [N: nat] :
      ~ ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ ( suc @ N ) ) ).

% not_bit_Suc_0_Suc
thf(fact_6872_bit__push__bit__iff__nat,axiom,
    ! [M: nat,Q4: nat,N: nat] :
      ( ( bit_se1148574629649215175it_nat @ ( bit_se547839408752420682it_nat @ M @ Q4 ) @ N )
      = ( ( ord_less_eq_nat @ M @ N )
        & ( bit_se1148574629649215175it_nat @ Q4 @ ( minus_minus_nat @ N @ M ) ) ) ) ).

% bit_push_bit_iff_nat
thf(fact_6873_OR__lower,axiom,
    ! [X: int,Y3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
       => ( ord_less_eq_int @ zero_zero_int @ ( bit_se1409905431419307370or_int @ X @ Y3 ) ) ) ) ).

% OR_lower
thf(fact_6874_or__greater__eq,axiom,
    ! [L: int,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ L )
     => ( ord_less_eq_int @ K @ ( bit_se1409905431419307370or_int @ K @ L ) ) ) ).

% or_greater_eq
thf(fact_6875_set__bit__int__def,axiom,
    ( bit_se7879613467334960850it_int
    = ( ^ [N3: nat,K3: int] : ( bit_se1409905431419307370or_int @ K3 @ ( bit_se545348938243370406it_int @ N3 @ one_one_int ) ) ) ) ).

% set_bit_int_def
thf(fact_6876_take__bit__eq__self__iff__drop__bit__eq__0,axiom,
    ! [N: nat,A: int] :
      ( ( ( bit_se2923211474154528505it_int @ N @ A )
        = A )
      = ( ( bit_se8568078237143864401it_int @ N @ A )
        = zero_zero_int ) ) ).

% take_bit_eq_self_iff_drop_bit_eq_0
thf(fact_6877_take__bit__eq__self__iff__drop__bit__eq__0,axiom,
    ! [N: nat,A: nat] :
      ( ( ( bit_se2925701944663578781it_nat @ N @ A )
        = A )
      = ( ( bit_se8570568707652914677it_nat @ N @ A )
        = zero_zero_nat ) ) ).

% take_bit_eq_self_iff_drop_bit_eq_0
thf(fact_6878_take__bit__eq__self__iff__drop__bit__eq__0,axiom,
    ! [N: nat,A: code_integer] :
      ( ( ( bit_se1745604003318907178nteger @ N @ A )
        = A )
      = ( ( bit_se3928097537394005634nteger @ N @ A )
        = zero_z3403309356797280102nteger ) ) ).

% take_bit_eq_self_iff_drop_bit_eq_0
thf(fact_6879_take__bit__eq__self__iff__drop__bit__eq__0,axiom,
    ! [N: nat,A: code_natural] :
      ( ( ( bit_se569199155075624693atural @ N @ A )
        = A )
      = ( ( bit_se2751692689150723149atural @ N @ A )
        = zero_z2226904508553997617atural ) ) ).

% take_bit_eq_self_iff_drop_bit_eq_0
thf(fact_6880_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_6881_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_6882_set__bit__eq__or,axiom,
    ( bit_se7879613467334960850it_int
    = ( ^ [N3: nat,A2: int] : ( bit_se1409905431419307370or_int @ A2 @ ( bit_se545348938243370406it_int @ N3 @ one_one_int ) ) ) ) ).

% set_bit_eq_or
thf(fact_6883_set__bit__eq__or,axiom,
    ( bit_se7882103937844011126it_nat
    = ( ^ [N3: nat,A2: nat] : ( bit_se1412395901928357646or_nat @ A2 @ ( bit_se547839408752420682it_nat @ N3 @ one_one_nat ) ) ) ) ).

% set_bit_eq_or
thf(fact_6884_set__bit__eq__or,axiom,
    ( bit_se2793503036327961859nteger
    = ( ^ [N3: nat,A2: code_integer] : ( bit_se1080825931792720795nteger @ A2 @ ( bit_se7788150548672797655nteger @ N3 @ one_one_Code_integer ) ) ) ) ).

% set_bit_eq_or
thf(fact_6885_set__bit__eq__or,axiom,
    ( bit_se1617098188084679374atural
    = ( ^ [N3: nat,A2: code_natural] : ( bit_se9127793120404214118atural @ A2 @ ( bit_se6611745700429515170atural @ N3 @ one_one_Code_natural ) ) ) ) ).

% set_bit_eq_or
thf(fact_6886_div__push__bit__of__1__eq__drop__bit,axiom,
    ! [A: int,N: nat] :
      ( ( divide_divide_int @ A @ ( bit_se545348938243370406it_int @ N @ one_one_int ) )
      = ( bit_se8568078237143864401it_int @ N @ A ) ) ).

% div_push_bit_of_1_eq_drop_bit
thf(fact_6887_div__push__bit__of__1__eq__drop__bit,axiom,
    ! [A: nat,N: nat] :
      ( ( divide_divide_nat @ A @ ( bit_se547839408752420682it_nat @ N @ one_one_nat ) )
      = ( bit_se8570568707652914677it_nat @ N @ A ) ) ).

% div_push_bit_of_1_eq_drop_bit
thf(fact_6888_div__push__bit__of__1__eq__drop__bit,axiom,
    ! [A: code_integer,N: nat] :
      ( ( divide6298287555418463151nteger @ A @ ( bit_se7788150548672797655nteger @ N @ one_one_Code_integer ) )
      = ( bit_se3928097537394005634nteger @ N @ A ) ) ).

% div_push_bit_of_1_eq_drop_bit
thf(fact_6889_div__push__bit__of__1__eq__drop__bit,axiom,
    ! [A: code_natural,N: nat] :
      ( ( divide5121882707175180666atural @ A @ ( bit_se6611745700429515170atural @ N @ one_one_Code_natural ) )
      = ( bit_se2751692689150723149atural @ N @ A ) ) ).

% div_push_bit_of_1_eq_drop_bit
thf(fact_6890_bit__iff__and__drop__bit__eq__1,axiom,
    ( bit_se1146084159140164899it_int
    = ( ^ [A2: int,N3: nat] :
          ( ( bit_se725231765392027082nd_int @ ( bit_se8568078237143864401it_int @ N3 @ A2 ) @ one_one_int )
          = one_one_int ) ) ) ).

% bit_iff_and_drop_bit_eq_1
thf(fact_6891_bit__iff__and__drop__bit__eq__1,axiom,
    ( bit_se1148574629649215175it_nat
    = ( ^ [A2: nat,N3: nat] :
          ( ( bit_se727722235901077358nd_nat @ ( bit_se8570568707652914677it_nat @ N3 @ A2 ) @ one_one_nat )
          = one_one_nat ) ) ) ).

% bit_iff_and_drop_bit_eq_1
thf(fact_6892_bit__iff__and__drop__bit__eq__1,axiom,
    ( bit_se9216721137139052372nteger
    = ( ^ [A2: code_integer,N3: nat] :
          ( ( bit_se3949692690581998587nteger @ ( bit_se3928097537394005634nteger @ N3 @ A2 ) @ one_one_Code_integer )
          = one_one_Code_integer ) ) ) ).

% bit_iff_and_drop_bit_eq_1
thf(fact_6893_bit__iff__and__drop__bit__eq__1,axiom,
    ( bit_se8040316288895769887atural
    = ( ^ [A2: code_natural,N3: nat] :
          ( ( bit_se2773287842338716102atural @ ( bit_se2751692689150723149atural @ N3 @ A2 ) @ one_one_Code_natural )
          = one_one_Code_natural ) ) ) ).

% bit_iff_and_drop_bit_eq_1
thf(fact_6894_bit_Ocomplement__unique,axiom,
    ! [A: int,X: int,Y3: int] :
      ( ( ( bit_se725231765392027082nd_int @ A @ X )
        = zero_zero_int )
     => ( ( ( bit_se1409905431419307370or_int @ A @ X )
          = ( uminus_uminus_int @ one_one_int ) )
       => ( ( ( bit_se725231765392027082nd_int @ A @ Y3 )
            = zero_zero_int )
         => ( ( ( bit_se1409905431419307370or_int @ A @ Y3 )
              = ( uminus_uminus_int @ one_one_int ) )
           => ( X = Y3 ) ) ) ) ) ).

% bit.complement_unique
thf(fact_6895_bit_Ocomplement__unique,axiom,
    ! [A: code_integer,X: code_integer,Y3: code_integer] :
      ( ( ( bit_se3949692690581998587nteger @ A @ X )
        = zero_z3403309356797280102nteger )
     => ( ( ( bit_se1080825931792720795nteger @ A @ X )
          = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
       => ( ( ( bit_se3949692690581998587nteger @ A @ Y3 )
            = zero_z3403309356797280102nteger )
         => ( ( ( bit_se1080825931792720795nteger @ A @ Y3 )
              = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
           => ( X = Y3 ) ) ) ) ) ).

% bit.complement_unique
thf(fact_6896_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_6897_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_6898_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_6899_num_Osize_I4_J,axiom,
    ( ( size_size_num @ one )
    = zero_zero_nat ) ).

% num.size(4)
thf(fact_6900_bit_Ocompl__unique,axiom,
    ! [X: int,Y3: int] :
      ( ( ( bit_se725231765392027082nd_int @ X @ Y3 )
        = zero_zero_int )
     => ( ( ( bit_se1409905431419307370or_int @ X @ Y3 )
          = ( uminus_uminus_int @ one_one_int ) )
       => ( ( bit_ri7919022796975470100ot_int @ X )
          = Y3 ) ) ) ).

% bit.compl_unique
thf(fact_6901_bit_Ocompl__unique,axiom,
    ! [X: code_integer,Y3: code_integer] :
      ( ( ( bit_se3949692690581998587nteger @ X @ Y3 )
        = zero_z3403309356797280102nteger )
     => ( ( ( bit_se1080825931792720795nteger @ X @ Y3 )
          = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
       => ( ( bit_ri7632146776885996613nteger @ X )
          = Y3 ) ) ) ).

% bit.compl_unique
thf(fact_6902_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_6903_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_6904_drop__bit__Suc,axiom,
    ! [N: nat,A: int] :
      ( ( bit_se8568078237143864401it_int @ ( suc @ N ) @ A )
      = ( bit_se8568078237143864401it_int @ N @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).

% drop_bit_Suc
thf(fact_6905_drop__bit__Suc,axiom,
    ! [N: nat,A: nat] :
      ( ( bit_se8570568707652914677it_nat @ ( suc @ N ) @ A )
      = ( bit_se8570568707652914677it_nat @ N @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% drop_bit_Suc
thf(fact_6906_drop__bit__Suc,axiom,
    ! [N: nat,A: code_integer] :
      ( ( bit_se3928097537394005634nteger @ ( suc @ N ) @ A )
      = ( bit_se3928097537394005634nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).

% drop_bit_Suc
thf(fact_6907_drop__bit__Suc,axiom,
    ! [N: nat,A: code_natural] :
      ( ( bit_se2751692689150723149atural @ ( suc @ N ) @ A )
      = ( bit_se2751692689150723149atural @ N @ ( divide5121882707175180666atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ) ).

% drop_bit_Suc
thf(fact_6908_mask__Suc__exp,axiom,
    ! [N: nat] :
      ( ( bit_se2000444600071755411sk_int @ ( suc @ N ) )
      = ( bit_se1409905431419307370or_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ ( bit_se2000444600071755411sk_int @ N ) ) ) ).

% mask_Suc_exp
thf(fact_6909_mask__Suc__exp,axiom,
    ! [N: nat] :
      ( ( bit_se2002935070580805687sk_nat @ ( suc @ N ) )
      = ( bit_se1412395901928357646or_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ ( bit_se2002935070580805687sk_nat @ N ) ) ) ).

% mask_Suc_exp
thf(fact_6910_mask__Suc__exp,axiom,
    ! [N: nat] :
      ( ( bit_se2119862282449309892nteger @ ( suc @ N ) )
      = ( bit_se1080825931792720795nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) @ ( bit_se2119862282449309892nteger @ N ) ) ) ).

% mask_Suc_exp
thf(fact_6911_mask__Suc__exp,axiom,
    ! [N: nat] :
      ( ( bit_se943457434206027407atural @ ( suc @ N ) )
      = ( bit_se9127793120404214118atural @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) @ ( bit_se943457434206027407atural @ N ) ) ) ).

% mask_Suc_exp
thf(fact_6912_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_6913_one__or__eq,axiom,
    ! [A: int] :
      ( ( bit_se1409905431419307370or_int @ one_one_int @ A )
      = ( plus_plus_int @ A @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ) ).

% one_or_eq
thf(fact_6914_one__or__eq,axiom,
    ! [A: nat] :
      ( ( bit_se1412395901928357646or_nat @ one_one_nat @ A )
      = ( plus_plus_nat @ A @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ) ).

% one_or_eq
thf(fact_6915_one__or__eq,axiom,
    ! [A: code_integer] :
      ( ( bit_se1080825931792720795nteger @ one_one_Code_integer @ A )
      = ( plus_p5714425477246183910nteger @ A @ ( zero_n356916108424825756nteger @ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ) ).

% one_or_eq
thf(fact_6916_one__or__eq,axiom,
    ! [A: code_natural] :
      ( ( bit_se9127793120404214118atural @ one_one_Code_natural @ A )
      = ( plus_p4538020629002901425atural @ A @ ( zero_n8403883297036319079atural @ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A ) ) ) ) ).

% one_or_eq
thf(fact_6917_or__one__eq,axiom,
    ! [A: int] :
      ( ( bit_se1409905431419307370or_int @ A @ one_one_int )
      = ( plus_plus_int @ A @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ) ).

% or_one_eq
thf(fact_6918_or__one__eq,axiom,
    ! [A: nat] :
      ( ( bit_se1412395901928357646or_nat @ A @ one_one_nat )
      = ( plus_plus_nat @ A @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ) ).

% or_one_eq
thf(fact_6919_or__one__eq,axiom,
    ! [A: code_integer] :
      ( ( bit_se1080825931792720795nteger @ A @ one_one_Code_integer )
      = ( plus_p5714425477246183910nteger @ A @ ( zero_n356916108424825756nteger @ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ) ).

% or_one_eq
thf(fact_6920_or__one__eq,axiom,
    ! [A: code_natural] :
      ( ( bit_se9127793120404214118atural @ A @ one_one_Code_natural )
      = ( plus_p4538020629002901425atural @ A @ ( zero_n8403883297036319079atural @ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A ) ) ) ) ).

% or_one_eq
thf(fact_6921_mask__Suc__double,axiom,
    ! [N: nat] :
      ( ( bit_se2000444600071755411sk_int @ ( suc @ N ) )
      = ( bit_se1409905431419307370or_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2000444600071755411sk_int @ N ) ) ) ) ).

% mask_Suc_double
thf(fact_6922_mask__Suc__double,axiom,
    ! [N: nat] :
      ( ( bit_se2002935070580805687sk_nat @ ( suc @ N ) )
      = ( bit_se1412395901928357646or_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2002935070580805687sk_nat @ N ) ) ) ) ).

% mask_Suc_double
thf(fact_6923_mask__Suc__double,axiom,
    ! [N: nat] :
      ( ( bit_se2119862282449309892nteger @ ( suc @ N ) )
      = ( bit_se1080825931792720795nteger @ one_one_Code_integer @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se2119862282449309892nteger @ N ) ) ) ) ).

% mask_Suc_double
thf(fact_6924_mask__Suc__double,axiom,
    ! [N: nat] :
      ( ( bit_se943457434206027407atural @ ( suc @ N ) )
      = ( bit_se9127793120404214118atural @ one_one_Code_natural @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( bit_se943457434206027407atural @ N ) ) ) ) ).

% mask_Suc_double
thf(fact_6925_OR__upper,axiom,
    ! [X: int,N: nat,Y3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_int @ X @ ( 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 @ X @ Y3 ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).

% OR_upper
thf(fact_6926_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_6927_signed__take__bit__def,axiom,
    ( bit_ri6519982836138164636nteger
    = ( ^ [N3: nat,A2: code_integer] : ( bit_se1080825931792720795nteger @ ( bit_se1745604003318907178nteger @ N3 @ A2 ) @ ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ ( bit_se9216721137139052372nteger @ A2 @ N3 ) ) @ ( bit_ri7632146776885996613nteger @ ( bit_se2119862282449309892nteger @ N3 ) ) ) ) ) ) ).

% signed_take_bit_def
thf(fact_6928_signed__take__bit__def,axiom,
    ( bit_ri631733984087533419it_int
    = ( ^ [N3: nat,A2: int] : ( bit_se1409905431419307370or_int @ ( bit_se2923211474154528505it_int @ N3 @ A2 ) @ ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ A2 @ N3 ) ) @ ( bit_ri7919022796975470100ot_int @ ( bit_se2000444600071755411sk_int @ N3 ) ) ) ) ) ) ).

% signed_take_bit_def
thf(fact_6929_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_6930_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_6931_drop__bit__rec,axiom,
    ( bit_se8568078237143864401it_int
    = ( ^ [N3: nat,A2: int] : ( if_int @ ( N3 = zero_zero_nat ) @ A2 @ ( bit_se8568078237143864401it_int @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( divide_divide_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).

% drop_bit_rec
thf(fact_6932_drop__bit__rec,axiom,
    ( bit_se8570568707652914677it_nat
    = ( ^ [N3: nat,A2: nat] : ( if_nat @ ( N3 = zero_zero_nat ) @ A2 @ ( bit_se8570568707652914677it_nat @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( divide_divide_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% drop_bit_rec
thf(fact_6933_drop__bit__rec,axiom,
    ( bit_se3928097537394005634nteger
    = ( ^ [N3: nat,A2: code_integer] : ( if_Code_integer @ ( N3 = zero_zero_nat ) @ A2 @ ( bit_se3928097537394005634nteger @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( divide6298287555418463151nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% drop_bit_rec
thf(fact_6934_drop__bit__rec,axiom,
    ( bit_se2751692689150723149atural
    = ( ^ [N3: nat,A2: code_natural] : ( if_Code_natural @ ( N3 = zero_zero_nat ) @ A2 @ ( bit_se2751692689150723149atural @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( divide5121882707175180666atural @ A2 @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ) ) ) ).

% drop_bit_rec
thf(fact_6935_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_6936_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_6937_or__nat__rec,axiom,
    ( bit_se1412395901928357646or_nat
    = ( ^ [M2: nat,N3: nat] :
          ( plus_plus_nat
          @ ( zero_n2687167440665602831ol_nat
            @ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 )
              | ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) )
          @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( divide_divide_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% or_nat_rec
thf(fact_6938_or__int__unfold,axiom,
    ( bit_se1409905431419307370or_int
    = ( ^ [K3: int,L3: int] :
          ( if_int
          @ ( ( K3
              = ( uminus_uminus_int @ one_one_int ) )
            | ( L3
              = ( uminus_uminus_int @ one_one_int ) ) )
          @ ( uminus_uminus_int @ one_one_int )
          @ ( if_int @ ( K3 = zero_zero_int ) @ L3 @ ( if_int @ ( L3 = zero_zero_int ) @ K3 @ ( plus_plus_int @ ( ord_max_int @ ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L3 @ ( 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 @ L3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).

% or_int_unfold
thf(fact_6939_bit_Oabstract__boolean__algebra__sym__diff__axioms,axiom,
    boolea8527374999097803216ff_int @ bit_se725231765392027082nd_int @ bit_se1409905431419307370or_int @ bit_ri7919022796975470100ot_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) @ bit_se6526347334894502574or_int ).

% bit.abstract_boolean_algebra_sym_diff_axioms
thf(fact_6940_bit_Oabstract__boolean__algebra__sym__diff__axioms,axiom,
    boolea2445317508997433345nteger @ bit_se3949692690581998587nteger @ bit_se1080825931792720795nteger @ bit_ri7632146776885996613nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ bit_se3222712562003087583nteger ).

% bit.abstract_boolean_algebra_sym_diff_axioms
thf(fact_6941_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_6942_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_6943_or__nat__unfold,axiom,
    ( bit_se1412395901928357646or_nat
    = ( ^ [M2: nat,N3: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ N3 @ ( if_nat @ ( N3 = zero_zero_nat ) @ M2 @ ( plus_plus_nat @ ( ord_max_nat @ ( modulo_modulo_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( divide_divide_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% or_nat_unfold
thf(fact_6944_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_6945_max_Oright__idem,axiom,
    ! [A: int,B: int] :
      ( ( ord_max_int @ ( ord_max_int @ A @ B ) @ B )
      = ( ord_max_int @ A @ B ) ) ).

% max.right_idem
thf(fact_6946_max_Oright__idem,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_max_nat @ ( ord_max_nat @ A @ B ) @ B )
      = ( ord_max_nat @ A @ B ) ) ).

% max.right_idem
thf(fact_6947_max_Oright__idem,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_max_Code_integer @ ( ord_max_Code_integer @ A @ B ) @ B )
      = ( ord_max_Code_integer @ A @ B ) ) ).

% max.right_idem
thf(fact_6948_max_Oright__idem,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( ord_max_Code_natural @ ( ord_max_Code_natural @ A @ B ) @ B )
      = ( ord_max_Code_natural @ A @ B ) ) ).

% max.right_idem
thf(fact_6949_max_Oleft__idem,axiom,
    ! [A: int,B: int] :
      ( ( ord_max_int @ A @ ( ord_max_int @ A @ B ) )
      = ( ord_max_int @ A @ B ) ) ).

% max.left_idem
thf(fact_6950_max_Oleft__idem,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_max_nat @ A @ ( ord_max_nat @ A @ B ) )
      = ( ord_max_nat @ A @ B ) ) ).

% max.left_idem
thf(fact_6951_max_Oleft__idem,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_max_Code_integer @ A @ ( ord_max_Code_integer @ A @ B ) )
      = ( ord_max_Code_integer @ A @ B ) ) ).

% max.left_idem
thf(fact_6952_max_Oleft__idem,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( ord_max_Code_natural @ A @ ( ord_max_Code_natural @ A @ B ) )
      = ( ord_max_Code_natural @ A @ B ) ) ).

% max.left_idem
thf(fact_6953_max_Oidem,axiom,
    ! [A: int] :
      ( ( ord_max_int @ A @ A )
      = A ) ).

% max.idem
thf(fact_6954_max_Oidem,axiom,
    ! [A: nat] :
      ( ( ord_max_nat @ A @ A )
      = A ) ).

% max.idem
thf(fact_6955_max_Oidem,axiom,
    ! [A: code_integer] :
      ( ( ord_max_Code_integer @ A @ A )
      = A ) ).

% max.idem
thf(fact_6956_max_Oidem,axiom,
    ! [A: code_natural] :
      ( ( ord_max_Code_natural @ A @ A )
      = A ) ).

% max.idem
thf(fact_6957_max_Oabsorb1,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ B @ A )
     => ( ( ord_max_Code_integer @ A @ B )
        = A ) ) ).

% max.absorb1
thf(fact_6958_max_Oabsorb1,axiom,
    ! [B: code_natural,A: code_natural] :
      ( ( ord_le1926595141338095240atural @ B @ A )
     => ( ( ord_max_Code_natural @ A @ B )
        = A ) ) ).

% max.absorb1
thf(fact_6959_max_Oabsorb1,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( ord_max_rat @ A @ B )
        = A ) ) ).

% max.absorb1
thf(fact_6960_max_Oabsorb1,axiom,
    ! [B: num,A: num] :
      ( ( ord_less_eq_num @ B @ A )
     => ( ( ord_max_num @ A @ B )
        = A ) ) ).

% max.absorb1
thf(fact_6961_max_Oabsorb1,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_max_nat @ A @ B )
        = A ) ) ).

% max.absorb1
thf(fact_6962_max_Oabsorb1,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_max_int @ A @ B )
        = A ) ) ).

% max.absorb1
thf(fact_6963_max_Oabsorb2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ B )
     => ( ( ord_max_Code_integer @ A @ B )
        = B ) ) ).

% max.absorb2
thf(fact_6964_max_Oabsorb2,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( ord_le1926595141338095240atural @ A @ B )
     => ( ( ord_max_Code_natural @ A @ B )
        = B ) ) ).

% max.absorb2
thf(fact_6965_max_Oabsorb2,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_max_rat @ A @ B )
        = B ) ) ).

% max.absorb2
thf(fact_6966_max_Oabsorb2,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_max_num @ A @ B )
        = B ) ) ).

% max.absorb2
thf(fact_6967_max_Oabsorb2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_max_nat @ A @ B )
        = B ) ) ).

% max.absorb2
thf(fact_6968_max_Oabsorb2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_max_int @ A @ B )
        = B ) ) ).

% max.absorb2
thf(fact_6969_max_Obounded__iff,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( ord_max_Code_integer @ B @ C ) @ A )
      = ( ( ord_le3102999989581377725nteger @ B @ A )
        & ( ord_le3102999989581377725nteger @ C @ A ) ) ) ).

% max.bounded_iff
thf(fact_6970_max_Obounded__iff,axiom,
    ! [B: code_natural,C: code_natural,A: code_natural] :
      ( ( ord_le1926595141338095240atural @ ( ord_max_Code_natural @ B @ C ) @ A )
      = ( ( ord_le1926595141338095240atural @ B @ A )
        & ( ord_le1926595141338095240atural @ C @ A ) ) ) ).

% max.bounded_iff
thf(fact_6971_max_Obounded__iff,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( ord_max_rat @ B @ C ) @ A )
      = ( ( ord_less_eq_rat @ B @ A )
        & ( ord_less_eq_rat @ C @ A ) ) ) ).

% max.bounded_iff
thf(fact_6972_max_Obounded__iff,axiom,
    ! [B: num,C: num,A: num] :
      ( ( ord_less_eq_num @ ( ord_max_num @ B @ C ) @ A )
      = ( ( ord_less_eq_num @ B @ A )
        & ( ord_less_eq_num @ C @ A ) ) ) ).

% max.bounded_iff
thf(fact_6973_max_Obounded__iff,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( ord_less_eq_nat @ ( ord_max_nat @ B @ C ) @ A )
      = ( ( ord_less_eq_nat @ B @ A )
        & ( ord_less_eq_nat @ C @ A ) ) ) ).

% max.bounded_iff
thf(fact_6974_max_Obounded__iff,axiom,
    ! [B: int,C: int,A: int] :
      ( ( ord_less_eq_int @ ( ord_max_int @ B @ C ) @ A )
      = ( ( ord_less_eq_int @ B @ A )
        & ( ord_less_eq_int @ C @ A ) ) ) ).

% max.bounded_iff
thf(fact_6975_max_Oabsorb3,axiom,
    ! [B: code_natural,A: code_natural] :
      ( ( ord_le5570908160329646204atural @ B @ A )
     => ( ( ord_max_Code_natural @ A @ B )
        = A ) ) ).

% max.absorb3
thf(fact_6976_max_Oabsorb3,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ B @ A )
     => ( ( ord_max_Code_integer @ A @ B )
        = A ) ) ).

% max.absorb3
thf(fact_6977_max_Oabsorb3,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( ord_max_rat @ A @ B )
        = A ) ) ).

% max.absorb3
thf(fact_6978_max_Oabsorb3,axiom,
    ! [B: num,A: num] :
      ( ( ord_less_num @ B @ A )
     => ( ( ord_max_num @ A @ B )
        = A ) ) ).

% max.absorb3
thf(fact_6979_max_Oabsorb3,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( ord_max_nat @ A @ B )
        = A ) ) ).

% max.absorb3
thf(fact_6980_max_Oabsorb3,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( ord_max_int @ A @ B )
        = A ) ) ).

% max.absorb3
thf(fact_6981_max_Oabsorb4,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( ord_le5570908160329646204atural @ A @ B )
     => ( ( ord_max_Code_natural @ A @ B )
        = B ) ) ).

% max.absorb4
thf(fact_6982_max_Oabsorb4,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ B )
     => ( ( ord_max_Code_integer @ A @ B )
        = B ) ) ).

% max.absorb4
thf(fact_6983_max_Oabsorb4,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_max_rat @ A @ B )
        = B ) ) ).

% max.absorb4
thf(fact_6984_max_Oabsorb4,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_num @ A @ B )
     => ( ( ord_max_num @ A @ B )
        = B ) ) ).

% max.absorb4
thf(fact_6985_max_Oabsorb4,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_max_nat @ A @ B )
        = B ) ) ).

% max.absorb4
thf(fact_6986_max_Oabsorb4,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_max_int @ A @ B )
        = B ) ) ).

% max.absorb4
thf(fact_6987_max__less__iff__conj,axiom,
    ! [X: code_natural,Y3: code_natural,Z2: code_natural] :
      ( ( ord_le5570908160329646204atural @ ( ord_max_Code_natural @ X @ Y3 ) @ Z2 )
      = ( ( ord_le5570908160329646204atural @ X @ Z2 )
        & ( ord_le5570908160329646204atural @ Y3 @ Z2 ) ) ) ).

% max_less_iff_conj
thf(fact_6988_max__less__iff__conj,axiom,
    ! [X: code_integer,Y3: code_integer,Z2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( ord_max_Code_integer @ X @ Y3 ) @ Z2 )
      = ( ( ord_le6747313008572928689nteger @ X @ Z2 )
        & ( ord_le6747313008572928689nteger @ Y3 @ Z2 ) ) ) ).

% max_less_iff_conj
thf(fact_6989_max__less__iff__conj,axiom,
    ! [X: rat,Y3: rat,Z2: rat] :
      ( ( ord_less_rat @ ( ord_max_rat @ X @ Y3 ) @ Z2 )
      = ( ( ord_less_rat @ X @ Z2 )
        & ( ord_less_rat @ Y3 @ Z2 ) ) ) ).

% max_less_iff_conj
thf(fact_6990_max__less__iff__conj,axiom,
    ! [X: num,Y3: num,Z2: num] :
      ( ( ord_less_num @ ( ord_max_num @ X @ Y3 ) @ Z2 )
      = ( ( ord_less_num @ X @ Z2 )
        & ( ord_less_num @ Y3 @ Z2 ) ) ) ).

% max_less_iff_conj
thf(fact_6991_max__less__iff__conj,axiom,
    ! [X: nat,Y3: nat,Z2: nat] :
      ( ( ord_less_nat @ ( ord_max_nat @ X @ Y3 ) @ Z2 )
      = ( ( ord_less_nat @ X @ Z2 )
        & ( ord_less_nat @ Y3 @ Z2 ) ) ) ).

% max_less_iff_conj
thf(fact_6992_max__less__iff__conj,axiom,
    ! [X: int,Y3: int,Z2: int] :
      ( ( ord_less_int @ ( ord_max_int @ X @ Y3 ) @ Z2 )
      = ( ( ord_less_int @ X @ Z2 )
        & ( ord_less_int @ Y3 @ Z2 ) ) ) ).

% max_less_iff_conj
thf(fact_6993_of__bool__or__iff,axiom,
    ! [P: $o,Q: $o] :
      ( ( zero_n356916108424825756nteger
        @ ( P
          | Q ) )
      = ( ord_max_Code_integer @ ( zero_n356916108424825756nteger @ P ) @ ( zero_n356916108424825756nteger @ Q ) ) ) ).

% of_bool_or_iff
thf(fact_6994_of__bool__or__iff,axiom,
    ! [P: $o,Q: $o] :
      ( ( zero_n8403883297036319079atural
        @ ( P
          | Q ) )
      = ( ord_max_Code_natural @ ( zero_n8403883297036319079atural @ P ) @ ( zero_n8403883297036319079atural @ Q ) ) ) ).

% of_bool_or_iff
thf(fact_6995_of__bool__or__iff,axiom,
    ! [P: $o,Q: $o] :
      ( ( zero_n2687167440665602831ol_nat
        @ ( P
          | Q ) )
      = ( ord_max_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q ) ) ) ).

% of_bool_or_iff
thf(fact_6996_of__bool__or__iff,axiom,
    ! [P: $o,Q: $o] :
      ( ( zero_n2684676970156552555ol_int
        @ ( P
          | Q ) )
      = ( ord_max_int @ ( zero_n2684676970156552555ol_int @ P ) @ ( zero_n2684676970156552555ol_int @ Q ) ) ) ).

% of_bool_or_iff
thf(fact_6997_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_6998_max__0R,axiom,
    ! [N: nat] :
      ( ( ord_max_nat @ N @ zero_zero_nat )
      = N ) ).

% max_0R
thf(fact_6999_max__0L,axiom,
    ! [N: nat] :
      ( ( ord_max_nat @ zero_zero_nat @ N )
      = N ) ).

% max_0L
thf(fact_7000_max__nat_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( ord_max_nat @ A @ zero_zero_nat )
      = A ) ).

% max_nat.right_neutral
thf(fact_7001_max__nat_Oneutr__eq__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( zero_zero_nat
        = ( ord_max_nat @ A @ B ) )
      = ( ( A = zero_zero_nat )
        & ( B = zero_zero_nat ) ) ) ).

% max_nat.neutr_eq_iff
thf(fact_7002_max__nat_Oleft__neutral,axiom,
    ! [A: nat] :
      ( ( ord_max_nat @ zero_zero_nat @ A )
      = A ) ).

% max_nat.left_neutral
thf(fact_7003_max__nat_Oeq__neutr__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( ( ord_max_nat @ A @ B )
        = zero_zero_nat )
      = ( ( A = zero_zero_nat )
        & ( B = zero_zero_nat ) ) ) ).

% max_nat.eq_neutr_iff
thf(fact_7004_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ U ) @ ( numera6620942414471956472nteger @ V ) )
       => ( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ U ) @ ( numera6620942414471956472nteger @ V ) )
          = ( numera6620942414471956472nteger @ V ) ) )
      & ( ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ U ) @ ( numera6620942414471956472nteger @ V ) )
       => ( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ U ) @ ( numera6620942414471956472nteger @ V ) )
          = ( numera6620942414471956472nteger @ U ) ) ) ) ).

% max_number_of(1)
thf(fact_7005_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_le1926595141338095240atural @ ( numera5444537566228673987atural @ U ) @ ( numera5444537566228673987atural @ V ) )
       => ( ( ord_max_Code_natural @ ( numera5444537566228673987atural @ U ) @ ( numera5444537566228673987atural @ V ) )
          = ( numera5444537566228673987atural @ V ) ) )
      & ( ~ ( ord_le1926595141338095240atural @ ( numera5444537566228673987atural @ U ) @ ( numera5444537566228673987atural @ V ) )
       => ( ( ord_max_Code_natural @ ( numera5444537566228673987atural @ U ) @ ( numera5444537566228673987atural @ V ) )
          = ( numera5444537566228673987atural @ U ) ) ) ) ).

% max_number_of(1)
thf(fact_7006_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
       => ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
          = ( numeral_numeral_rat @ V ) ) )
      & ( ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
       => ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
          = ( numeral_numeral_rat @ U ) ) ) ) ).

% max_number_of(1)
thf(fact_7007_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
       => ( ( ord_max_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
          = ( numeral_numeral_nat @ V ) ) )
      & ( ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
       => ( ( ord_max_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
          = ( numeral_numeral_nat @ U ) ) ) ) ).

% max_number_of(1)
thf(fact_7008_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
       => ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
          = ( numeral_numeral_int @ V ) ) )
      & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
       => ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
          = ( numeral_numeral_int @ U ) ) ) ) ).

% max_number_of(1)
thf(fact_7009_max__0__1_I4_J,axiom,
    ! [X: num] :
      ( ( ord_max_int @ ( numeral_numeral_int @ X ) @ zero_zero_int )
      = ( numeral_numeral_int @ X ) ) ).

% max_0_1(4)
thf(fact_7010_max__0__1_I4_J,axiom,
    ! [X: num] :
      ( ( ord_max_nat @ ( numeral_numeral_nat @ X ) @ zero_zero_nat )
      = ( numeral_numeral_nat @ X ) ) ).

% max_0_1(4)
thf(fact_7011_max__0__1_I4_J,axiom,
    ! [X: num] :
      ( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ X ) @ zero_z3403309356797280102nteger )
      = ( numera6620942414471956472nteger @ X ) ) ).

% max_0_1(4)
thf(fact_7012_max__0__1_I4_J,axiom,
    ! [X: num] :
      ( ( ord_max_rat @ ( numeral_numeral_rat @ X ) @ zero_zero_rat )
      = ( numeral_numeral_rat @ X ) ) ).

% max_0_1(4)
thf(fact_7013_max__0__1_I4_J,axiom,
    ! [X: num] :
      ( ( ord_max_Code_natural @ ( numera5444537566228673987atural @ X ) @ zero_z2226904508553997617atural )
      = ( numera5444537566228673987atural @ X ) ) ).

% max_0_1(4)
thf(fact_7014_max__0__1_I3_J,axiom,
    ! [X: num] :
      ( ( ord_max_int @ zero_zero_int @ ( numeral_numeral_int @ X ) )
      = ( numeral_numeral_int @ X ) ) ).

% max_0_1(3)
thf(fact_7015_max__0__1_I3_J,axiom,
    ! [X: num] :
      ( ( ord_max_nat @ zero_zero_nat @ ( numeral_numeral_nat @ X ) )
      = ( numeral_numeral_nat @ X ) ) ).

% max_0_1(3)
thf(fact_7016_max__0__1_I3_J,axiom,
    ! [X: num] :
      ( ( ord_max_Code_integer @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ X ) )
      = ( numera6620942414471956472nteger @ X ) ) ).

% max_0_1(3)
thf(fact_7017_max__0__1_I3_J,axiom,
    ! [X: num] :
      ( ( ord_max_rat @ zero_zero_rat @ ( numeral_numeral_rat @ X ) )
      = ( numeral_numeral_rat @ X ) ) ).

% max_0_1(3)
thf(fact_7018_max__0__1_I3_J,axiom,
    ! [X: num] :
      ( ( ord_max_Code_natural @ zero_z2226904508553997617atural @ ( numera5444537566228673987atural @ X ) )
      = ( numera5444537566228673987atural @ X ) ) ).

% max_0_1(3)
thf(fact_7019_max__0__1_I2_J,axiom,
    ( ( ord_max_rat @ one_one_rat @ zero_zero_rat )
    = one_one_rat ) ).

% max_0_1(2)
thf(fact_7020_max__0__1_I2_J,axiom,
    ( ( ord_max_int @ one_one_int @ zero_zero_int )
    = one_one_int ) ).

% max_0_1(2)
thf(fact_7021_max__0__1_I2_J,axiom,
    ( ( ord_max_nat @ one_one_nat @ zero_zero_nat )
    = one_one_nat ) ).

% max_0_1(2)
thf(fact_7022_max__0__1_I2_J,axiom,
    ( ( ord_max_Code_integer @ one_one_Code_integer @ zero_z3403309356797280102nteger )
    = one_one_Code_integer ) ).

% max_0_1(2)
thf(fact_7023_max__0__1_I2_J,axiom,
    ( ( ord_max_Code_natural @ one_one_Code_natural @ zero_z2226904508553997617atural )
    = one_one_Code_natural ) ).

% max_0_1(2)
thf(fact_7024_max__0__1_I1_J,axiom,
    ( ( ord_max_rat @ zero_zero_rat @ one_one_rat )
    = one_one_rat ) ).

% max_0_1(1)
thf(fact_7025_max__0__1_I1_J,axiom,
    ( ( ord_max_int @ zero_zero_int @ one_one_int )
    = one_one_int ) ).

% max_0_1(1)
thf(fact_7026_max__0__1_I1_J,axiom,
    ( ( ord_max_nat @ zero_zero_nat @ one_one_nat )
    = one_one_nat ) ).

% max_0_1(1)
thf(fact_7027_max__0__1_I1_J,axiom,
    ( ( ord_max_Code_integer @ zero_z3403309356797280102nteger @ one_one_Code_integer )
    = one_one_Code_integer ) ).

% max_0_1(1)
thf(fact_7028_max__0__1_I1_J,axiom,
    ( ( ord_max_Code_natural @ zero_z2226904508553997617atural @ one_one_Code_natural )
    = one_one_Code_natural ) ).

% max_0_1(1)
thf(fact_7029_max__0__1_I5_J,axiom,
    ! [X: num] :
      ( ( ord_max_int @ one_one_int @ ( numeral_numeral_int @ X ) )
      = ( numeral_numeral_int @ X ) ) ).

% max_0_1(5)
thf(fact_7030_max__0__1_I5_J,axiom,
    ! [X: num] :
      ( ( ord_max_nat @ one_one_nat @ ( numeral_numeral_nat @ X ) )
      = ( numeral_numeral_nat @ X ) ) ).

% max_0_1(5)
thf(fact_7031_max__0__1_I5_J,axiom,
    ! [X: num] :
      ( ( ord_max_Code_integer @ one_one_Code_integer @ ( numera6620942414471956472nteger @ X ) )
      = ( numera6620942414471956472nteger @ X ) ) ).

% max_0_1(5)
thf(fact_7032_max__0__1_I5_J,axiom,
    ! [X: num] :
      ( ( ord_max_rat @ one_one_rat @ ( numeral_numeral_rat @ X ) )
      = ( numeral_numeral_rat @ X ) ) ).

% max_0_1(5)
thf(fact_7033_max__0__1_I5_J,axiom,
    ! [X: num] :
      ( ( ord_max_Code_natural @ one_one_Code_natural @ ( numera5444537566228673987atural @ X ) )
      = ( numera5444537566228673987atural @ X ) ) ).

% max_0_1(5)
thf(fact_7034_max__0__1_I6_J,axiom,
    ! [X: num] :
      ( ( ord_max_int @ ( numeral_numeral_int @ X ) @ one_one_int )
      = ( numeral_numeral_int @ X ) ) ).

% max_0_1(6)
thf(fact_7035_max__0__1_I6_J,axiom,
    ! [X: num] :
      ( ( ord_max_nat @ ( numeral_numeral_nat @ X ) @ one_one_nat )
      = ( numeral_numeral_nat @ X ) ) ).

% max_0_1(6)
thf(fact_7036_max__0__1_I6_J,axiom,
    ! [X: num] :
      ( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ X ) @ one_one_Code_integer )
      = ( numera6620942414471956472nteger @ X ) ) ).

% max_0_1(6)
thf(fact_7037_max__0__1_I6_J,axiom,
    ! [X: num] :
      ( ( ord_max_rat @ ( numeral_numeral_rat @ X ) @ one_one_rat )
      = ( numeral_numeral_rat @ X ) ) ).

% max_0_1(6)
thf(fact_7038_max__0__1_I6_J,axiom,
    ! [X: num] :
      ( ( ord_max_Code_natural @ ( numera5444537566228673987atural @ X ) @ one_one_Code_natural )
      = ( numera5444537566228673987atural @ X ) ) ).

% max_0_1(6)
thf(fact_7039_sub__num__simps_I1_J,axiom,
    ( ( neg_numeral_sub_rat @ one @ one )
    = zero_zero_rat ) ).

% sub_num_simps(1)
thf(fact_7040_sub__num__simps_I1_J,axiom,
    ( ( neg_numeral_sub_int @ one @ one )
    = zero_zero_int ) ).

% sub_num_simps(1)
thf(fact_7041_sub__num__simps_I1_J,axiom,
    ( ( neg_nu5755505904847501662nteger @ one @ one )
    = zero_z3403309356797280102nteger ) ).

% sub_num_simps(1)
thf(fact_7042_diff__numeral__simps_I1_J,axiom,
    ! [M: num,N: num] :
      ( ( minus_minus_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
      = ( neg_numeral_sub_int @ M @ N ) ) ).

% diff_numeral_simps(1)
thf(fact_7043_diff__numeral__simps_I1_J,axiom,
    ! [M: num,N: num] :
      ( ( minus_8373710615458151222nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ N ) )
      = ( neg_nu5755505904847501662nteger @ M @ N ) ) ).

% diff_numeral_simps(1)
thf(fact_7044_diff__numeral__simps_I1_J,axiom,
    ! [M: num,N: num] :
      ( ( minus_minus_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
      = ( neg_numeral_sub_rat @ M @ N ) ) ).

% diff_numeral_simps(1)
thf(fact_7045_sub__num__simps_I6_J,axiom,
    ! [K: num,L: num] :
      ( ( neg_numeral_sub_int @ ( bit0 @ K ) @ ( bit0 @ L ) )
      = ( neg_numeral_dbl_int @ ( neg_numeral_sub_int @ K @ L ) ) ) ).

% sub_num_simps(6)
thf(fact_7046_sub__num__simps_I6_J,axiom,
    ! [K: num,L: num] :
      ( ( neg_nu5755505904847501662nteger @ ( bit0 @ K ) @ ( bit0 @ L ) )
      = ( neg_nu8804712462038260780nteger @ ( neg_nu5755505904847501662nteger @ K @ L ) ) ) ).

% sub_num_simps(6)
thf(fact_7047_sub__num__simps_I9_J,axiom,
    ! [K: num,L: num] :
      ( ( neg_numeral_sub_int @ ( bit1 @ K ) @ ( bit1 @ L ) )
      = ( neg_numeral_dbl_int @ ( neg_numeral_sub_int @ K @ L ) ) ) ).

% sub_num_simps(9)
thf(fact_7048_sub__num__simps_I9_J,axiom,
    ! [K: num,L: num] :
      ( ( neg_nu5755505904847501662nteger @ ( bit1 @ K ) @ ( bit1 @ L ) )
      = ( neg_nu8804712462038260780nteger @ ( neg_nu5755505904847501662nteger @ K @ L ) ) ) ).

% sub_num_simps(9)
thf(fact_7049_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
       => ( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
          = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) ) )
      & ( ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
       => ( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
          = ( numera6620942414471956472nteger @ U ) ) ) ) ).

% max_number_of(2)
thf(fact_7050_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
       => ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
          = ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) )
      & ( ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
       => ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
          = ( numeral_numeral_rat @ U ) ) ) ) ).

% max_number_of(2)
thf(fact_7051_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
       => ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
          = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
       => ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
          = ( numeral_numeral_int @ U ) ) ) ) ).

% max_number_of(2)
thf(fact_7052_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
       => ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
          = ( numera6620942414471956472nteger @ V ) ) )
      & ( ~ ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
       => ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
          = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) ) ) ) ).

% max_number_of(3)
thf(fact_7053_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
       => ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
          = ( numeral_numeral_rat @ V ) ) )
      & ( ~ ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
       => ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
          = ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) ) ) ) ).

% max_number_of(3)
thf(fact_7054_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
       => ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
          = ( numeral_numeral_int @ V ) ) )
      & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
       => ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
          = ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) ) ) ) ).

% max_number_of(3)
thf(fact_7055_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
       => ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
          = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) ) )
      & ( ~ ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
       => ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
          = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) ) ) ) ).

% max_number_of(4)
thf(fact_7056_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
       => ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
          = ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) )
      & ( ~ ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
       => ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
          = ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) ) ) ) ).

% max_number_of(4)
thf(fact_7057_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
       => ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
          = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
       => ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
          = ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) ) ) ) ).

% max_number_of(4)
thf(fact_7058_add__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
      = ( neg_numeral_sub_int @ N @ M ) ) ).

% add_neg_numeral_simps(2)
thf(fact_7059_add__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) )
      = ( neg_nu5755505904847501662nteger @ N @ M ) ) ).

% add_neg_numeral_simps(2)
thf(fact_7060_add__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) )
      = ( neg_numeral_sub_rat @ N @ M ) ) ).

% add_neg_numeral_simps(2)
thf(fact_7061_add__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N: num] :
      ( ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( neg_numeral_sub_int @ M @ N ) ) ).

% add_neg_numeral_simps(1)
thf(fact_7062_add__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N: num] :
      ( ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
      = ( neg_nu5755505904847501662nteger @ M @ N ) ) ).

% add_neg_numeral_simps(1)
thf(fact_7063_add__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N: num] :
      ( ( plus_plus_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
      = ( neg_numeral_sub_rat @ M @ N ) ) ).

% add_neg_numeral_simps(1)
thf(fact_7064_semiring__norm_I166_J,axiom,
    ! [V: num,W2: num,Y3: int] :
      ( ( plus_plus_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W2 ) ) @ Y3 ) )
      = ( plus_plus_int @ ( neg_numeral_sub_int @ V @ W2 ) @ Y3 ) ) ).

% semiring_norm(166)
thf(fact_7065_semiring__norm_I166_J,axiom,
    ! [V: num,W2: num,Y3: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ V ) @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W2 ) ) @ Y3 ) )
      = ( plus_p5714425477246183910nteger @ ( neg_nu5755505904847501662nteger @ V @ W2 ) @ Y3 ) ) ).

% semiring_norm(166)
thf(fact_7066_semiring__norm_I166_J,axiom,
    ! [V: num,W2: num,Y3: rat] :
      ( ( plus_plus_rat @ ( numeral_numeral_rat @ V ) @ ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ Y3 ) )
      = ( plus_plus_rat @ ( neg_numeral_sub_rat @ V @ W2 ) @ Y3 ) ) ).

% semiring_norm(166)
thf(fact_7067_semiring__norm_I167_J,axiom,
    ! [V: num,W2: num,Y3: int] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( plus_plus_int @ ( numeral_numeral_int @ W2 ) @ Y3 ) )
      = ( plus_plus_int @ ( neg_numeral_sub_int @ W2 @ V ) @ Y3 ) ) ).

% semiring_norm(167)
thf(fact_7068_semiring__norm_I167_J,axiom,
    ! [V: num,W2: num,Y3: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) @ ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ W2 ) @ Y3 ) )
      = ( plus_p5714425477246183910nteger @ ( neg_nu5755505904847501662nteger @ W2 @ V ) @ Y3 ) ) ).

% semiring_norm(167)
thf(fact_7069_semiring__norm_I167_J,axiom,
    ! [V: num,W2: num,Y3: rat] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( plus_plus_rat @ ( numeral_numeral_rat @ W2 ) @ Y3 ) )
      = ( plus_plus_rat @ ( neg_numeral_sub_rat @ W2 @ V ) @ Y3 ) ) ).

% semiring_norm(167)
thf(fact_7070_diff__numeral__simps_I4_J,axiom,
    ! [M: num,N: num] :
      ( ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( neg_numeral_sub_int @ N @ M ) ) ).

% diff_numeral_simps(4)
thf(fact_7071_diff__numeral__simps_I4_J,axiom,
    ! [M: num,N: num] :
      ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
      = ( neg_nu5755505904847501662nteger @ N @ M ) ) ).

% diff_numeral_simps(4)
thf(fact_7072_diff__numeral__simps_I4_J,axiom,
    ! [M: num,N: num] :
      ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
      = ( neg_numeral_sub_rat @ N @ M ) ) ).

% diff_numeral_simps(4)
thf(fact_7073_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_7074_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_7075_sub__num__simps_I7_J,axiom,
    ! [K: num,L: num] :
      ( ( neg_numeral_sub_int @ ( bit0 @ K ) @ ( bit1 @ L ) )
      = ( neg_nu3811975205180677377ec_int @ ( neg_numeral_sub_int @ K @ L ) ) ) ).

% sub_num_simps(7)
thf(fact_7076_sub__num__simps_I7_J,axiom,
    ! [K: num,L: num] :
      ( ( neg_nu5755505904847501662nteger @ ( bit0 @ K ) @ ( bit1 @ L ) )
      = ( neg_nu7757733837767384882nteger @ ( neg_nu5755505904847501662nteger @ K @ L ) ) ) ).

% sub_num_simps(7)
thf(fact_7077_sub__num__simps_I8_J,axiom,
    ! [K: num,L: num] :
      ( ( neg_numeral_sub_int @ ( bit1 @ K ) @ ( bit0 @ L ) )
      = ( neg_nu5851722552734809277nc_int @ ( neg_numeral_sub_int @ K @ L ) ) ) ).

% sub_num_simps(8)
thf(fact_7078_sub__num__simps_I8_J,axiom,
    ! [K: num,L: num] :
      ( ( neg_nu5755505904847501662nteger @ ( bit1 @ K ) @ ( bit0 @ L ) )
      = ( neg_nu5831290666863070958nteger @ ( neg_nu5755505904847501662nteger @ K @ L ) ) ) ).

% sub_num_simps(8)
thf(fact_7079_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_7080_diff__numeral__special_I2_J,axiom,
    ! [M: num] :
      ( ( minus_minus_int @ ( numeral_numeral_int @ M ) @ one_one_int )
      = ( neg_numeral_sub_int @ M @ one ) ) ).

% diff_numeral_special(2)
thf(fact_7081_diff__numeral__special_I2_J,axiom,
    ! [M: num] :
      ( ( minus_8373710615458151222nteger @ ( numera6620942414471956472nteger @ M ) @ one_one_Code_integer )
      = ( neg_nu5755505904847501662nteger @ M @ one ) ) ).

% diff_numeral_special(2)
thf(fact_7082_diff__numeral__special_I2_J,axiom,
    ! [M: num] :
      ( ( minus_minus_rat @ ( numeral_numeral_rat @ M ) @ one_one_rat )
      = ( neg_numeral_sub_rat @ M @ one ) ) ).

% diff_numeral_special(2)
thf(fact_7083_diff__numeral__special_I1_J,axiom,
    ! [N: num] :
      ( ( minus_minus_int @ one_one_int @ ( numeral_numeral_int @ N ) )
      = ( neg_numeral_sub_int @ one @ N ) ) ).

% diff_numeral_special(1)
thf(fact_7084_diff__numeral__special_I1_J,axiom,
    ! [N: num] :
      ( ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ N ) )
      = ( neg_nu5755505904847501662nteger @ one @ N ) ) ).

% diff_numeral_special(1)
thf(fact_7085_diff__numeral__special_I1_J,axiom,
    ! [N: num] :
      ( ( minus_minus_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) )
      = ( neg_numeral_sub_rat @ one @ N ) ) ).

% diff_numeral_special(1)
thf(fact_7086_sub__num__simps_I5_J,axiom,
    ! [K: num] :
      ( ( neg_numeral_sub_int @ ( bit1 @ K ) @ one )
      = ( numeral_numeral_int @ ( bit0 @ K ) ) ) ).

% sub_num_simps(5)
thf(fact_7087_sub__num__simps_I5_J,axiom,
    ! [K: num] :
      ( ( neg_nu5755505904847501662nteger @ ( bit1 @ K ) @ one )
      = ( numera6620942414471956472nteger @ ( bit0 @ K ) ) ) ).

% sub_num_simps(5)
thf(fact_7088_sub__num__simps_I5_J,axiom,
    ! [K: num] :
      ( ( neg_numeral_sub_rat @ ( bit1 @ K ) @ one )
      = ( numeral_numeral_rat @ ( bit0 @ K ) ) ) ).

% sub_num_simps(5)
thf(fact_7089_not__minus__numeral__eq,axiom,
    ! [N: num] :
      ( ( bit_ri7919022796975470100ot_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( neg_numeral_sub_int @ N @ one ) ) ).

% not_minus_numeral_eq
thf(fact_7090_not__minus__numeral__eq,axiom,
    ! [N: num] :
      ( ( bit_ri7632146776885996613nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
      = ( neg_nu5755505904847501662nteger @ N @ one ) ) ).

% not_minus_numeral_eq
thf(fact_7091_add__neg__numeral__special_I1_J,axiom,
    ! [M: num] :
      ( ( plus_plus_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) )
      = ( neg_numeral_sub_int @ one @ M ) ) ).

% add_neg_numeral_special(1)
thf(fact_7092_add__neg__numeral__special_I1_J,axiom,
    ! [M: num] :
      ( ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) )
      = ( neg_nu5755505904847501662nteger @ one @ M ) ) ).

% add_neg_numeral_special(1)
thf(fact_7093_add__neg__numeral__special_I1_J,axiom,
    ! [M: num] :
      ( ( plus_plus_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) )
      = ( neg_numeral_sub_rat @ one @ M ) ) ).

% add_neg_numeral_special(1)
thf(fact_7094_add__neg__numeral__special_I2_J,axiom,
    ! [M: num] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int )
      = ( neg_numeral_sub_int @ one @ M ) ) ).

% add_neg_numeral_special(2)
thf(fact_7095_add__neg__numeral__special_I2_J,axiom,
    ! [M: num] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer )
      = ( neg_nu5755505904847501662nteger @ one @ M ) ) ).

% add_neg_numeral_special(2)
thf(fact_7096_add__neg__numeral__special_I2_J,axiom,
    ! [M: num] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ one_one_rat )
      = ( neg_numeral_sub_rat @ one @ M ) ) ).

% add_neg_numeral_special(2)
thf(fact_7097_add__neg__numeral__special_I3_J,axiom,
    ! [M: num] :
      ( ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) )
      = ( neg_numeral_sub_int @ M @ one ) ) ).

% add_neg_numeral_special(3)
thf(fact_7098_add__neg__numeral__special_I3_J,axiom,
    ! [M: num] :
      ( ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( neg_nu5755505904847501662nteger @ M @ one ) ) ).

% add_neg_numeral_special(3)
thf(fact_7099_add__neg__numeral__special_I3_J,axiom,
    ! [M: num] :
      ( ( plus_plus_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ one_one_rat ) )
      = ( neg_numeral_sub_rat @ M @ one ) ) ).

% add_neg_numeral_special(3)
thf(fact_7100_add__neg__numeral__special_I4_J,axiom,
    ! [N: num] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N ) )
      = ( neg_numeral_sub_int @ N @ one ) ) ).

% add_neg_numeral_special(4)
thf(fact_7101_add__neg__numeral__special_I4_J,axiom,
    ! [N: num] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ N ) )
      = ( neg_nu5755505904847501662nteger @ N @ one ) ) ).

% add_neg_numeral_special(4)
thf(fact_7102_add__neg__numeral__special_I4_J,axiom,
    ! [N: num] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ N ) )
      = ( neg_numeral_sub_rat @ N @ one ) ) ).

% add_neg_numeral_special(4)
thf(fact_7103_minus__sub__one__diff__one,axiom,
    ! [M: num] :
      ( ( minus_minus_int @ ( uminus_uminus_int @ ( neg_numeral_sub_int @ M @ one ) ) @ one_one_int )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).

% minus_sub_one_diff_one
thf(fact_7104_minus__sub__one__diff__one,axiom,
    ! [M: num] :
      ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ ( neg_nu5755505904847501662nteger @ M @ one ) ) @ one_one_Code_integer )
      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).

% minus_sub_one_diff_one
thf(fact_7105_minus__sub__one__diff__one,axiom,
    ! [M: num] :
      ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( neg_numeral_sub_rat @ M @ one ) ) @ one_one_rat )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).

% minus_sub_one_diff_one
thf(fact_7106_diff__numeral__special_I7_J,axiom,
    ! [N: num] :
      ( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( neg_numeral_sub_int @ N @ one ) ) ).

% diff_numeral_special(7)
thf(fact_7107_diff__numeral__special_I7_J,axiom,
    ! [N: num] :
      ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
      = ( neg_nu5755505904847501662nteger @ N @ one ) ) ).

% diff_numeral_special(7)
thf(fact_7108_diff__numeral__special_I7_J,axiom,
    ! [N: num] :
      ( ( minus_minus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
      = ( neg_numeral_sub_rat @ N @ one ) ) ).

% diff_numeral_special(7)
thf(fact_7109_diff__numeral__special_I8_J,axiom,
    ! [M: num] :
      ( ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) )
      = ( neg_numeral_sub_int @ one @ M ) ) ).

% diff_numeral_special(8)
thf(fact_7110_diff__numeral__special_I8_J,axiom,
    ! [M: num] :
      ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( neg_nu5755505904847501662nteger @ one @ M ) ) ).

% diff_numeral_special(8)
thf(fact_7111_diff__numeral__special_I8_J,axiom,
    ! [M: num] :
      ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ one_one_rat ) )
      = ( neg_numeral_sub_rat @ one @ M ) ) ).

% diff_numeral_special(8)
thf(fact_7112_sub__num__simps_I3_J,axiom,
    ! [L: num] :
      ( ( neg_numeral_sub_int @ one @ ( bit1 @ L ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ L ) ) ) ) ).

% sub_num_simps(3)
thf(fact_7113_sub__num__simps_I3_J,axiom,
    ! [L: num] :
      ( ( neg_nu5755505904847501662nteger @ one @ ( bit1 @ L ) )
      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ L ) ) ) ) ).

% sub_num_simps(3)
thf(fact_7114_sub__num__simps_I3_J,axiom,
    ! [L: num] :
      ( ( neg_numeral_sub_rat @ one @ ( bit1 @ L ) )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit0 @ L ) ) ) ) ).

% sub_num_simps(3)
thf(fact_7115_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_7116_max_Ostrict__coboundedI2,axiom,
    ! [C: code_natural,B: code_natural,A: code_natural] :
      ( ( ord_le5570908160329646204atural @ C @ B )
     => ( ord_le5570908160329646204atural @ C @ ( ord_max_Code_natural @ A @ B ) ) ) ).

% max.strict_coboundedI2
thf(fact_7117_max_Ostrict__coboundedI2,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ C @ B )
     => ( ord_le6747313008572928689nteger @ C @ ( ord_max_Code_integer @ A @ B ) ) ) ).

% max.strict_coboundedI2
thf(fact_7118_max_Ostrict__coboundedI2,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ C @ B )
     => ( ord_less_rat @ C @ ( ord_max_rat @ A @ B ) ) ) ).

% max.strict_coboundedI2
thf(fact_7119_max_Ostrict__coboundedI2,axiom,
    ! [C: num,B: num,A: num] :
      ( ( ord_less_num @ C @ B )
     => ( ord_less_num @ C @ ( ord_max_num @ A @ B ) ) ) ).

% max.strict_coboundedI2
thf(fact_7120_max_Ostrict__coboundedI2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( ord_less_nat @ C @ B )
     => ( ord_less_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).

% max.strict_coboundedI2
thf(fact_7121_max_Ostrict__coboundedI2,axiom,
    ! [C: int,B: int,A: int] :
      ( ( ord_less_int @ C @ B )
     => ( ord_less_int @ C @ ( ord_max_int @ A @ B ) ) ) ).

% max.strict_coboundedI2
thf(fact_7122_max_Ostrict__coboundedI1,axiom,
    ! [C: code_natural,A: code_natural,B: code_natural] :
      ( ( ord_le5570908160329646204atural @ C @ A )
     => ( ord_le5570908160329646204atural @ C @ ( ord_max_Code_natural @ A @ B ) ) ) ).

% max.strict_coboundedI1
thf(fact_7123_max_Ostrict__coboundedI1,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ C @ A )
     => ( ord_le6747313008572928689nteger @ C @ ( ord_max_Code_integer @ A @ B ) ) ) ).

% max.strict_coboundedI1
thf(fact_7124_max_Ostrict__coboundedI1,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ C @ A )
     => ( ord_less_rat @ C @ ( ord_max_rat @ A @ B ) ) ) ).

% max.strict_coboundedI1
thf(fact_7125_max_Ostrict__coboundedI1,axiom,
    ! [C: num,A: num,B: num] :
      ( ( ord_less_num @ C @ A )
     => ( ord_less_num @ C @ ( ord_max_num @ A @ B ) ) ) ).

% max.strict_coboundedI1
thf(fact_7126_max_Ostrict__coboundedI1,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ C @ A )
     => ( ord_less_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).

% max.strict_coboundedI1
thf(fact_7127_max_Ostrict__coboundedI1,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ C @ A )
     => ( ord_less_int @ C @ ( ord_max_int @ A @ B ) ) ) ).

% max.strict_coboundedI1
thf(fact_7128_max_Ostrict__order__iff,axiom,
    ( ord_le5570908160329646204atural
    = ( ^ [B2: code_natural,A2: code_natural] :
          ( ( A2
            = ( ord_max_Code_natural @ A2 @ B2 ) )
          & ( A2 != B2 ) ) ) ) ).

% max.strict_order_iff
thf(fact_7129_max_Ostrict__order__iff,axiom,
    ( ord_le6747313008572928689nteger
    = ( ^ [B2: code_integer,A2: code_integer] :
          ( ( A2
            = ( ord_max_Code_integer @ A2 @ B2 ) )
          & ( A2 != B2 ) ) ) ) ).

% max.strict_order_iff
thf(fact_7130_max_Ostrict__order__iff,axiom,
    ( ord_less_rat
    = ( ^ [B2: rat,A2: rat] :
          ( ( A2
            = ( ord_max_rat @ A2 @ B2 ) )
          & ( A2 != B2 ) ) ) ) ).

% max.strict_order_iff
thf(fact_7131_max_Ostrict__order__iff,axiom,
    ( ord_less_num
    = ( ^ [B2: num,A2: num] :
          ( ( A2
            = ( ord_max_num @ A2 @ B2 ) )
          & ( A2 != B2 ) ) ) ) ).

% max.strict_order_iff
thf(fact_7132_max_Ostrict__order__iff,axiom,
    ( ord_less_nat
    = ( ^ [B2: nat,A2: nat] :
          ( ( A2
            = ( ord_max_nat @ A2 @ B2 ) )
          & ( A2 != B2 ) ) ) ) ).

% max.strict_order_iff
thf(fact_7133_max_Ostrict__order__iff,axiom,
    ( ord_less_int
    = ( ^ [B2: int,A2: int] :
          ( ( A2
            = ( ord_max_int @ A2 @ B2 ) )
          & ( A2 != B2 ) ) ) ) ).

% max.strict_order_iff
thf(fact_7134_max_Ostrict__boundedE,axiom,
    ! [B: code_natural,C: code_natural,A: code_natural] :
      ( ( ord_le5570908160329646204atural @ ( ord_max_Code_natural @ B @ C ) @ A )
     => ~ ( ( ord_le5570908160329646204atural @ B @ A )
         => ~ ( ord_le5570908160329646204atural @ C @ A ) ) ) ).

% max.strict_boundedE
thf(fact_7135_max_Ostrict__boundedE,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( ord_max_Code_integer @ B @ C ) @ A )
     => ~ ( ( ord_le6747313008572928689nteger @ B @ A )
         => ~ ( ord_le6747313008572928689nteger @ C @ A ) ) ) ).

% max.strict_boundedE
thf(fact_7136_max_Ostrict__boundedE,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( ord_less_rat @ ( ord_max_rat @ B @ C ) @ A )
     => ~ ( ( ord_less_rat @ B @ A )
         => ~ ( ord_less_rat @ C @ A ) ) ) ).

% max.strict_boundedE
thf(fact_7137_max_Ostrict__boundedE,axiom,
    ! [B: num,C: num,A: num] :
      ( ( ord_less_num @ ( ord_max_num @ B @ C ) @ A )
     => ~ ( ( ord_less_num @ B @ A )
         => ~ ( ord_less_num @ C @ A ) ) ) ).

% max.strict_boundedE
thf(fact_7138_max_Ostrict__boundedE,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( ord_less_nat @ ( ord_max_nat @ B @ C ) @ A )
     => ~ ( ( ord_less_nat @ B @ A )
         => ~ ( ord_less_nat @ C @ A ) ) ) ).

% max.strict_boundedE
thf(fact_7139_max_Ostrict__boundedE,axiom,
    ! [B: int,C: int,A: int] :
      ( ( ord_less_int @ ( ord_max_int @ B @ C ) @ A )
     => ~ ( ( ord_less_int @ B @ A )
         => ~ ( ord_less_int @ C @ A ) ) ) ).

% max.strict_boundedE
thf(fact_7140_less__max__iff__disj,axiom,
    ! [Z2: code_natural,X: code_natural,Y3: code_natural] :
      ( ( ord_le5570908160329646204atural @ Z2 @ ( ord_max_Code_natural @ X @ Y3 ) )
      = ( ( ord_le5570908160329646204atural @ Z2 @ X )
        | ( ord_le5570908160329646204atural @ Z2 @ Y3 ) ) ) ).

% less_max_iff_disj
thf(fact_7141_less__max__iff__disj,axiom,
    ! [Z2: code_integer,X: code_integer,Y3: code_integer] :
      ( ( ord_le6747313008572928689nteger @ Z2 @ ( ord_max_Code_integer @ X @ Y3 ) )
      = ( ( ord_le6747313008572928689nteger @ Z2 @ X )
        | ( ord_le6747313008572928689nteger @ Z2 @ Y3 ) ) ) ).

% less_max_iff_disj
thf(fact_7142_less__max__iff__disj,axiom,
    ! [Z2: rat,X: rat,Y3: rat] :
      ( ( ord_less_rat @ Z2 @ ( ord_max_rat @ X @ Y3 ) )
      = ( ( ord_less_rat @ Z2 @ X )
        | ( ord_less_rat @ Z2 @ Y3 ) ) ) ).

% less_max_iff_disj
thf(fact_7143_less__max__iff__disj,axiom,
    ! [Z2: num,X: num,Y3: num] :
      ( ( ord_less_num @ Z2 @ ( ord_max_num @ X @ Y3 ) )
      = ( ( ord_less_num @ Z2 @ X )
        | ( ord_less_num @ Z2 @ Y3 ) ) ) ).

% less_max_iff_disj
thf(fact_7144_less__max__iff__disj,axiom,
    ! [Z2: nat,X: nat,Y3: nat] :
      ( ( ord_less_nat @ Z2 @ ( ord_max_nat @ X @ Y3 ) )
      = ( ( ord_less_nat @ Z2 @ X )
        | ( ord_less_nat @ Z2 @ Y3 ) ) ) ).

% less_max_iff_disj
thf(fact_7145_less__max__iff__disj,axiom,
    ! [Z2: int,X: int,Y3: int] :
      ( ( ord_less_int @ Z2 @ ( ord_max_int @ X @ Y3 ) )
      = ( ( ord_less_int @ Z2 @ X )
        | ( ord_less_int @ Z2 @ Y3 ) ) ) ).

% less_max_iff_disj
thf(fact_7146_max__add__distrib__left,axiom,
    ! [X: rat,Y3: rat,Z2: rat] :
      ( ( plus_plus_rat @ ( ord_max_rat @ X @ Y3 ) @ Z2 )
      = ( ord_max_rat @ ( plus_plus_rat @ X @ Z2 ) @ ( plus_plus_rat @ Y3 @ Z2 ) ) ) ).

% max_add_distrib_left
thf(fact_7147_max__add__distrib__left,axiom,
    ! [X: int,Y3: int,Z2: int] :
      ( ( plus_plus_int @ ( ord_max_int @ X @ Y3 ) @ Z2 )
      = ( ord_max_int @ ( plus_plus_int @ X @ Z2 ) @ ( plus_plus_int @ Y3 @ Z2 ) ) ) ).

% max_add_distrib_left
thf(fact_7148_max__add__distrib__left,axiom,
    ! [X: nat,Y3: nat,Z2: nat] :
      ( ( plus_plus_nat @ ( ord_max_nat @ X @ Y3 ) @ Z2 )
      = ( ord_max_nat @ ( plus_plus_nat @ X @ Z2 ) @ ( plus_plus_nat @ Y3 @ Z2 ) ) ) ).

% max_add_distrib_left
thf(fact_7149_max__add__distrib__left,axiom,
    ! [X: code_integer,Y3: code_integer,Z2: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( ord_max_Code_integer @ X @ Y3 ) @ Z2 )
      = ( ord_max_Code_integer @ ( plus_p5714425477246183910nteger @ X @ Z2 ) @ ( plus_p5714425477246183910nteger @ Y3 @ Z2 ) ) ) ).

% max_add_distrib_left
thf(fact_7150_max__add__distrib__left,axiom,
    ! [X: code_natural,Y3: code_natural,Z2: code_natural] :
      ( ( plus_p4538020629002901425atural @ ( ord_max_Code_natural @ X @ Y3 ) @ Z2 )
      = ( ord_max_Code_natural @ ( plus_p4538020629002901425atural @ X @ Z2 ) @ ( plus_p4538020629002901425atural @ Y3 @ Z2 ) ) ) ).

% max_add_distrib_left
thf(fact_7151_max__add__distrib__right,axiom,
    ! [X: rat,Y3: rat,Z2: rat] :
      ( ( plus_plus_rat @ X @ ( ord_max_rat @ Y3 @ Z2 ) )
      = ( ord_max_rat @ ( plus_plus_rat @ X @ Y3 ) @ ( plus_plus_rat @ X @ Z2 ) ) ) ).

% max_add_distrib_right
thf(fact_7152_max__add__distrib__right,axiom,
    ! [X: int,Y3: int,Z2: int] :
      ( ( plus_plus_int @ X @ ( ord_max_int @ Y3 @ Z2 ) )
      = ( ord_max_int @ ( plus_plus_int @ X @ Y3 ) @ ( plus_plus_int @ X @ Z2 ) ) ) ).

% max_add_distrib_right
thf(fact_7153_max__add__distrib__right,axiom,
    ! [X: nat,Y3: nat,Z2: nat] :
      ( ( plus_plus_nat @ X @ ( ord_max_nat @ Y3 @ Z2 ) )
      = ( ord_max_nat @ ( plus_plus_nat @ X @ Y3 ) @ ( plus_plus_nat @ X @ Z2 ) ) ) ).

% max_add_distrib_right
thf(fact_7154_max__add__distrib__right,axiom,
    ! [X: code_integer,Y3: code_integer,Z2: code_integer] :
      ( ( plus_p5714425477246183910nteger @ X @ ( ord_max_Code_integer @ Y3 @ Z2 ) )
      = ( ord_max_Code_integer @ ( plus_p5714425477246183910nteger @ X @ Y3 ) @ ( plus_p5714425477246183910nteger @ X @ Z2 ) ) ) ).

% max_add_distrib_right
thf(fact_7155_max__add__distrib__right,axiom,
    ! [X: code_natural,Y3: code_natural,Z2: code_natural] :
      ( ( plus_p4538020629002901425atural @ X @ ( ord_max_Code_natural @ Y3 @ Z2 ) )
      = ( ord_max_Code_natural @ ( plus_p4538020629002901425atural @ X @ Y3 ) @ ( plus_p4538020629002901425atural @ X @ Z2 ) ) ) ).

% max_add_distrib_right
thf(fact_7156_max__diff__distrib__left,axiom,
    ! [X: code_integer,Y3: code_integer,Z2: code_integer] :
      ( ( minus_8373710615458151222nteger @ ( ord_max_Code_integer @ X @ Y3 ) @ Z2 )
      = ( ord_max_Code_integer @ ( minus_8373710615458151222nteger @ X @ Z2 ) @ ( minus_8373710615458151222nteger @ Y3 @ Z2 ) ) ) ).

% max_diff_distrib_left
thf(fact_7157_max__diff__distrib__left,axiom,
    ! [X: rat,Y3: rat,Z2: rat] :
      ( ( minus_minus_rat @ ( ord_max_rat @ X @ Y3 ) @ Z2 )
      = ( ord_max_rat @ ( minus_minus_rat @ X @ Z2 ) @ ( minus_minus_rat @ Y3 @ Z2 ) ) ) ).

% max_diff_distrib_left
thf(fact_7158_max__diff__distrib__left,axiom,
    ! [X: int,Y3: int,Z2: int] :
      ( ( minus_minus_int @ ( ord_max_int @ X @ Y3 ) @ Z2 )
      = ( ord_max_int @ ( minus_minus_int @ X @ Z2 ) @ ( minus_minus_int @ Y3 @ Z2 ) ) ) ).

% max_diff_distrib_left
thf(fact_7159_of__nat__max,axiom,
    ! [X: nat,Y3: nat] :
      ( ( semiri1314217659103216013at_int @ ( ord_max_nat @ X @ Y3 ) )
      = ( ord_max_int @ ( semiri1314217659103216013at_int @ X ) @ ( semiri1314217659103216013at_int @ Y3 ) ) ) ).

% of_nat_max
thf(fact_7160_of__nat__max,axiom,
    ! [X: nat,Y3: nat] :
      ( ( semiri4939895301339042750nteger @ ( ord_max_nat @ X @ Y3 ) )
      = ( ord_max_Code_integer @ ( semiri4939895301339042750nteger @ X ) @ ( semiri4939895301339042750nteger @ Y3 ) ) ) ).

% of_nat_max
thf(fact_7161_of__nat__max,axiom,
    ! [X: nat,Y3: nat] :
      ( ( semiri1316708129612266289at_nat @ ( ord_max_nat @ X @ Y3 ) )
      = ( ord_max_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( semiri1316708129612266289at_nat @ Y3 ) ) ) ).

% of_nat_max
thf(fact_7162_of__nat__max,axiom,
    ! [X: nat,Y3: nat] :
      ( ( semiri681578069525770553at_rat @ ( ord_max_nat @ X @ Y3 ) )
      = ( ord_max_rat @ ( semiri681578069525770553at_rat @ X ) @ ( semiri681578069525770553at_rat @ Y3 ) ) ) ).

% of_nat_max
thf(fact_7163_of__nat__max,axiom,
    ! [X: nat,Y3: nat] :
      ( ( semiri3763490453095760265atural @ ( ord_max_nat @ X @ Y3 ) )
      = ( ord_max_Code_natural @ ( semiri3763490453095760265atural @ X ) @ ( semiri3763490453095760265atural @ Y3 ) ) ) ).

% of_nat_max
thf(fact_7164_max_Oleft__commute,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_max_int @ B @ ( ord_max_int @ A @ C ) )
      = ( ord_max_int @ A @ ( ord_max_int @ B @ C ) ) ) ).

% max.left_commute
thf(fact_7165_max_Oleft__commute,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ord_max_nat @ B @ ( ord_max_nat @ A @ C ) )
      = ( ord_max_nat @ A @ ( ord_max_nat @ B @ C ) ) ) ).

% max.left_commute
thf(fact_7166_max_Oleft__commute,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( ord_max_Code_integer @ B @ ( ord_max_Code_integer @ A @ C ) )
      = ( ord_max_Code_integer @ A @ ( ord_max_Code_integer @ B @ C ) ) ) ).

% max.left_commute
thf(fact_7167_max_Oleft__commute,axiom,
    ! [B: code_natural,A: code_natural,C: code_natural] :
      ( ( ord_max_Code_natural @ B @ ( ord_max_Code_natural @ A @ C ) )
      = ( ord_max_Code_natural @ A @ ( ord_max_Code_natural @ B @ C ) ) ) ).

% max.left_commute
thf(fact_7168_max_Ocommute,axiom,
    ( ord_max_int
    = ( ^ [A2: int,B2: int] : ( ord_max_int @ B2 @ A2 ) ) ) ).

% max.commute
thf(fact_7169_max_Ocommute,axiom,
    ( ord_max_nat
    = ( ^ [A2: nat,B2: nat] : ( ord_max_nat @ B2 @ A2 ) ) ) ).

% max.commute
thf(fact_7170_max_Ocommute,axiom,
    ( ord_max_Code_integer
    = ( ^ [A2: code_integer,B2: code_integer] : ( ord_max_Code_integer @ B2 @ A2 ) ) ) ).

% max.commute
thf(fact_7171_max_Ocommute,axiom,
    ( ord_max_Code_natural
    = ( ^ [A2: code_natural,B2: code_natural] : ( ord_max_Code_natural @ B2 @ A2 ) ) ) ).

% max.commute
thf(fact_7172_max_Oassoc,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_max_int @ ( ord_max_int @ A @ B ) @ C )
      = ( ord_max_int @ A @ ( ord_max_int @ B @ C ) ) ) ).

% max.assoc
thf(fact_7173_max_Oassoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_max_nat @ ( ord_max_nat @ A @ B ) @ C )
      = ( ord_max_nat @ A @ ( ord_max_nat @ B @ C ) ) ) ).

% max.assoc
thf(fact_7174_max_Oassoc,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( ord_max_Code_integer @ ( ord_max_Code_integer @ A @ B ) @ C )
      = ( ord_max_Code_integer @ A @ ( ord_max_Code_integer @ B @ C ) ) ) ).

% max.assoc
thf(fact_7175_max_Oassoc,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( ( ord_max_Code_natural @ ( ord_max_Code_natural @ A @ B ) @ C )
      = ( ord_max_Code_natural @ A @ ( ord_max_Code_natural @ B @ C ) ) ) ).

% max.assoc
thf(fact_7176_of__int__max,axiom,
    ! [X: int,Y3: int] :
      ( ( ring_1_of_int_rat @ ( ord_max_int @ X @ Y3 ) )
      = ( ord_max_rat @ ( ring_1_of_int_rat @ X ) @ ( ring_1_of_int_rat @ Y3 ) ) ) ).

% of_int_max
thf(fact_7177_of__int__max,axiom,
    ! [X: int,Y3: int] :
      ( ( ring_1_of_int_int @ ( ord_max_int @ X @ Y3 ) )
      = ( ord_max_int @ ( ring_1_of_int_int @ X ) @ ( ring_1_of_int_int @ Y3 ) ) ) ).

% of_int_max
thf(fact_7178_of__int__max,axiom,
    ! [X: int,Y3: int] :
      ( ( ring_18347121197199848620nteger @ ( ord_max_int @ X @ Y3 ) )
      = ( ord_max_Code_integer @ ( ring_18347121197199848620nteger @ X ) @ ( ring_18347121197199848620nteger @ Y3 ) ) ) ).

% of_int_max
thf(fact_7179_max__absorb2,axiom,
    ! [X: code_integer,Y3: code_integer] :
      ( ( ord_le3102999989581377725nteger @ X @ Y3 )
     => ( ( ord_max_Code_integer @ X @ Y3 )
        = Y3 ) ) ).

% max_absorb2
thf(fact_7180_max__absorb2,axiom,
    ! [X: code_natural,Y3: code_natural] :
      ( ( ord_le1926595141338095240atural @ X @ Y3 )
     => ( ( ord_max_Code_natural @ X @ Y3 )
        = Y3 ) ) ).

% max_absorb2
thf(fact_7181_max__absorb2,axiom,
    ! [X: set_nat,Y3: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ Y3 )
     => ( ( ord_max_set_nat @ X @ Y3 )
        = Y3 ) ) ).

% max_absorb2
thf(fact_7182_max__absorb2,axiom,
    ! [X: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ X @ Y3 )
     => ( ( ord_max_rat @ X @ Y3 )
        = Y3 ) ) ).

% max_absorb2
thf(fact_7183_max__absorb2,axiom,
    ! [X: num,Y3: num] :
      ( ( ord_less_eq_num @ X @ Y3 )
     => ( ( ord_max_num @ X @ Y3 )
        = Y3 ) ) ).

% max_absorb2
thf(fact_7184_max__absorb2,axiom,
    ! [X: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ X @ Y3 )
     => ( ( ord_max_nat @ X @ Y3 )
        = Y3 ) ) ).

% max_absorb2
thf(fact_7185_max__absorb2,axiom,
    ! [X: int,Y3: int] :
      ( ( ord_less_eq_int @ X @ Y3 )
     => ( ( ord_max_int @ X @ Y3 )
        = Y3 ) ) ).

% max_absorb2
thf(fact_7186_max__absorb1,axiom,
    ! [Y3: code_integer,X: code_integer] :
      ( ( ord_le3102999989581377725nteger @ Y3 @ X )
     => ( ( ord_max_Code_integer @ X @ Y3 )
        = X ) ) ).

% max_absorb1
thf(fact_7187_max__absorb1,axiom,
    ! [Y3: code_natural,X: code_natural] :
      ( ( ord_le1926595141338095240atural @ Y3 @ X )
     => ( ( ord_max_Code_natural @ X @ Y3 )
        = X ) ) ).

% max_absorb1
thf(fact_7188_max__absorb1,axiom,
    ! [Y3: set_nat,X: set_nat] :
      ( ( ord_less_eq_set_nat @ Y3 @ X )
     => ( ( ord_max_set_nat @ X @ Y3 )
        = X ) ) ).

% max_absorb1
thf(fact_7189_max__absorb1,axiom,
    ! [Y3: rat,X: rat] :
      ( ( ord_less_eq_rat @ Y3 @ X )
     => ( ( ord_max_rat @ X @ Y3 )
        = X ) ) ).

% max_absorb1
thf(fact_7190_max__absorb1,axiom,
    ! [Y3: num,X: num] :
      ( ( ord_less_eq_num @ Y3 @ X )
     => ( ( ord_max_num @ X @ Y3 )
        = X ) ) ).

% max_absorb1
thf(fact_7191_max__absorb1,axiom,
    ! [Y3: nat,X: nat] :
      ( ( ord_less_eq_nat @ Y3 @ X )
     => ( ( ord_max_nat @ X @ Y3 )
        = X ) ) ).

% max_absorb1
thf(fact_7192_max__absorb1,axiom,
    ! [Y3: int,X: int] :
      ( ( ord_less_eq_int @ Y3 @ X )
     => ( ( ord_max_int @ X @ Y3 )
        = X ) ) ).

% max_absorb1
thf(fact_7193_max_OcoboundedI2,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ C @ B )
     => ( ord_le3102999989581377725nteger @ C @ ( ord_max_Code_integer @ A @ B ) ) ) ).

% max.coboundedI2
thf(fact_7194_max_OcoboundedI2,axiom,
    ! [C: code_natural,B: code_natural,A: code_natural] :
      ( ( ord_le1926595141338095240atural @ C @ B )
     => ( ord_le1926595141338095240atural @ C @ ( ord_max_Code_natural @ A @ B ) ) ) ).

% max.coboundedI2
thf(fact_7195_max_OcoboundedI2,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_eq_rat @ C @ B )
     => ( ord_less_eq_rat @ C @ ( ord_max_rat @ A @ B ) ) ) ).

% max.coboundedI2
thf(fact_7196_max_OcoboundedI2,axiom,
    ! [C: num,B: num,A: num] :
      ( ( ord_less_eq_num @ C @ B )
     => ( ord_less_eq_num @ C @ ( ord_max_num @ A @ B ) ) ) ).

% max.coboundedI2
thf(fact_7197_max_OcoboundedI2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( ord_less_eq_nat @ C @ B )
     => ( ord_less_eq_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).

% max.coboundedI2
thf(fact_7198_max_OcoboundedI2,axiom,
    ! [C: int,B: int,A: int] :
      ( ( ord_less_eq_int @ C @ B )
     => ( ord_less_eq_int @ C @ ( ord_max_int @ A @ B ) ) ) ).

% max.coboundedI2
thf(fact_7199_max_OcoboundedI1,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ C @ A )
     => ( ord_le3102999989581377725nteger @ C @ ( ord_max_Code_integer @ A @ B ) ) ) ).

% max.coboundedI1
thf(fact_7200_max_OcoboundedI1,axiom,
    ! [C: code_natural,A: code_natural,B: code_natural] :
      ( ( ord_le1926595141338095240atural @ C @ A )
     => ( ord_le1926595141338095240atural @ C @ ( ord_max_Code_natural @ A @ B ) ) ) ).

% max.coboundedI1
thf(fact_7201_max_OcoboundedI1,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_eq_rat @ C @ A )
     => ( ord_less_eq_rat @ C @ ( ord_max_rat @ A @ B ) ) ) ).

% max.coboundedI1
thf(fact_7202_max_OcoboundedI1,axiom,
    ! [C: num,A: num,B: num] :
      ( ( ord_less_eq_num @ C @ A )
     => ( ord_less_eq_num @ C @ ( ord_max_num @ A @ B ) ) ) ).

% max.coboundedI1
thf(fact_7203_max_OcoboundedI1,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ C @ A )
     => ( ord_less_eq_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).

% max.coboundedI1
thf(fact_7204_max_OcoboundedI1,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ C @ A )
     => ( ord_less_eq_int @ C @ ( ord_max_int @ A @ B ) ) ) ).

% max.coboundedI1
thf(fact_7205_max_Oabsorb__iff2,axiom,
    ( ord_le3102999989581377725nteger
    = ( ^ [A2: code_integer,B2: code_integer] :
          ( ( ord_max_Code_integer @ A2 @ B2 )
          = B2 ) ) ) ).

% max.absorb_iff2
thf(fact_7206_max_Oabsorb__iff2,axiom,
    ( ord_le1926595141338095240atural
    = ( ^ [A2: code_natural,B2: code_natural] :
          ( ( ord_max_Code_natural @ A2 @ B2 )
          = B2 ) ) ) ).

% max.absorb_iff2
thf(fact_7207_max_Oabsorb__iff2,axiom,
    ( ord_less_eq_rat
    = ( ^ [A2: rat,B2: rat] :
          ( ( ord_max_rat @ A2 @ B2 )
          = B2 ) ) ) ).

% max.absorb_iff2
thf(fact_7208_max_Oabsorb__iff2,axiom,
    ( ord_less_eq_num
    = ( ^ [A2: num,B2: num] :
          ( ( ord_max_num @ A2 @ B2 )
          = B2 ) ) ) ).

% max.absorb_iff2
thf(fact_7209_max_Oabsorb__iff2,axiom,
    ( ord_less_eq_nat
    = ( ^ [A2: nat,B2: nat] :
          ( ( ord_max_nat @ A2 @ B2 )
          = B2 ) ) ) ).

% max.absorb_iff2
thf(fact_7210_max_Oabsorb__iff2,axiom,
    ( ord_less_eq_int
    = ( ^ [A2: int,B2: int] :
          ( ( ord_max_int @ A2 @ B2 )
          = B2 ) ) ) ).

% max.absorb_iff2
thf(fact_7211_max_Oabsorb__iff1,axiom,
    ( ord_le3102999989581377725nteger
    = ( ^ [B2: code_integer,A2: code_integer] :
          ( ( ord_max_Code_integer @ A2 @ B2 )
          = A2 ) ) ) ).

% max.absorb_iff1
thf(fact_7212_max_Oabsorb__iff1,axiom,
    ( ord_le1926595141338095240atural
    = ( ^ [B2: code_natural,A2: code_natural] :
          ( ( ord_max_Code_natural @ A2 @ B2 )
          = A2 ) ) ) ).

% max.absorb_iff1
thf(fact_7213_max_Oabsorb__iff1,axiom,
    ( ord_less_eq_rat
    = ( ^ [B2: rat,A2: rat] :
          ( ( ord_max_rat @ A2 @ B2 )
          = A2 ) ) ) ).

% max.absorb_iff1
thf(fact_7214_max_Oabsorb__iff1,axiom,
    ( ord_less_eq_num
    = ( ^ [B2: num,A2: num] :
          ( ( ord_max_num @ A2 @ B2 )
          = A2 ) ) ) ).

% max.absorb_iff1
thf(fact_7215_max_Oabsorb__iff1,axiom,
    ( ord_less_eq_nat
    = ( ^ [B2: nat,A2: nat] :
          ( ( ord_max_nat @ A2 @ B2 )
          = A2 ) ) ) ).

% max.absorb_iff1
thf(fact_7216_max_Oabsorb__iff1,axiom,
    ( ord_less_eq_int
    = ( ^ [B2: int,A2: int] :
          ( ( ord_max_int @ A2 @ B2 )
          = A2 ) ) ) ).

% max.absorb_iff1
thf(fact_7217_le__max__iff__disj,axiom,
    ! [Z2: code_integer,X: code_integer,Y3: code_integer] :
      ( ( ord_le3102999989581377725nteger @ Z2 @ ( ord_max_Code_integer @ X @ Y3 ) )
      = ( ( ord_le3102999989581377725nteger @ Z2 @ X )
        | ( ord_le3102999989581377725nteger @ Z2 @ Y3 ) ) ) ).

% le_max_iff_disj
thf(fact_7218_le__max__iff__disj,axiom,
    ! [Z2: code_natural,X: code_natural,Y3: code_natural] :
      ( ( ord_le1926595141338095240atural @ Z2 @ ( ord_max_Code_natural @ X @ Y3 ) )
      = ( ( ord_le1926595141338095240atural @ Z2 @ X )
        | ( ord_le1926595141338095240atural @ Z2 @ Y3 ) ) ) ).

% le_max_iff_disj
thf(fact_7219_le__max__iff__disj,axiom,
    ! [Z2: rat,X: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ Z2 @ ( ord_max_rat @ X @ Y3 ) )
      = ( ( ord_less_eq_rat @ Z2 @ X )
        | ( ord_less_eq_rat @ Z2 @ Y3 ) ) ) ).

% le_max_iff_disj
thf(fact_7220_le__max__iff__disj,axiom,
    ! [Z2: num,X: num,Y3: num] :
      ( ( ord_less_eq_num @ Z2 @ ( ord_max_num @ X @ Y3 ) )
      = ( ( ord_less_eq_num @ Z2 @ X )
        | ( ord_less_eq_num @ Z2 @ Y3 ) ) ) ).

% le_max_iff_disj
thf(fact_7221_le__max__iff__disj,axiom,
    ! [Z2: nat,X: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ Z2 @ ( ord_max_nat @ X @ Y3 ) )
      = ( ( ord_less_eq_nat @ Z2 @ X )
        | ( ord_less_eq_nat @ Z2 @ Y3 ) ) ) ).

% le_max_iff_disj
thf(fact_7222_le__max__iff__disj,axiom,
    ! [Z2: int,X: int,Y3: int] :
      ( ( ord_less_eq_int @ Z2 @ ( ord_max_int @ X @ Y3 ) )
      = ( ( ord_less_eq_int @ Z2 @ X )
        | ( ord_less_eq_int @ Z2 @ Y3 ) ) ) ).

% le_max_iff_disj
thf(fact_7223_max_Ocobounded2,axiom,
    ! [B: code_integer,A: code_integer] : ( ord_le3102999989581377725nteger @ B @ ( ord_max_Code_integer @ A @ B ) ) ).

% max.cobounded2
thf(fact_7224_max_Ocobounded2,axiom,
    ! [B: code_natural,A: code_natural] : ( ord_le1926595141338095240atural @ B @ ( ord_max_Code_natural @ A @ B ) ) ).

% max.cobounded2
thf(fact_7225_max_Ocobounded2,axiom,
    ! [B: rat,A: rat] : ( ord_less_eq_rat @ B @ ( ord_max_rat @ A @ B ) ) ).

% max.cobounded2
thf(fact_7226_max_Ocobounded2,axiom,
    ! [B: num,A: num] : ( ord_less_eq_num @ B @ ( ord_max_num @ A @ B ) ) ).

% max.cobounded2
thf(fact_7227_max_Ocobounded2,axiom,
    ! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( ord_max_nat @ A @ B ) ) ).

% max.cobounded2
thf(fact_7228_max_Ocobounded2,axiom,
    ! [B: int,A: int] : ( ord_less_eq_int @ B @ ( ord_max_int @ A @ B ) ) ).

% max.cobounded2
thf(fact_7229_max_Ocobounded1,axiom,
    ! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ A @ ( ord_max_Code_integer @ A @ B ) ) ).

% max.cobounded1
thf(fact_7230_max_Ocobounded1,axiom,
    ! [A: code_natural,B: code_natural] : ( ord_le1926595141338095240atural @ A @ ( ord_max_Code_natural @ A @ B ) ) ).

% max.cobounded1
thf(fact_7231_max_Ocobounded1,axiom,
    ! [A: rat,B: rat] : ( ord_less_eq_rat @ A @ ( ord_max_rat @ A @ B ) ) ).

% max.cobounded1
thf(fact_7232_max_Ocobounded1,axiom,
    ! [A: num,B: num] : ( ord_less_eq_num @ A @ ( ord_max_num @ A @ B ) ) ).

% max.cobounded1
thf(fact_7233_max_Ocobounded1,axiom,
    ! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( ord_max_nat @ A @ B ) ) ).

% max.cobounded1
thf(fact_7234_max_Ocobounded1,axiom,
    ! [A: int,B: int] : ( ord_less_eq_int @ A @ ( ord_max_int @ A @ B ) ) ).

% max.cobounded1
thf(fact_7235_max_Oorder__iff,axiom,
    ( ord_le3102999989581377725nteger
    = ( ^ [B2: code_integer,A2: code_integer] :
          ( A2
          = ( ord_max_Code_integer @ A2 @ B2 ) ) ) ) ).

% max.order_iff
thf(fact_7236_max_Oorder__iff,axiom,
    ( ord_le1926595141338095240atural
    = ( ^ [B2: code_natural,A2: code_natural] :
          ( A2
          = ( ord_max_Code_natural @ A2 @ B2 ) ) ) ) ).

% max.order_iff
thf(fact_7237_max_Oorder__iff,axiom,
    ( ord_less_eq_rat
    = ( ^ [B2: rat,A2: rat] :
          ( A2
          = ( ord_max_rat @ A2 @ B2 ) ) ) ) ).

% max.order_iff
thf(fact_7238_max_Oorder__iff,axiom,
    ( ord_less_eq_num
    = ( ^ [B2: num,A2: num] :
          ( A2
          = ( ord_max_num @ A2 @ B2 ) ) ) ) ).

% max.order_iff
thf(fact_7239_max_Oorder__iff,axiom,
    ( ord_less_eq_nat
    = ( ^ [B2: nat,A2: nat] :
          ( A2
          = ( ord_max_nat @ A2 @ B2 ) ) ) ) ).

% max.order_iff
thf(fact_7240_max_Oorder__iff,axiom,
    ( ord_less_eq_int
    = ( ^ [B2: int,A2: int] :
          ( A2
          = ( ord_max_int @ A2 @ B2 ) ) ) ) ).

% max.order_iff
thf(fact_7241_max_OboundedI,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( ord_le3102999989581377725nteger @ B @ A )
     => ( ( ord_le3102999989581377725nteger @ C @ A )
       => ( ord_le3102999989581377725nteger @ ( ord_max_Code_integer @ B @ C ) @ A ) ) ) ).

% max.boundedI
thf(fact_7242_max_OboundedI,axiom,
    ! [B: code_natural,A: code_natural,C: code_natural] :
      ( ( ord_le1926595141338095240atural @ B @ A )
     => ( ( ord_le1926595141338095240atural @ C @ A )
       => ( ord_le1926595141338095240atural @ ( ord_max_Code_natural @ B @ C ) @ A ) ) ) ).

% max.boundedI
thf(fact_7243_max_OboundedI,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( ord_less_eq_rat @ C @ A )
       => ( ord_less_eq_rat @ ( ord_max_rat @ B @ C ) @ A ) ) ) ).

% max.boundedI
thf(fact_7244_max_OboundedI,axiom,
    ! [B: num,A: num,C: num] :
      ( ( ord_less_eq_num @ B @ A )
     => ( ( ord_less_eq_num @ C @ A )
       => ( ord_less_eq_num @ ( ord_max_num @ B @ C ) @ A ) ) ) ).

% max.boundedI
thf(fact_7245_max_OboundedI,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_eq_nat @ C @ A )
       => ( ord_less_eq_nat @ ( ord_max_nat @ B @ C ) @ A ) ) ) ).

% max.boundedI
thf(fact_7246_max_OboundedI,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_less_eq_int @ C @ A )
       => ( ord_less_eq_int @ ( ord_max_int @ B @ C ) @ A ) ) ) ).

% max.boundedI
thf(fact_7247_max_OboundedE,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( ord_max_Code_integer @ B @ C ) @ A )
     => ~ ( ( ord_le3102999989581377725nteger @ B @ A )
         => ~ ( ord_le3102999989581377725nteger @ C @ A ) ) ) ).

% max.boundedE
thf(fact_7248_max_OboundedE,axiom,
    ! [B: code_natural,C: code_natural,A: code_natural] :
      ( ( ord_le1926595141338095240atural @ ( ord_max_Code_natural @ B @ C ) @ A )
     => ~ ( ( ord_le1926595141338095240atural @ B @ A )
         => ~ ( ord_le1926595141338095240atural @ C @ A ) ) ) ).

% max.boundedE
thf(fact_7249_max_OboundedE,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( ord_max_rat @ B @ C ) @ A )
     => ~ ( ( ord_less_eq_rat @ B @ A )
         => ~ ( ord_less_eq_rat @ C @ A ) ) ) ).

% max.boundedE
thf(fact_7250_max_OboundedE,axiom,
    ! [B: num,C: num,A: num] :
      ( ( ord_less_eq_num @ ( ord_max_num @ B @ C ) @ A )
     => ~ ( ( ord_less_eq_num @ B @ A )
         => ~ ( ord_less_eq_num @ C @ A ) ) ) ).

% max.boundedE
thf(fact_7251_max_OboundedE,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( ord_less_eq_nat @ ( ord_max_nat @ B @ C ) @ A )
     => ~ ( ( ord_less_eq_nat @ B @ A )
         => ~ ( ord_less_eq_nat @ C @ A ) ) ) ).

% max.boundedE
thf(fact_7252_max_OboundedE,axiom,
    ! [B: int,C: int,A: int] :
      ( ( ord_less_eq_int @ ( ord_max_int @ B @ C ) @ A )
     => ~ ( ( ord_less_eq_int @ B @ A )
         => ~ ( ord_less_eq_int @ C @ A ) ) ) ).

% max.boundedE
thf(fact_7253_max_OorderI,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A
        = ( ord_max_Code_integer @ A @ B ) )
     => ( ord_le3102999989581377725nteger @ B @ A ) ) ).

% max.orderI
thf(fact_7254_max_OorderI,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( A
        = ( ord_max_Code_natural @ A @ B ) )
     => ( ord_le1926595141338095240atural @ B @ A ) ) ).

% max.orderI
thf(fact_7255_max_OorderI,axiom,
    ! [A: rat,B: rat] :
      ( ( A
        = ( ord_max_rat @ A @ B ) )
     => ( ord_less_eq_rat @ B @ A ) ) ).

% max.orderI
thf(fact_7256_max_OorderI,axiom,
    ! [A: num,B: num] :
      ( ( A
        = ( ord_max_num @ A @ B ) )
     => ( ord_less_eq_num @ B @ A ) ) ).

% max.orderI
thf(fact_7257_max_OorderI,axiom,
    ! [A: nat,B: nat] :
      ( ( A
        = ( ord_max_nat @ A @ B ) )
     => ( ord_less_eq_nat @ B @ A ) ) ).

% max.orderI
thf(fact_7258_max_OorderI,axiom,
    ! [A: int,B: int] :
      ( ( A
        = ( ord_max_int @ A @ B ) )
     => ( ord_less_eq_int @ B @ A ) ) ).

% max.orderI
thf(fact_7259_max_OorderE,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ B @ A )
     => ( A
        = ( ord_max_Code_integer @ A @ B ) ) ) ).

% max.orderE
thf(fact_7260_max_OorderE,axiom,
    ! [B: code_natural,A: code_natural] :
      ( ( ord_le1926595141338095240atural @ B @ A )
     => ( A
        = ( ord_max_Code_natural @ A @ B ) ) ) ).

% max.orderE
thf(fact_7261_max_OorderE,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( A
        = ( ord_max_rat @ A @ B ) ) ) ).

% max.orderE
thf(fact_7262_max_OorderE,axiom,
    ! [B: num,A: num] :
      ( ( ord_less_eq_num @ B @ A )
     => ( A
        = ( ord_max_num @ A @ B ) ) ) ).

% max.orderE
thf(fact_7263_max_OorderE,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( A
        = ( ord_max_nat @ A @ B ) ) ) ).

% max.orderE
thf(fact_7264_max_OorderE,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( A
        = ( ord_max_int @ A @ B ) ) ) ).

% max.orderE
thf(fact_7265_max_Omono,axiom,
    ! [C: code_integer,A: code_integer,D: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ C @ A )
     => ( ( ord_le3102999989581377725nteger @ D @ B )
       => ( ord_le3102999989581377725nteger @ ( ord_max_Code_integer @ C @ D ) @ ( ord_max_Code_integer @ A @ B ) ) ) ) ).

% max.mono
thf(fact_7266_max_Omono,axiom,
    ! [C: code_natural,A: code_natural,D: code_natural,B: code_natural] :
      ( ( ord_le1926595141338095240atural @ C @ A )
     => ( ( ord_le1926595141338095240atural @ D @ B )
       => ( ord_le1926595141338095240atural @ ( ord_max_Code_natural @ C @ D ) @ ( ord_max_Code_natural @ A @ B ) ) ) ) ).

% max.mono
thf(fact_7267_max_Omono,axiom,
    ! [C: rat,A: rat,D: rat,B: rat] :
      ( ( ord_less_eq_rat @ C @ A )
     => ( ( ord_less_eq_rat @ D @ B )
       => ( ord_less_eq_rat @ ( ord_max_rat @ C @ D ) @ ( ord_max_rat @ A @ B ) ) ) ) ).

% max.mono
thf(fact_7268_max_Omono,axiom,
    ! [C: num,A: num,D: num,B: num] :
      ( ( ord_less_eq_num @ C @ A )
     => ( ( ord_less_eq_num @ D @ B )
       => ( ord_less_eq_num @ ( ord_max_num @ C @ D ) @ ( ord_max_num @ A @ B ) ) ) ) ).

% max.mono
thf(fact_7269_max_Omono,axiom,
    ! [C: nat,A: nat,D: nat,B: nat] :
      ( ( ord_less_eq_nat @ C @ A )
     => ( ( ord_less_eq_nat @ D @ B )
       => ( ord_less_eq_nat @ ( ord_max_nat @ C @ D ) @ ( ord_max_nat @ A @ B ) ) ) ) ).

% max.mono
thf(fact_7270_max_Omono,axiom,
    ! [C: int,A: int,D: int,B: int] :
      ( ( ord_less_eq_int @ C @ A )
     => ( ( ord_less_eq_int @ D @ B )
       => ( ord_less_eq_int @ ( ord_max_int @ C @ D ) @ ( ord_max_int @ A @ B ) ) ) ) ).

% max.mono
thf(fact_7271_max__def,axiom,
    ( ord_max_Code_integer
    = ( ^ [A2: code_integer,B2: code_integer] : ( if_Code_integer @ ( ord_le3102999989581377725nteger @ A2 @ B2 ) @ B2 @ A2 ) ) ) ).

% max_def
thf(fact_7272_max__def,axiom,
    ( ord_max_Code_natural
    = ( ^ [A2: code_natural,B2: code_natural] : ( if_Code_natural @ ( ord_le1926595141338095240atural @ A2 @ B2 ) @ B2 @ A2 ) ) ) ).

% max_def
thf(fact_7273_max__def,axiom,
    ( ord_max_set_nat
    = ( ^ [A2: set_nat,B2: set_nat] : ( if_set_nat @ ( ord_less_eq_set_nat @ A2 @ B2 ) @ B2 @ A2 ) ) ) ).

% max_def
thf(fact_7274_max__def,axiom,
    ( ord_max_rat
    = ( ^ [A2: rat,B2: rat] : ( if_rat @ ( ord_less_eq_rat @ A2 @ B2 ) @ B2 @ A2 ) ) ) ).

% max_def
thf(fact_7275_max__def,axiom,
    ( ord_max_num
    = ( ^ [A2: num,B2: num] : ( if_num @ ( ord_less_eq_num @ A2 @ B2 ) @ B2 @ A2 ) ) ) ).

% max_def
thf(fact_7276_max__def,axiom,
    ( ord_max_nat
    = ( ^ [A2: nat,B2: nat] : ( if_nat @ ( ord_less_eq_nat @ A2 @ B2 ) @ B2 @ A2 ) ) ) ).

% max_def
thf(fact_7277_max__def,axiom,
    ( ord_max_int
    = ( ^ [A2: int,B2: int] : ( if_int @ ( ord_less_eq_int @ A2 @ B2 ) @ B2 @ A2 ) ) ) ).

% max_def
thf(fact_7278_nat__add__max__left,axiom,
    ! [M: nat,N: nat,Q4: nat] :
      ( ( plus_plus_nat @ ( ord_max_nat @ M @ N ) @ Q4 )
      = ( ord_max_nat @ ( plus_plus_nat @ M @ Q4 ) @ ( plus_plus_nat @ N @ Q4 ) ) ) ).

% nat_add_max_left
thf(fact_7279_nat__add__max__right,axiom,
    ! [M: nat,N: nat,Q4: nat] :
      ( ( plus_plus_nat @ M @ ( ord_max_nat @ N @ Q4 ) )
      = ( ord_max_nat @ ( plus_plus_nat @ M @ N ) @ ( plus_plus_nat @ M @ Q4 ) ) ) ).

% nat_add_max_right
thf(fact_7280_nat__mult__max__left,axiom,
    ! [M: nat,N: nat,Q4: nat] :
      ( ( times_times_nat @ ( ord_max_nat @ M @ N ) @ Q4 )
      = ( ord_max_nat @ ( times_times_nat @ M @ Q4 ) @ ( times_times_nat @ N @ Q4 ) ) ) ).

% nat_mult_max_left
thf(fact_7281_nat__mult__max__right,axiom,
    ! [M: nat,N: nat,Q4: nat] :
      ( ( times_times_nat @ M @ ( ord_max_nat @ N @ Q4 ) )
      = ( ord_max_nat @ ( times_times_nat @ M @ N ) @ ( times_times_nat @ M @ Q4 ) ) ) ).

% nat_mult_max_right
thf(fact_7282_length__induct,axiom,
    ! [P: list_o > $o,Xs: list_o] :
      ( ! [Xs2: list_o] :
          ( ! [Ys: list_o] :
              ( ( ord_less_nat @ ( size_size_list_o @ Ys ) @ ( size_size_list_o @ Xs2 ) )
             => ( P @ Ys ) )
         => ( P @ Xs2 ) )
     => ( P @ Xs ) ) ).

% length_induct
thf(fact_7283_length__induct,axiom,
    ! [P: list_nat > $o,Xs: list_nat] :
      ( ! [Xs2: list_nat] :
          ( ! [Ys: list_nat] :
              ( ( ord_less_nat @ ( size_size_list_nat @ Ys ) @ ( size_size_list_nat @ Xs2 ) )
             => ( P @ Ys ) )
         => ( P @ Xs2 ) )
     => ( P @ Xs ) ) ).

% length_induct
thf(fact_7284_length__induct,axiom,
    ! [P: list_int > $o,Xs: list_int] :
      ( ! [Xs2: list_int] :
          ( ! [Ys: list_int] :
              ( ( ord_less_nat @ ( size_size_list_int @ Ys ) @ ( size_size_list_int @ Xs2 ) )
             => ( P @ Ys ) )
         => ( P @ Xs2 ) )
     => ( P @ Xs ) ) ).

% length_induct
thf(fact_7285_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_7286_neg__numeral__class_Osub__def,axiom,
    ( neg_numeral_sub_int
    = ( ^ [K3: num,L3: num] : ( minus_minus_int @ ( numeral_numeral_int @ K3 ) @ ( numeral_numeral_int @ L3 ) ) ) ) ).

% neg_numeral_class.sub_def
thf(fact_7287_neg__numeral__class_Osub__def,axiom,
    ( neg_nu5755505904847501662nteger
    = ( ^ [K3: num,L3: num] : ( minus_8373710615458151222nteger @ ( numera6620942414471956472nteger @ K3 ) @ ( numera6620942414471956472nteger @ L3 ) ) ) ) ).

% neg_numeral_class.sub_def
thf(fact_7288_neg__numeral__class_Osub__def,axiom,
    ( neg_numeral_sub_rat
    = ( ^ [K3: num,L3: num] : ( minus_minus_rat @ ( numeral_numeral_rat @ K3 ) @ ( numeral_numeral_rat @ L3 ) ) ) ) ).

% neg_numeral_class.sub_def
thf(fact_7289_sub__non__negative,axiom,
    ! [N: num,M: num] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( neg_nu5755505904847501662nteger @ N @ M ) )
      = ( ord_less_eq_num @ M @ N ) ) ).

% sub_non_negative
thf(fact_7290_sub__non__negative,axiom,
    ! [N: num,M: num] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( neg_numeral_sub_rat @ N @ M ) )
      = ( ord_less_eq_num @ M @ N ) ) ).

% sub_non_negative
thf(fact_7291_sub__non__negative,axiom,
    ! [N: num,M: num] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( neg_numeral_sub_int @ N @ M ) )
      = ( ord_less_eq_num @ M @ N ) ) ).

% sub_non_negative
thf(fact_7292_sub__non__positive,axiom,
    ! [N: num,M: num] :
      ( ( ord_le3102999989581377725nteger @ ( neg_nu5755505904847501662nteger @ N @ M ) @ zero_z3403309356797280102nteger )
      = ( ord_less_eq_num @ N @ M ) ) ).

% sub_non_positive
thf(fact_7293_sub__non__positive,axiom,
    ! [N: num,M: num] :
      ( ( ord_less_eq_rat @ ( neg_numeral_sub_rat @ N @ M ) @ zero_zero_rat )
      = ( ord_less_eq_num @ N @ M ) ) ).

% sub_non_positive
thf(fact_7294_sub__non__positive,axiom,
    ! [N: num,M: num] :
      ( ( ord_less_eq_int @ ( neg_numeral_sub_int @ N @ M ) @ zero_zero_int )
      = ( ord_less_eq_num @ N @ M ) ) ).

% sub_non_positive
thf(fact_7295_sub__negative,axiom,
    ! [N: num,M: num] :
      ( ( ord_less_rat @ ( neg_numeral_sub_rat @ N @ M ) @ zero_zero_rat )
      = ( ord_less_num @ N @ M ) ) ).

% sub_negative
thf(fact_7296_sub__negative,axiom,
    ! [N: num,M: num] :
      ( ( ord_less_int @ ( neg_numeral_sub_int @ N @ M ) @ zero_zero_int )
      = ( ord_less_num @ N @ M ) ) ).

% sub_negative
thf(fact_7297_sub__negative,axiom,
    ! [N: num,M: num] :
      ( ( ord_le6747313008572928689nteger @ ( neg_nu5755505904847501662nteger @ N @ M ) @ zero_z3403309356797280102nteger )
      = ( ord_less_num @ N @ M ) ) ).

% sub_negative
thf(fact_7298_sub__positive,axiom,
    ! [N: num,M: num] :
      ( ( ord_less_rat @ zero_zero_rat @ ( neg_numeral_sub_rat @ N @ M ) )
      = ( ord_less_num @ M @ N ) ) ).

% sub_positive
thf(fact_7299_sub__positive,axiom,
    ! [N: num,M: num] :
      ( ( ord_less_int @ zero_zero_int @ ( neg_numeral_sub_int @ N @ M ) )
      = ( ord_less_num @ M @ N ) ) ).

% sub_positive
thf(fact_7300_sub__positive,axiom,
    ! [N: num,M: num] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( neg_nu5755505904847501662nteger @ N @ M ) )
      = ( ord_less_num @ M @ N ) ) ).

% sub_positive
thf(fact_7301_sub__inc__One__eq,axiom,
    ! [N: num] :
      ( ( neg_numeral_sub_int @ ( inc @ N ) @ one )
      = ( numeral_numeral_int @ N ) ) ).

% sub_inc_One_eq
thf(fact_7302_sub__inc__One__eq,axiom,
    ! [N: num] :
      ( ( neg_nu5755505904847501662nteger @ ( inc @ N ) @ one )
      = ( numera6620942414471956472nteger @ N ) ) ).

% sub_inc_One_eq
thf(fact_7303_sub__inc__One__eq,axiom,
    ! [N: num] :
      ( ( neg_numeral_sub_rat @ ( inc @ N ) @ one )
      = ( numeral_numeral_rat @ N ) ) ).

% sub_inc_One_eq
thf(fact_7304_minus__numeral__eq__not__sub__one,axiom,
    ! [N: num] :
      ( ( uminus_uminus_int @ ( numeral_numeral_int @ N ) )
      = ( bit_ri7919022796975470100ot_int @ ( neg_numeral_sub_int @ N @ one ) ) ) ).

% minus_numeral_eq_not_sub_one
thf(fact_7305_minus__numeral__eq__not__sub__one,axiom,
    ! [N: num] :
      ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) )
      = ( bit_ri7632146776885996613nteger @ ( neg_nu5755505904847501662nteger @ N @ one ) ) ) ).

% minus_numeral_eq_not_sub_one
thf(fact_7306_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_7307_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_7308_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_7309_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_7310_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_7311_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_7312_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_7313_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_7314_dbl__dec__simps_I5_J,axiom,
    ! [K: num] :
      ( ( neg_nu3811975205180677377ec_int @ ( numeral_numeral_int @ K ) )
      = ( numeral_numeral_int @ ( bitM @ K ) ) ) ).

% dbl_dec_simps(5)
thf(fact_7315_dbl__dec__simps_I5_J,axiom,
    ! [K: num] :
      ( ( neg_nu7757733837767384882nteger @ ( numera6620942414471956472nteger @ K ) )
      = ( numera6620942414471956472nteger @ ( bitM @ K ) ) ) ).

% dbl_dec_simps(5)
thf(fact_7316_dbl__dec__simps_I5_J,axiom,
    ! [K: num] :
      ( ( neg_nu3179335615603231917ec_rat @ ( numeral_numeral_rat @ K ) )
      = ( numeral_numeral_rat @ ( bitM @ K ) ) ) ).

% dbl_dec_simps(5)
thf(fact_7317_pred__numeral__simps_I2_J,axiom,
    ! [K: num] :
      ( ( pred_numeral @ ( bit0 @ K ) )
      = ( numeral_numeral_nat @ ( bitM @ K ) ) ) ).

% pred_numeral_simps(2)
thf(fact_7318_sub__num__simps_I4_J,axiom,
    ! [K: num] :
      ( ( neg_numeral_sub_int @ ( bit0 @ K ) @ one )
      = ( numeral_numeral_int @ ( bitM @ K ) ) ) ).

% sub_num_simps(4)
thf(fact_7319_sub__num__simps_I4_J,axiom,
    ! [K: num] :
      ( ( neg_nu5755505904847501662nteger @ ( bit0 @ K ) @ one )
      = ( numera6620942414471956472nteger @ ( bitM @ K ) ) ) ).

% sub_num_simps(4)
thf(fact_7320_sub__num__simps_I4_J,axiom,
    ! [K: num] :
      ( ( neg_numeral_sub_rat @ ( bit0 @ K ) @ one )
      = ( numeral_numeral_rat @ ( bitM @ K ) ) ) ).

% sub_num_simps(4)
thf(fact_7321_sub__num__simps_I2_J,axiom,
    ! [L: num] :
      ( ( neg_numeral_sub_int @ one @ ( bit0 @ L ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bitM @ L ) ) ) ) ).

% sub_num_simps(2)
thf(fact_7322_sub__num__simps_I2_J,axiom,
    ! [L: num] :
      ( ( neg_nu5755505904847501662nteger @ one @ ( bit0 @ L ) )
      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bitM @ L ) ) ) ) ).

% sub_num_simps(2)
thf(fact_7323_sub__num__simps_I2_J,axiom,
    ! [L: num] :
      ( ( neg_numeral_sub_rat @ one @ ( bit0 @ L ) )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bitM @ L ) ) ) ) ).

% sub_num_simps(2)
thf(fact_7324_semiring__norm_I26_J,axiom,
    ( ( bitM @ one )
    = one ) ).

% semiring_norm(26)
thf(fact_7325_semiring__norm_I27_J,axiom,
    ! [N: num] :
      ( ( bitM @ ( bit0 @ N ) )
      = ( bit1 @ ( bitM @ N ) ) ) ).

% semiring_norm(27)
thf(fact_7326_semiring__norm_I28_J,axiom,
    ! [N: num] :
      ( ( bitM @ ( bit1 @ N ) )
      = ( bit1 @ ( bit0 @ N ) ) ) ).

% semiring_norm(28)
thf(fact_7327_inc__BitM__eq,axiom,
    ! [N: num] :
      ( ( inc @ ( bitM @ N ) )
      = ( bit0 @ N ) ) ).

% inc_BitM_eq
thf(fact_7328_BitM__inc__eq,axiom,
    ! [N: num] :
      ( ( bitM @ ( inc @ N ) )
      = ( bit1 @ N ) ) ).

% BitM_inc_eq
thf(fact_7329_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_7330_one__plus__BitM,axiom,
    ! [N: num] :
      ( ( plus_plus_num @ one @ ( bitM @ N ) )
      = ( bit0 @ N ) ) ).

% one_plus_BitM
thf(fact_7331_BitM__plus__one,axiom,
    ! [N: num] :
      ( ( plus_plus_num @ ( bitM @ N ) @ one )
      = ( bit0 @ N ) ) ).

% BitM_plus_one
thf(fact_7332_numeral__BitM,axiom,
    ! [N: num] :
      ( ( numeral_numeral_int @ ( bitM @ N ) )
      = ( minus_minus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ one_one_int ) ) ).

% numeral_BitM
thf(fact_7333_numeral__BitM,axiom,
    ! [N: num] :
      ( ( numera6620942414471956472nteger @ ( bitM @ N ) )
      = ( minus_8373710615458151222nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ one_one_Code_integer ) ) ).

% numeral_BitM
thf(fact_7334_numeral__BitM,axiom,
    ! [N: num] :
      ( ( numeral_numeral_rat @ ( bitM @ N ) )
      = ( minus_minus_rat @ ( numeral_numeral_rat @ ( bit0 @ N ) ) @ one_one_rat ) ) ).

% numeral_BitM
thf(fact_7335_not__numeral__BitM__eq,axiom,
    ! [N: num] :
      ( ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bitM @ N ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) ) ).

% not_numeral_BitM_eq
thf(fact_7336_not__numeral__BitM__eq,axiom,
    ! [N: num] :
      ( ( bit_ri7632146776885996613nteger @ ( numera6620942414471956472nteger @ ( bitM @ N ) ) )
      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) ) ) ).

% not_numeral_BitM_eq
thf(fact_7337_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_7338_bit__horner__sum__bit__iff,axiom,
    ! [Bs: list_o,N: nat] :
      ( ( bit_se1148574629649215175it_nat @ ( groups9119017779487936845_o_nat @ zero_n2687167440665602831ol_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Bs ) @ N )
      = ( ( ord_less_nat @ N @ ( size_size_list_o @ Bs ) )
        & ( nth_o @ Bs @ N ) ) ) ).

% bit_horner_sum_bit_iff
thf(fact_7339_bit__horner__sum__bit__iff,axiom,
    ! [Bs: list_o,N: nat] :
      ( ( bit_se9216721137139052372nteger @ ( groups3417619833198082522nteger @ zero_n356916108424825756nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Bs ) @ N )
      = ( ( ord_less_nat @ N @ ( size_size_list_o @ Bs ) )
        & ( nth_o @ Bs @ N ) ) ) ).

% bit_horner_sum_bit_iff
thf(fact_7340_bit__horner__sum__bit__iff,axiom,
    ! [Bs: list_o,N: nat] :
      ( ( bit_se8040316288895769887atural @ ( groups2241214984954800037atural @ zero_n8403883297036319079atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ Bs ) @ N )
      = ( ( ord_less_nat @ N @ ( size_size_list_o @ Bs ) )
        & ( nth_o @ Bs @ N ) ) ) ).

% bit_horner_sum_bit_iff
thf(fact_7341_bit__horner__sum__bit__iff,axiom,
    ! [Bs: list_o,N: nat] :
      ( ( bit_se1146084159140164899it_int @ ( groups9116527308978886569_o_int @ zero_n2684676970156552555ol_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Bs ) @ N )
      = ( ( ord_less_nat @ N @ ( size_size_list_o @ Bs ) )
        & ( nth_o @ Bs @ N ) ) ) ).

% bit_horner_sum_bit_iff
thf(fact_7342_bit_Oabstract__boolean__algebra__axioms,axiom,
    boolea5469654029284665142ra_int @ bit_se725231765392027082nd_int @ bit_se1409905431419307370or_int @ bit_ri7919022796975470100ot_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ).

% bit.abstract_boolean_algebra_axioms
thf(fact_7343_bit_Oabstract__boolean__algebra__axioms,axiom,
    boolea5643492148547655015nteger @ bit_se3949692690581998587nteger @ bit_se1080825931792720795nteger @ bit_ri7632146776885996613nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ).

% bit.abstract_boolean_algebra_axioms
thf(fact_7344_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_7345_eq__numeral__iff__iszero_I7_J,axiom,
    ! [X: num] :
      ( ( ( uminus_uminus_int @ ( numeral_numeral_int @ X ) )
        = one_one_int )
      = ( ring_1_iszero_int @ ( numeral_numeral_int @ ( plus_plus_num @ X @ one ) ) ) ) ).

% eq_numeral_iff_iszero(7)
thf(fact_7346_eq__numeral__iff__iszero_I7_J,axiom,
    ! [X: num] :
      ( ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) )
        = one_one_Code_integer )
      = ( ring_16219924574208605041nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ X @ one ) ) ) ) ).

% eq_numeral_iff_iszero(7)
thf(fact_7347_eq__numeral__iff__iszero_I7_J,axiom,
    ! [X: num] :
      ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) )
        = one_one_rat )
      = ( ring_1_iszero_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ X @ one ) ) ) ) ).

% eq_numeral_iff_iszero(7)
thf(fact_7348_eq__numeral__iff__iszero_I8_J,axiom,
    ! [Y3: num] :
      ( ( one_one_int
        = ( uminus_uminus_int @ ( numeral_numeral_int @ Y3 ) ) )
      = ( ring_1_iszero_int @ ( numeral_numeral_int @ ( plus_plus_num @ one @ Y3 ) ) ) ) ).

% eq_numeral_iff_iszero(8)
thf(fact_7349_eq__numeral__iff__iszero_I8_J,axiom,
    ! [Y3: num] :
      ( ( one_one_Code_integer
        = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ Y3 ) ) )
      = ( ring_16219924574208605041nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ one @ Y3 ) ) ) ) ).

% eq_numeral_iff_iszero(8)
thf(fact_7350_eq__numeral__iff__iszero_I8_J,axiom,
    ! [Y3: num] :
      ( ( one_one_rat
        = ( uminus_uminus_rat @ ( numeral_numeral_rat @ Y3 ) ) )
      = ( ring_1_iszero_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ one @ Y3 ) ) ) ) ).

% eq_numeral_iff_iszero(8)
thf(fact_7351_num__of__nat__numeral__eq,axiom,
    ! [Q4: num] :
      ( ( num_of_nat @ ( numeral_numeral_nat @ Q4 ) )
      = Q4 ) ).

% num_of_nat_numeral_eq
thf(fact_7352_iszero__neg__numeral,axiom,
    ! [W2: num] :
      ( ( ring_1_iszero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W2 ) ) )
      = ( ring_1_iszero_int @ ( numeral_numeral_int @ W2 ) ) ) ).

% iszero_neg_numeral
thf(fact_7353_iszero__neg__numeral,axiom,
    ! [W2: num] :
      ( ( ring_16219924574208605041nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W2 ) ) )
      = ( ring_16219924574208605041nteger @ ( numera6620942414471956472nteger @ W2 ) ) ) ).

% iszero_neg_numeral
thf(fact_7354_iszero__neg__numeral,axiom,
    ! [W2: num] :
      ( ( ring_1_iszero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) )
      = ( ring_1_iszero_rat @ ( numeral_numeral_rat @ W2 ) ) ) ).

% iszero_neg_numeral
thf(fact_7355_iszero__def,axiom,
    ( ring_1_iszero_int
    = ( ^ [Z6: int] : ( Z6 = zero_zero_int ) ) ) ).

% iszero_def
thf(fact_7356_iszero__def,axiom,
    ( ring_16219924574208605041nteger
    = ( ^ [Z6: code_integer] : ( Z6 = zero_z3403309356797280102nteger ) ) ) ).

% iszero_def
thf(fact_7357_iszero__def,axiom,
    ( ring_1_iszero_rat
    = ( ^ [Z6: rat] : ( Z6 = zero_zero_rat ) ) ) ).

% iszero_def
thf(fact_7358_iszero__0,axiom,
    ring_1_iszero_int @ zero_zero_int ).

% iszero_0
thf(fact_7359_iszero__0,axiom,
    ring_16219924574208605041nteger @ zero_z3403309356797280102nteger ).

% iszero_0
thf(fact_7360_iszero__0,axiom,
    ring_1_iszero_rat @ zero_zero_rat ).

% iszero_0
thf(fact_7361_not__iszero__numeral,axiom,
    ! [W2: num] :
      ~ ( ring_1_iszero_int @ ( numeral_numeral_int @ W2 ) ) ).

% not_iszero_numeral
thf(fact_7362_not__iszero__numeral,axiom,
    ! [W2: num] :
      ~ ( ring_16219924574208605041nteger @ ( numera6620942414471956472nteger @ W2 ) ) ).

% not_iszero_numeral
thf(fact_7363_not__iszero__numeral,axiom,
    ! [W2: num] :
      ~ ( ring_1_iszero_rat @ ( numeral_numeral_rat @ W2 ) ) ).

% not_iszero_numeral
thf(fact_7364_not__iszero__1,axiom,
    ~ ( ring_1_iszero_int @ one_one_int ) ).

% not_iszero_1
thf(fact_7365_not__iszero__1,axiom,
    ~ ( ring_16219924574208605041nteger @ one_one_Code_integer ) ).

% not_iszero_1
thf(fact_7366_not__iszero__1,axiom,
    ~ ( ring_1_iszero_rat @ one_one_rat ) ).

% not_iszero_1
thf(fact_7367_eq__iff__iszero__diff,axiom,
    ( ( ^ [Y: code_integer,Z: code_integer] : ( Y = Z ) )
    = ( ^ [X2: code_integer,Y6: code_integer] : ( ring_16219924574208605041nteger @ ( minus_8373710615458151222nteger @ X2 @ Y6 ) ) ) ) ).

% eq_iff_iszero_diff
thf(fact_7368_eq__iff__iszero__diff,axiom,
    ( ( ^ [Y: rat,Z: rat] : ( Y = Z ) )
    = ( ^ [X2: rat,Y6: rat] : ( ring_1_iszero_rat @ ( minus_minus_rat @ X2 @ Y6 ) ) ) ) ).

% eq_iff_iszero_diff
thf(fact_7369_eq__iff__iszero__diff,axiom,
    ( ( ^ [Y: int,Z: int] : ( Y = Z ) )
    = ( ^ [X2: int,Y6: int] : ( ring_1_iszero_int @ ( minus_minus_int @ X2 @ Y6 ) ) ) ) ).

% eq_iff_iszero_diff
thf(fact_7370_nth__equalityI,axiom,
    ! [Xs: list_o,Ys2: list_o] :
      ( ( ( size_size_list_o @ Xs )
        = ( size_size_list_o @ Ys2 ) )
     => ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_o @ Xs ) )
           => ( ( nth_o @ Xs @ I3 )
              = ( nth_o @ Ys2 @ I3 ) ) )
       => ( Xs = Ys2 ) ) ) ).

% nth_equalityI
thf(fact_7371_nth__equalityI,axiom,
    ! [Xs: list_nat,Ys2: list_nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys2 ) )
     => ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
           => ( ( nth_nat @ Xs @ I3 )
              = ( nth_nat @ Ys2 @ I3 ) ) )
       => ( Xs = Ys2 ) ) ) ).

% nth_equalityI
thf(fact_7372_nth__equalityI,axiom,
    ! [Xs: list_int,Ys2: list_int] :
      ( ( ( size_size_list_int @ Xs )
        = ( size_size_list_int @ Ys2 ) )
     => ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs ) )
           => ( ( nth_int @ Xs @ I3 )
              = ( nth_int @ Ys2 @ I3 ) ) )
       => ( Xs = Ys2 ) ) ) ).

% nth_equalityI
thf(fact_7373_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > $o > $o] :
      ( ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ K )
           => ? [X7: $o] : ( P @ I4 @ X7 ) ) )
      = ( ? [Xs3: list_o] :
            ( ( ( size_size_list_o @ Xs3 )
              = K )
            & ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ K )
               => ( P @ I4 @ ( nth_o @ Xs3 @ I4 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_7374_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > nat > $o] :
      ( ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ K )
           => ? [X7: nat] : ( P @ I4 @ X7 ) ) )
      = ( ? [Xs3: list_nat] :
            ( ( ( size_size_list_nat @ Xs3 )
              = K )
            & ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ K )
               => ( P @ I4 @ ( nth_nat @ Xs3 @ I4 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_7375_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > int > $o] :
      ( ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ K )
           => ? [X7: int] : ( P @ I4 @ X7 ) ) )
      = ( ? [Xs3: list_int] :
            ( ( ( size_size_list_int @ Xs3 )
              = K )
            & ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ K )
               => ( P @ I4 @ ( nth_int @ Xs3 @ I4 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_7376_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y: list_o,Z: list_o] : ( Y = Z ) )
    = ( ^ [Xs3: list_o,Ys3: list_o] :
          ( ( ( size_size_list_o @ Xs3 )
            = ( size_size_list_o @ Ys3 ) )
          & ! [I4: nat] :
              ( ( ord_less_nat @ I4 @ ( size_size_list_o @ Xs3 ) )
             => ( ( nth_o @ Xs3 @ I4 )
                = ( nth_o @ Ys3 @ I4 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_7377_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y: list_nat,Z: list_nat] : ( Y = Z ) )
    = ( ^ [Xs3: list_nat,Ys3: list_nat] :
          ( ( ( size_size_list_nat @ Xs3 )
            = ( size_size_list_nat @ Ys3 ) )
          & ! [I4: nat] :
              ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs3 ) )
             => ( ( nth_nat @ Xs3 @ I4 )
                = ( nth_nat @ Ys3 @ I4 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_7378_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y: list_int,Z: list_int] : ( Y = Z ) )
    = ( ^ [Xs3: list_int,Ys3: list_int] :
          ( ( ( size_size_list_int @ Xs3 )
            = ( size_size_list_int @ Ys3 ) )
          & ! [I4: nat] :
              ( ( ord_less_nat @ I4 @ ( size_size_list_int @ Xs3 ) )
             => ( ( nth_int @ Xs3 @ I4 )
                = ( nth_int @ Ys3 @ I4 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_7379_eq__numeral__iff__iszero_I10_J,axiom,
    ! [Y3: num] :
      ( ( zero_zero_int
        = ( numeral_numeral_int @ Y3 ) )
      = ( ring_1_iszero_int @ ( numeral_numeral_int @ Y3 ) ) ) ).

% eq_numeral_iff_iszero(10)
thf(fact_7380_eq__numeral__iff__iszero_I10_J,axiom,
    ! [Y3: num] :
      ( ( zero_z3403309356797280102nteger
        = ( numera6620942414471956472nteger @ Y3 ) )
      = ( ring_16219924574208605041nteger @ ( numera6620942414471956472nteger @ Y3 ) ) ) ).

% eq_numeral_iff_iszero(10)
thf(fact_7381_eq__numeral__iff__iszero_I10_J,axiom,
    ! [Y3: num] :
      ( ( zero_zero_rat
        = ( numeral_numeral_rat @ Y3 ) )
      = ( ring_1_iszero_rat @ ( numeral_numeral_rat @ Y3 ) ) ) ).

% eq_numeral_iff_iszero(10)
thf(fact_7382_eq__numeral__iff__iszero_I9_J,axiom,
    ! [X: num] :
      ( ( ( numeral_numeral_int @ X )
        = zero_zero_int )
      = ( ring_1_iszero_int @ ( numeral_numeral_int @ X ) ) ) ).

% eq_numeral_iff_iszero(9)
thf(fact_7383_eq__numeral__iff__iszero_I9_J,axiom,
    ! [X: num] :
      ( ( ( numera6620942414471956472nteger @ X )
        = zero_z3403309356797280102nteger )
      = ( ring_16219924574208605041nteger @ ( numera6620942414471956472nteger @ X ) ) ) ).

% eq_numeral_iff_iszero(9)
thf(fact_7384_eq__numeral__iff__iszero_I9_J,axiom,
    ! [X: num] :
      ( ( ( numeral_numeral_rat @ X )
        = zero_zero_rat )
      = ( ring_1_iszero_rat @ ( numeral_numeral_rat @ X ) ) ) ).

% eq_numeral_iff_iszero(9)
thf(fact_7385_num__of__nat_Osimps_I1_J,axiom,
    ( ( num_of_nat @ zero_zero_nat )
    = one ) ).

% num_of_nat.simps(1)
thf(fact_7386_not__iszero__Numeral1,axiom,
    ~ ( ring_1_iszero_int @ ( numeral_numeral_int @ one ) ) ).

% not_iszero_Numeral1
thf(fact_7387_not__iszero__Numeral1,axiom,
    ~ ( ring_16219924574208605041nteger @ ( numera6620942414471956472nteger @ one ) ) ).

% not_iszero_Numeral1
thf(fact_7388_not__iszero__Numeral1,axiom,
    ~ ( ring_1_iszero_rat @ ( numeral_numeral_rat @ one ) ) ).

% not_iszero_Numeral1
thf(fact_7389_not__iszero__neg__1,axiom,
    ~ ( ring_1_iszero_int @ ( uminus_uminus_int @ one_one_int ) ) ).

% not_iszero_neg_1
thf(fact_7390_not__iszero__neg__1,axiom,
    ~ ( ring_16219924574208605041nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% not_iszero_neg_1
thf(fact_7391_not__iszero__neg__1,axiom,
    ~ ( ring_1_iszero_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).

% not_iszero_neg_1
thf(fact_7392_eq__numeral__iff__iszero_I1_J,axiom,
    ! [X: num,Y3: num] :
      ( ( ( numeral_numeral_int @ X )
        = ( numeral_numeral_int @ Y3 ) )
      = ( ring_1_iszero_int @ ( neg_numeral_sub_int @ X @ Y3 ) ) ) ).

% eq_numeral_iff_iszero(1)
thf(fact_7393_eq__numeral__iff__iszero_I1_J,axiom,
    ! [X: num,Y3: num] :
      ( ( ( numera6620942414471956472nteger @ X )
        = ( numera6620942414471956472nteger @ Y3 ) )
      = ( ring_16219924574208605041nteger @ ( neg_nu5755505904847501662nteger @ X @ Y3 ) ) ) ).

% eq_numeral_iff_iszero(1)
thf(fact_7394_eq__numeral__iff__iszero_I1_J,axiom,
    ! [X: num,Y3: num] :
      ( ( ( numeral_numeral_rat @ X )
        = ( numeral_numeral_rat @ Y3 ) )
      = ( ring_1_iszero_rat @ ( neg_numeral_sub_rat @ X @ Y3 ) ) ) ).

% eq_numeral_iff_iszero(1)
thf(fact_7395_eq__numeral__iff__iszero_I12_J,axiom,
    ! [Y3: num] :
      ( ( zero_zero_int
        = ( uminus_uminus_int @ ( numeral_numeral_int @ Y3 ) ) )
      = ( ring_1_iszero_int @ ( numeral_numeral_int @ Y3 ) ) ) ).

% eq_numeral_iff_iszero(12)
thf(fact_7396_eq__numeral__iff__iszero_I12_J,axiom,
    ! [Y3: num] :
      ( ( zero_z3403309356797280102nteger
        = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ Y3 ) ) )
      = ( ring_16219924574208605041nteger @ ( numera6620942414471956472nteger @ Y3 ) ) ) ).

% eq_numeral_iff_iszero(12)
thf(fact_7397_eq__numeral__iff__iszero_I12_J,axiom,
    ! [Y3: num] :
      ( ( zero_zero_rat
        = ( uminus_uminus_rat @ ( numeral_numeral_rat @ Y3 ) ) )
      = ( ring_1_iszero_rat @ ( numeral_numeral_rat @ Y3 ) ) ) ).

% eq_numeral_iff_iszero(12)
thf(fact_7398_eq__numeral__iff__iszero_I11_J,axiom,
    ! [X: num] :
      ( ( ( uminus_uminus_int @ ( numeral_numeral_int @ X ) )
        = zero_zero_int )
      = ( ring_1_iszero_int @ ( numeral_numeral_int @ X ) ) ) ).

% eq_numeral_iff_iszero(11)
thf(fact_7399_eq__numeral__iff__iszero_I11_J,axiom,
    ! [X: num] :
      ( ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) )
        = zero_z3403309356797280102nteger )
      = ( ring_16219924574208605041nteger @ ( numera6620942414471956472nteger @ X ) ) ) ).

% eq_numeral_iff_iszero(11)
thf(fact_7400_eq__numeral__iff__iszero_I11_J,axiom,
    ! [X: num] :
      ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) )
        = zero_zero_rat )
      = ( ring_1_iszero_rat @ ( numeral_numeral_rat @ X ) ) ) ).

% eq_numeral_iff_iszero(11)
thf(fact_7401_not__iszero__neg__Numeral1,axiom,
    ~ ( ring_1_iszero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ one ) ) ) ).

% not_iszero_neg_Numeral1
thf(fact_7402_not__iszero__neg__Numeral1,axiom,
    ~ ( ring_16219924574208605041nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) ) ) ).

% not_iszero_neg_Numeral1
thf(fact_7403_not__iszero__neg__Numeral1,axiom,
    ~ ( ring_1_iszero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) ) ) ).

% not_iszero_neg_Numeral1
thf(fact_7404_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_7405_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_7406_eq__numeral__iff__iszero_I3_J,axiom,
    ! [X: num,Y3: num] :
      ( ( ( uminus_uminus_int @ ( numeral_numeral_int @ X ) )
        = ( numeral_numeral_int @ Y3 ) )
      = ( ring_1_iszero_int @ ( numeral_numeral_int @ ( plus_plus_num @ X @ Y3 ) ) ) ) ).

% eq_numeral_iff_iszero(3)
thf(fact_7407_eq__numeral__iff__iszero_I3_J,axiom,
    ! [X: num,Y3: num] :
      ( ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) )
        = ( numera6620942414471956472nteger @ Y3 ) )
      = ( ring_16219924574208605041nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ X @ Y3 ) ) ) ) ).

% eq_numeral_iff_iszero(3)
thf(fact_7408_eq__numeral__iff__iszero_I3_J,axiom,
    ! [X: num,Y3: num] :
      ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) )
        = ( numeral_numeral_rat @ Y3 ) )
      = ( ring_1_iszero_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ X @ Y3 ) ) ) ) ).

% eq_numeral_iff_iszero(3)
thf(fact_7409_eq__numeral__iff__iszero_I2_J,axiom,
    ! [X: num,Y3: num] :
      ( ( ( numeral_numeral_int @ X )
        = ( uminus_uminus_int @ ( numeral_numeral_int @ Y3 ) ) )
      = ( ring_1_iszero_int @ ( numeral_numeral_int @ ( plus_plus_num @ X @ Y3 ) ) ) ) ).

% eq_numeral_iff_iszero(2)
thf(fact_7410_eq__numeral__iff__iszero_I2_J,axiom,
    ! [X: num,Y3: num] :
      ( ( ( numera6620942414471956472nteger @ X )
        = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ Y3 ) ) )
      = ( ring_16219924574208605041nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ X @ Y3 ) ) ) ) ).

% eq_numeral_iff_iszero(2)
thf(fact_7411_eq__numeral__iff__iszero_I2_J,axiom,
    ! [X: num,Y3: num] :
      ( ( ( numeral_numeral_rat @ X )
        = ( uminus_uminus_rat @ ( numeral_numeral_rat @ Y3 ) ) )
      = ( ring_1_iszero_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ X @ Y3 ) ) ) ) ).

% eq_numeral_iff_iszero(2)
thf(fact_7412_eq__numeral__iff__iszero_I4_J,axiom,
    ! [X: num,Y3: num] :
      ( ( ( uminus_uminus_int @ ( numeral_numeral_int @ X ) )
        = ( uminus_uminus_int @ ( numeral_numeral_int @ Y3 ) ) )
      = ( ring_1_iszero_int @ ( neg_numeral_sub_int @ Y3 @ X ) ) ) ).

% eq_numeral_iff_iszero(4)
thf(fact_7413_eq__numeral__iff__iszero_I4_J,axiom,
    ! [X: num,Y3: num] :
      ( ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) )
        = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ Y3 ) ) )
      = ( ring_16219924574208605041nteger @ ( neg_nu5755505904847501662nteger @ Y3 @ X ) ) ) ).

% eq_numeral_iff_iszero(4)
thf(fact_7414_eq__numeral__iff__iszero_I4_J,axiom,
    ! [X: num,Y3: num] :
      ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) )
        = ( uminus_uminus_rat @ ( numeral_numeral_rat @ Y3 ) ) )
      = ( ring_1_iszero_rat @ ( neg_numeral_sub_rat @ Y3 @ X ) ) ) ).

% eq_numeral_iff_iszero(4)
thf(fact_7415_numeral__num__of__nat__unfold,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( numeral_numeral_int @ ( num_of_nat @ N ) )
          = one_one_int ) )
      & ( ( N != zero_zero_nat )
       => ( ( numeral_numeral_int @ ( num_of_nat @ N ) )
          = ( semiri1314217659103216013at_int @ N ) ) ) ) ).

% numeral_num_of_nat_unfold
thf(fact_7416_numeral__num__of__nat__unfold,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( numera6620942414471956472nteger @ ( num_of_nat @ N ) )
          = one_one_Code_integer ) )
      & ( ( N != zero_zero_nat )
       => ( ( numera6620942414471956472nteger @ ( num_of_nat @ N ) )
          = ( semiri4939895301339042750nteger @ N ) ) ) ) ).

% numeral_num_of_nat_unfold
thf(fact_7417_numeral__num__of__nat__unfold,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( numeral_numeral_nat @ ( num_of_nat @ N ) )
          = one_one_nat ) )
      & ( ( N != zero_zero_nat )
       => ( ( numeral_numeral_nat @ ( num_of_nat @ N ) )
          = ( semiri1316708129612266289at_nat @ N ) ) ) ) ).

% numeral_num_of_nat_unfold
thf(fact_7418_numeral__num__of__nat__unfold,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( numeral_numeral_rat @ ( num_of_nat @ N ) )
          = one_one_rat ) )
      & ( ( N != zero_zero_nat )
       => ( ( numeral_numeral_rat @ ( num_of_nat @ N ) )
          = ( semiri681578069525770553at_rat @ N ) ) ) ) ).

% numeral_num_of_nat_unfold
thf(fact_7419_numeral__num__of__nat__unfold,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( numera5444537566228673987atural @ ( num_of_nat @ N ) )
          = one_one_Code_natural ) )
      & ( ( N != zero_zero_nat )
       => ( ( numera5444537566228673987atural @ ( num_of_nat @ N ) )
          = ( semiri3763490453095760265atural @ N ) ) ) ) ).

% numeral_num_of_nat_unfold
thf(fact_7420_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_7421_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_7422_eq__numeral__iff__iszero_I5_J,axiom,
    ! [X: num] :
      ( ( ( numeral_numeral_int @ X )
        = one_one_int )
      = ( ring_1_iszero_int @ ( neg_numeral_sub_int @ X @ one ) ) ) ).

% eq_numeral_iff_iszero(5)
thf(fact_7423_eq__numeral__iff__iszero_I5_J,axiom,
    ! [X: num] :
      ( ( ( numera6620942414471956472nteger @ X )
        = one_one_Code_integer )
      = ( ring_16219924574208605041nteger @ ( neg_nu5755505904847501662nteger @ X @ one ) ) ) ).

% eq_numeral_iff_iszero(5)
thf(fact_7424_eq__numeral__iff__iszero_I5_J,axiom,
    ! [X: num] :
      ( ( ( numeral_numeral_rat @ X )
        = one_one_rat )
      = ( ring_1_iszero_rat @ ( neg_numeral_sub_rat @ X @ one ) ) ) ).

% eq_numeral_iff_iszero(5)
thf(fact_7425_eq__numeral__iff__iszero_I6_J,axiom,
    ! [Y3: num] :
      ( ( one_one_int
        = ( numeral_numeral_int @ Y3 ) )
      = ( ring_1_iszero_int @ ( neg_numeral_sub_int @ one @ Y3 ) ) ) ).

% eq_numeral_iff_iszero(6)
thf(fact_7426_eq__numeral__iff__iszero_I6_J,axiom,
    ! [Y3: num] :
      ( ( one_one_Code_integer
        = ( numera6620942414471956472nteger @ Y3 ) )
      = ( ring_16219924574208605041nteger @ ( neg_nu5755505904847501662nteger @ one @ Y3 ) ) ) ).

% eq_numeral_iff_iszero(6)
thf(fact_7427_eq__numeral__iff__iszero_I6_J,axiom,
    ! [Y3: num] :
      ( ( one_one_rat
        = ( numeral_numeral_rat @ Y3 ) )
      = ( ring_1_iszero_rat @ ( neg_numeral_sub_rat @ one @ Y3 ) ) ) ).

% eq_numeral_iff_iszero(6)
thf(fact_7428_nth__rotate1,axiom,
    ! [N: nat,Xs: list_o] :
      ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs ) )
     => ( ( nth_o @ ( rotate1_o @ Xs ) @ N )
        = ( nth_o @ Xs @ ( modulo_modulo_nat @ ( suc @ N ) @ ( size_size_list_o @ Xs ) ) ) ) ) ).

% nth_rotate1
thf(fact_7429_nth__rotate1,axiom,
    ! [N: nat,Xs: list_nat] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
     => ( ( nth_nat @ ( rotate1_nat @ Xs ) @ N )
        = ( nth_nat @ Xs @ ( modulo_modulo_nat @ ( suc @ N ) @ ( size_size_list_nat @ Xs ) ) ) ) ) ).

% nth_rotate1
thf(fact_7430_nth__rotate1,axiom,
    ! [N: nat,Xs: list_int] :
      ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
     => ( ( nth_int @ ( rotate1_int @ Xs ) @ N )
        = ( nth_int @ Xs @ ( modulo_modulo_nat @ ( suc @ N ) @ ( size_size_list_int @ Xs ) ) ) ) ) ).

% nth_rotate1
thf(fact_7431_mask__mod__exp,axiom,
    ! [N: nat,M: nat] :
      ( ( modulo_modulo_nat @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
      = ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( ord_min_nat @ M @ N ) ) @ one_one_nat ) ) ).

% mask_mod_exp
thf(fact_7432_mask__mod__exp,axiom,
    ! [N: nat,M: nat] :
      ( ( modulo_modulo_int @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
      = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( ord_min_nat @ M @ N ) ) @ one_one_int ) ) ).

% mask_mod_exp
thf(fact_7433_mask__mod__exp,axiom,
    ! [N: nat,M: nat] :
      ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) @ one_one_Code_integer ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
      = ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( ord_min_nat @ M @ N ) ) @ one_one_Code_integer ) ) ).

% mask_mod_exp
thf(fact_7434_mask__mod__exp,axiom,
    ! [N: nat,M: nat] :
      ( ( modulo8411746178871703098atural @ ( minus_7197305767214868737atural @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) @ one_one_Code_natural ) @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ M ) )
      = ( minus_7197305767214868737atural @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( ord_min_nat @ M @ N ) ) @ one_one_Code_natural ) ) ).

% mask_mod_exp
thf(fact_7435_set__decode__plus__power__2,axiom,
    ! [N: nat,Z2: nat] :
      ( ~ ( member_nat @ N @ ( nat_set_decode @ Z2 ) )
     => ( ( nat_set_decode @ ( plus_plus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ Z2 ) )
        = ( insert_nat @ N @ ( nat_set_decode @ Z2 ) ) ) ) ).

% set_decode_plus_power_2
thf(fact_7436_nth__rotate,axiom,
    ! [N: nat,Xs: list_o,M: nat] :
      ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs ) )
     => ( ( nth_o @ ( rotate_o @ M @ Xs ) @ N )
        = ( nth_o @ Xs @ ( modulo_modulo_nat @ ( plus_plus_nat @ M @ N ) @ ( size_size_list_o @ Xs ) ) ) ) ) ).

% nth_rotate
thf(fact_7437_nth__rotate,axiom,
    ! [N: nat,Xs: list_nat,M: nat] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
     => ( ( nth_nat @ ( rotate_nat @ M @ Xs ) @ N )
        = ( nth_nat @ Xs @ ( modulo_modulo_nat @ ( plus_plus_nat @ M @ N ) @ ( size_size_list_nat @ Xs ) ) ) ) ) ).

% nth_rotate
thf(fact_7438_nth__rotate,axiom,
    ! [N: nat,Xs: list_int,M: nat] :
      ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
     => ( ( nth_int @ ( rotate_int @ M @ Xs ) @ N )
        = ( nth_int @ Xs @ ( modulo_modulo_nat @ ( plus_plus_nat @ M @ N ) @ ( size_size_list_int @ Xs ) ) ) ) ) ).

% nth_rotate
thf(fact_7439_min_Oidem,axiom,
    ! [A: nat] :
      ( ( ord_min_nat @ A @ A )
      = A ) ).

% min.idem
thf(fact_7440_min_Oidem,axiom,
    ! [A: int] :
      ( ( ord_min_int @ A @ A )
      = A ) ).

% min.idem
thf(fact_7441_min_Oidem,axiom,
    ! [A: code_integer] :
      ( ( ord_min_Code_integer @ A @ A )
      = A ) ).

% min.idem
thf(fact_7442_min_Oidem,axiom,
    ! [A: code_natural] :
      ( ( ord_min_Code_natural @ A @ A )
      = A ) ).

% min.idem
thf(fact_7443_min_Oleft__idem,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_min_nat @ A @ ( ord_min_nat @ A @ B ) )
      = ( ord_min_nat @ A @ B ) ) ).

% min.left_idem
thf(fact_7444_min_Oleft__idem,axiom,
    ! [A: int,B: int] :
      ( ( ord_min_int @ A @ ( ord_min_int @ A @ B ) )
      = ( ord_min_int @ A @ B ) ) ).

% min.left_idem
thf(fact_7445_min_Oleft__idem,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_min_Code_integer @ A @ ( ord_min_Code_integer @ A @ B ) )
      = ( ord_min_Code_integer @ A @ B ) ) ).

% min.left_idem
thf(fact_7446_min_Oleft__idem,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( ord_min_Code_natural @ A @ ( ord_min_Code_natural @ A @ B ) )
      = ( ord_min_Code_natural @ A @ B ) ) ).

% min.left_idem
thf(fact_7447_min_Oright__idem,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_min_nat @ ( ord_min_nat @ A @ B ) @ B )
      = ( ord_min_nat @ A @ B ) ) ).

% min.right_idem
thf(fact_7448_min_Oright__idem,axiom,
    ! [A: int,B: int] :
      ( ( ord_min_int @ ( ord_min_int @ A @ B ) @ B )
      = ( ord_min_int @ A @ B ) ) ).

% min.right_idem
thf(fact_7449_min_Oright__idem,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_min_Code_integer @ ( ord_min_Code_integer @ A @ B ) @ B )
      = ( ord_min_Code_integer @ A @ B ) ) ).

% min.right_idem
thf(fact_7450_min_Oright__idem,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( ord_min_Code_natural @ ( ord_min_Code_natural @ A @ B ) @ B )
      = ( ord_min_Code_natural @ A @ B ) ) ).

% min.right_idem
thf(fact_7451_min_Obounded__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ ( ord_min_Code_integer @ B @ C ) )
      = ( ( ord_le3102999989581377725nteger @ A @ B )
        & ( ord_le3102999989581377725nteger @ A @ C ) ) ) ).

% min.bounded_iff
thf(fact_7452_min_Obounded__iff,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( ( ord_le1926595141338095240atural @ A @ ( ord_min_Code_natural @ B @ C ) )
      = ( ( ord_le1926595141338095240atural @ A @ B )
        & ( ord_le1926595141338095240atural @ A @ C ) ) ) ).

% min.bounded_iff
thf(fact_7453_min_Obounded__iff,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ ( ord_min_rat @ B @ C ) )
      = ( ( ord_less_eq_rat @ A @ B )
        & ( ord_less_eq_rat @ A @ C ) ) ) ).

% min.bounded_iff
thf(fact_7454_min_Obounded__iff,axiom,
    ! [A: num,B: num,C: num] :
      ( ( ord_less_eq_num @ A @ ( ord_min_num @ B @ C ) )
      = ( ( ord_less_eq_num @ A @ B )
        & ( ord_less_eq_num @ A @ C ) ) ) ).

% min.bounded_iff
thf(fact_7455_min_Obounded__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ ( ord_min_nat @ B @ C ) )
      = ( ( ord_less_eq_nat @ A @ B )
        & ( ord_less_eq_nat @ A @ C ) ) ) ).

% min.bounded_iff
thf(fact_7456_min_Obounded__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ ( ord_min_int @ B @ C ) )
      = ( ( ord_less_eq_int @ A @ B )
        & ( ord_less_eq_int @ A @ C ) ) ) ).

% min.bounded_iff
thf(fact_7457_min_Oabsorb2,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ B @ A )
     => ( ( ord_min_Code_integer @ A @ B )
        = B ) ) ).

% min.absorb2
thf(fact_7458_min_Oabsorb2,axiom,
    ! [B: code_natural,A: code_natural] :
      ( ( ord_le1926595141338095240atural @ B @ A )
     => ( ( ord_min_Code_natural @ A @ B )
        = B ) ) ).

% min.absorb2
thf(fact_7459_min_Oabsorb2,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( ord_min_rat @ A @ B )
        = B ) ) ).

% min.absorb2
thf(fact_7460_min_Oabsorb2,axiom,
    ! [B: num,A: num] :
      ( ( ord_less_eq_num @ B @ A )
     => ( ( ord_min_num @ A @ B )
        = B ) ) ).

% min.absorb2
thf(fact_7461_min_Oabsorb2,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_min_nat @ A @ B )
        = B ) ) ).

% min.absorb2
thf(fact_7462_min_Oabsorb2,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_min_int @ A @ B )
        = B ) ) ).

% min.absorb2
thf(fact_7463_min_Oabsorb1,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ B )
     => ( ( ord_min_Code_integer @ A @ B )
        = A ) ) ).

% min.absorb1
thf(fact_7464_min_Oabsorb1,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( ord_le1926595141338095240atural @ A @ B )
     => ( ( ord_min_Code_natural @ A @ B )
        = A ) ) ).

% min.absorb1
thf(fact_7465_min_Oabsorb1,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_min_rat @ A @ B )
        = A ) ) ).

% min.absorb1
thf(fact_7466_min_Oabsorb1,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_min_num @ A @ B )
        = A ) ) ).

% min.absorb1
thf(fact_7467_min_Oabsorb1,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_min_nat @ A @ B )
        = A ) ) ).

% min.absorb1
thf(fact_7468_min_Oabsorb1,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_min_int @ A @ B )
        = A ) ) ).

% min.absorb1
thf(fact_7469_min__less__iff__conj,axiom,
    ! [Z2: code_natural,X: code_natural,Y3: code_natural] :
      ( ( ord_le5570908160329646204atural @ Z2 @ ( ord_min_Code_natural @ X @ Y3 ) )
      = ( ( ord_le5570908160329646204atural @ Z2 @ X )
        & ( ord_le5570908160329646204atural @ Z2 @ Y3 ) ) ) ).

% min_less_iff_conj
thf(fact_7470_min__less__iff__conj,axiom,
    ! [Z2: code_integer,X: code_integer,Y3: code_integer] :
      ( ( ord_le6747313008572928689nteger @ Z2 @ ( ord_min_Code_integer @ X @ Y3 ) )
      = ( ( ord_le6747313008572928689nteger @ Z2 @ X )
        & ( ord_le6747313008572928689nteger @ Z2 @ Y3 ) ) ) ).

% min_less_iff_conj
thf(fact_7471_min__less__iff__conj,axiom,
    ! [Z2: rat,X: rat,Y3: rat] :
      ( ( ord_less_rat @ Z2 @ ( ord_min_rat @ X @ Y3 ) )
      = ( ( ord_less_rat @ Z2 @ X )
        & ( ord_less_rat @ Z2 @ Y3 ) ) ) ).

% min_less_iff_conj
thf(fact_7472_min__less__iff__conj,axiom,
    ! [Z2: num,X: num,Y3: num] :
      ( ( ord_less_num @ Z2 @ ( ord_min_num @ X @ Y3 ) )
      = ( ( ord_less_num @ Z2 @ X )
        & ( ord_less_num @ Z2 @ Y3 ) ) ) ).

% min_less_iff_conj
thf(fact_7473_min__less__iff__conj,axiom,
    ! [Z2: nat,X: nat,Y3: nat] :
      ( ( ord_less_nat @ Z2 @ ( ord_min_nat @ X @ Y3 ) )
      = ( ( ord_less_nat @ Z2 @ X )
        & ( ord_less_nat @ Z2 @ Y3 ) ) ) ).

% min_less_iff_conj
thf(fact_7474_min__less__iff__conj,axiom,
    ! [Z2: int,X: int,Y3: int] :
      ( ( ord_less_int @ Z2 @ ( ord_min_int @ X @ Y3 ) )
      = ( ( ord_less_int @ Z2 @ X )
        & ( ord_less_int @ Z2 @ Y3 ) ) ) ).

% min_less_iff_conj
thf(fact_7475_min_Oabsorb4,axiom,
    ! [B: code_natural,A: code_natural] :
      ( ( ord_le5570908160329646204atural @ B @ A )
     => ( ( ord_min_Code_natural @ A @ B )
        = B ) ) ).

% min.absorb4
thf(fact_7476_min_Oabsorb4,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ B @ A )
     => ( ( ord_min_Code_integer @ A @ B )
        = B ) ) ).

% min.absorb4
thf(fact_7477_min_Oabsorb4,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( ord_min_rat @ A @ B )
        = B ) ) ).

% min.absorb4
thf(fact_7478_min_Oabsorb4,axiom,
    ! [B: num,A: num] :
      ( ( ord_less_num @ B @ A )
     => ( ( ord_min_num @ A @ B )
        = B ) ) ).

% min.absorb4
thf(fact_7479_min_Oabsorb4,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( ord_min_nat @ A @ B )
        = B ) ) ).

% min.absorb4
thf(fact_7480_min_Oabsorb4,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( ord_min_int @ A @ B )
        = B ) ) ).

% min.absorb4
thf(fact_7481_min_Oabsorb3,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( ord_le5570908160329646204atural @ A @ B )
     => ( ( ord_min_Code_natural @ A @ B )
        = A ) ) ).

% min.absorb3
thf(fact_7482_min_Oabsorb3,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ B )
     => ( ( ord_min_Code_integer @ A @ B )
        = A ) ) ).

% min.absorb3
thf(fact_7483_min_Oabsorb3,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_min_rat @ A @ B )
        = A ) ) ).

% min.absorb3
thf(fact_7484_min_Oabsorb3,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_num @ A @ B )
     => ( ( ord_min_num @ A @ B )
        = A ) ) ).

% min.absorb3
thf(fact_7485_min_Oabsorb3,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_min_nat @ A @ B )
        = A ) ) ).

% min.absorb3
thf(fact_7486_min_Oabsorb3,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_min_int @ A @ B )
        = A ) ) ).

% min.absorb3
thf(fact_7487_max__min__same_I4_J,axiom,
    ! [Y3: nat,X: nat] :
      ( ( ord_max_nat @ Y3 @ ( ord_min_nat @ X @ Y3 ) )
      = Y3 ) ).

% max_min_same(4)
thf(fact_7488_max__min__same_I4_J,axiom,
    ! [Y3: int,X: int] :
      ( ( ord_max_int @ Y3 @ ( ord_min_int @ X @ Y3 ) )
      = Y3 ) ).

% max_min_same(4)
thf(fact_7489_max__min__same_I4_J,axiom,
    ! [Y3: code_integer,X: code_integer] :
      ( ( ord_max_Code_integer @ Y3 @ ( ord_min_Code_integer @ X @ Y3 ) )
      = Y3 ) ).

% max_min_same(4)
thf(fact_7490_max__min__same_I4_J,axiom,
    ! [Y3: code_natural,X: code_natural] :
      ( ( ord_max_Code_natural @ Y3 @ ( ord_min_Code_natural @ X @ Y3 ) )
      = Y3 ) ).

% max_min_same(4)
thf(fact_7491_max__min__same_I3_J,axiom,
    ! [X: nat,Y3: nat] :
      ( ( ord_max_nat @ ( ord_min_nat @ X @ Y3 ) @ Y3 )
      = Y3 ) ).

% max_min_same(3)
thf(fact_7492_max__min__same_I3_J,axiom,
    ! [X: int,Y3: int] :
      ( ( ord_max_int @ ( ord_min_int @ X @ Y3 ) @ Y3 )
      = Y3 ) ).

% max_min_same(3)
thf(fact_7493_max__min__same_I3_J,axiom,
    ! [X: code_integer,Y3: code_integer] :
      ( ( ord_max_Code_integer @ ( ord_min_Code_integer @ X @ Y3 ) @ Y3 )
      = Y3 ) ).

% max_min_same(3)
thf(fact_7494_max__min__same_I3_J,axiom,
    ! [X: code_natural,Y3: code_natural] :
      ( ( ord_max_Code_natural @ ( ord_min_Code_natural @ X @ Y3 ) @ Y3 )
      = Y3 ) ).

% max_min_same(3)
thf(fact_7495_max__min__same_I2_J,axiom,
    ! [X: nat,Y3: nat] :
      ( ( ord_max_nat @ ( ord_min_nat @ X @ Y3 ) @ X )
      = X ) ).

% max_min_same(2)
thf(fact_7496_max__min__same_I2_J,axiom,
    ! [X: int,Y3: int] :
      ( ( ord_max_int @ ( ord_min_int @ X @ Y3 ) @ X )
      = X ) ).

% max_min_same(2)
thf(fact_7497_max__min__same_I2_J,axiom,
    ! [X: code_integer,Y3: code_integer] :
      ( ( ord_max_Code_integer @ ( ord_min_Code_integer @ X @ Y3 ) @ X )
      = X ) ).

% max_min_same(2)
thf(fact_7498_max__min__same_I2_J,axiom,
    ! [X: code_natural,Y3: code_natural] :
      ( ( ord_max_Code_natural @ ( ord_min_Code_natural @ X @ Y3 ) @ X )
      = X ) ).

% max_min_same(2)
thf(fact_7499_max__min__same_I1_J,axiom,
    ! [X: nat,Y3: nat] :
      ( ( ord_max_nat @ X @ ( ord_min_nat @ X @ Y3 ) )
      = X ) ).

% max_min_same(1)
thf(fact_7500_max__min__same_I1_J,axiom,
    ! [X: int,Y3: int] :
      ( ( ord_max_int @ X @ ( ord_min_int @ X @ Y3 ) )
      = X ) ).

% max_min_same(1)
thf(fact_7501_max__min__same_I1_J,axiom,
    ! [X: code_integer,Y3: code_integer] :
      ( ( ord_max_Code_integer @ X @ ( ord_min_Code_integer @ X @ Y3 ) )
      = X ) ).

% max_min_same(1)
thf(fact_7502_max__min__same_I1_J,axiom,
    ! [X: code_natural,Y3: code_natural] :
      ( ( ord_max_Code_natural @ X @ ( ord_min_Code_natural @ X @ Y3 ) )
      = X ) ).

% max_min_same(1)
thf(fact_7503_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_7504_min__0L,axiom,
    ! [N: nat] :
      ( ( ord_min_nat @ zero_zero_nat @ N )
      = zero_zero_nat ) ).

% min_0L
thf(fact_7505_min__0R,axiom,
    ! [N: nat] :
      ( ( ord_min_nat @ N @ zero_zero_nat )
      = zero_zero_nat ) ).

% min_0R
thf(fact_7506_min__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ U ) @ ( numera6620942414471956472nteger @ V ) )
       => ( ( ord_min_Code_integer @ ( numera6620942414471956472nteger @ U ) @ ( numera6620942414471956472nteger @ V ) )
          = ( numera6620942414471956472nteger @ U ) ) )
      & ( ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ U ) @ ( numera6620942414471956472nteger @ V ) )
       => ( ( ord_min_Code_integer @ ( numera6620942414471956472nteger @ U ) @ ( numera6620942414471956472nteger @ V ) )
          = ( numera6620942414471956472nteger @ V ) ) ) ) ).

% min_number_of(1)
thf(fact_7507_min__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_le1926595141338095240atural @ ( numera5444537566228673987atural @ U ) @ ( numera5444537566228673987atural @ V ) )
       => ( ( ord_min_Code_natural @ ( numera5444537566228673987atural @ U ) @ ( numera5444537566228673987atural @ V ) )
          = ( numera5444537566228673987atural @ U ) ) )
      & ( ~ ( ord_le1926595141338095240atural @ ( numera5444537566228673987atural @ U ) @ ( numera5444537566228673987atural @ V ) )
       => ( ( ord_min_Code_natural @ ( numera5444537566228673987atural @ U ) @ ( numera5444537566228673987atural @ V ) )
          = ( numera5444537566228673987atural @ V ) ) ) ) ).

% min_number_of(1)
thf(fact_7508_min__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
       => ( ( ord_min_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
          = ( numeral_numeral_rat @ U ) ) )
      & ( ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
       => ( ( ord_min_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
          = ( numeral_numeral_rat @ V ) ) ) ) ).

% min_number_of(1)
thf(fact_7509_min__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
       => ( ( ord_min_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
          = ( numeral_numeral_nat @ U ) ) )
      & ( ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
       => ( ( ord_min_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
          = ( numeral_numeral_nat @ V ) ) ) ) ).

% min_number_of(1)
thf(fact_7510_min__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
       => ( ( ord_min_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
          = ( numeral_numeral_int @ U ) ) )
      & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
       => ( ( ord_min_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
          = ( numeral_numeral_int @ V ) ) ) ) ).

% min_number_of(1)
thf(fact_7511_min__0__1_I3_J,axiom,
    ! [X: num] :
      ( ( ord_min_int @ zero_zero_int @ ( numeral_numeral_int @ X ) )
      = zero_zero_int ) ).

% min_0_1(3)
thf(fact_7512_min__0__1_I3_J,axiom,
    ! [X: num] :
      ( ( ord_min_nat @ zero_zero_nat @ ( numeral_numeral_nat @ X ) )
      = zero_zero_nat ) ).

% min_0_1(3)
thf(fact_7513_min__0__1_I3_J,axiom,
    ! [X: num] :
      ( ( ord_min_Code_integer @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ X ) )
      = zero_z3403309356797280102nteger ) ).

% min_0_1(3)
thf(fact_7514_min__0__1_I3_J,axiom,
    ! [X: num] :
      ( ( ord_min_rat @ zero_zero_rat @ ( numeral_numeral_rat @ X ) )
      = zero_zero_rat ) ).

% min_0_1(3)
thf(fact_7515_min__0__1_I3_J,axiom,
    ! [X: num] :
      ( ( ord_min_Code_natural @ zero_z2226904508553997617atural @ ( numera5444537566228673987atural @ X ) )
      = zero_z2226904508553997617atural ) ).

% min_0_1(3)
thf(fact_7516_min__0__1_I4_J,axiom,
    ! [X: num] :
      ( ( ord_min_int @ ( numeral_numeral_int @ X ) @ zero_zero_int )
      = zero_zero_int ) ).

% min_0_1(4)
thf(fact_7517_min__0__1_I4_J,axiom,
    ! [X: num] :
      ( ( ord_min_nat @ ( numeral_numeral_nat @ X ) @ zero_zero_nat )
      = zero_zero_nat ) ).

% min_0_1(4)
thf(fact_7518_min__0__1_I4_J,axiom,
    ! [X: num] :
      ( ( ord_min_Code_integer @ ( numera6620942414471956472nteger @ X ) @ zero_z3403309356797280102nteger )
      = zero_z3403309356797280102nteger ) ).

% min_0_1(4)
thf(fact_7519_min__0__1_I4_J,axiom,
    ! [X: num] :
      ( ( ord_min_rat @ ( numeral_numeral_rat @ X ) @ zero_zero_rat )
      = zero_zero_rat ) ).

% min_0_1(4)
thf(fact_7520_min__0__1_I4_J,axiom,
    ! [X: num] :
      ( ( ord_min_Code_natural @ ( numera5444537566228673987atural @ X ) @ zero_z2226904508553997617atural )
      = zero_z2226904508553997617atural ) ).

% min_0_1(4)
thf(fact_7521_min__0__1_I1_J,axiom,
    ( ( ord_min_rat @ zero_zero_rat @ one_one_rat )
    = zero_zero_rat ) ).

% min_0_1(1)
thf(fact_7522_min__0__1_I1_J,axiom,
    ( ( ord_min_nat @ zero_zero_nat @ one_one_nat )
    = zero_zero_nat ) ).

% min_0_1(1)
thf(fact_7523_min__0__1_I1_J,axiom,
    ( ( ord_min_int @ zero_zero_int @ one_one_int )
    = zero_zero_int ) ).

% min_0_1(1)
thf(fact_7524_min__0__1_I1_J,axiom,
    ( ( ord_min_Code_integer @ zero_z3403309356797280102nteger @ one_one_Code_integer )
    = zero_z3403309356797280102nteger ) ).

% min_0_1(1)
thf(fact_7525_min__0__1_I1_J,axiom,
    ( ( ord_min_Code_natural @ zero_z2226904508553997617atural @ one_one_Code_natural )
    = zero_z2226904508553997617atural ) ).

% min_0_1(1)
thf(fact_7526_min__0__1_I2_J,axiom,
    ( ( ord_min_rat @ one_one_rat @ zero_zero_rat )
    = zero_zero_rat ) ).

% min_0_1(2)
thf(fact_7527_min__0__1_I2_J,axiom,
    ( ( ord_min_nat @ one_one_nat @ zero_zero_nat )
    = zero_zero_nat ) ).

% min_0_1(2)
thf(fact_7528_min__0__1_I2_J,axiom,
    ( ( ord_min_int @ one_one_int @ zero_zero_int )
    = zero_zero_int ) ).

% min_0_1(2)
thf(fact_7529_min__0__1_I2_J,axiom,
    ( ( ord_min_Code_integer @ one_one_Code_integer @ zero_z3403309356797280102nteger )
    = zero_z3403309356797280102nteger ) ).

% min_0_1(2)
thf(fact_7530_min__0__1_I2_J,axiom,
    ( ( ord_min_Code_natural @ one_one_Code_natural @ zero_z2226904508553997617atural )
    = zero_z2226904508553997617atural ) ).

% min_0_1(2)
thf(fact_7531_min__0__1_I6_J,axiom,
    ! [X: num] :
      ( ( ord_min_int @ ( numeral_numeral_int @ X ) @ one_one_int )
      = one_one_int ) ).

% min_0_1(6)
thf(fact_7532_min__0__1_I6_J,axiom,
    ! [X: num] :
      ( ( ord_min_nat @ ( numeral_numeral_nat @ X ) @ one_one_nat )
      = one_one_nat ) ).

% min_0_1(6)
thf(fact_7533_min__0__1_I6_J,axiom,
    ! [X: num] :
      ( ( ord_min_Code_integer @ ( numera6620942414471956472nteger @ X ) @ one_one_Code_integer )
      = one_one_Code_integer ) ).

% min_0_1(6)
thf(fact_7534_min__0__1_I6_J,axiom,
    ! [X: num] :
      ( ( ord_min_rat @ ( numeral_numeral_rat @ X ) @ one_one_rat )
      = one_one_rat ) ).

% min_0_1(6)
thf(fact_7535_min__0__1_I6_J,axiom,
    ! [X: num] :
      ( ( ord_min_Code_natural @ ( numera5444537566228673987atural @ X ) @ one_one_Code_natural )
      = one_one_Code_natural ) ).

% min_0_1(6)
thf(fact_7536_min__0__1_I5_J,axiom,
    ! [X: num] :
      ( ( ord_min_int @ one_one_int @ ( numeral_numeral_int @ X ) )
      = one_one_int ) ).

% min_0_1(5)
thf(fact_7537_min__0__1_I5_J,axiom,
    ! [X: num] :
      ( ( ord_min_nat @ one_one_nat @ ( numeral_numeral_nat @ X ) )
      = one_one_nat ) ).

% min_0_1(5)
thf(fact_7538_min__0__1_I5_J,axiom,
    ! [X: num] :
      ( ( ord_min_Code_integer @ one_one_Code_integer @ ( numera6620942414471956472nteger @ X ) )
      = one_one_Code_integer ) ).

% min_0_1(5)
thf(fact_7539_min__0__1_I5_J,axiom,
    ! [X: num] :
      ( ( ord_min_rat @ one_one_rat @ ( numeral_numeral_rat @ X ) )
      = one_one_rat ) ).

% min_0_1(5)
thf(fact_7540_min__0__1_I5_J,axiom,
    ! [X: num] :
      ( ( ord_min_Code_natural @ one_one_Code_natural @ ( numera5444537566228673987atural @ X ) )
      = one_one_Code_natural ) ).

% min_0_1(5)
thf(fact_7541_get__update__elem__neqIndex,axiom,
    ! [I2: nat,J: nat,A: array_nat,V: nat,H2: heap_e7401611519738050253t_unit] :
      ( ( I2 != J )
     => ( ( nth_nat @ ( array_get_nat @ ( array_update_nat @ A @ J @ V @ H2 ) @ A ) @ I2 )
        = ( nth_nat @ ( array_get_nat @ H2 @ A ) @ I2 ) ) ) ).

% get_update_elem_neqIndex
thf(fact_7542_get__update__elem__neqIndex,axiom,
    ! [I2: nat,J: nat,A: array_int,V: int,H2: heap_e7401611519738050253t_unit] :
      ( ( I2 != J )
     => ( ( nth_int @ ( array_get_int @ ( array_update_int @ A @ J @ V @ H2 ) @ A ) @ I2 )
        = ( nth_int @ ( array_get_int @ H2 @ A ) @ I2 ) ) ) ).

% get_update_elem_neqIndex
thf(fact_7543_min__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
       => ( ( ord_min_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
          = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) ) )
      & ( ~ ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
       => ( ( ord_min_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
          = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) ) ) ) ).

% min_number_of(4)
thf(fact_7544_min__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
       => ( ( ord_min_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
          = ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) ) )
      & ( ~ ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
       => ( ( ord_min_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
          = ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) ) ) ).

% min_number_of(4)
thf(fact_7545_min__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
       => ( ( ord_min_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
          = ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
       => ( ( ord_min_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
          = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) ) ) ).

% min_number_of(4)
thf(fact_7546_min__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
       => ( ( ord_min_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
          = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) ) )
      & ( ~ ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
       => ( ( ord_min_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
          = ( numera6620942414471956472nteger @ V ) ) ) ) ).

% min_number_of(3)
thf(fact_7547_min__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
       => ( ( ord_min_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
          = ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) ) )
      & ( ~ ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
       => ( ( ord_min_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
          = ( numeral_numeral_rat @ V ) ) ) ) ).

% min_number_of(3)
thf(fact_7548_min__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
       => ( ( ord_min_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
          = ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
       => ( ( ord_min_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
          = ( numeral_numeral_int @ V ) ) ) ) ).

% min_number_of(3)
thf(fact_7549_min__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
       => ( ( ord_min_Code_integer @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
          = ( numera6620942414471956472nteger @ U ) ) )
      & ( ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
       => ( ( ord_min_Code_integer @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
          = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) ) ) ) ).

% min_number_of(2)
thf(fact_7550_min__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
       => ( ( ord_min_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
          = ( numeral_numeral_rat @ U ) ) )
      & ( ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
       => ( ( ord_min_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
          = ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) ) ) ).

% min_number_of(2)
thf(fact_7551_min__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
       => ( ( ord_min_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
          = ( numeral_numeral_int @ U ) ) )
      & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
       => ( ( ord_min_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
          = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) ) ) ).

% min_number_of(2)
thf(fact_7552_rotate__length01,axiom,
    ! [Xs: list_o,N: nat] :
      ( ( ord_less_eq_nat @ ( size_size_list_o @ Xs ) @ one_one_nat )
     => ( ( rotate_o @ N @ Xs )
        = Xs ) ) ).

% rotate_length01
thf(fact_7553_rotate__length01,axiom,
    ! [Xs: list_nat,N: nat] :
      ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat )
     => ( ( rotate_nat @ N @ Xs )
        = Xs ) ) ).

% rotate_length01
thf(fact_7554_rotate__length01,axiom,
    ! [Xs: list_int,N: nat] :
      ( ( ord_less_eq_nat @ ( size_size_list_int @ Xs ) @ one_one_nat )
     => ( ( rotate_int @ N @ Xs )
        = Xs ) ) ).

% rotate_length01
thf(fact_7555_rotate__id,axiom,
    ! [N: nat,Xs: list_o] :
      ( ( ( modulo_modulo_nat @ N @ ( size_size_list_o @ Xs ) )
        = zero_zero_nat )
     => ( ( rotate_o @ N @ Xs )
        = Xs ) ) ).

% rotate_id
thf(fact_7556_rotate__id,axiom,
    ! [N: nat,Xs: list_nat] :
      ( ( ( modulo_modulo_nat @ N @ ( size_size_list_nat @ Xs ) )
        = zero_zero_nat )
     => ( ( rotate_nat @ N @ Xs )
        = Xs ) ) ).

% rotate_id
thf(fact_7557_rotate__id,axiom,
    ! [N: nat,Xs: list_int] :
      ( ( ( modulo_modulo_nat @ N @ ( size_size_list_int @ Xs ) )
        = zero_zero_nat )
     => ( ( rotate_int @ N @ Xs )
        = Xs ) ) ).

% rotate_id
thf(fact_7558_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_7559_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_7560_rotate1__length01,axiom,
    ! [Xs: list_o] :
      ( ( ord_less_eq_nat @ ( size_size_list_o @ Xs ) @ one_one_nat )
     => ( ( rotate1_o @ Xs )
        = Xs ) ) ).

% rotate1_length01
thf(fact_7561_rotate1__length01,axiom,
    ! [Xs: list_nat] :
      ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat )
     => ( ( rotate1_nat @ Xs )
        = Xs ) ) ).

% rotate1_length01
thf(fact_7562_rotate1__length01,axiom,
    ! [Xs: list_int] :
      ( ( ord_less_eq_nat @ ( size_size_list_int @ Xs ) @ one_one_nat )
     => ( ( rotate1_int @ Xs )
        = Xs ) ) ).

% rotate1_length01
thf(fact_7563_min__max__distrib2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_min_nat @ A @ ( ord_max_nat @ B @ C ) )
      = ( ord_max_nat @ ( ord_min_nat @ A @ B ) @ ( ord_min_nat @ A @ C ) ) ) ).

% min_max_distrib2
thf(fact_7564_min__max__distrib2,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_min_int @ A @ ( ord_max_int @ B @ C ) )
      = ( ord_max_int @ ( ord_min_int @ A @ B ) @ ( ord_min_int @ A @ C ) ) ) ).

% min_max_distrib2
thf(fact_7565_min__max__distrib2,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( ord_min_Code_integer @ A @ ( ord_max_Code_integer @ B @ C ) )
      = ( ord_max_Code_integer @ ( ord_min_Code_integer @ A @ B ) @ ( ord_min_Code_integer @ A @ C ) ) ) ).

% min_max_distrib2
thf(fact_7566_min__max__distrib2,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( ( ord_min_Code_natural @ A @ ( ord_max_Code_natural @ B @ C ) )
      = ( ord_max_Code_natural @ ( ord_min_Code_natural @ A @ B ) @ ( ord_min_Code_natural @ A @ C ) ) ) ).

% min_max_distrib2
thf(fact_7567_min__max__distrib1,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( ord_min_nat @ ( ord_max_nat @ B @ C ) @ A )
      = ( ord_max_nat @ ( ord_min_nat @ B @ A ) @ ( ord_min_nat @ C @ A ) ) ) ).

% min_max_distrib1
thf(fact_7568_min__max__distrib1,axiom,
    ! [B: int,C: int,A: int] :
      ( ( ord_min_int @ ( ord_max_int @ B @ C ) @ A )
      = ( ord_max_int @ ( ord_min_int @ B @ A ) @ ( ord_min_int @ C @ A ) ) ) ).

% min_max_distrib1
thf(fact_7569_min__max__distrib1,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] :
      ( ( ord_min_Code_integer @ ( ord_max_Code_integer @ B @ C ) @ A )
      = ( ord_max_Code_integer @ ( ord_min_Code_integer @ B @ A ) @ ( ord_min_Code_integer @ C @ A ) ) ) ).

% min_max_distrib1
thf(fact_7570_min__max__distrib1,axiom,
    ! [B: code_natural,C: code_natural,A: code_natural] :
      ( ( ord_min_Code_natural @ ( ord_max_Code_natural @ B @ C ) @ A )
      = ( ord_max_Code_natural @ ( ord_min_Code_natural @ B @ A ) @ ( ord_min_Code_natural @ C @ A ) ) ) ).

% min_max_distrib1
thf(fact_7571_max__min__distrib2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_max_nat @ A @ ( ord_min_nat @ B @ C ) )
      = ( ord_min_nat @ ( ord_max_nat @ A @ B ) @ ( ord_max_nat @ A @ C ) ) ) ).

% max_min_distrib2
thf(fact_7572_max__min__distrib2,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_max_int @ A @ ( ord_min_int @ B @ C ) )
      = ( ord_min_int @ ( ord_max_int @ A @ B ) @ ( ord_max_int @ A @ C ) ) ) ).

% max_min_distrib2
thf(fact_7573_max__min__distrib2,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( ord_max_Code_integer @ A @ ( ord_min_Code_integer @ B @ C ) )
      = ( ord_min_Code_integer @ ( ord_max_Code_integer @ A @ B ) @ ( ord_max_Code_integer @ A @ C ) ) ) ).

% max_min_distrib2
thf(fact_7574_max__min__distrib2,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( ( ord_max_Code_natural @ A @ ( ord_min_Code_natural @ B @ C ) )
      = ( ord_min_Code_natural @ ( ord_max_Code_natural @ A @ B ) @ ( ord_max_Code_natural @ A @ C ) ) ) ).

% max_min_distrib2
thf(fact_7575_max__min__distrib1,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( ord_max_nat @ ( ord_min_nat @ B @ C ) @ A )
      = ( ord_min_nat @ ( ord_max_nat @ B @ A ) @ ( ord_max_nat @ C @ A ) ) ) ).

% max_min_distrib1
thf(fact_7576_max__min__distrib1,axiom,
    ! [B: int,C: int,A: int] :
      ( ( ord_max_int @ ( ord_min_int @ B @ C ) @ A )
      = ( ord_min_int @ ( ord_max_int @ B @ A ) @ ( ord_max_int @ C @ A ) ) ) ).

% max_min_distrib1
thf(fact_7577_max__min__distrib1,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] :
      ( ( ord_max_Code_integer @ ( ord_min_Code_integer @ B @ C ) @ A )
      = ( ord_min_Code_integer @ ( ord_max_Code_integer @ B @ A ) @ ( ord_max_Code_integer @ C @ A ) ) ) ).

% max_min_distrib1
thf(fact_7578_max__min__distrib1,axiom,
    ! [B: code_natural,C: code_natural,A: code_natural] :
      ( ( ord_max_Code_natural @ ( ord_min_Code_natural @ B @ C ) @ A )
      = ( ord_min_Code_natural @ ( ord_max_Code_natural @ B @ A ) @ ( ord_max_Code_natural @ C @ A ) ) ) ).

% max_min_distrib1
thf(fact_7579_min__def,axiom,
    ( ord_min_Code_integer
    = ( ^ [A2: code_integer,B2: code_integer] : ( if_Code_integer @ ( ord_le3102999989581377725nteger @ A2 @ B2 ) @ A2 @ B2 ) ) ) ).

% min_def
thf(fact_7580_min__def,axiom,
    ( ord_min_Code_natural
    = ( ^ [A2: code_natural,B2: code_natural] : ( if_Code_natural @ ( ord_le1926595141338095240atural @ A2 @ B2 ) @ A2 @ B2 ) ) ) ).

% min_def
thf(fact_7581_min__def,axiom,
    ( ord_min_set_nat
    = ( ^ [A2: set_nat,B2: set_nat] : ( if_set_nat @ ( ord_less_eq_set_nat @ A2 @ B2 ) @ A2 @ B2 ) ) ) ).

% min_def
thf(fact_7582_min__def,axiom,
    ( ord_min_rat
    = ( ^ [A2: rat,B2: rat] : ( if_rat @ ( ord_less_eq_rat @ A2 @ B2 ) @ A2 @ B2 ) ) ) ).

% min_def
thf(fact_7583_min__def,axiom,
    ( ord_min_num
    = ( ^ [A2: num,B2: num] : ( if_num @ ( ord_less_eq_num @ A2 @ B2 ) @ A2 @ B2 ) ) ) ).

% min_def
thf(fact_7584_min__def,axiom,
    ( ord_min_nat
    = ( ^ [A2: nat,B2: nat] : ( if_nat @ ( ord_less_eq_nat @ A2 @ B2 ) @ A2 @ B2 ) ) ) ).

% min_def
thf(fact_7585_min__def,axiom,
    ( ord_min_int
    = ( ^ [A2: int,B2: int] : ( if_int @ ( ord_less_eq_int @ A2 @ B2 ) @ A2 @ B2 ) ) ) ).

% min_def
thf(fact_7586_min_Omono,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer,D: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ C )
     => ( ( ord_le3102999989581377725nteger @ B @ D )
       => ( ord_le3102999989581377725nteger @ ( ord_min_Code_integer @ A @ B ) @ ( ord_min_Code_integer @ C @ D ) ) ) ) ).

% min.mono
thf(fact_7587_min_Omono,axiom,
    ! [A: code_natural,C: code_natural,B: code_natural,D: code_natural] :
      ( ( ord_le1926595141338095240atural @ A @ C )
     => ( ( ord_le1926595141338095240atural @ B @ D )
       => ( ord_le1926595141338095240atural @ ( ord_min_Code_natural @ A @ B ) @ ( ord_min_Code_natural @ C @ D ) ) ) ) ).

% min.mono
thf(fact_7588_min_Omono,axiom,
    ! [A: rat,C: rat,B: rat,D: rat] :
      ( ( ord_less_eq_rat @ A @ C )
     => ( ( ord_less_eq_rat @ B @ D )
       => ( ord_less_eq_rat @ ( ord_min_rat @ A @ B ) @ ( ord_min_rat @ C @ D ) ) ) ) ).

% min.mono
thf(fact_7589_min_Omono,axiom,
    ! [A: num,C: num,B: num,D: num] :
      ( ( ord_less_eq_num @ A @ C )
     => ( ( ord_less_eq_num @ B @ D )
       => ( ord_less_eq_num @ ( ord_min_num @ A @ B ) @ ( ord_min_num @ C @ D ) ) ) ) ).

% min.mono
thf(fact_7590_min_Omono,axiom,
    ! [A: nat,C: nat,B: nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ C )
     => ( ( ord_less_eq_nat @ B @ D )
       => ( ord_less_eq_nat @ ( ord_min_nat @ A @ B ) @ ( ord_min_nat @ C @ D ) ) ) ) ).

% min.mono
thf(fact_7591_min_Omono,axiom,
    ! [A: int,C: int,B: int,D: int] :
      ( ( ord_less_eq_int @ A @ C )
     => ( ( ord_less_eq_int @ B @ D )
       => ( ord_less_eq_int @ ( ord_min_int @ A @ B ) @ ( ord_min_int @ C @ D ) ) ) ) ).

% min.mono
thf(fact_7592_min_OorderE,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ B )
     => ( A
        = ( ord_min_Code_integer @ A @ B ) ) ) ).

% min.orderE
thf(fact_7593_min_OorderE,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( ord_le1926595141338095240atural @ A @ B )
     => ( A
        = ( ord_min_Code_natural @ A @ B ) ) ) ).

% min.orderE
thf(fact_7594_min_OorderE,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( A
        = ( ord_min_rat @ A @ B ) ) ) ).

% min.orderE
thf(fact_7595_min_OorderE,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( A
        = ( ord_min_num @ A @ B ) ) ) ).

% min.orderE
thf(fact_7596_min_OorderE,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( A
        = ( ord_min_nat @ A @ B ) ) ) ).

% min.orderE
thf(fact_7597_min_OorderE,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( A
        = ( ord_min_int @ A @ B ) ) ) ).

% min.orderE
thf(fact_7598_min_OorderI,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A
        = ( ord_min_Code_integer @ A @ B ) )
     => ( ord_le3102999989581377725nteger @ A @ B ) ) ).

% min.orderI
thf(fact_7599_min_OorderI,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( A
        = ( ord_min_Code_natural @ A @ B ) )
     => ( ord_le1926595141338095240atural @ A @ B ) ) ).

% min.orderI
thf(fact_7600_min_OorderI,axiom,
    ! [A: rat,B: rat] :
      ( ( A
        = ( ord_min_rat @ A @ B ) )
     => ( ord_less_eq_rat @ A @ B ) ) ).

% min.orderI
thf(fact_7601_min_OorderI,axiom,
    ! [A: num,B: num] :
      ( ( A
        = ( ord_min_num @ A @ B ) )
     => ( ord_less_eq_num @ A @ B ) ) ).

% min.orderI
thf(fact_7602_min_OorderI,axiom,
    ! [A: nat,B: nat] :
      ( ( A
        = ( ord_min_nat @ A @ B ) )
     => ( ord_less_eq_nat @ A @ B ) ) ).

% min.orderI
thf(fact_7603_min_OorderI,axiom,
    ! [A: int,B: int] :
      ( ( A
        = ( ord_min_int @ A @ B ) )
     => ( ord_less_eq_int @ A @ B ) ) ).

% min.orderI
thf(fact_7604_min_OboundedE,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ ( ord_min_Code_integer @ B @ C ) )
     => ~ ( ( ord_le3102999989581377725nteger @ A @ B )
         => ~ ( ord_le3102999989581377725nteger @ A @ C ) ) ) ).

% min.boundedE
thf(fact_7605_min_OboundedE,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( ( ord_le1926595141338095240atural @ A @ ( ord_min_Code_natural @ B @ C ) )
     => ~ ( ( ord_le1926595141338095240atural @ A @ B )
         => ~ ( ord_le1926595141338095240atural @ A @ C ) ) ) ).

% min.boundedE
thf(fact_7606_min_OboundedE,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ ( ord_min_rat @ B @ C ) )
     => ~ ( ( ord_less_eq_rat @ A @ B )
         => ~ ( ord_less_eq_rat @ A @ C ) ) ) ).

% min.boundedE
thf(fact_7607_min_OboundedE,axiom,
    ! [A: num,B: num,C: num] :
      ( ( ord_less_eq_num @ A @ ( ord_min_num @ B @ C ) )
     => ~ ( ( ord_less_eq_num @ A @ B )
         => ~ ( ord_less_eq_num @ A @ C ) ) ) ).

% min.boundedE
thf(fact_7608_min_OboundedE,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ ( ord_min_nat @ B @ C ) )
     => ~ ( ( ord_less_eq_nat @ A @ B )
         => ~ ( ord_less_eq_nat @ A @ C ) ) ) ).

% min.boundedE
thf(fact_7609_min_OboundedE,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ ( ord_min_int @ B @ C ) )
     => ~ ( ( ord_less_eq_int @ A @ B )
         => ~ ( ord_less_eq_int @ A @ C ) ) ) ).

% min.boundedE
thf(fact_7610_min_OboundedI,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ B )
     => ( ( ord_le3102999989581377725nteger @ A @ C )
       => ( ord_le3102999989581377725nteger @ A @ ( ord_min_Code_integer @ B @ C ) ) ) ) ).

% min.boundedI
thf(fact_7611_min_OboundedI,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( ( ord_le1926595141338095240atural @ A @ B )
     => ( ( ord_le1926595141338095240atural @ A @ C )
       => ( ord_le1926595141338095240atural @ A @ ( ord_min_Code_natural @ B @ C ) ) ) ) ).

% min.boundedI
thf(fact_7612_min_OboundedI,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ A @ C )
       => ( ord_less_eq_rat @ A @ ( ord_min_rat @ B @ C ) ) ) ) ).

% min.boundedI
thf(fact_7613_min_OboundedI,axiom,
    ! [A: num,B: num,C: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_eq_num @ A @ C )
       => ( ord_less_eq_num @ A @ ( ord_min_num @ B @ C ) ) ) ) ).

% min.boundedI
thf(fact_7614_min_OboundedI,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ A @ C )
       => ( ord_less_eq_nat @ A @ ( ord_min_nat @ B @ C ) ) ) ) ).

% min.boundedI
thf(fact_7615_min_OboundedI,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ A @ C )
       => ( ord_less_eq_int @ A @ ( ord_min_int @ B @ C ) ) ) ) ).

% min.boundedI
thf(fact_7616_min_Oorder__iff,axiom,
    ( ord_le3102999989581377725nteger
    = ( ^ [A2: code_integer,B2: code_integer] :
          ( A2
          = ( ord_min_Code_integer @ A2 @ B2 ) ) ) ) ).

% min.order_iff
thf(fact_7617_min_Oorder__iff,axiom,
    ( ord_le1926595141338095240atural
    = ( ^ [A2: code_natural,B2: code_natural] :
          ( A2
          = ( ord_min_Code_natural @ A2 @ B2 ) ) ) ) ).

% min.order_iff
thf(fact_7618_min_Oorder__iff,axiom,
    ( ord_less_eq_rat
    = ( ^ [A2: rat,B2: rat] :
          ( A2
          = ( ord_min_rat @ A2 @ B2 ) ) ) ) ).

% min.order_iff
thf(fact_7619_min_Oorder__iff,axiom,
    ( ord_less_eq_num
    = ( ^ [A2: num,B2: num] :
          ( A2
          = ( ord_min_num @ A2 @ B2 ) ) ) ) ).

% min.order_iff
thf(fact_7620_min_Oorder__iff,axiom,
    ( ord_less_eq_nat
    = ( ^ [A2: nat,B2: nat] :
          ( A2
          = ( ord_min_nat @ A2 @ B2 ) ) ) ) ).

% min.order_iff
thf(fact_7621_min_Oorder__iff,axiom,
    ( ord_less_eq_int
    = ( ^ [A2: int,B2: int] :
          ( A2
          = ( ord_min_int @ A2 @ B2 ) ) ) ) ).

% min.order_iff
thf(fact_7622_min_Ocobounded1,axiom,
    ! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( ord_min_Code_integer @ A @ B ) @ A ) ).

% min.cobounded1
thf(fact_7623_min_Ocobounded1,axiom,
    ! [A: code_natural,B: code_natural] : ( ord_le1926595141338095240atural @ ( ord_min_Code_natural @ A @ B ) @ A ) ).

% min.cobounded1
thf(fact_7624_min_Ocobounded1,axiom,
    ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( ord_min_rat @ A @ B ) @ A ) ).

% min.cobounded1
thf(fact_7625_min_Ocobounded1,axiom,
    ! [A: num,B: num] : ( ord_less_eq_num @ ( ord_min_num @ A @ B ) @ A ) ).

% min.cobounded1
thf(fact_7626_min_Ocobounded1,axiom,
    ! [A: nat,B: nat] : ( ord_less_eq_nat @ ( ord_min_nat @ A @ B ) @ A ) ).

% min.cobounded1
thf(fact_7627_min_Ocobounded1,axiom,
    ! [A: int,B: int] : ( ord_less_eq_int @ ( ord_min_int @ A @ B ) @ A ) ).

% min.cobounded1
thf(fact_7628_min_Ocobounded2,axiom,
    ! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( ord_min_Code_integer @ A @ B ) @ B ) ).

% min.cobounded2
thf(fact_7629_min_Ocobounded2,axiom,
    ! [A: code_natural,B: code_natural] : ( ord_le1926595141338095240atural @ ( ord_min_Code_natural @ A @ B ) @ B ) ).

% min.cobounded2
thf(fact_7630_min_Ocobounded2,axiom,
    ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( ord_min_rat @ A @ B ) @ B ) ).

% min.cobounded2
thf(fact_7631_min_Ocobounded2,axiom,
    ! [A: num,B: num] : ( ord_less_eq_num @ ( ord_min_num @ A @ B ) @ B ) ).

% min.cobounded2
thf(fact_7632_min_Ocobounded2,axiom,
    ! [A: nat,B: nat] : ( ord_less_eq_nat @ ( ord_min_nat @ A @ B ) @ B ) ).

% min.cobounded2
thf(fact_7633_min_Ocobounded2,axiom,
    ! [A: int,B: int] : ( ord_less_eq_int @ ( ord_min_int @ A @ B ) @ B ) ).

% min.cobounded2
thf(fact_7634_min_Oabsorb__iff1,axiom,
    ( ord_le3102999989581377725nteger
    = ( ^ [A2: code_integer,B2: code_integer] :
          ( ( ord_min_Code_integer @ A2 @ B2 )
          = A2 ) ) ) ).

% min.absorb_iff1
thf(fact_7635_min_Oabsorb__iff1,axiom,
    ( ord_le1926595141338095240atural
    = ( ^ [A2: code_natural,B2: code_natural] :
          ( ( ord_min_Code_natural @ A2 @ B2 )
          = A2 ) ) ) ).

% min.absorb_iff1
thf(fact_7636_min_Oabsorb__iff1,axiom,
    ( ord_less_eq_rat
    = ( ^ [A2: rat,B2: rat] :
          ( ( ord_min_rat @ A2 @ B2 )
          = A2 ) ) ) ).

% min.absorb_iff1
thf(fact_7637_min_Oabsorb__iff1,axiom,
    ( ord_less_eq_num
    = ( ^ [A2: num,B2: num] :
          ( ( ord_min_num @ A2 @ B2 )
          = A2 ) ) ) ).

% min.absorb_iff1
thf(fact_7638_min_Oabsorb__iff1,axiom,
    ( ord_less_eq_nat
    = ( ^ [A2: nat,B2: nat] :
          ( ( ord_min_nat @ A2 @ B2 )
          = A2 ) ) ) ).

% min.absorb_iff1
thf(fact_7639_min_Oabsorb__iff1,axiom,
    ( ord_less_eq_int
    = ( ^ [A2: int,B2: int] :
          ( ( ord_min_int @ A2 @ B2 )
          = A2 ) ) ) ).

% min.absorb_iff1
thf(fact_7640_min_Oabsorb__iff2,axiom,
    ( ord_le3102999989581377725nteger
    = ( ^ [B2: code_integer,A2: code_integer] :
          ( ( ord_min_Code_integer @ A2 @ B2 )
          = B2 ) ) ) ).

% min.absorb_iff2
thf(fact_7641_min_Oabsorb__iff2,axiom,
    ( ord_le1926595141338095240atural
    = ( ^ [B2: code_natural,A2: code_natural] :
          ( ( ord_min_Code_natural @ A2 @ B2 )
          = B2 ) ) ) ).

% min.absorb_iff2
thf(fact_7642_min_Oabsorb__iff2,axiom,
    ( ord_less_eq_rat
    = ( ^ [B2: rat,A2: rat] :
          ( ( ord_min_rat @ A2 @ B2 )
          = B2 ) ) ) ).

% min.absorb_iff2
thf(fact_7643_min_Oabsorb__iff2,axiom,
    ( ord_less_eq_num
    = ( ^ [B2: num,A2: num] :
          ( ( ord_min_num @ A2 @ B2 )
          = B2 ) ) ) ).

% min.absorb_iff2
thf(fact_7644_min_Oabsorb__iff2,axiom,
    ( ord_less_eq_nat
    = ( ^ [B2: nat,A2: nat] :
          ( ( ord_min_nat @ A2 @ B2 )
          = B2 ) ) ) ).

% min.absorb_iff2
thf(fact_7645_min_Oabsorb__iff2,axiom,
    ( ord_less_eq_int
    = ( ^ [B2: int,A2: int] :
          ( ( ord_min_int @ A2 @ B2 )
          = B2 ) ) ) ).

% min.absorb_iff2
thf(fact_7646_min_OcoboundedI1,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ C )
     => ( ord_le3102999989581377725nteger @ ( ord_min_Code_integer @ A @ B ) @ C ) ) ).

% min.coboundedI1
thf(fact_7647_min_OcoboundedI1,axiom,
    ! [A: code_natural,C: code_natural,B: code_natural] :
      ( ( ord_le1926595141338095240atural @ A @ C )
     => ( ord_le1926595141338095240atural @ ( ord_min_Code_natural @ A @ B ) @ C ) ) ).

% min.coboundedI1
thf(fact_7648_min_OcoboundedI1,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ C )
     => ( ord_less_eq_rat @ ( ord_min_rat @ A @ B ) @ C ) ) ).

% min.coboundedI1
thf(fact_7649_min_OcoboundedI1,axiom,
    ! [A: num,C: num,B: num] :
      ( ( ord_less_eq_num @ A @ C )
     => ( ord_less_eq_num @ ( ord_min_num @ A @ B ) @ C ) ) ).

% min.coboundedI1
thf(fact_7650_min_OcoboundedI1,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ C )
     => ( ord_less_eq_nat @ ( ord_min_nat @ A @ B ) @ C ) ) ).

% min.coboundedI1
thf(fact_7651_min_OcoboundedI1,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_eq_int @ A @ C )
     => ( ord_less_eq_int @ ( ord_min_int @ A @ B ) @ C ) ) ).

% min.coboundedI1
thf(fact_7652_min_OcoboundedI2,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ B @ C )
     => ( ord_le3102999989581377725nteger @ ( ord_min_Code_integer @ A @ B ) @ C ) ) ).

% min.coboundedI2
thf(fact_7653_min_OcoboundedI2,axiom,
    ! [B: code_natural,C: code_natural,A: code_natural] :
      ( ( ord_le1926595141338095240atural @ B @ C )
     => ( ord_le1926595141338095240atural @ ( ord_min_Code_natural @ A @ B ) @ C ) ) ).

% min.coboundedI2
thf(fact_7654_min_OcoboundedI2,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( ord_less_eq_rat @ B @ C )
     => ( ord_less_eq_rat @ ( ord_min_rat @ A @ B ) @ C ) ) ).

% min.coboundedI2
thf(fact_7655_min_OcoboundedI2,axiom,
    ! [B: num,C: num,A: num] :
      ( ( ord_less_eq_num @ B @ C )
     => ( ord_less_eq_num @ ( ord_min_num @ A @ B ) @ C ) ) ).

% min.coboundedI2
thf(fact_7656_min_OcoboundedI2,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ C )
     => ( ord_less_eq_nat @ ( ord_min_nat @ A @ B ) @ C ) ) ).

% min.coboundedI2
thf(fact_7657_min_OcoboundedI2,axiom,
    ! [B: int,C: int,A: int] :
      ( ( ord_less_eq_int @ B @ C )
     => ( ord_less_eq_int @ ( ord_min_int @ A @ B ) @ C ) ) ).

% min.coboundedI2
thf(fact_7658_min__le__iff__disj,axiom,
    ! [X: code_integer,Y3: code_integer,Z2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( ord_min_Code_integer @ X @ Y3 ) @ Z2 )
      = ( ( ord_le3102999989581377725nteger @ X @ Z2 )
        | ( ord_le3102999989581377725nteger @ Y3 @ Z2 ) ) ) ).

% min_le_iff_disj
thf(fact_7659_min__le__iff__disj,axiom,
    ! [X: code_natural,Y3: code_natural,Z2: code_natural] :
      ( ( ord_le1926595141338095240atural @ ( ord_min_Code_natural @ X @ Y3 ) @ Z2 )
      = ( ( ord_le1926595141338095240atural @ X @ Z2 )
        | ( ord_le1926595141338095240atural @ Y3 @ Z2 ) ) ) ).

% min_le_iff_disj
thf(fact_7660_min__le__iff__disj,axiom,
    ! [X: rat,Y3: rat,Z2: rat] :
      ( ( ord_less_eq_rat @ ( ord_min_rat @ X @ Y3 ) @ Z2 )
      = ( ( ord_less_eq_rat @ X @ Z2 )
        | ( ord_less_eq_rat @ Y3 @ Z2 ) ) ) ).

% min_le_iff_disj
thf(fact_7661_min__le__iff__disj,axiom,
    ! [X: num,Y3: num,Z2: num] :
      ( ( ord_less_eq_num @ ( ord_min_num @ X @ Y3 ) @ Z2 )
      = ( ( ord_less_eq_num @ X @ Z2 )
        | ( ord_less_eq_num @ Y3 @ Z2 ) ) ) ).

% min_le_iff_disj
thf(fact_7662_min__le__iff__disj,axiom,
    ! [X: nat,Y3: nat,Z2: nat] :
      ( ( ord_less_eq_nat @ ( ord_min_nat @ X @ Y3 ) @ Z2 )
      = ( ( ord_less_eq_nat @ X @ Z2 )
        | ( ord_less_eq_nat @ Y3 @ Z2 ) ) ) ).

% min_le_iff_disj
thf(fact_7663_min__le__iff__disj,axiom,
    ! [X: int,Y3: int,Z2: int] :
      ( ( ord_less_eq_int @ ( ord_min_int @ X @ Y3 ) @ Z2 )
      = ( ( ord_less_eq_int @ X @ Z2 )
        | ( ord_less_eq_int @ Y3 @ Z2 ) ) ) ).

% min_le_iff_disj
thf(fact_7664_min__absorb1,axiom,
    ! [X: code_integer,Y3: code_integer] :
      ( ( ord_le3102999989581377725nteger @ X @ Y3 )
     => ( ( ord_min_Code_integer @ X @ Y3 )
        = X ) ) ).

% min_absorb1
thf(fact_7665_min__absorb1,axiom,
    ! [X: code_natural,Y3: code_natural] :
      ( ( ord_le1926595141338095240atural @ X @ Y3 )
     => ( ( ord_min_Code_natural @ X @ Y3 )
        = X ) ) ).

% min_absorb1
thf(fact_7666_min__absorb1,axiom,
    ! [X: set_nat,Y3: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ Y3 )
     => ( ( ord_min_set_nat @ X @ Y3 )
        = X ) ) ).

% min_absorb1
thf(fact_7667_min__absorb1,axiom,
    ! [X: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ X @ Y3 )
     => ( ( ord_min_rat @ X @ Y3 )
        = X ) ) ).

% min_absorb1
thf(fact_7668_min__absorb1,axiom,
    ! [X: num,Y3: num] :
      ( ( ord_less_eq_num @ X @ Y3 )
     => ( ( ord_min_num @ X @ Y3 )
        = X ) ) ).

% min_absorb1
thf(fact_7669_min__absorb1,axiom,
    ! [X: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ X @ Y3 )
     => ( ( ord_min_nat @ X @ Y3 )
        = X ) ) ).

% min_absorb1
thf(fact_7670_min__absorb1,axiom,
    ! [X: int,Y3: int] :
      ( ( ord_less_eq_int @ X @ Y3 )
     => ( ( ord_min_int @ X @ Y3 )
        = X ) ) ).

% min_absorb1
thf(fact_7671_min__absorb2,axiom,
    ! [Y3: code_integer,X: code_integer] :
      ( ( ord_le3102999989581377725nteger @ Y3 @ X )
     => ( ( ord_min_Code_integer @ X @ Y3 )
        = Y3 ) ) ).

% min_absorb2
thf(fact_7672_min__absorb2,axiom,
    ! [Y3: code_natural,X: code_natural] :
      ( ( ord_le1926595141338095240atural @ Y3 @ X )
     => ( ( ord_min_Code_natural @ X @ Y3 )
        = Y3 ) ) ).

% min_absorb2
thf(fact_7673_min__absorb2,axiom,
    ! [Y3: set_nat,X: set_nat] :
      ( ( ord_less_eq_set_nat @ Y3 @ X )
     => ( ( ord_min_set_nat @ X @ Y3 )
        = Y3 ) ) ).

% min_absorb2
thf(fact_7674_min__absorb2,axiom,
    ! [Y3: rat,X: rat] :
      ( ( ord_less_eq_rat @ Y3 @ X )
     => ( ( ord_min_rat @ X @ Y3 )
        = Y3 ) ) ).

% min_absorb2
thf(fact_7675_min__absorb2,axiom,
    ! [Y3: num,X: num] :
      ( ( ord_less_eq_num @ Y3 @ X )
     => ( ( ord_min_num @ X @ Y3 )
        = Y3 ) ) ).

% min_absorb2
thf(fact_7676_min__absorb2,axiom,
    ! [Y3: nat,X: nat] :
      ( ( ord_less_eq_nat @ Y3 @ X )
     => ( ( ord_min_nat @ X @ Y3 )
        = Y3 ) ) ).

% min_absorb2
thf(fact_7677_min__absorb2,axiom,
    ! [Y3: int,X: int] :
      ( ( ord_less_eq_int @ Y3 @ X )
     => ( ( ord_min_int @ X @ Y3 )
        = Y3 ) ) ).

% min_absorb2
thf(fact_7678_of__int__min,axiom,
    ! [X: int,Y3: int] :
      ( ( ring_1_of_int_rat @ ( ord_min_int @ X @ Y3 ) )
      = ( ord_min_rat @ ( ring_1_of_int_rat @ X ) @ ( ring_1_of_int_rat @ Y3 ) ) ) ).

% of_int_min
thf(fact_7679_of__int__min,axiom,
    ! [X: int,Y3: int] :
      ( ( ring_1_of_int_int @ ( ord_min_int @ X @ Y3 ) )
      = ( ord_min_int @ ( ring_1_of_int_int @ X ) @ ( ring_1_of_int_int @ Y3 ) ) ) ).

% of_int_min
thf(fact_7680_of__int__min,axiom,
    ! [X: int,Y3: int] :
      ( ( ring_18347121197199848620nteger @ ( ord_min_int @ X @ Y3 ) )
      = ( ord_min_Code_integer @ ( ring_18347121197199848620nteger @ X ) @ ( ring_18347121197199848620nteger @ Y3 ) ) ) ).

% of_int_min
thf(fact_7681_min_Oassoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_min_nat @ ( ord_min_nat @ A @ B ) @ C )
      = ( ord_min_nat @ A @ ( ord_min_nat @ B @ C ) ) ) ).

% min.assoc
thf(fact_7682_min_Oassoc,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_min_int @ ( ord_min_int @ A @ B ) @ C )
      = ( ord_min_int @ A @ ( ord_min_int @ B @ C ) ) ) ).

% min.assoc
thf(fact_7683_min_Oassoc,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( ord_min_Code_integer @ ( ord_min_Code_integer @ A @ B ) @ C )
      = ( ord_min_Code_integer @ A @ ( ord_min_Code_integer @ B @ C ) ) ) ).

% min.assoc
thf(fact_7684_min_Oassoc,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( ( ord_min_Code_natural @ ( ord_min_Code_natural @ A @ B ) @ C )
      = ( ord_min_Code_natural @ A @ ( ord_min_Code_natural @ B @ C ) ) ) ).

% min.assoc
thf(fact_7685_min_Ocommute,axiom,
    ( ord_min_nat
    = ( ^ [A2: nat,B2: nat] : ( ord_min_nat @ B2 @ A2 ) ) ) ).

% min.commute
thf(fact_7686_min_Ocommute,axiom,
    ( ord_min_int
    = ( ^ [A2: int,B2: int] : ( ord_min_int @ B2 @ A2 ) ) ) ).

% min.commute
thf(fact_7687_min_Ocommute,axiom,
    ( ord_min_Code_integer
    = ( ^ [A2: code_integer,B2: code_integer] : ( ord_min_Code_integer @ B2 @ A2 ) ) ) ).

% min.commute
thf(fact_7688_min_Ocommute,axiom,
    ( ord_min_Code_natural
    = ( ^ [A2: code_natural,B2: code_natural] : ( ord_min_Code_natural @ B2 @ A2 ) ) ) ).

% min.commute
thf(fact_7689_min_Oleft__commute,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ord_min_nat @ B @ ( ord_min_nat @ A @ C ) )
      = ( ord_min_nat @ A @ ( ord_min_nat @ B @ C ) ) ) ).

% min.left_commute
thf(fact_7690_min_Oleft__commute,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_min_int @ B @ ( ord_min_int @ A @ C ) )
      = ( ord_min_int @ A @ ( ord_min_int @ B @ C ) ) ) ).

% min.left_commute
thf(fact_7691_min_Oleft__commute,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( ord_min_Code_integer @ B @ ( ord_min_Code_integer @ A @ C ) )
      = ( ord_min_Code_integer @ A @ ( ord_min_Code_integer @ B @ C ) ) ) ).

% min.left_commute
thf(fact_7692_min_Oleft__commute,axiom,
    ! [B: code_natural,A: code_natural,C: code_natural] :
      ( ( ord_min_Code_natural @ B @ ( ord_min_Code_natural @ A @ C ) )
      = ( ord_min_Code_natural @ A @ ( ord_min_Code_natural @ B @ C ) ) ) ).

% min.left_commute
thf(fact_7693_of__nat__min,axiom,
    ! [X: nat,Y3: nat] :
      ( ( semiri1314217659103216013at_int @ ( ord_min_nat @ X @ Y3 ) )
      = ( ord_min_int @ ( semiri1314217659103216013at_int @ X ) @ ( semiri1314217659103216013at_int @ Y3 ) ) ) ).

% of_nat_min
thf(fact_7694_of__nat__min,axiom,
    ! [X: nat,Y3: nat] :
      ( ( semiri4939895301339042750nteger @ ( ord_min_nat @ X @ Y3 ) )
      = ( ord_min_Code_integer @ ( semiri4939895301339042750nteger @ X ) @ ( semiri4939895301339042750nteger @ Y3 ) ) ) ).

% of_nat_min
thf(fact_7695_of__nat__min,axiom,
    ! [X: nat,Y3: nat] :
      ( ( semiri1316708129612266289at_nat @ ( ord_min_nat @ X @ Y3 ) )
      = ( ord_min_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( semiri1316708129612266289at_nat @ Y3 ) ) ) ).

% of_nat_min
thf(fact_7696_of__nat__min,axiom,
    ! [X: nat,Y3: nat] :
      ( ( semiri681578069525770553at_rat @ ( ord_min_nat @ X @ Y3 ) )
      = ( ord_min_rat @ ( semiri681578069525770553at_rat @ X ) @ ( semiri681578069525770553at_rat @ Y3 ) ) ) ).

% of_nat_min
thf(fact_7697_of__nat__min,axiom,
    ! [X: nat,Y3: nat] :
      ( ( semiri3763490453095760265atural @ ( ord_min_nat @ X @ Y3 ) )
      = ( ord_min_Code_natural @ ( semiri3763490453095760265atural @ X ) @ ( semiri3763490453095760265atural @ Y3 ) ) ) ).

% of_nat_min
thf(fact_7698_min__diff__distrib__left,axiom,
    ! [X: code_integer,Y3: code_integer,Z2: code_integer] :
      ( ( minus_8373710615458151222nteger @ ( ord_min_Code_integer @ X @ Y3 ) @ Z2 )
      = ( ord_min_Code_integer @ ( minus_8373710615458151222nteger @ X @ Z2 ) @ ( minus_8373710615458151222nteger @ Y3 @ Z2 ) ) ) ).

% min_diff_distrib_left
thf(fact_7699_min__diff__distrib__left,axiom,
    ! [X: rat,Y3: rat,Z2: rat] :
      ( ( minus_minus_rat @ ( ord_min_rat @ X @ Y3 ) @ Z2 )
      = ( ord_min_rat @ ( minus_minus_rat @ X @ Z2 ) @ ( minus_minus_rat @ Y3 @ Z2 ) ) ) ).

% min_diff_distrib_left
thf(fact_7700_min__diff__distrib__left,axiom,
    ! [X: int,Y3: int,Z2: int] :
      ( ( minus_minus_int @ ( ord_min_int @ X @ Y3 ) @ Z2 )
      = ( ord_min_int @ ( minus_minus_int @ X @ Z2 ) @ ( minus_minus_int @ Y3 @ Z2 ) ) ) ).

% min_diff_distrib_left
thf(fact_7701_min__add__distrib__right,axiom,
    ! [X: rat,Y3: rat,Z2: rat] :
      ( ( plus_plus_rat @ X @ ( ord_min_rat @ Y3 @ Z2 ) )
      = ( ord_min_rat @ ( plus_plus_rat @ X @ Y3 ) @ ( plus_plus_rat @ X @ Z2 ) ) ) ).

% min_add_distrib_right
thf(fact_7702_min__add__distrib__right,axiom,
    ! [X: nat,Y3: nat,Z2: nat] :
      ( ( plus_plus_nat @ X @ ( ord_min_nat @ Y3 @ Z2 ) )
      = ( ord_min_nat @ ( plus_plus_nat @ X @ Y3 ) @ ( plus_plus_nat @ X @ Z2 ) ) ) ).

% min_add_distrib_right
thf(fact_7703_min__add__distrib__right,axiom,
    ! [X: int,Y3: int,Z2: int] :
      ( ( plus_plus_int @ X @ ( ord_min_int @ Y3 @ Z2 ) )
      = ( ord_min_int @ ( plus_plus_int @ X @ Y3 ) @ ( plus_plus_int @ X @ Z2 ) ) ) ).

% min_add_distrib_right
thf(fact_7704_min__add__distrib__right,axiom,
    ! [X: code_integer,Y3: code_integer,Z2: code_integer] :
      ( ( plus_p5714425477246183910nteger @ X @ ( ord_min_Code_integer @ Y3 @ Z2 ) )
      = ( ord_min_Code_integer @ ( plus_p5714425477246183910nteger @ X @ Y3 ) @ ( plus_p5714425477246183910nteger @ X @ Z2 ) ) ) ).

% min_add_distrib_right
thf(fact_7705_min__add__distrib__right,axiom,
    ! [X: code_natural,Y3: code_natural,Z2: code_natural] :
      ( ( plus_p4538020629002901425atural @ X @ ( ord_min_Code_natural @ Y3 @ Z2 ) )
      = ( ord_min_Code_natural @ ( plus_p4538020629002901425atural @ X @ Y3 ) @ ( plus_p4538020629002901425atural @ X @ Z2 ) ) ) ).

% min_add_distrib_right
thf(fact_7706_min__add__distrib__left,axiom,
    ! [X: rat,Y3: rat,Z2: rat] :
      ( ( plus_plus_rat @ ( ord_min_rat @ X @ Y3 ) @ Z2 )
      = ( ord_min_rat @ ( plus_plus_rat @ X @ Z2 ) @ ( plus_plus_rat @ Y3 @ Z2 ) ) ) ).

% min_add_distrib_left
thf(fact_7707_min__add__distrib__left,axiom,
    ! [X: nat,Y3: nat,Z2: nat] :
      ( ( plus_plus_nat @ ( ord_min_nat @ X @ Y3 ) @ Z2 )
      = ( ord_min_nat @ ( plus_plus_nat @ X @ Z2 ) @ ( plus_plus_nat @ Y3 @ Z2 ) ) ) ).

% min_add_distrib_left
thf(fact_7708_min__add__distrib__left,axiom,
    ! [X: int,Y3: int,Z2: int] :
      ( ( plus_plus_int @ ( ord_min_int @ X @ Y3 ) @ Z2 )
      = ( ord_min_int @ ( plus_plus_int @ X @ Z2 ) @ ( plus_plus_int @ Y3 @ Z2 ) ) ) ).

% min_add_distrib_left
thf(fact_7709_min__add__distrib__left,axiom,
    ! [X: code_integer,Y3: code_integer,Z2: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( ord_min_Code_integer @ X @ Y3 ) @ Z2 )
      = ( ord_min_Code_integer @ ( plus_p5714425477246183910nteger @ X @ Z2 ) @ ( plus_p5714425477246183910nteger @ Y3 @ Z2 ) ) ) ).

% min_add_distrib_left
thf(fact_7710_min__add__distrib__left,axiom,
    ! [X: code_natural,Y3: code_natural,Z2: code_natural] :
      ( ( plus_p4538020629002901425atural @ ( ord_min_Code_natural @ X @ Y3 ) @ Z2 )
      = ( ord_min_Code_natural @ ( plus_p4538020629002901425atural @ X @ Z2 ) @ ( plus_p4538020629002901425atural @ Y3 @ Z2 ) ) ) ).

% min_add_distrib_left
thf(fact_7711_min__less__iff__disj,axiom,
    ! [X: code_natural,Y3: code_natural,Z2: code_natural] :
      ( ( ord_le5570908160329646204atural @ ( ord_min_Code_natural @ X @ Y3 ) @ Z2 )
      = ( ( ord_le5570908160329646204atural @ X @ Z2 )
        | ( ord_le5570908160329646204atural @ Y3 @ Z2 ) ) ) ).

% min_less_iff_disj
thf(fact_7712_min__less__iff__disj,axiom,
    ! [X: code_integer,Y3: code_integer,Z2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( ord_min_Code_integer @ X @ Y3 ) @ Z2 )
      = ( ( ord_le6747313008572928689nteger @ X @ Z2 )
        | ( ord_le6747313008572928689nteger @ Y3 @ Z2 ) ) ) ).

% min_less_iff_disj
thf(fact_7713_min__less__iff__disj,axiom,
    ! [X: rat,Y3: rat,Z2: rat] :
      ( ( ord_less_rat @ ( ord_min_rat @ X @ Y3 ) @ Z2 )
      = ( ( ord_less_rat @ X @ Z2 )
        | ( ord_less_rat @ Y3 @ Z2 ) ) ) ).

% min_less_iff_disj
thf(fact_7714_min__less__iff__disj,axiom,
    ! [X: num,Y3: num,Z2: num] :
      ( ( ord_less_num @ ( ord_min_num @ X @ Y3 ) @ Z2 )
      = ( ( ord_less_num @ X @ Z2 )
        | ( ord_less_num @ Y3 @ Z2 ) ) ) ).

% min_less_iff_disj
thf(fact_7715_min__less__iff__disj,axiom,
    ! [X: nat,Y3: nat,Z2: nat] :
      ( ( ord_less_nat @ ( ord_min_nat @ X @ Y3 ) @ Z2 )
      = ( ( ord_less_nat @ X @ Z2 )
        | ( ord_less_nat @ Y3 @ Z2 ) ) ) ).

% min_less_iff_disj
thf(fact_7716_min__less__iff__disj,axiom,
    ! [X: int,Y3: int,Z2: int] :
      ( ( ord_less_int @ ( ord_min_int @ X @ Y3 ) @ Z2 )
      = ( ( ord_less_int @ X @ Z2 )
        | ( ord_less_int @ Y3 @ Z2 ) ) ) ).

% min_less_iff_disj
thf(fact_7717_min_Ostrict__boundedE,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( ( ord_le5570908160329646204atural @ A @ ( ord_min_Code_natural @ B @ C ) )
     => ~ ( ( ord_le5570908160329646204atural @ A @ B )
         => ~ ( ord_le5570908160329646204atural @ A @ C ) ) ) ).

% min.strict_boundedE
thf(fact_7718_min_Ostrict__boundedE,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ ( ord_min_Code_integer @ B @ C ) )
     => ~ ( ( ord_le6747313008572928689nteger @ A @ B )
         => ~ ( ord_le6747313008572928689nteger @ A @ C ) ) ) ).

% min.strict_boundedE
thf(fact_7719_min_Ostrict__boundedE,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ ( ord_min_rat @ B @ C ) )
     => ~ ( ( ord_less_rat @ A @ B )
         => ~ ( ord_less_rat @ A @ C ) ) ) ).

% min.strict_boundedE
thf(fact_7720_min_Ostrict__boundedE,axiom,
    ! [A: num,B: num,C: num] :
      ( ( ord_less_num @ A @ ( ord_min_num @ B @ C ) )
     => ~ ( ( ord_less_num @ A @ B )
         => ~ ( ord_less_num @ A @ C ) ) ) ).

% min.strict_boundedE
thf(fact_7721_min_Ostrict__boundedE,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ ( ord_min_nat @ B @ C ) )
     => ~ ( ( ord_less_nat @ A @ B )
         => ~ ( ord_less_nat @ A @ C ) ) ) ).

% min.strict_boundedE
thf(fact_7722_min_Ostrict__boundedE,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ ( ord_min_int @ B @ C ) )
     => ~ ( ( ord_less_int @ A @ B )
         => ~ ( ord_less_int @ A @ C ) ) ) ).

% min.strict_boundedE
thf(fact_7723_min_Ostrict__order__iff,axiom,
    ( ord_le5570908160329646204atural
    = ( ^ [A2: code_natural,B2: code_natural] :
          ( ( A2
            = ( ord_min_Code_natural @ A2 @ B2 ) )
          & ( A2 != B2 ) ) ) ) ).

% min.strict_order_iff
thf(fact_7724_min_Ostrict__order__iff,axiom,
    ( ord_le6747313008572928689nteger
    = ( ^ [A2: code_integer,B2: code_integer] :
          ( ( A2
            = ( ord_min_Code_integer @ A2 @ B2 ) )
          & ( A2 != B2 ) ) ) ) ).

% min.strict_order_iff
thf(fact_7725_min_Ostrict__order__iff,axiom,
    ( ord_less_rat
    = ( ^ [A2: rat,B2: rat] :
          ( ( A2
            = ( ord_min_rat @ A2 @ B2 ) )
          & ( A2 != B2 ) ) ) ) ).

% min.strict_order_iff
thf(fact_7726_min_Ostrict__order__iff,axiom,
    ( ord_less_num
    = ( ^ [A2: num,B2: num] :
          ( ( A2
            = ( ord_min_num @ A2 @ B2 ) )
          & ( A2 != B2 ) ) ) ) ).

% min.strict_order_iff
thf(fact_7727_min_Ostrict__order__iff,axiom,
    ( ord_less_nat
    = ( ^ [A2: nat,B2: nat] :
          ( ( A2
            = ( ord_min_nat @ A2 @ B2 ) )
          & ( A2 != B2 ) ) ) ) ).

% min.strict_order_iff
thf(fact_7728_min_Ostrict__order__iff,axiom,
    ( ord_less_int
    = ( ^ [A2: int,B2: int] :
          ( ( A2
            = ( ord_min_int @ A2 @ B2 ) )
          & ( A2 != B2 ) ) ) ) ).

% min.strict_order_iff
thf(fact_7729_min_Ostrict__coboundedI1,axiom,
    ! [A: code_natural,C: code_natural,B: code_natural] :
      ( ( ord_le5570908160329646204atural @ A @ C )
     => ( ord_le5570908160329646204atural @ ( ord_min_Code_natural @ A @ B ) @ C ) ) ).

% min.strict_coboundedI1
thf(fact_7730_min_Ostrict__coboundedI1,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ C )
     => ( ord_le6747313008572928689nteger @ ( ord_min_Code_integer @ A @ B ) @ C ) ) ).

% min.strict_coboundedI1
thf(fact_7731_min_Ostrict__coboundedI1,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_rat @ A @ C )
     => ( ord_less_rat @ ( ord_min_rat @ A @ B ) @ C ) ) ).

% min.strict_coboundedI1
thf(fact_7732_min_Ostrict__coboundedI1,axiom,
    ! [A: num,C: num,B: num] :
      ( ( ord_less_num @ A @ C )
     => ( ord_less_num @ ( ord_min_num @ A @ B ) @ C ) ) ).

% min.strict_coboundedI1
thf(fact_7733_min_Ostrict__coboundedI1,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_nat @ A @ C )
     => ( ord_less_nat @ ( ord_min_nat @ A @ B ) @ C ) ) ).

% min.strict_coboundedI1
thf(fact_7734_min_Ostrict__coboundedI1,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_int @ A @ C )
     => ( ord_less_int @ ( ord_min_int @ A @ B ) @ C ) ) ).

% min.strict_coboundedI1
thf(fact_7735_min_Ostrict__coboundedI2,axiom,
    ! [B: code_natural,C: code_natural,A: code_natural] :
      ( ( ord_le5570908160329646204atural @ B @ C )
     => ( ord_le5570908160329646204atural @ ( ord_min_Code_natural @ A @ B ) @ C ) ) ).

% min.strict_coboundedI2
thf(fact_7736_min_Ostrict__coboundedI2,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ B @ C )
     => ( ord_le6747313008572928689nteger @ ( ord_min_Code_integer @ A @ B ) @ C ) ) ).

% min.strict_coboundedI2
thf(fact_7737_min_Ostrict__coboundedI2,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( ord_less_rat @ B @ C )
     => ( ord_less_rat @ ( ord_min_rat @ A @ B ) @ C ) ) ).

% min.strict_coboundedI2
thf(fact_7738_min_Ostrict__coboundedI2,axiom,
    ! [B: num,C: num,A: num] :
      ( ( ord_less_num @ B @ C )
     => ( ord_less_num @ ( ord_min_num @ A @ B ) @ C ) ) ).

% min.strict_coboundedI2
thf(fact_7739_min_Ostrict__coboundedI2,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( ord_less_nat @ B @ C )
     => ( ord_less_nat @ ( ord_min_nat @ A @ B ) @ C ) ) ).

% min.strict_coboundedI2
thf(fact_7740_min_Ostrict__coboundedI2,axiom,
    ! [B: int,C: int,A: int] :
      ( ( ord_less_int @ B @ C )
     => ( ord_less_int @ ( ord_min_int @ A @ B ) @ C ) ) ).

% min.strict_coboundedI2
thf(fact_7741_nat__mult__min__left,axiom,
    ! [M: nat,N: nat,Q4: nat] :
      ( ( times_times_nat @ ( ord_min_nat @ M @ N ) @ Q4 )
      = ( ord_min_nat @ ( times_times_nat @ M @ Q4 ) @ ( times_times_nat @ N @ Q4 ) ) ) ).

% nat_mult_min_left
thf(fact_7742_nat__mult__min__right,axiom,
    ! [M: nat,N: nat,Q4: nat] :
      ( ( times_times_nat @ M @ ( ord_min_nat @ N @ Q4 ) )
      = ( ord_min_nat @ ( times_times_nat @ M @ N ) @ ( times_times_nat @ M @ Q4 ) ) ) ).

% nat_mult_min_right
thf(fact_7743_min__diff,axiom,
    ! [M: nat,I2: nat,N: nat] :
      ( ( ord_min_nat @ ( minus_minus_nat @ M @ I2 ) @ ( minus_minus_nat @ N @ I2 ) )
      = ( minus_minus_nat @ ( ord_min_nat @ M @ N ) @ I2 ) ) ).

% min_diff
thf(fact_7744_minus__min__eq__max,axiom,
    ! [X: int,Y3: int] :
      ( ( uminus_uminus_int @ ( ord_min_int @ X @ Y3 ) )
      = ( ord_max_int @ ( uminus_uminus_int @ X ) @ ( uminus_uminus_int @ Y3 ) ) ) ).

% minus_min_eq_max
thf(fact_7745_minus__min__eq__max,axiom,
    ! [X: code_integer,Y3: code_integer] :
      ( ( uminus1351360451143612070nteger @ ( ord_min_Code_integer @ X @ Y3 ) )
      = ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ X ) @ ( uminus1351360451143612070nteger @ Y3 ) ) ) ).

% minus_min_eq_max
thf(fact_7746_minus__min__eq__max,axiom,
    ! [X: rat,Y3: rat] :
      ( ( uminus_uminus_rat @ ( ord_min_rat @ X @ Y3 ) )
      = ( ord_max_rat @ ( uminus_uminus_rat @ X ) @ ( uminus_uminus_rat @ Y3 ) ) ) ).

% minus_min_eq_max
thf(fact_7747_minus__max__eq__min,axiom,
    ! [X: int,Y3: int] :
      ( ( uminus_uminus_int @ ( ord_max_int @ X @ Y3 ) )
      = ( ord_min_int @ ( uminus_uminus_int @ X ) @ ( uminus_uminus_int @ Y3 ) ) ) ).

% minus_max_eq_min
thf(fact_7748_minus__max__eq__min,axiom,
    ! [X: code_integer,Y3: code_integer] :
      ( ( uminus1351360451143612070nteger @ ( ord_max_Code_integer @ X @ Y3 ) )
      = ( ord_min_Code_integer @ ( uminus1351360451143612070nteger @ X ) @ ( uminus1351360451143612070nteger @ Y3 ) ) ) ).

% minus_max_eq_min
thf(fact_7749_minus__max__eq__min,axiom,
    ! [X: rat,Y3: rat] :
      ( ( uminus_uminus_rat @ ( ord_max_rat @ X @ Y3 ) )
      = ( ord_min_rat @ ( uminus_uminus_rat @ X ) @ ( uminus_uminus_rat @ Y3 ) ) ) ).

% minus_max_eq_min
thf(fact_7750_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_7751_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_7752_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_7753_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_7754_length__def,axiom,
    ( array_length_a
    = ( ^ [H4: heap_e7401611519738050253t_unit,A2: array_a] : ( size_size_list_a @ ( array_get_a @ H4 @ A2 ) ) ) ) ).

% length_def
thf(fact_7755_length__def,axiom,
    ( array_length_o
    = ( ^ [H4: heap_e7401611519738050253t_unit,A2: array_o] : ( size_size_list_o @ ( array_get_o @ H4 @ A2 ) ) ) ) ).

% length_def
thf(fact_7756_length__def,axiom,
    ( array_length_nat
    = ( ^ [H4: heap_e7401611519738050253t_unit,A2: array_nat] : ( size_size_list_nat @ ( array_get_nat @ H4 @ A2 ) ) ) ) ).

% length_def
thf(fact_7757_length__def,axiom,
    ( array_length_int
    = ( ^ [H4: heap_e7401611519738050253t_unit,A2: array_int] : ( size_size_list_int @ ( array_get_int @ H4 @ A2 ) ) ) ) ).

% length_def
thf(fact_7758_max__mult__distrib__left,axiom,
    ! [P5: code_integer,X: code_integer,Y3: code_integer] :
      ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ P5 )
       => ( ( times_3573771949741848930nteger @ P5 @ ( ord_max_Code_integer @ X @ Y3 ) )
          = ( ord_max_Code_integer @ ( times_3573771949741848930nteger @ P5 @ X ) @ ( times_3573771949741848930nteger @ P5 @ Y3 ) ) ) )
      & ( ~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ P5 )
       => ( ( times_3573771949741848930nteger @ P5 @ ( ord_max_Code_integer @ X @ Y3 ) )
          = ( ord_min_Code_integer @ ( times_3573771949741848930nteger @ P5 @ X ) @ ( times_3573771949741848930nteger @ P5 @ Y3 ) ) ) ) ) ).

% max_mult_distrib_left
thf(fact_7759_max__mult__distrib__left,axiom,
    ! [P5: rat,X: rat,Y3: rat] :
      ( ( ( ord_less_eq_rat @ zero_zero_rat @ P5 )
       => ( ( times_times_rat @ P5 @ ( ord_max_rat @ X @ Y3 ) )
          = ( ord_max_rat @ ( times_times_rat @ P5 @ X ) @ ( times_times_rat @ P5 @ Y3 ) ) ) )
      & ( ~ ( ord_less_eq_rat @ zero_zero_rat @ P5 )
       => ( ( times_times_rat @ P5 @ ( ord_max_rat @ X @ Y3 ) )
          = ( ord_min_rat @ ( times_times_rat @ P5 @ X ) @ ( times_times_rat @ P5 @ Y3 ) ) ) ) ) ).

% max_mult_distrib_left
thf(fact_7760_max__mult__distrib__left,axiom,
    ! [P5: int,X: int,Y3: int] :
      ( ( ( ord_less_eq_int @ zero_zero_int @ P5 )
       => ( ( times_times_int @ P5 @ ( ord_max_int @ X @ Y3 ) )
          = ( ord_max_int @ ( times_times_int @ P5 @ X ) @ ( times_times_int @ P5 @ Y3 ) ) ) )
      & ( ~ ( ord_less_eq_int @ zero_zero_int @ P5 )
       => ( ( times_times_int @ P5 @ ( ord_max_int @ X @ Y3 ) )
          = ( ord_min_int @ ( times_times_int @ P5 @ X ) @ ( times_times_int @ P5 @ Y3 ) ) ) ) ) ).

% max_mult_distrib_left
thf(fact_7761_min__mult__distrib__left,axiom,
    ! [P5: code_integer,X: code_integer,Y3: code_integer] :
      ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ P5 )
       => ( ( times_3573771949741848930nteger @ P5 @ ( ord_min_Code_integer @ X @ Y3 ) )
          = ( ord_min_Code_integer @ ( times_3573771949741848930nteger @ P5 @ X ) @ ( times_3573771949741848930nteger @ P5 @ Y3 ) ) ) )
      & ( ~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ P5 )
       => ( ( times_3573771949741848930nteger @ P5 @ ( ord_min_Code_integer @ X @ Y3 ) )
          = ( ord_max_Code_integer @ ( times_3573771949741848930nteger @ P5 @ X ) @ ( times_3573771949741848930nteger @ P5 @ Y3 ) ) ) ) ) ).

% min_mult_distrib_left
thf(fact_7762_min__mult__distrib__left,axiom,
    ! [P5: rat,X: rat,Y3: rat] :
      ( ( ( ord_less_eq_rat @ zero_zero_rat @ P5 )
       => ( ( times_times_rat @ P5 @ ( ord_min_rat @ X @ Y3 ) )
          = ( ord_min_rat @ ( times_times_rat @ P5 @ X ) @ ( times_times_rat @ P5 @ Y3 ) ) ) )
      & ( ~ ( ord_less_eq_rat @ zero_zero_rat @ P5 )
       => ( ( times_times_rat @ P5 @ ( ord_min_rat @ X @ Y3 ) )
          = ( ord_max_rat @ ( times_times_rat @ P5 @ X ) @ ( times_times_rat @ P5 @ Y3 ) ) ) ) ) ).

% min_mult_distrib_left
thf(fact_7763_min__mult__distrib__left,axiom,
    ! [P5: int,X: int,Y3: int] :
      ( ( ( ord_less_eq_int @ zero_zero_int @ P5 )
       => ( ( times_times_int @ P5 @ ( ord_min_int @ X @ Y3 ) )
          = ( ord_min_int @ ( times_times_int @ P5 @ X ) @ ( times_times_int @ P5 @ Y3 ) ) ) )
      & ( ~ ( ord_less_eq_int @ zero_zero_int @ P5 )
       => ( ( times_times_int @ P5 @ ( ord_min_int @ X @ Y3 ) )
          = ( ord_max_int @ ( times_times_int @ P5 @ X ) @ ( times_times_int @ P5 @ Y3 ) ) ) ) ) ).

% min_mult_distrib_left
thf(fact_7764_max__mult__distrib__right,axiom,
    ! [P5: code_integer,X: code_integer,Y3: code_integer] :
      ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ P5 )
       => ( ( times_3573771949741848930nteger @ ( ord_max_Code_integer @ X @ Y3 ) @ P5 )
          = ( ord_max_Code_integer @ ( times_3573771949741848930nteger @ X @ P5 ) @ ( times_3573771949741848930nteger @ Y3 @ P5 ) ) ) )
      & ( ~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ P5 )
       => ( ( times_3573771949741848930nteger @ ( ord_max_Code_integer @ X @ Y3 ) @ P5 )
          = ( ord_min_Code_integer @ ( times_3573771949741848930nteger @ X @ P5 ) @ ( times_3573771949741848930nteger @ Y3 @ P5 ) ) ) ) ) ).

% max_mult_distrib_right
thf(fact_7765_max__mult__distrib__right,axiom,
    ! [P5: rat,X: rat,Y3: rat] :
      ( ( ( ord_less_eq_rat @ zero_zero_rat @ P5 )
       => ( ( times_times_rat @ ( ord_max_rat @ X @ Y3 ) @ P5 )
          = ( ord_max_rat @ ( times_times_rat @ X @ P5 ) @ ( times_times_rat @ Y3 @ P5 ) ) ) )
      & ( ~ ( ord_less_eq_rat @ zero_zero_rat @ P5 )
       => ( ( times_times_rat @ ( ord_max_rat @ X @ Y3 ) @ P5 )
          = ( ord_min_rat @ ( times_times_rat @ X @ P5 ) @ ( times_times_rat @ Y3 @ P5 ) ) ) ) ) ).

% max_mult_distrib_right
thf(fact_7766_max__mult__distrib__right,axiom,
    ! [P5: int,X: int,Y3: int] :
      ( ( ( ord_less_eq_int @ zero_zero_int @ P5 )
       => ( ( times_times_int @ ( ord_max_int @ X @ Y3 ) @ P5 )
          = ( ord_max_int @ ( times_times_int @ X @ P5 ) @ ( times_times_int @ Y3 @ P5 ) ) ) )
      & ( ~ ( ord_less_eq_int @ zero_zero_int @ P5 )
       => ( ( times_times_int @ ( ord_max_int @ X @ Y3 ) @ P5 )
          = ( ord_min_int @ ( times_times_int @ X @ P5 ) @ ( times_times_int @ Y3 @ P5 ) ) ) ) ) ).

% max_mult_distrib_right
thf(fact_7767_min__mult__distrib__right,axiom,
    ! [P5: code_integer,X: code_integer,Y3: code_integer] :
      ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ P5 )
       => ( ( times_3573771949741848930nteger @ ( ord_min_Code_integer @ X @ Y3 ) @ P5 )
          = ( ord_min_Code_integer @ ( times_3573771949741848930nteger @ X @ P5 ) @ ( times_3573771949741848930nteger @ Y3 @ P5 ) ) ) )
      & ( ~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ P5 )
       => ( ( times_3573771949741848930nteger @ ( ord_min_Code_integer @ X @ Y3 ) @ P5 )
          = ( ord_max_Code_integer @ ( times_3573771949741848930nteger @ X @ P5 ) @ ( times_3573771949741848930nteger @ Y3 @ P5 ) ) ) ) ) ).

% min_mult_distrib_right
thf(fact_7768_min__mult__distrib__right,axiom,
    ! [P5: rat,X: rat,Y3: rat] :
      ( ( ( ord_less_eq_rat @ zero_zero_rat @ P5 )
       => ( ( times_times_rat @ ( ord_min_rat @ X @ Y3 ) @ P5 )
          = ( ord_min_rat @ ( times_times_rat @ X @ P5 ) @ ( times_times_rat @ Y3 @ P5 ) ) ) )
      & ( ~ ( ord_less_eq_rat @ zero_zero_rat @ P5 )
       => ( ( times_times_rat @ ( ord_min_rat @ X @ Y3 ) @ P5 )
          = ( ord_max_rat @ ( times_times_rat @ X @ P5 ) @ ( times_times_rat @ Y3 @ P5 ) ) ) ) ) ).

% min_mult_distrib_right
thf(fact_7769_min__mult__distrib__right,axiom,
    ! [P5: int,X: int,Y3: int] :
      ( ( ( ord_less_eq_int @ zero_zero_int @ P5 )
       => ( ( times_times_int @ ( ord_min_int @ X @ Y3 ) @ P5 )
          = ( ord_min_int @ ( times_times_int @ X @ P5 ) @ ( times_times_int @ Y3 @ P5 ) ) ) )
      & ( ~ ( ord_less_eq_int @ zero_zero_int @ P5 )
       => ( ( times_times_int @ ( ord_min_int @ X @ Y3 ) @ P5 )
          = ( ord_max_int @ ( times_times_int @ X @ P5 ) @ ( times_times_int @ Y3 @ P5 ) ) ) ) ) ).

% min_mult_distrib_right
thf(fact_7770_min__divide__distrib__right,axiom,
    ! [P5: rat,X: rat,Y3: rat] :
      ( ( ( ord_less_eq_rat @ zero_zero_rat @ P5 )
       => ( ( divide_divide_rat @ ( ord_min_rat @ X @ Y3 ) @ P5 )
          = ( ord_min_rat @ ( divide_divide_rat @ X @ P5 ) @ ( divide_divide_rat @ Y3 @ P5 ) ) ) )
      & ( ~ ( ord_less_eq_rat @ zero_zero_rat @ P5 )
       => ( ( divide_divide_rat @ ( ord_min_rat @ X @ Y3 ) @ P5 )
          = ( ord_max_rat @ ( divide_divide_rat @ X @ P5 ) @ ( divide_divide_rat @ Y3 @ P5 ) ) ) ) ) ).

% min_divide_distrib_right
thf(fact_7771_max__divide__distrib__right,axiom,
    ! [P5: rat,X: rat,Y3: rat] :
      ( ( ( ord_less_eq_rat @ zero_zero_rat @ P5 )
       => ( ( divide_divide_rat @ ( ord_max_rat @ X @ Y3 ) @ P5 )
          = ( ord_max_rat @ ( divide_divide_rat @ X @ P5 ) @ ( divide_divide_rat @ Y3 @ P5 ) ) ) )
      & ( ~ ( ord_less_eq_rat @ zero_zero_rat @ P5 )
       => ( ( divide_divide_rat @ ( ord_max_rat @ X @ Y3 ) @ P5 )
          = ( ord_min_rat @ ( divide_divide_rat @ X @ P5 ) @ ( divide_divide_rat @ Y3 @ P5 ) ) ) ) ) ).

% max_divide_distrib_right
thf(fact_7772_and__int_Oelims,axiom,
    ! [X: int,Xa3: int,Y3: int] :
      ( ( ( bit_se725231765392027082nd_int @ X @ Xa3 )
        = Y3 )
     => ( ( ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
            & ( member_int @ Xa3 @ ( 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 ) ) @ X )
                  & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa3 ) ) ) ) ) )
        & ( ~ ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
              & ( member_int @ Xa3 @ ( 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 ) ) @ X )
                  & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa3 ) ) )
              @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ X @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ Xa3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% and_int.elims
thf(fact_7773_and__int_Osimps,axiom,
    ( bit_se725231765392027082nd_int
    = ( ^ [K3: int,L3: 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 @ L3 @ ( 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 ) ) @ L3 ) ) ) )
          @ ( plus_plus_int
            @ ( zero_n2684676970156552555ol_int
              @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 )
                & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L3 ) ) )
            @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% and_int.simps
thf(fact_7774_nth__Cons__pos,axiom,
    ! [N: nat,X: int,Xs: list_int] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( nth_int @ ( cons_int @ X @ Xs ) @ N )
        = ( nth_int @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).

% nth_Cons_pos
thf(fact_7775_nth__Cons__pos,axiom,
    ! [N: nat,X: nat,Xs: list_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ N )
        = ( nth_nat @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).

% nth_Cons_pos
thf(fact_7776_divmod__step__eq,axiom,
    ! [L: num,R: nat,Q4: nat] :
      ( ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ L ) @ R )
       => ( ( unique5026877609467782581ep_nat @ L @ ( product_Pair_nat_nat @ Q4 @ R ) )
          = ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q4 ) @ one_one_nat ) @ ( minus_minus_nat @ R @ ( numeral_numeral_nat @ L ) ) ) ) )
      & ( ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L ) @ R )
       => ( ( unique5026877609467782581ep_nat @ L @ ( product_Pair_nat_nat @ Q4 @ R ) )
          = ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q4 ) @ R ) ) ) ) ).

% divmod_step_eq
thf(fact_7777_divmod__step__eq,axiom,
    ! [L: num,R: int,Q4: int] :
      ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ L ) @ R )
       => ( ( unique5024387138958732305ep_int @ L @ ( product_Pair_int_int @ Q4 @ R ) )
          = ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q4 ) @ one_one_int ) @ ( minus_minus_int @ R @ ( numeral_numeral_int @ L ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ L ) @ R )
       => ( ( unique5024387138958732305ep_int @ L @ ( product_Pair_int_int @ Q4 @ R ) )
          = ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q4 ) @ R ) ) ) ) ).

% divmod_step_eq
thf(fact_7778_divmod__step__eq,axiom,
    ! [L: num,R: code_integer,Q4: code_integer] :
      ( ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L ) @ R )
       => ( ( unique4921790084139445826nteger @ L @ ( produc1086072967326762835nteger @ Q4 @ R ) )
          = ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q4 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R @ ( numera6620942414471956472nteger @ L ) ) ) ) )
      & ( ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L ) @ R )
       => ( ( unique4921790084139445826nteger @ L @ ( produc1086072967326762835nteger @ Q4 @ R ) )
          = ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q4 ) @ R ) ) ) ) ).

% divmod_step_eq
thf(fact_7779_max__bot2,axiom,
    ! [X: set_int] :
      ( ( ord_max_set_int @ X @ bot_bot_set_int )
      = X ) ).

% max_bot2
thf(fact_7780_max__bot2,axiom,
    ! [X: set_nat] :
      ( ( ord_max_set_nat @ X @ bot_bot_set_nat )
      = X ) ).

% max_bot2
thf(fact_7781_max__bot2,axiom,
    ! [X: set_Pr1261947904930325089at_nat] :
      ( ( ord_ma7524802468073614006at_nat @ X @ bot_bo2099793752762293965at_nat )
      = X ) ).

% max_bot2
thf(fact_7782_max__bot2,axiom,
    ! [X: set_Pr4329608150637261639at_nat] :
      ( ( ord_ma6735246370732646428at_nat @ X @ bot_bo228742789529271731at_nat )
      = X ) ).

% max_bot2
thf(fact_7783_max__bot2,axiom,
    ! [X: nat] :
      ( ( ord_max_nat @ X @ bot_bot_nat )
      = X ) ).

% max_bot2
thf(fact_7784_max__bot,axiom,
    ! [X: set_int] :
      ( ( ord_max_set_int @ bot_bot_set_int @ X )
      = X ) ).

% max_bot
thf(fact_7785_max__bot,axiom,
    ! [X: set_nat] :
      ( ( ord_max_set_nat @ bot_bot_set_nat @ X )
      = X ) ).

% max_bot
thf(fact_7786_max__bot,axiom,
    ! [X: set_Pr1261947904930325089at_nat] :
      ( ( ord_ma7524802468073614006at_nat @ bot_bo2099793752762293965at_nat @ X )
      = X ) ).

% max_bot
thf(fact_7787_max__bot,axiom,
    ! [X: set_Pr4329608150637261639at_nat] :
      ( ( ord_ma6735246370732646428at_nat @ bot_bo228742789529271731at_nat @ X )
      = X ) ).

% max_bot
thf(fact_7788_max__bot,axiom,
    ! [X: nat] :
      ( ( ord_max_nat @ bot_bot_nat @ X )
      = X ) ).

% max_bot
thf(fact_7789_min__bot2,axiom,
    ! [X: set_int] :
      ( ( ord_min_set_int @ X @ bot_bot_set_int )
      = bot_bot_set_int ) ).

% min_bot2
thf(fact_7790_min__bot2,axiom,
    ! [X: set_nat] :
      ( ( ord_min_set_nat @ X @ bot_bot_set_nat )
      = bot_bot_set_nat ) ).

% min_bot2
thf(fact_7791_min__bot2,axiom,
    ! [X: set_Pr1261947904930325089at_nat] :
      ( ( ord_mi4194197829855655752at_nat @ X @ bot_bo2099793752762293965at_nat )
      = bot_bo2099793752762293965at_nat ) ).

% min_bot2
thf(fact_7792_min__bot2,axiom,
    ! [X: set_Pr4329608150637261639at_nat] :
      ( ( ord_mi6693596345299712430at_nat @ X @ bot_bo228742789529271731at_nat )
      = bot_bo228742789529271731at_nat ) ).

% min_bot2
thf(fact_7793_min__bot2,axiom,
    ! [X: nat] :
      ( ( ord_min_nat @ X @ bot_bot_nat )
      = bot_bot_nat ) ).

% min_bot2
thf(fact_7794_min__bot,axiom,
    ! [X: set_int] :
      ( ( ord_min_set_int @ bot_bot_set_int @ X )
      = bot_bot_set_int ) ).

% min_bot
thf(fact_7795_min__bot,axiom,
    ! [X: set_nat] :
      ( ( ord_min_set_nat @ bot_bot_set_nat @ X )
      = bot_bot_set_nat ) ).

% min_bot
thf(fact_7796_min__bot,axiom,
    ! [X: set_Pr1261947904930325089at_nat] :
      ( ( ord_mi4194197829855655752at_nat @ bot_bo2099793752762293965at_nat @ X )
      = bot_bo2099793752762293965at_nat ) ).

% min_bot
thf(fact_7797_min__bot,axiom,
    ! [X: set_Pr4329608150637261639at_nat] :
      ( ( ord_mi6693596345299712430at_nat @ bot_bo228742789529271731at_nat @ X )
      = bot_bo228742789529271731at_nat ) ).

% min_bot
thf(fact_7798_min__bot,axiom,
    ! [X: nat] :
      ( ( ord_min_nat @ bot_bot_nat @ X )
      = bot_bot_nat ) ).

% min_bot
thf(fact_7799_atLeastatMost__empty__iff2,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( bot_bot_set_set_nat
        = ( set_or4548717258645045905et_nat @ A @ B ) )
      = ( ~ ( ord_less_eq_set_nat @ A @ B ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_7800_atLeastatMost__empty__iff2,axiom,
    ! [A: rat,B: rat] :
      ( ( bot_bot_set_rat
        = ( set_or633870826150836451st_rat @ A @ B ) )
      = ( ~ ( ord_less_eq_rat @ A @ B ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_7801_atLeastatMost__empty__iff2,axiom,
    ! [A: num,B: num] :
      ( ( bot_bot_set_num
        = ( set_or7049704709247886629st_num @ A @ B ) )
      = ( ~ ( ord_less_eq_num @ A @ B ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_7802_atLeastatMost__empty__iff2,axiom,
    ! [A: int,B: int] :
      ( ( bot_bot_set_int
        = ( set_or1266510415728281911st_int @ A @ B ) )
      = ( ~ ( ord_less_eq_int @ A @ B ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_7803_atLeastatMost__empty__iff2,axiom,
    ! [A: nat,B: nat] :
      ( ( bot_bot_set_nat
        = ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( ~ ( ord_less_eq_nat @ A @ B ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_7804_atLeastatMost__empty__iff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ( set_or4548717258645045905et_nat @ A @ B )
        = bot_bot_set_set_nat )
      = ( ~ ( ord_less_eq_set_nat @ A @ B ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_7805_atLeastatMost__empty__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ( set_or633870826150836451st_rat @ A @ B )
        = bot_bot_set_rat )
      = ( ~ ( ord_less_eq_rat @ A @ B ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_7806_atLeastatMost__empty__iff,axiom,
    ! [A: num,B: num] :
      ( ( ( set_or7049704709247886629st_num @ A @ B )
        = bot_bot_set_num )
      = ( ~ ( ord_less_eq_num @ A @ B ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_7807_atLeastatMost__empty__iff,axiom,
    ! [A: int,B: int] :
      ( ( ( set_or1266510415728281911st_int @ A @ B )
        = bot_bot_set_int )
      = ( ~ ( ord_less_eq_int @ A @ B ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_7808_atLeastatMost__empty__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( ( set_or1269000886237332187st_nat @ A @ B )
        = bot_bot_set_nat )
      = ( ~ ( ord_less_eq_nat @ A @ B ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_7809_atLeastatMost__empty,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ B @ A )
     => ( ( set_or189985376899183464nteger @ A @ B )
        = bot_bo3990330152332043303nteger ) ) ).

% atLeastatMost_empty
thf(fact_7810_atLeastatMost__empty,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( set_or633870826150836451st_rat @ A @ B )
        = bot_bot_set_rat ) ) ).

% atLeastatMost_empty
thf(fact_7811_atLeastatMost__empty,axiom,
    ! [B: num,A: num] :
      ( ( ord_less_num @ B @ A )
     => ( ( set_or7049704709247886629st_num @ A @ B )
        = bot_bot_set_num ) ) ).

% atLeastatMost_empty
thf(fact_7812_atLeastatMost__empty,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( set_or1266510415728281911st_int @ A @ B )
        = bot_bot_set_int ) ) ).

% atLeastatMost_empty
thf(fact_7813_atLeastatMost__empty,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( set_or1269000886237332187st_nat @ A @ B )
        = bot_bot_set_nat ) ) ).

% atLeastatMost_empty
thf(fact_7814_nth__Cons__Suc,axiom,
    ! [X: int,Xs: list_int,N: nat] :
      ( ( nth_int @ ( cons_int @ X @ Xs ) @ ( suc @ N ) )
      = ( nth_int @ Xs @ N ) ) ).

% nth_Cons_Suc
thf(fact_7815_nth__Cons__Suc,axiom,
    ! [X: nat,Xs: list_nat,N: nat] :
      ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ ( suc @ N ) )
      = ( nth_nat @ Xs @ N ) ) ).

% nth_Cons_Suc
thf(fact_7816_nth__Cons__0,axiom,
    ! [X: int,Xs: list_int] :
      ( ( nth_int @ ( cons_int @ X @ Xs ) @ zero_zero_nat )
      = X ) ).

% nth_Cons_0
thf(fact_7817_nth__Cons__0,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ zero_zero_nat )
      = X ) ).

% nth_Cons_0
thf(fact_7818_subset__Compl__singleton,axiom,
    ! [A3: set_se7855581050983116737at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( ord_le2077887516847798113at_nat @ A3 @ ( uminus8018730102153342520at_nat @ ( insert9200635055090092081at_nat @ B @ bot_bo3083307316010499117at_nat ) ) )
      = ( ~ ( member2643936169264416010at_nat @ B @ A3 ) ) ) ).

% subset_Compl_singleton
thf(fact_7819_subset__Compl__singleton,axiom,
    ! [A3: set_set_nat,B: set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A3 @ ( uminus613421341184616069et_nat @ ( insert_set_nat @ B @ bot_bot_set_set_nat ) ) )
      = ( ~ ( member_set_nat @ B @ A3 ) ) ) ).

% subset_Compl_singleton
thf(fact_7820_subset__Compl__singleton,axiom,
    ! [A3: set_int,B: int] :
      ( ( ord_less_eq_set_int @ A3 @ ( uminus1532241313380277803et_int @ ( insert_int @ B @ bot_bot_set_int ) ) )
      = ( ~ ( member_int @ B @ A3 ) ) ) ).

% subset_Compl_singleton
thf(fact_7821_subset__Compl__singleton,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat] :
      ( ( ord_le3146513528884898305at_nat @ A3 @ ( uminus6524753893492686040at_nat @ ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat ) ) )
      = ( ~ ( member8440522571783428010at_nat @ B @ A3 ) ) ) ).

% subset_Compl_singleton
thf(fact_7822_subset__Compl__singleton,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B: produc3843707927480180839at_nat] :
      ( ( ord_le1268244103169919719at_nat @ A3 @ ( uminus935396558254630718at_nat @ ( insert9069300056098147895at_nat @ B @ bot_bo228742789529271731at_nat ) ) )
      = ( ~ ( member8757157785044589968at_nat @ B @ A3 ) ) ) ).

% subset_Compl_singleton
thf(fact_7823_subset__Compl__singleton,axiom,
    ! [A3: set_nat,B: nat] :
      ( ( ord_less_eq_set_nat @ A3 @ ( uminus5710092332889474511et_nat @ ( insert_nat @ B @ bot_bot_set_nat ) ) )
      = ( ~ ( member_nat @ B @ A3 ) ) ) ).

% subset_Compl_singleton
thf(fact_7824_nth__Cons__numeral,axiom,
    ! [X: int,Xs: list_int,V: num] :
      ( ( nth_int @ ( cons_int @ X @ Xs ) @ ( numeral_numeral_nat @ V ) )
      = ( nth_int @ Xs @ ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ one_one_nat ) ) ) ).

% nth_Cons_numeral
thf(fact_7825_nth__Cons__numeral,axiom,
    ! [X: nat,Xs: list_nat,V: num] :
      ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ ( numeral_numeral_nat @ V ) )
      = ( nth_nat @ Xs @ ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ one_one_nat ) ) ) ).

% nth_Cons_numeral
thf(fact_7826_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_7827_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_7828_bot_Onot__eq__extremum,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( ( A != bot_bo2099793752762293965at_nat )
      = ( ord_le7866589430770878221at_nat @ bot_bo2099793752762293965at_nat @ A ) ) ).

% bot.not_eq_extremum
thf(fact_7829_bot_Onot__eq__extremum,axiom,
    ! [A: set_Pr4329608150637261639at_nat] :
      ( ( A != bot_bo228742789529271731at_nat )
      = ( ord_le2604355607129572851at_nat @ bot_bo228742789529271731at_nat @ A ) ) ).

% bot.not_eq_extremum
thf(fact_7830_bot_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != bot_bot_nat )
      = ( ord_less_nat @ bot_bot_nat @ A ) ) ).

% bot.not_eq_extremum
thf(fact_7831_bot_Oextremum__strict,axiom,
    ! [A: set_int] :
      ~ ( ord_less_set_int @ A @ bot_bot_set_int ) ).

% bot.extremum_strict
thf(fact_7832_bot_Oextremum__strict,axiom,
    ! [A: set_nat] :
      ~ ( ord_less_set_nat @ A @ bot_bot_set_nat ) ).

% bot.extremum_strict
thf(fact_7833_bot_Oextremum__strict,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ~ ( ord_le7866589430770878221at_nat @ A @ bot_bo2099793752762293965at_nat ) ).

% bot.extremum_strict
thf(fact_7834_bot_Oextremum__strict,axiom,
    ! [A: set_Pr4329608150637261639at_nat] :
      ~ ( ord_le2604355607129572851at_nat @ A @ bot_bo228742789529271731at_nat ) ).

% bot.extremum_strict
thf(fact_7835_bot_Oextremum__strict,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ bot_bot_nat ) ).

% bot.extremum_strict
thf(fact_7836_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_7837_bot_Oextremum__uniqueI,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A @ bot_bo2099793752762293965at_nat )
     => ( A = bot_bo2099793752762293965at_nat ) ) ).

% bot.extremum_uniqueI
thf(fact_7838_bot_Oextremum__uniqueI,axiom,
    ! [A: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ A @ bot_bo228742789529271731at_nat )
     => ( A = bot_bo228742789529271731at_nat ) ) ).

% bot.extremum_uniqueI
thf(fact_7839_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_7840_bot_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ bot_bot_nat )
     => ( A = bot_bot_nat ) ) ).

% bot.extremum_uniqueI
thf(fact_7841_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_7842_bot_Oextremum__unique,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A @ bot_bo2099793752762293965at_nat )
      = ( A = bot_bo2099793752762293965at_nat ) ) ).

% bot.extremum_unique
thf(fact_7843_bot_Oextremum__unique,axiom,
    ! [A: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ A @ bot_bo228742789529271731at_nat )
      = ( A = bot_bo228742789529271731at_nat ) ) ).

% bot.extremum_unique
thf(fact_7844_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_7845_bot_Oextremum__unique,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ bot_bot_nat )
      = ( A = bot_bot_nat ) ) ).

% bot.extremum_unique
thf(fact_7846_bot_Oextremum,axiom,
    ! [A: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A ) ).

% bot.extremum
thf(fact_7847_bot_Oextremum,axiom,
    ! [A: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ bot_bo2099793752762293965at_nat @ A ) ).

% bot.extremum
thf(fact_7848_bot_Oextremum,axiom,
    ! [A: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ bot_bo228742789529271731at_nat @ A ) ).

% bot.extremum
thf(fact_7849_bot_Oextremum,axiom,
    ! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).

% bot.extremum
thf(fact_7850_bot_Oextremum,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).

% bot.extremum
thf(fact_7851_not__psubset__empty,axiom,
    ! [A3: set_int] :
      ~ ( ord_less_set_int @ A3 @ bot_bot_set_int ) ).

% not_psubset_empty
thf(fact_7852_not__psubset__empty,axiom,
    ! [A3: set_nat] :
      ~ ( ord_less_set_nat @ A3 @ bot_bot_set_nat ) ).

% not_psubset_empty
thf(fact_7853_not__psubset__empty,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ~ ( ord_le7866589430770878221at_nat @ A3 @ bot_bo2099793752762293965at_nat ) ).

% not_psubset_empty
thf(fact_7854_not__psubset__empty,axiom,
    ! [A3: set_Pr4329608150637261639at_nat] :
      ~ ( ord_le2604355607129572851at_nat @ A3 @ bot_bo228742789529271731at_nat ) ).

% not_psubset_empty
thf(fact_7855_diff__shunt__var,axiom,
    ! [X: set_int,Y3: set_int] :
      ( ( ( minus_minus_set_int @ X @ Y3 )
        = bot_bot_set_int )
      = ( ord_less_eq_set_int @ X @ Y3 ) ) ).

% diff_shunt_var
thf(fact_7856_diff__shunt__var,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
      ( ( ( minus_1356011639430497352at_nat @ X @ Y3 )
        = bot_bo2099793752762293965at_nat )
      = ( ord_le3146513528884898305at_nat @ X @ Y3 ) ) ).

% diff_shunt_var
thf(fact_7857_diff__shunt__var,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat] :
      ( ( ( minus_3314409938677909166at_nat @ X @ Y3 )
        = bot_bo228742789529271731at_nat )
      = ( ord_le1268244103169919719at_nat @ X @ Y3 ) ) ).

% diff_shunt_var
thf(fact_7858_diff__shunt__var,axiom,
    ! [X: set_nat,Y3: set_nat] :
      ( ( ( minus_minus_set_nat @ X @ Y3 )
        = bot_bot_set_nat )
      = ( ord_less_eq_set_nat @ X @ Y3 ) ) ).

% diff_shunt_var
thf(fact_7859_simp__from__to,axiom,
    ( set_or1266510415728281911st_int
    = ( ^ [I4: int,J3: int] : ( if_set_int @ ( ord_less_int @ J3 @ I4 ) @ bot_bot_set_int @ ( insert_int @ I4 @ ( set_or1266510415728281911st_int @ ( plus_plus_int @ I4 @ one_one_int ) @ J3 ) ) ) ) ) ).

% simp_from_to
thf(fact_7860_length__Suc__conv,axiom,
    ! [Xs: list_o,N: nat] :
      ( ( ( size_size_list_o @ Xs )
        = ( suc @ N ) )
      = ( ? [Y6: $o,Ys3: list_o] :
            ( ( Xs
              = ( cons_o @ Y6 @ Ys3 ) )
            & ( ( size_size_list_o @ Ys3 )
              = N ) ) ) ) ).

% length_Suc_conv
thf(fact_7861_length__Suc__conv,axiom,
    ! [Xs: list_nat,N: nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( suc @ N ) )
      = ( ? [Y6: nat,Ys3: list_nat] :
            ( ( Xs
              = ( cons_nat @ Y6 @ Ys3 ) )
            & ( ( size_size_list_nat @ Ys3 )
              = N ) ) ) ) ).

% length_Suc_conv
thf(fact_7862_length__Suc__conv,axiom,
    ! [Xs: list_int,N: nat] :
      ( ( ( size_size_list_int @ Xs )
        = ( suc @ N ) )
      = ( ? [Y6: int,Ys3: list_int] :
            ( ( Xs
              = ( cons_int @ Y6 @ Ys3 ) )
            & ( ( size_size_list_int @ Ys3 )
              = N ) ) ) ) ).

% length_Suc_conv
thf(fact_7863_Suc__length__conv,axiom,
    ! [N: nat,Xs: list_o] :
      ( ( ( suc @ N )
        = ( size_size_list_o @ Xs ) )
      = ( ? [Y6: $o,Ys3: list_o] :
            ( ( Xs
              = ( cons_o @ Y6 @ Ys3 ) )
            & ( ( size_size_list_o @ Ys3 )
              = N ) ) ) ) ).

% Suc_length_conv
thf(fact_7864_Suc__length__conv,axiom,
    ! [N: nat,Xs: list_nat] :
      ( ( ( suc @ N )
        = ( size_size_list_nat @ Xs ) )
      = ( ? [Y6: nat,Ys3: list_nat] :
            ( ( Xs
              = ( cons_nat @ Y6 @ Ys3 ) )
            & ( ( size_size_list_nat @ Ys3 )
              = N ) ) ) ) ).

% Suc_length_conv
thf(fact_7865_Suc__length__conv,axiom,
    ! [N: nat,Xs: list_int] :
      ( ( ( suc @ N )
        = ( size_size_list_int @ Xs ) )
      = ( ? [Y6: int,Ys3: list_int] :
            ( ( Xs
              = ( cons_int @ Y6 @ Ys3 ) )
            & ( ( size_size_list_int @ Ys3 )
              = N ) ) ) ) ).

% Suc_length_conv
thf(fact_7866_impossible__Cons,axiom,
    ! [Xs: list_o,Ys2: list_o,X: $o] :
      ( ( ord_less_eq_nat @ ( size_size_list_o @ Xs ) @ ( size_size_list_o @ Ys2 ) )
     => ( Xs
       != ( cons_o @ X @ Ys2 ) ) ) ).

% impossible_Cons
thf(fact_7867_impossible__Cons,axiom,
    ! [Xs: list_nat,Ys2: list_nat,X: nat] :
      ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys2 ) )
     => ( Xs
       != ( cons_nat @ X @ Ys2 ) ) ) ).

% impossible_Cons
thf(fact_7868_impossible__Cons,axiom,
    ! [Xs: list_int,Ys2: list_int,X: int] :
      ( ( ord_less_eq_nat @ ( size_size_list_int @ Xs ) @ ( size_size_list_int @ Ys2 ) )
     => ( Xs
       != ( cons_int @ X @ Ys2 ) ) ) ).

% impossible_Cons
thf(fact_7869_subset__Compl__self__eq,axiom,
    ! [A3: set_int] :
      ( ( ord_less_eq_set_int @ A3 @ ( uminus1532241313380277803et_int @ A3 ) )
      = ( A3 = bot_bot_set_int ) ) ).

% subset_Compl_self_eq
thf(fact_7870_subset__Compl__self__eq,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A3 @ ( uminus6524753893492686040at_nat @ A3 ) )
      = ( A3 = bot_bo2099793752762293965at_nat ) ) ).

% subset_Compl_self_eq
thf(fact_7871_subset__Compl__self__eq,axiom,
    ! [A3: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ A3 @ ( uminus935396558254630718at_nat @ A3 ) )
      = ( A3 = bot_bo228742789529271731at_nat ) ) ).

% subset_Compl_self_eq
thf(fact_7872_subset__Compl__self__eq,axiom,
    ! [A3: set_nat] :
      ( ( ord_less_eq_set_nat @ A3 @ ( uminus5710092332889474511et_nat @ A3 ) )
      = ( A3 = bot_bot_set_nat ) ) ).

% subset_Compl_self_eq
thf(fact_7873_Suc__le__length__iff,axiom,
    ! [N: nat,Xs: list_o] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_o @ Xs ) )
      = ( ? [X2: $o,Ys3: list_o] :
            ( ( Xs
              = ( cons_o @ X2 @ Ys3 ) )
            & ( ord_less_eq_nat @ N @ ( size_size_list_o @ Ys3 ) ) ) ) ) ).

% Suc_le_length_iff
thf(fact_7874_Suc__le__length__iff,axiom,
    ! [N: nat,Xs: list_nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_nat @ Xs ) )
      = ( ? [X2: nat,Ys3: list_nat] :
            ( ( Xs
              = ( cons_nat @ X2 @ Ys3 ) )
            & ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Ys3 ) ) ) ) ) ).

% Suc_le_length_iff
thf(fact_7875_Suc__le__length__iff,axiom,
    ! [N: nat,Xs: list_int] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_int @ Xs ) )
      = ( ? [X2: int,Ys3: list_int] :
            ( ( Xs
              = ( cons_int @ X2 @ Ys3 ) )
            & ( ord_less_eq_nat @ N @ ( size_size_list_int @ Ys3 ) ) ) ) ) ).

% Suc_le_length_iff
thf(fact_7876_Compl__insert,axiom,
    ! [X: int,A3: set_int] :
      ( ( uminus1532241313380277803et_int @ ( insert_int @ X @ A3 ) )
      = ( minus_minus_set_int @ ( uminus1532241313380277803et_int @ A3 ) @ ( insert_int @ X @ bot_bot_set_int ) ) ) ).

% Compl_insert
thf(fact_7877_Compl__insert,axiom,
    ! [X: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( uminus6524753893492686040at_nat @ ( insert8211810215607154385at_nat @ X @ A3 ) )
      = ( minus_1356011639430497352at_nat @ ( uminus6524753893492686040at_nat @ A3 ) @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) ) ).

% Compl_insert
thf(fact_7878_Compl__insert,axiom,
    ! [X: produc3843707927480180839at_nat,A3: set_Pr4329608150637261639at_nat] :
      ( ( uminus935396558254630718at_nat @ ( insert9069300056098147895at_nat @ X @ A3 ) )
      = ( minus_3314409938677909166at_nat @ ( uminus935396558254630718at_nat @ A3 ) @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) ) ).

% Compl_insert
thf(fact_7879_Compl__insert,axiom,
    ! [X: nat,A3: set_nat] :
      ( ( uminus5710092332889474511et_nat @ ( insert_nat @ X @ A3 ) )
      = ( minus_minus_set_nat @ ( uminus5710092332889474511et_nat @ A3 ) @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ).

% Compl_insert
thf(fact_7880_list_Osize_I4_J,axiom,
    ! [X21: $o,X222: list_o] :
      ( ( size_size_list_o @ ( cons_o @ X21 @ X222 ) )
      = ( plus_plus_nat @ ( size_size_list_o @ X222 ) @ ( suc @ zero_zero_nat ) ) ) ).

% list.size(4)
thf(fact_7881_list_Osize_I4_J,axiom,
    ! [X21: nat,X222: list_nat] :
      ( ( size_size_list_nat @ ( cons_nat @ X21 @ X222 ) )
      = ( plus_plus_nat @ ( size_size_list_nat @ X222 ) @ ( suc @ zero_zero_nat ) ) ) ).

% list.size(4)
thf(fact_7882_list_Osize_I4_J,axiom,
    ! [X21: int,X222: list_int] :
      ( ( size_size_list_int @ ( cons_int @ X21 @ X222 ) )
      = ( plus_plus_nat @ ( size_size_list_int @ X222 ) @ ( suc @ zero_zero_nat ) ) ) ).

% list.size(4)
thf(fact_7883_nth__Cons_H,axiom,
    ! [N: nat,X: int,Xs: list_int] :
      ( ( ( N = zero_zero_nat )
       => ( ( nth_int @ ( cons_int @ X @ Xs ) @ N )
          = X ) )
      & ( ( N != zero_zero_nat )
       => ( ( nth_int @ ( cons_int @ X @ Xs ) @ N )
          = ( nth_int @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).

% nth_Cons'
thf(fact_7884_nth__Cons_H,axiom,
    ! [N: nat,X: nat,Xs: list_nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ N )
          = X ) )
      & ( ( N != zero_zero_nat )
       => ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ N )
          = ( nth_nat @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).

% nth_Cons'
thf(fact_7885_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_7886_psubset__insert__iff,axiom,
    ! [A3: set_se7855581050983116737at_nat,X: set_Pr1261947904930325089at_nat,B5: set_se7855581050983116737at_nat] :
      ( ( ord_le3519150613208228973at_nat @ A3 @ ( insert9200635055090092081at_nat @ X @ B5 ) )
      = ( ( ( member2643936169264416010at_nat @ X @ B5 )
         => ( ord_le3519150613208228973at_nat @ A3 @ B5 ) )
        & ( ~ ( member2643936169264416010at_nat @ X @ B5 )
         => ( ( ( member2643936169264416010at_nat @ X @ A3 )
             => ( ord_le3519150613208228973at_nat @ ( minus_4207664762107033000at_nat @ A3 @ ( insert9200635055090092081at_nat @ X @ bot_bo3083307316010499117at_nat ) ) @ B5 ) )
            & ( ~ ( member2643936169264416010at_nat @ X @ A3 )
             => ( ord_le2077887516847798113at_nat @ A3 @ B5 ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_7887_psubset__insert__iff,axiom,
    ! [A3: set_set_nat,X: set_nat,B5: set_set_nat] :
      ( ( ord_less_set_set_nat @ A3 @ ( insert_set_nat @ X @ B5 ) )
      = ( ( ( member_set_nat @ X @ B5 )
         => ( ord_less_set_set_nat @ A3 @ B5 ) )
        & ( ~ ( member_set_nat @ X @ B5 )
         => ( ( ( member_set_nat @ X @ A3 )
             => ( ord_less_set_set_nat @ ( minus_2163939370556025621et_nat @ A3 @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) @ B5 ) )
            & ( ~ ( member_set_nat @ X @ A3 )
             => ( ord_le6893508408891458716et_nat @ A3 @ B5 ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_7888_psubset__insert__iff,axiom,
    ! [A3: set_int,X: int,B5: set_int] :
      ( ( ord_less_set_int @ A3 @ ( insert_int @ X @ B5 ) )
      = ( ( ( member_int @ X @ B5 )
         => ( ord_less_set_int @ A3 @ B5 ) )
        & ( ~ ( member_int @ X @ B5 )
         => ( ( ( member_int @ X @ A3 )
             => ( ord_less_set_int @ ( minus_minus_set_int @ A3 @ ( insert_int @ X @ bot_bot_set_int ) ) @ B5 ) )
            & ( ~ ( member_int @ X @ A3 )
             => ( ord_less_eq_set_int @ A3 @ B5 ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_7889_psubset__insert__iff,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( ord_le7866589430770878221at_nat @ A3 @ ( insert8211810215607154385at_nat @ X @ B5 ) )
      = ( ( ( member8440522571783428010at_nat @ X @ B5 )
         => ( ord_le7866589430770878221at_nat @ A3 @ B5 ) )
        & ( ~ ( member8440522571783428010at_nat @ X @ B5 )
         => ( ( ( member8440522571783428010at_nat @ X @ A3 )
             => ( ord_le7866589430770878221at_nat @ ( minus_1356011639430497352at_nat @ A3 @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) @ B5 ) )
            & ( ~ ( member8440522571783428010at_nat @ X @ A3 )
             => ( ord_le3146513528884898305at_nat @ A3 @ B5 ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_7890_psubset__insert__iff,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat,B5: set_Pr4329608150637261639at_nat] :
      ( ( ord_le2604355607129572851at_nat @ A3 @ ( insert9069300056098147895at_nat @ X @ B5 ) )
      = ( ( ( member8757157785044589968at_nat @ X @ B5 )
         => ( ord_le2604355607129572851at_nat @ A3 @ B5 ) )
        & ( ~ ( member8757157785044589968at_nat @ X @ B5 )
         => ( ( ( member8757157785044589968at_nat @ X @ A3 )
             => ( ord_le2604355607129572851at_nat @ ( minus_3314409938677909166at_nat @ A3 @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) @ B5 ) )
            & ( ~ ( member8757157785044589968at_nat @ X @ A3 )
             => ( ord_le1268244103169919719at_nat @ A3 @ B5 ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_7891_psubset__insert__iff,axiom,
    ! [A3: set_nat,X: nat,B5: set_nat] :
      ( ( ord_less_set_nat @ A3 @ ( insert_nat @ X @ B5 ) )
      = ( ( ( member_nat @ X @ B5 )
         => ( ord_less_set_nat @ A3 @ B5 ) )
        & ( ~ ( member_nat @ X @ B5 )
         => ( ( ( member_nat @ X @ A3 )
             => ( ord_less_set_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B5 ) )
            & ( ~ ( member_nat @ X @ A3 )
             => ( ord_less_eq_set_nat @ A3 @ B5 ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_7892_nth__non__equal__first__eq,axiom,
    ! [X: int,Y3: int,Xs: list_int,N: nat] :
      ( ( X != Y3 )
     => ( ( ( nth_int @ ( cons_int @ X @ Xs ) @ N )
          = Y3 )
        = ( ( ( nth_int @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) )
            = Y3 )
          & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).

% nth_non_equal_first_eq
thf(fact_7893_nth__non__equal__first__eq,axiom,
    ! [X: nat,Y3: nat,Xs: list_nat,N: nat] :
      ( ( X != Y3 )
     => ( ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ N )
          = Y3 )
        = ( ( ( nth_nat @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) )
            = Y3 )
          & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).

% nth_non_equal_first_eq
thf(fact_7894_horner__sum__simps_I2_J,axiom,
    ! [F: int > code_integer,A: code_integer,X: int,Xs: list_int] :
      ( ( groups8594936613892548326nteger @ F @ A @ ( cons_int @ X @ Xs ) )
      = ( plus_p5714425477246183910nteger @ ( F @ X ) @ ( times_3573771949741848930nteger @ A @ ( groups8594936613892548326nteger @ F @ A @ Xs ) ) ) ) ).

% horner_sum_simps(2)
thf(fact_7895_horner__sum__simps_I2_J,axiom,
    ! [F: nat > code_integer,A: code_integer,X: nat,Xs: list_nat] :
      ( ( groups8223283053655703690nteger @ F @ A @ ( cons_nat @ X @ Xs ) )
      = ( plus_p5714425477246183910nteger @ ( F @ X ) @ ( times_3573771949741848930nteger @ A @ ( groups8223283053655703690nteger @ F @ A @ Xs ) ) ) ) ).

% horner_sum_simps(2)
thf(fact_7896_horner__sum__simps_I2_J,axiom,
    ! [F: int > rat,A: rat,X: int,Xs: list_int] :
      ( ( groups7852591826665563233nt_rat @ F @ A @ ( cons_int @ X @ Xs ) )
      = ( plus_plus_rat @ ( F @ X ) @ ( times_times_rat @ A @ ( groups7852591826665563233nt_rat @ F @ A @ Xs ) ) ) ) ).

% horner_sum_simps(2)
thf(fact_7897_horner__sum__simps_I2_J,axiom,
    ! [F: nat > rat,A: rat,X: nat,Xs: list_nat] :
      ( ( groups6853238114764508677at_rat @ F @ A @ ( cons_nat @ X @ Xs ) )
      = ( plus_plus_rat @ ( F @ X ) @ ( times_times_rat @ A @ ( groups6853238114764508677at_rat @ F @ A @ Xs ) ) ) ) ).

% horner_sum_simps(2)
thf(fact_7898_horner__sum__simps_I2_J,axiom,
    ! [F: int > nat,A: nat,X: int,Xs: list_int] :
      ( ( groups8487721886752058969nt_nat @ F @ A @ ( cons_int @ X @ Xs ) )
      = ( plus_plus_nat @ ( F @ X ) @ ( times_times_nat @ A @ ( groups8487721886752058969nt_nat @ F @ A @ Xs ) ) ) ) ).

% horner_sum_simps(2)
thf(fact_7899_horner__sum__simps_I2_J,axiom,
    ! [F: nat > nat,A: nat,X: nat,Xs: list_nat] :
      ( ( groups7488368174851004413at_nat @ F @ A @ ( cons_nat @ X @ Xs ) )
      = ( plus_plus_nat @ ( F @ X ) @ ( times_times_nat @ A @ ( groups7488368174851004413at_nat @ F @ A @ Xs ) ) ) ) ).

% horner_sum_simps(2)
thf(fact_7900_horner__sum__simps_I2_J,axiom,
    ! [F: int > int,A: int,X: int,Xs: list_int] :
      ( ( groups8485231416243008693nt_int @ F @ A @ ( cons_int @ X @ Xs ) )
      = ( plus_plus_int @ ( F @ X ) @ ( times_times_int @ A @ ( groups8485231416243008693nt_int @ F @ A @ Xs ) ) ) ) ).

% horner_sum_simps(2)
thf(fact_7901_horner__sum__simps_I2_J,axiom,
    ! [F: nat > int,A: int,X: nat,Xs: list_nat] :
      ( ( groups7485877704341954137at_int @ F @ A @ ( cons_nat @ X @ Xs ) )
      = ( plus_plus_int @ ( F @ X ) @ ( times_times_int @ A @ ( groups7485877704341954137at_int @ F @ A @ Xs ) ) ) ) ).

% horner_sum_simps(2)
thf(fact_7902_horner__sum__simps_I2_J,axiom,
    ! [F: $o > int,A: int,X: $o,Xs: list_o] :
      ( ( groups9116527308978886569_o_int @ F @ A @ ( cons_o @ X @ Xs ) )
      = ( plus_plus_int @ ( F @ X ) @ ( times_times_int @ A @ ( groups9116527308978886569_o_int @ F @ A @ Xs ) ) ) ) ).

% horner_sum_simps(2)
thf(fact_7903_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_7904_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_7905_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_7906_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_7907_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_7908_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_7909_divides__aux__eq,axiom,
    ! [Q4: nat,R: nat] :
      ( ( unique6322359934112328802ux_nat @ ( product_Pair_nat_nat @ Q4 @ R ) )
      = ( R = zero_zero_nat ) ) ).

% divides_aux_eq
thf(fact_7910_divides__aux__eq,axiom,
    ! [Q4: int,R: int] :
      ( ( unique6319869463603278526ux_int @ ( product_Pair_int_int @ Q4 @ R ) )
      = ( R = zero_zero_int ) ) ).

% divides_aux_eq
thf(fact_7911_divides__aux__eq,axiom,
    ! [Q4: code_integer,R: code_integer] :
      ( ( unique5706413561485394159nteger @ ( produc1086072967326762835nteger @ Q4 @ R ) )
      = ( R = zero_z3403309356797280102nteger ) ) ).

% divides_aux_eq
thf(fact_7912_product__nth,axiom,
    ! [N: nat,Xs: list_o,Ys2: list_o] :
      ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_size_list_o @ Ys2 ) ) )
     => ( ( nth_Product_prod_o_o @ ( product_o_o @ Xs @ Ys2 ) @ N )
        = ( product_Pair_o_o @ ( nth_o @ Xs @ ( 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_7913_product__nth,axiom,
    ! [N: nat,Xs: list_o,Ys2: list_nat] :
      ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_size_list_nat @ Ys2 ) ) )
     => ( ( nth_Pr5826913651314560976_o_nat @ ( product_o_nat @ Xs @ Ys2 ) @ N )
        = ( product_Pair_o_nat @ ( nth_o @ Xs @ ( 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_7914_product__nth,axiom,
    ! [N: nat,Xs: list_o,Ys2: list_int] :
      ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_size_list_int @ Ys2 ) ) )
     => ( ( nth_Pr1649062631805364268_o_int @ ( product_o_int @ Xs @ Ys2 ) @ N )
        = ( product_Pair_o_int @ ( nth_o @ Xs @ ( 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_7915_product__nth,axiom,
    ! [N: nat,Xs: list_nat,Ys2: list_o] :
      ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_o @ Ys2 ) ) )
     => ( ( nth_Pr112076138515278198_nat_o @ ( product_nat_o @ Xs @ Ys2 ) @ N )
        = ( product_Pair_nat_o @ ( nth_nat @ Xs @ ( 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_7916_product__nth,axiom,
    ! [N: nat,Xs: list_nat,Ys2: list_nat] :
      ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys2 ) ) )
     => ( ( nth_Pr7617993195940197384at_nat @ ( product_nat_nat @ Xs @ Ys2 ) @ N )
        = ( product_Pair_nat_nat @ ( nth_nat @ Xs @ ( 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_7917_product__nth,axiom,
    ! [N: nat,Xs: list_nat,Ys2: list_int] :
      ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_int @ Ys2 ) ) )
     => ( ( nth_Pr3440142176431000676at_int @ ( product_nat_int @ Xs @ Ys2 ) @ N )
        = ( product_Pair_nat_int @ ( nth_nat @ Xs @ ( 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_7918_product__nth,axiom,
    ! [N: nat,Xs: list_int,Ys2: list_o] :
      ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_int @ Xs ) @ ( size_size_list_o @ Ys2 ) ) )
     => ( ( nth_Pr7514405829937366042_int_o @ ( product_int_o @ Xs @ Ys2 ) @ N )
        = ( product_Pair_int_o @ ( nth_int @ Xs @ ( 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_7919_product__nth,axiom,
    ! [N: nat,Xs: list_int,Ys2: list_nat] :
      ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_int @ Xs ) @ ( size_size_list_nat @ Ys2 ) ) )
     => ( ( nth_Pr8617346907841251940nt_nat @ ( product_int_nat @ Xs @ Ys2 ) @ N )
        = ( product_Pair_int_nat @ ( nth_int @ Xs @ ( 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_7920_product__nth,axiom,
    ! [N: nat,Xs: list_int,Ys2: list_int] :
      ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_int @ Xs ) @ ( size_size_list_int @ Ys2 ) ) )
     => ( ( nth_Pr4439495888332055232nt_int @ ( product_int_int @ Xs @ Ys2 ) @ N )
        = ( product_Pair_int_int @ ( nth_int @ Xs @ ( 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_7921_product__nth,axiom,
    ! [N: nat,Xs: list_P6011104703257516679at_nat,Ys2: list_P6011104703257516679at_nat] :
      ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s5460976970255530739at_nat @ Xs ) @ ( size_s5460976970255530739at_nat @ Ys2 ) ) )
     => ( ( nth_Pr6744343527793145070at_nat @ ( produc3544356994491977349at_nat @ Xs @ Ys2 ) @ N )
        = ( produc6161850002892822231at_nat @ ( nth_Pr7617993195940197384at_nat @ Xs @ ( divide_divide_nat @ N @ ( size_s5460976970255530739at_nat @ Ys2 ) ) ) @ ( nth_Pr7617993195940197384at_nat @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_s5460976970255530739at_nat @ Ys2 ) ) ) ) ) ) ).

% product_nth
thf(fact_7922_bit__0__eq,axiom,
    ( ( bit_se1146084159140164899it_int @ zero_zero_int )
    = bot_bot_nat_o ) ).

% bit_0_eq
thf(fact_7923_bit__0__eq,axiom,
    ( ( bit_se1148574629649215175it_nat @ zero_zero_nat )
    = bot_bot_nat_o ) ).

% bit_0_eq
thf(fact_7924_bit__0__eq,axiom,
    ( ( bit_se9216721137139052372nteger @ zero_z3403309356797280102nteger )
    = bot_bot_nat_o ) ).

% bit_0_eq
thf(fact_7925_bit__0__eq,axiom,
    ( ( bit_se8040316288895769887atural @ zero_z2226904508553997617atural )
    = bot_bot_nat_o ) ).

% bit_0_eq
thf(fact_7926_set__decode__zero,axiom,
    ( ( nat_set_decode @ zero_zero_nat )
    = bot_bot_set_nat ) ).

% set_decode_zero
thf(fact_7927_length__product,axiom,
    ! [Xs: list_o,Ys2: list_o] :
      ( ( size_s1515746228057227161od_o_o @ ( product_o_o @ Xs @ Ys2 ) )
      = ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_size_list_o @ Ys2 ) ) ) ).

% length_product
thf(fact_7928_length__product,axiom,
    ! [Xs: list_o,Ys2: list_nat] :
      ( ( size_s5443766701097040955_o_nat @ ( product_o_nat @ Xs @ Ys2 ) )
      = ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_size_list_nat @ Ys2 ) ) ) ).

% length_product
thf(fact_7929_length__product,axiom,
    ! [Xs: list_o,Ys2: list_int] :
      ( ( size_s2953683556165314199_o_int @ ( product_o_int @ Xs @ Ys2 ) )
      = ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_size_list_int @ Ys2 ) ) ) ).

% length_product
thf(fact_7930_length__product,axiom,
    ! [Xs: list_nat,Ys2: list_o] :
      ( ( size_s6491369823275344609_nat_o @ ( product_nat_o @ Xs @ Ys2 ) )
      = ( times_times_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_o @ Ys2 ) ) ) ).

% length_product
thf(fact_7931_length__product,axiom,
    ! [Xs: list_nat,Ys2: list_nat] :
      ( ( size_s5460976970255530739at_nat @ ( product_nat_nat @ Xs @ Ys2 ) )
      = ( times_times_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys2 ) ) ) ).

% length_product
thf(fact_7932_length__product,axiom,
    ! [Xs: list_nat,Ys2: list_int] :
      ( ( size_s2970893825323803983at_int @ ( product_nat_int @ Xs @ Ys2 ) )
      = ( times_times_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_int @ Ys2 ) ) ) ).

% length_product
thf(fact_7933_length__product,axiom,
    ! [Xs: list_int,Ys2: list_o] :
      ( ( size_s4246224855604898693_int_o @ ( product_int_o @ Xs @ Ys2 ) )
      = ( times_times_nat @ ( size_size_list_int @ Xs ) @ ( size_size_list_o @ Ys2 ) ) ) ).

% length_product
thf(fact_7934_length__product,axiom,
    ! [Xs: list_int,Ys2: list_nat] :
      ( ( size_s7647898544948552527nt_nat @ ( product_int_nat @ Xs @ Ys2 ) )
      = ( times_times_nat @ ( size_size_list_int @ Xs ) @ ( size_size_list_nat @ Ys2 ) ) ) ).

% length_product
thf(fact_7935_length__product,axiom,
    ! [Xs: list_int,Ys2: list_int] :
      ( ( size_s5157815400016825771nt_int @ ( product_int_int @ Xs @ Ys2 ) )
      = ( times_times_nat @ ( size_size_list_int @ Xs ) @ ( size_size_list_int @ Ys2 ) ) ) ).

% length_product
thf(fact_7936_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_7937_divmod__algorithm__code_I2_J,axiom,
    ! [M: num] :
      ( ( unique3479559517661332726nteger @ M @ one )
      = ( produc1086072967326762835nteger @ ( numera6620942414471956472nteger @ M ) @ zero_z3403309356797280102nteger ) ) ).

% divmod_algorithm_code(2)
thf(fact_7938_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_7939_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_7940_divmod__algorithm__code_I3_J,axiom,
    ! [N: num] :
      ( ( unique3479559517661332726nteger @ one @ ( bit0 @ N ) )
      = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ one ) ) ) ).

% divmod_algorithm_code(3)
thf(fact_7941_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_7942_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_7943_divmod__algorithm__code_I4_J,axiom,
    ! [N: num] :
      ( ( unique3479559517661332726nteger @ one @ ( bit1 @ N ) )
      = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ one ) ) ) ).

% divmod_algorithm_code(4)
thf(fact_7944_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_7945_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_7946_bot__nat__def,axiom,
    bot_bot_nat = zero_zero_nat ).

% bot_nat_def
thf(fact_7947_divmod__divmod__step,axiom,
    ( unique5055182867167087721od_nat
    = ( ^ [M2: num,N3: num] : ( if_Pro6206227464963214023at_nat @ ( ord_less_num @ M2 @ N3 ) @ ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ M2 ) ) @ ( unique5026877609467782581ep_nat @ N3 @ ( unique5055182867167087721od_nat @ M2 @ ( bit0 @ N3 ) ) ) ) ) ) ).

% divmod_divmod_step
thf(fact_7948_divmod__divmod__step,axiom,
    ( unique5052692396658037445od_int
    = ( ^ [M2: num,N3: num] : ( if_Pro3027730157355071871nt_int @ ( ord_less_num @ M2 @ N3 ) @ ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ M2 ) ) @ ( unique5024387138958732305ep_int @ N3 @ ( unique5052692396658037445od_int @ M2 @ ( bit0 @ N3 ) ) ) ) ) ) ).

% divmod_divmod_step
thf(fact_7949_divmod__divmod__step,axiom,
    ( unique3479559517661332726nteger
    = ( ^ [M2: num,N3: num] : ( if_Pro6119634080678213985nteger @ ( ord_less_num @ M2 @ N3 ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ M2 ) ) @ ( unique4921790084139445826nteger @ N3 @ ( unique3479559517661332726nteger @ M2 @ ( bit0 @ N3 ) ) ) ) ) ) ).

% divmod_divmod_step
thf(fact_7950_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_7951_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_7952_neg__eucl__rel__int__mult__2,axiom,
    ! [B: int,A: int,Q4: int,R: int] :
      ( ( ord_less_eq_int @ B @ zero_zero_int )
     => ( ( eucl_rel_int @ ( plus_plus_int @ A @ one_one_int ) @ B @ ( product_Pair_int_int @ Q4 @ R ) )
       => ( 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 ) ) @ B ) @ ( product_Pair_int_int @ Q4 @ ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R ) @ one_one_int ) ) ) ) ) ).

% neg_eucl_rel_int_mult_2
thf(fact_7953_Divides_Oadjust__div__eq,axiom,
    ! [Q4: int,R: int] :
      ( ( adjust_div @ ( product_Pair_int_int @ Q4 @ R ) )
      = ( plus_plus_int @ Q4 @ ( zero_n2684676970156552555ol_int @ ( R != zero_zero_int ) ) ) ) ).

% Divides.adjust_div_eq
thf(fact_7954_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_7955_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_7956_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_7957_eucl__rel__int__dividesI,axiom,
    ! [L: int,K: int,Q4: int] :
      ( ( L != zero_zero_int )
     => ( ( K
          = ( times_times_int @ Q4 @ L ) )
       => ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q4 @ zero_zero_int ) ) ) ) ).

% eucl_rel_int_dividesI
thf(fact_7958_zminus1__lemma,axiom,
    ! [A: int,B: int,Q4: int,R: int] :
      ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q4 @ R ) )
     => ( ( B != zero_zero_int )
       => ( eucl_rel_int @ ( uminus_uminus_int @ A ) @ B @ ( product_Pair_int_int @ ( if_int @ ( R = zero_zero_int ) @ ( uminus_uminus_int @ Q4 ) @ ( minus_minus_int @ ( uminus_uminus_int @ Q4 ) @ one_one_int ) ) @ ( if_int @ ( R = zero_zero_int ) @ zero_zero_int @ ( minus_minus_int @ B @ R ) ) ) ) ) ) ).

% zminus1_lemma
thf(fact_7959_eucl__rel__int__iff,axiom,
    ! [K: int,L: int,Q4: int,R: int] :
      ( ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q4 @ R ) )
      = ( ( K
          = ( plus_plus_int @ ( times_times_int @ L @ Q4 ) @ R ) )
        & ( ( ord_less_int @ zero_zero_int @ L )
         => ( ( ord_less_eq_int @ zero_zero_int @ R )
            & ( ord_less_int @ R @ L ) ) )
        & ( ~ ( ord_less_int @ zero_zero_int @ L )
         => ( ( ( ord_less_int @ L @ zero_zero_int )
             => ( ( ord_less_int @ L @ R )
                & ( ord_less_eq_int @ R @ zero_zero_int ) ) )
            & ( ~ ( ord_less_int @ L @ zero_zero_int )
             => ( Q4 = zero_zero_int ) ) ) ) ) ) ).

% eucl_rel_int_iff
thf(fact_7960_eucl__rel__int__remainderI,axiom,
    ! [R: int,L: int,K: int,Q4: int] :
      ( ( ( sgn_sgn_int @ R )
        = ( sgn_sgn_int @ L ) )
     => ( ( ord_less_int @ ( abs_abs_int @ R ) @ ( abs_abs_int @ L ) )
       => ( ( K
            = ( plus_plus_int @ ( times_times_int @ Q4 @ L ) @ R ) )
         => ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q4 @ R ) ) ) ) ) ).

% eucl_rel_int_remainderI
thf(fact_7961_eucl__rel__int_Ocases,axiom,
    ! [A1: int,A22: int,A32: product_prod_int_int] :
      ( ( eucl_rel_int @ A1 @ A22 @ A32 )
     => ( ( ( A22 = zero_zero_int )
         => ( A32
           != ( product_Pair_int_int @ zero_zero_int @ A1 ) ) )
       => ( ! [Q5: int] :
              ( ( A32
                = ( product_Pair_int_int @ Q5 @ zero_zero_int ) )
             => ( ( A22 != zero_zero_int )
               => ( A1
                 != ( times_times_int @ Q5 @ A22 ) ) ) )
         => ~ ! [R3: int,Q5: int] :
                ( ( A32
                  = ( product_Pair_int_int @ Q5 @ R3 ) )
               => ( ( ( sgn_sgn_int @ R3 )
                    = ( sgn_sgn_int @ A22 ) )
                 => ( ( ord_less_int @ ( abs_abs_int @ R3 ) @ ( abs_abs_int @ A22 ) )
                   => ( A1
                     != ( plus_plus_int @ ( times_times_int @ Q5 @ A22 ) @ R3 ) ) ) ) ) ) ) ) ).

% eucl_rel_int.cases
thf(fact_7962_eucl__rel__int_Osimps,axiom,
    ( eucl_rel_int
    = ( ^ [A12: int,A23: int,A33: product_prod_int_int] :
          ( ? [K3: int] :
              ( ( A12 = K3 )
              & ( A23 = zero_zero_int )
              & ( A33
                = ( product_Pair_int_int @ zero_zero_int @ K3 ) ) )
          | ? [L3: int,K3: int,Q3: int] :
              ( ( A12 = K3 )
              & ( A23 = L3 )
              & ( A33
                = ( product_Pair_int_int @ Q3 @ zero_zero_int ) )
              & ( L3 != zero_zero_int )
              & ( K3
                = ( times_times_int @ Q3 @ L3 ) ) )
          | ? [R5: int,L3: int,K3: int,Q3: int] :
              ( ( A12 = K3 )
              & ( A23 = L3 )
              & ( A33
                = ( product_Pair_int_int @ Q3 @ R5 ) )
              & ( ( sgn_sgn_int @ R5 )
                = ( sgn_sgn_int @ L3 ) )
              & ( ord_less_int @ ( abs_abs_int @ R5 ) @ ( abs_abs_int @ L3 ) )
              & ( K3
                = ( plus_plus_int @ ( times_times_int @ Q3 @ L3 ) @ R5 ) ) ) ) ) ) ).

% eucl_rel_int.simps
thf(fact_7963_pos__eucl__rel__int__mult__2,axiom,
    ! [B: int,A: int,Q4: int,R: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ B )
     => ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q4 @ R ) )
       => ( 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 ) ) @ B ) @ ( product_Pair_int_int @ Q4 @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R ) ) ) ) ) ) ).

% pos_eucl_rel_int_mult_2
thf(fact_7964_and__int_Opelims,axiom,
    ! [X: int,Xa3: int,Y3: int] :
      ( ( ( bit_se725231765392027082nd_int @ X @ Xa3 )
        = Y3 )
     => ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ X @ Xa3 ) )
       => ~ ( ( ( ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
                  & ( member_int @ Xa3 @ ( 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 ) ) @ X )
                        & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa3 ) ) ) ) ) )
              & ( ~ ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
                    & ( member_int @ Xa3 @ ( 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 ) ) @ X )
                        & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa3 ) ) )
                    @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ X @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ Xa3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) )
           => ~ ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ X @ Xa3 ) ) ) ) ) ).

% and_int.pelims
thf(fact_7965_and__int_Opsimps,axiom,
    ! [K: int,L: int] :
      ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K @ L ) )
     => ( ( ( ( member_int @ K @ ( 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 ) ) ) )
         => ( ( bit_se725231765392027082nd_int @ K @ L )
            = ( uminus_uminus_int
              @ ( zero_n2684676970156552555ol_int
                @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
                  & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) ) ) ) )
        & ( ~ ( ( member_int @ K @ ( 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 ) ) ) )
         => ( ( bit_se725231765392027082nd_int @ K @ L )
            = ( plus_plus_int
              @ ( zero_n2684676970156552555ol_int
                @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
                  & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) )
              @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% and_int.psimps
thf(fact_7966_length__Cons,axiom,
    ! [X: $o,Xs: list_o] :
      ( ( size_size_list_o @ ( cons_o @ X @ Xs ) )
      = ( suc @ ( size_size_list_o @ Xs ) ) ) ).

% length_Cons
thf(fact_7967_length__Cons,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( size_size_list_nat @ ( cons_nat @ X @ Xs ) )
      = ( suc @ ( size_size_list_nat @ Xs ) ) ) ).

% length_Cons
thf(fact_7968_length__Cons,axiom,
    ! [X: int,Xs: list_int] :
      ( ( size_size_list_int @ ( cons_int @ X @ Xs ) )
      = ( suc @ ( size_size_list_int @ Xs ) ) ) ).

% length_Cons
thf(fact_7969_normalize__negative,axiom,
    ! [Q4: int,P5: int] :
      ( ( ord_less_int @ Q4 @ zero_zero_int )
     => ( ( normalize @ ( product_Pair_int_int @ P5 @ Q4 ) )
        = ( normalize @ ( product_Pair_int_int @ ( uminus_uminus_int @ P5 ) @ ( uminus_uminus_int @ Q4 ) ) ) ) ) ).

% normalize_negative
thf(fact_7970_upto__aux__rec,axiom,
    ( upto_aux
    = ( ^ [I4: int,J3: int,Js: list_int] : ( if_list_int @ ( ord_less_int @ J3 @ I4 ) @ Js @ ( upto_aux @ I4 @ ( minus_minus_int @ J3 @ one_one_int ) @ ( cons_int @ J3 @ Js ) ) ) ) ) ).

% upto_aux_rec
thf(fact_7971_normalize__denom__zero,axiom,
    ! [P5: int] :
      ( ( normalize @ ( product_Pair_int_int @ P5 @ zero_zero_int ) )
      = ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ).

% normalize_denom_zero
thf(fact_7972_normalize__denom__pos,axiom,
    ! [R: product_prod_int_int,P5: int,Q4: int] :
      ( ( ( normalize @ R )
        = ( product_Pair_int_int @ P5 @ Q4 ) )
     => ( ord_less_int @ zero_zero_int @ Q4 ) ) ).

% normalize_denom_pos
thf(fact_7973_normalize__crossproduct,axiom,
    ! [Q4: int,S: int,P5: int,R: int] :
      ( ( Q4 != zero_zero_int )
     => ( ( S != zero_zero_int )
       => ( ( ( normalize @ ( product_Pair_int_int @ P5 @ Q4 ) )
            = ( normalize @ ( product_Pair_int_int @ R @ S ) ) )
         => ( ( times_times_int @ P5 @ S )
            = ( times_times_int @ R @ Q4 ) ) ) ) ) ).

% normalize_crossproduct
thf(fact_7974_and__int_Opinduct,axiom,
    ! [A0: int,A1: int,P: int > int > $o] :
      ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ A0 @ A1 ) )
     => ( ! [K2: int,L2: int] :
            ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K2 @ L2 ) )
           => ( ( ~ ( ( member_int @ K2 @ ( 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 ) ) ) )
               => ( P @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
             => ( P @ K2 @ L2 ) ) )
       => ( P @ A0 @ A1 ) ) ) ).

% and_int.pinduct
thf(fact_7975_upto_Opinduct,axiom,
    ! [A0: int,A1: int,P: int > int > $o] :
      ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ A0 @ A1 ) )
     => ( ! [I3: int,J2: int] :
            ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I3 @ J2 ) )
           => ( ( ( ord_less_eq_int @ I3 @ J2 )
               => ( P @ ( plus_plus_int @ I3 @ one_one_int ) @ J2 ) )
             => ( P @ I3 @ J2 ) ) )
       => ( P @ A0 @ A1 ) ) ) ).

% upto.pinduct
thf(fact_7976_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_7977_bezw__0,axiom,
    ! [X: nat] :
      ( ( bezw @ X @ zero_zero_nat )
      = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) ).

% bezw_0
thf(fact_7978_nth__zip,axiom,
    ! [I2: nat,Xs: list_o,Ys2: list_o] :
      ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs ) )
     => ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Ys2 ) )
       => ( ( nth_Product_prod_o_o @ ( zip_o_o @ Xs @ Ys2 ) @ I2 )
          = ( product_Pair_o_o @ ( nth_o @ Xs @ I2 ) @ ( nth_o @ Ys2 @ I2 ) ) ) ) ) ).

% nth_zip
thf(fact_7979_nth__zip,axiom,
    ! [I2: nat,Xs: list_o,Ys2: list_nat] :
      ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs ) )
     => ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys2 ) )
       => ( ( nth_Pr5826913651314560976_o_nat @ ( zip_o_nat @ Xs @ Ys2 ) @ I2 )
          = ( product_Pair_o_nat @ ( nth_o @ Xs @ I2 ) @ ( nth_nat @ Ys2 @ I2 ) ) ) ) ) ).

% nth_zip
thf(fact_7980_nth__zip,axiom,
    ! [I2: nat,Xs: list_o,Ys2: list_int] :
      ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs ) )
     => ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Ys2 ) )
       => ( ( nth_Pr1649062631805364268_o_int @ ( zip_o_int @ Xs @ Ys2 ) @ I2 )
          = ( product_Pair_o_int @ ( nth_o @ Xs @ I2 ) @ ( nth_int @ Ys2 @ I2 ) ) ) ) ) ).

% nth_zip
thf(fact_7981_nth__zip,axiom,
    ! [I2: nat,Xs: list_nat,Ys2: list_o] :
      ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
     => ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Ys2 ) )
       => ( ( nth_Pr112076138515278198_nat_o @ ( zip_nat_o @ Xs @ Ys2 ) @ I2 )
          = ( product_Pair_nat_o @ ( nth_nat @ Xs @ I2 ) @ ( nth_o @ Ys2 @ I2 ) ) ) ) ) ).

% nth_zip
thf(fact_7982_nth__zip,axiom,
    ! [I2: nat,Xs: list_nat,Ys2: list_nat] :
      ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
     => ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys2 ) )
       => ( ( nth_Pr7617993195940197384at_nat @ ( zip_nat_nat @ Xs @ Ys2 ) @ I2 )
          = ( product_Pair_nat_nat @ ( nth_nat @ Xs @ I2 ) @ ( nth_nat @ Ys2 @ I2 ) ) ) ) ) ).

% nth_zip
thf(fact_7983_nth__zip,axiom,
    ! [I2: nat,Xs: list_nat,Ys2: list_int] :
      ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
     => ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Ys2 ) )
       => ( ( nth_Pr3440142176431000676at_int @ ( zip_nat_int @ Xs @ Ys2 ) @ I2 )
          = ( product_Pair_nat_int @ ( nth_nat @ Xs @ I2 ) @ ( nth_int @ Ys2 @ I2 ) ) ) ) ) ).

% nth_zip
thf(fact_7984_nth__zip,axiom,
    ! [I2: nat,Xs: list_int,Ys2: list_o] :
      ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs ) )
     => ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Ys2 ) )
       => ( ( nth_Pr7514405829937366042_int_o @ ( zip_int_o @ Xs @ Ys2 ) @ I2 )
          = ( product_Pair_int_o @ ( nth_int @ Xs @ I2 ) @ ( nth_o @ Ys2 @ I2 ) ) ) ) ) ).

% nth_zip
thf(fact_7985_nth__zip,axiom,
    ! [I2: nat,Xs: list_int,Ys2: list_nat] :
      ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs ) )
     => ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys2 ) )
       => ( ( nth_Pr8617346907841251940nt_nat @ ( zip_int_nat @ Xs @ Ys2 ) @ I2 )
          = ( product_Pair_int_nat @ ( nth_int @ Xs @ I2 ) @ ( nth_nat @ Ys2 @ I2 ) ) ) ) ) ).

% nth_zip
thf(fact_7986_nth__zip,axiom,
    ! [I2: nat,Xs: list_int,Ys2: list_int] :
      ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs ) )
     => ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Ys2 ) )
       => ( ( nth_Pr4439495888332055232nt_int @ ( zip_int_int @ Xs @ Ys2 ) @ I2 )
          = ( product_Pair_int_int @ ( nth_int @ Xs @ I2 ) @ ( nth_int @ Ys2 @ I2 ) ) ) ) ) ).

% nth_zip
thf(fact_7987_nth__zip,axiom,
    ! [I2: nat,Xs: list_P6011104703257516679at_nat,Ys2: list_P6011104703257516679at_nat] :
      ( ( ord_less_nat @ I2 @ ( size_s5460976970255530739at_nat @ Xs ) )
     => ( ( ord_less_nat @ I2 @ ( size_s5460976970255530739at_nat @ Ys2 ) )
       => ( ( nth_Pr6744343527793145070at_nat @ ( zip_Pr4664179122662387191at_nat @ Xs @ Ys2 ) @ I2 )
          = ( produc6161850002892822231at_nat @ ( nth_Pr7617993195940197384at_nat @ Xs @ I2 ) @ ( nth_Pr7617993195940197384at_nat @ Ys2 @ I2 ) ) ) ) ) ).

% nth_zip
thf(fact_7988_Gcd__0__iff,axiom,
    ! [A3: set_int] :
      ( ( ( gcd_Gcd_int @ A3 )
        = zero_zero_int )
      = ( ord_less_eq_set_int @ A3 @ ( insert_int @ zero_zero_int @ bot_bot_set_int ) ) ) ).

% Gcd_0_iff
thf(fact_7989_Gcd__0__iff,axiom,
    ! [A3: set_nat] :
      ( ( ( gcd_Gcd_nat @ A3 )
        = zero_zero_nat )
      = ( ord_less_eq_set_nat @ A3 @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).

% Gcd_0_iff
thf(fact_7990_Gcd__empty,axiom,
    ( ( gcd_Gcd_int @ bot_bot_set_int )
    = zero_zero_int ) ).

% Gcd_empty
thf(fact_7991_Gcd__empty,axiom,
    ( ( gcd_Gcd_nat @ bot_bot_set_nat )
    = zero_zero_nat ) ).

% Gcd_empty
thf(fact_7992_divmod__integer_H__def,axiom,
    ( unique3479559517661332726nteger
    = ( ^ [M2: num,N3: num] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( numera6620942414471956472nteger @ M2 ) @ ( numera6620942414471956472nteger @ N3 ) ) @ ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M2 ) @ ( numera6620942414471956472nteger @ N3 ) ) ) ) ) ).

% divmod_integer'_def
thf(fact_7993_Gcd__nat__eq__one,axiom,
    ! [N5: set_nat] :
      ( ( member_nat @ one_one_nat @ N5 )
     => ( ( gcd_Gcd_nat @ N5 )
        = one_one_nat ) ) ).

% Gcd_nat_eq_one
thf(fact_7994_Gcd__1,axiom,
    ! [A3: set_nat] :
      ( ( member_nat @ one_one_nat @ A3 )
     => ( ( gcd_Gcd_nat @ A3 )
        = one_one_nat ) ) ).

% Gcd_1
thf(fact_7995_Gcd__1,axiom,
    ! [A3: set_int] :
      ( ( member_int @ one_one_int @ A3 )
     => ( ( gcd_Gcd_int @ A3 )
        = one_one_int ) ) ).

% Gcd_1
thf(fact_7996_Gcd__int__greater__eq__0,axiom,
    ! [K5: set_int] : ( ord_less_eq_int @ zero_zero_int @ ( gcd_Gcd_int @ K5 ) ) ).

% Gcd_int_greater_eq_0
thf(fact_7997_Gcd__eq__1__I,axiom,
    ! [A: nat,A3: set_nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( member_nat @ A @ A3 )
       => ( ( gcd_Gcd_nat @ A3 )
          = one_one_nat ) ) ) ).

% Gcd_eq_1_I
thf(fact_7998_Gcd__eq__1__I,axiom,
    ! [A: int,A3: set_int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( member_int @ A @ A3 )
       => ( ( gcd_Gcd_int @ A3 )
          = one_one_int ) ) ) ).

% Gcd_eq_1_I
thf(fact_7999_Code__Numeral_Opositive__def,axiom,
    code_positive = numera6620942414471956472nteger ).

% Code_Numeral.positive_def
thf(fact_8000_integer__of__num__triv_I2_J,axiom,
    ( ( code_integer_of_num @ ( bit0 @ one ) )
    = ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ).

% integer_of_num_triv(2)
thf(fact_8001_card__atLeastAtMost__int,axiom,
    ! [L: int,U: int] :
      ( ( finite_card_int @ ( set_or1266510415728281911st_int @ L @ U ) )
      = ( nat2 @ ( plus_plus_int @ ( minus_minus_int @ U @ L ) @ one_one_int ) ) ) ).

% card_atLeastAtMost_int
thf(fact_8002_possible__bit__def,axiom,
    ( bit_se3746373301970924850it_int
    = ( ^ [Tyrep: itself_int,N3: nat] :
          ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 )
         != zero_zero_int ) ) ) ).

% possible_bit_def
thf(fact_8003_possible__bit__def,axiom,
    ( bit_se3748863772479975126it_nat
    = ( ^ [Tyrep: itself_nat,N3: nat] :
          ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
         != zero_zero_nat ) ) ) ).

% possible_bit_def
thf(fact_8004_possible__bit__def,axiom,
    ( bit_se6891987061192897379nteger
    = ( ^ [Tyrep: itself_Code_integer,N3: nat] :
          ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N3 )
         != zero_z3403309356797280102nteger ) ) ) ).

% possible_bit_def
thf(fact_8005_possible__bit__def,axiom,
    ( bit_se5715582212949614894atural
    = ( ^ [Tyrep: itself_Code_natural,N3: nat] :
          ( ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N3 )
         != zero_z2226904508553997617atural ) ) ) ).

% possible_bit_def
thf(fact_8006_integer__of__num__triv_I1_J,axiom,
    ( ( code_integer_of_num @ one )
    = one_one_Code_integer ) ).

% integer_of_num_triv(1)
thf(fact_8007_integer__of__num__def,axiom,
    code_integer_of_num = numera6620942414471956472nteger ).

% integer_of_num_def
thf(fact_8008_card__Diff__insert,axiom,
    ! [A: produc3843707927480180839at_nat,A3: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ A @ A3 )
     => ( ~ ( member8757157785044589968at_nat @ A @ B5 )
       => ( ( finite3771342082235030671at_nat @ ( minus_3314409938677909166at_nat @ A3 @ ( insert9069300056098147895at_nat @ A @ B5 ) ) )
          = ( minus_minus_nat @ ( finite3771342082235030671at_nat @ ( minus_3314409938677909166at_nat @ A3 @ B5 ) ) @ one_one_nat ) ) ) ) ).

% card_Diff_insert
thf(fact_8009_card__Diff__insert,axiom,
    ! [A: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ A @ A3 )
     => ( ~ ( member8440522571783428010at_nat @ A @ B5 )
       => ( ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ A3 @ ( insert8211810215607154385at_nat @ A @ B5 ) ) )
          = ( minus_minus_nat @ ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ A3 @ B5 ) ) @ one_one_nat ) ) ) ) ).

% card_Diff_insert
thf(fact_8010_card__Diff__insert,axiom,
    ! [A: set_Pr1261947904930325089at_nat,A3: set_se7855581050983116737at_nat,B5: set_se7855581050983116737at_nat] :
      ( ( member2643936169264416010at_nat @ A @ A3 )
     => ( ~ ( member2643936169264416010at_nat @ A @ B5 )
       => ( ( finite4356350796350151305at_nat @ ( minus_4207664762107033000at_nat @ A3 @ ( insert9200635055090092081at_nat @ A @ B5 ) ) )
          = ( minus_minus_nat @ ( finite4356350796350151305at_nat @ ( minus_4207664762107033000at_nat @ A3 @ B5 ) ) @ one_one_nat ) ) ) ) ).

% card_Diff_insert
thf(fact_8011_card__Diff__insert,axiom,
    ! [A: int,A3: set_int,B5: set_int] :
      ( ( member_int @ A @ A3 )
     => ( ~ ( member_int @ A @ B5 )
       => ( ( finite_card_int @ ( minus_minus_set_int @ A3 @ ( insert_int @ A @ B5 ) ) )
          = ( minus_minus_nat @ ( finite_card_int @ ( minus_minus_set_int @ A3 @ B5 ) ) @ one_one_nat ) ) ) ) ).

% card_Diff_insert
thf(fact_8012_card__Diff__insert,axiom,
    ! [A: list_nat,A3: set_list_nat,B5: set_list_nat] :
      ( ( member_list_nat @ A @ A3 )
     => ( ~ ( member_list_nat @ A @ B5 )
       => ( ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A3 @ ( insert_list_nat @ A @ B5 ) ) )
          = ( minus_minus_nat @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A3 @ B5 ) ) @ one_one_nat ) ) ) ) ).

% card_Diff_insert
thf(fact_8013_card__Diff__insert,axiom,
    ! [A: product_unit,A3: set_Product_unit,B5: set_Product_unit] :
      ( ( member_Product_unit @ A @ A3 )
     => ( ~ ( member_Product_unit @ A @ B5 )
       => ( ( finite410649719033368117t_unit @ ( minus_6452836326544984404t_unit @ A3 @ ( insert_Product_unit @ A @ B5 ) ) )
          = ( minus_minus_nat @ ( finite410649719033368117t_unit @ ( minus_6452836326544984404t_unit @ A3 @ B5 ) ) @ one_one_nat ) ) ) ) ).

% card_Diff_insert
thf(fact_8014_card__Diff__insert,axiom,
    ! [A: set_nat,A3: set_set_nat,B5: set_set_nat] :
      ( ( member_set_nat @ A @ A3 )
     => ( ~ ( member_set_nat @ A @ B5 )
       => ( ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A3 @ ( insert_set_nat @ A @ B5 ) ) )
          = ( minus_minus_nat @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A3 @ B5 ) ) @ one_one_nat ) ) ) ) ).

% card_Diff_insert
thf(fact_8015_card__Diff__insert,axiom,
    ! [A: nat,A3: set_nat,B5: set_nat] :
      ( ( member_nat @ A @ A3 )
     => ( ~ ( member_nat @ A @ B5 )
       => ( ( finite_card_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ A @ B5 ) ) )
          = ( minus_minus_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A3 @ B5 ) ) @ one_one_nat ) ) ) ) ).

% card_Diff_insert
thf(fact_8016_card_Oempty,axiom,
    ( ( finite_card_list_nat @ bot_bot_set_list_nat )
    = zero_zero_nat ) ).

% card.empty
thf(fact_8017_card_Oempty,axiom,
    ( ( finite410649719033368117t_unit @ bot_bo3957492148770167129t_unit )
    = zero_zero_nat ) ).

% card.empty
thf(fact_8018_card_Oempty,axiom,
    ( ( finite_card_set_nat @ bot_bot_set_set_nat )
    = zero_zero_nat ) ).

% card.empty
thf(fact_8019_card_Oempty,axiom,
    ( ( finite_card_int @ bot_bot_set_int )
    = zero_zero_nat ) ).

% card.empty
thf(fact_8020_card_Oempty,axiom,
    ( ( finite_card_nat @ bot_bot_set_nat )
    = zero_zero_nat ) ).

% card.empty
thf(fact_8021_card_Oempty,axiom,
    ( ( finite711546835091564841at_nat @ bot_bo2099793752762293965at_nat )
    = zero_zero_nat ) ).

% card.empty
thf(fact_8022_card_Oempty,axiom,
    ( ( finite3771342082235030671at_nat @ bot_bo228742789529271731at_nat )
    = zero_zero_nat ) ).

% card.empty
thf(fact_8023_card__insert__le__m1,axiom,
    ! [N: nat,Y3: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_nat @ ( finite711546835091564841at_nat @ Y3 ) @ ( minus_minus_nat @ N @ one_one_nat ) )
       => ( ord_less_eq_nat @ ( finite711546835091564841at_nat @ ( insert8211810215607154385at_nat @ X @ Y3 ) ) @ N ) ) ) ).

% card_insert_le_m1
thf(fact_8024_card__insert__le__m1,axiom,
    ! [N: nat,Y3: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_nat @ ( finite3771342082235030671at_nat @ Y3 ) @ ( minus_minus_nat @ N @ one_one_nat ) )
       => ( ord_less_eq_nat @ ( finite3771342082235030671at_nat @ ( insert9069300056098147895at_nat @ X @ Y3 ) ) @ N ) ) ) ).

% card_insert_le_m1
thf(fact_8025_card__insert__le__m1,axiom,
    ! [N: nat,Y3: set_int,X: int] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_nat @ ( finite_card_int @ Y3 ) @ ( minus_minus_nat @ N @ one_one_nat ) )
       => ( ord_less_eq_nat @ ( finite_card_int @ ( insert_int @ X @ Y3 ) ) @ N ) ) ) ).

% card_insert_le_m1
thf(fact_8026_card__insert__le__m1,axiom,
    ! [N: nat,Y3: set_nat,X: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_nat @ ( finite_card_nat @ Y3 ) @ ( minus_minus_nat @ N @ one_one_nat ) )
       => ( ord_less_eq_nat @ ( finite_card_nat @ ( insert_nat @ X @ Y3 ) ) @ N ) ) ) ).

% card_insert_le_m1
thf(fact_8027_card__insert__le__m1,axiom,
    ! [N: nat,Y3: set_list_nat,X: list_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_nat @ ( finite_card_list_nat @ Y3 ) @ ( minus_minus_nat @ N @ one_one_nat ) )
       => ( ord_less_eq_nat @ ( finite_card_list_nat @ ( insert_list_nat @ X @ Y3 ) ) @ N ) ) ) ).

% card_insert_le_m1
thf(fact_8028_card__insert__le__m1,axiom,
    ! [N: nat,Y3: set_Product_unit,X: product_unit] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ Y3 ) @ ( minus_minus_nat @ N @ one_one_nat ) )
       => ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ ( insert_Product_unit @ X @ Y3 ) ) @ N ) ) ) ).

% card_insert_le_m1
thf(fact_8029_card__insert__le__m1,axiom,
    ! [N: nat,Y3: set_set_nat,X: set_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_nat @ ( finite_card_set_nat @ Y3 ) @ ( minus_minus_nat @ N @ one_one_nat ) )
       => ( ord_less_eq_nat @ ( finite_card_set_nat @ ( insert_set_nat @ X @ Y3 ) ) @ N ) ) ) ).

% card_insert_le_m1
thf(fact_8030_card__Diff__singleton__if,axiom,
    ! [X: set_Pr1261947904930325089at_nat,A3: set_se7855581050983116737at_nat] :
      ( ( ( member2643936169264416010at_nat @ X @ A3 )
       => ( ( finite4356350796350151305at_nat @ ( minus_4207664762107033000at_nat @ A3 @ ( insert9200635055090092081at_nat @ X @ bot_bo3083307316010499117at_nat ) ) )
          = ( minus_minus_nat @ ( finite4356350796350151305at_nat @ A3 ) @ one_one_nat ) ) )
      & ( ~ ( member2643936169264416010at_nat @ X @ A3 )
       => ( ( finite4356350796350151305at_nat @ ( minus_4207664762107033000at_nat @ A3 @ ( insert9200635055090092081at_nat @ X @ bot_bo3083307316010499117at_nat ) ) )
          = ( finite4356350796350151305at_nat @ A3 ) ) ) ) ).

% card_Diff_singleton_if
thf(fact_8031_card__Diff__singleton__if,axiom,
    ! [X: list_nat,A3: set_list_nat] :
      ( ( ( member_list_nat @ X @ A3 )
       => ( ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A3 @ ( insert_list_nat @ X @ bot_bot_set_list_nat ) ) )
          = ( minus_minus_nat @ ( finite_card_list_nat @ A3 ) @ one_one_nat ) ) )
      & ( ~ ( member_list_nat @ X @ A3 )
       => ( ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A3 @ ( insert_list_nat @ X @ bot_bot_set_list_nat ) ) )
          = ( finite_card_list_nat @ A3 ) ) ) ) ).

% card_Diff_singleton_if
thf(fact_8032_card__Diff__singleton__if,axiom,
    ! [X: product_unit,A3: set_Product_unit] :
      ( ( ( member_Product_unit @ X @ A3 )
       => ( ( finite410649719033368117t_unit @ ( minus_6452836326544984404t_unit @ A3 @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) )
          = ( minus_minus_nat @ ( finite410649719033368117t_unit @ A3 ) @ one_one_nat ) ) )
      & ( ~ ( member_Product_unit @ X @ A3 )
       => ( ( finite410649719033368117t_unit @ ( minus_6452836326544984404t_unit @ A3 @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) )
          = ( finite410649719033368117t_unit @ A3 ) ) ) ) ).

% card_Diff_singleton_if
thf(fact_8033_card__Diff__singleton__if,axiom,
    ! [X: set_nat,A3: set_set_nat] :
      ( ( ( member_set_nat @ X @ A3 )
       => ( ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A3 @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) )
          = ( minus_minus_nat @ ( finite_card_set_nat @ A3 ) @ one_one_nat ) ) )
      & ( ~ ( member_set_nat @ X @ A3 )
       => ( ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A3 @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) )
          = ( finite_card_set_nat @ A3 ) ) ) ) ).

% card_Diff_singleton_if
thf(fact_8034_card__Diff__singleton__if,axiom,
    ! [X: int,A3: set_int] :
      ( ( ( member_int @ X @ A3 )
       => ( ( finite_card_int @ ( minus_minus_set_int @ A3 @ ( insert_int @ X @ bot_bot_set_int ) ) )
          = ( minus_minus_nat @ ( finite_card_int @ A3 ) @ one_one_nat ) ) )
      & ( ~ ( member_int @ X @ A3 )
       => ( ( finite_card_int @ ( minus_minus_set_int @ A3 @ ( insert_int @ X @ bot_bot_set_int ) ) )
          = ( finite_card_int @ A3 ) ) ) ) ).

% card_Diff_singleton_if
thf(fact_8035_card__Diff__singleton__if,axiom,
    ! [X: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( ( member8440522571783428010at_nat @ X @ A3 )
       => ( ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ A3 @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) )
          = ( minus_minus_nat @ ( finite711546835091564841at_nat @ A3 ) @ one_one_nat ) ) )
      & ( ~ ( member8440522571783428010at_nat @ X @ A3 )
       => ( ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ A3 @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) )
          = ( finite711546835091564841at_nat @ A3 ) ) ) ) ).

% card_Diff_singleton_if
thf(fact_8036_card__Diff__singleton__if,axiom,
    ! [X: produc3843707927480180839at_nat,A3: set_Pr4329608150637261639at_nat] :
      ( ( ( member8757157785044589968at_nat @ X @ A3 )
       => ( ( finite3771342082235030671at_nat @ ( minus_3314409938677909166at_nat @ A3 @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) )
          = ( minus_minus_nat @ ( finite3771342082235030671at_nat @ A3 ) @ one_one_nat ) ) )
      & ( ~ ( member8757157785044589968at_nat @ X @ A3 )
       => ( ( finite3771342082235030671at_nat @ ( minus_3314409938677909166at_nat @ A3 @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) )
          = ( finite3771342082235030671at_nat @ A3 ) ) ) ) ).

% card_Diff_singleton_if
thf(fact_8037_card__Diff__singleton__if,axiom,
    ! [X: nat,A3: set_nat] :
      ( ( ( member_nat @ X @ A3 )
       => ( ( finite_card_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ X @ bot_bot_set_nat ) ) )
          = ( minus_minus_nat @ ( finite_card_nat @ A3 ) @ one_one_nat ) ) )
      & ( ~ ( member_nat @ X @ A3 )
       => ( ( finite_card_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ X @ bot_bot_set_nat ) ) )
          = ( finite_card_nat @ A3 ) ) ) ) ).

% card_Diff_singleton_if
thf(fact_8038_card__Diff__singleton,axiom,
    ! [X: set_Pr1261947904930325089at_nat,A3: set_se7855581050983116737at_nat] :
      ( ( member2643936169264416010at_nat @ X @ A3 )
     => ( ( finite4356350796350151305at_nat @ ( minus_4207664762107033000at_nat @ A3 @ ( insert9200635055090092081at_nat @ X @ bot_bo3083307316010499117at_nat ) ) )
        = ( minus_minus_nat @ ( finite4356350796350151305at_nat @ A3 ) @ one_one_nat ) ) ) ).

% card_Diff_singleton
thf(fact_8039_card__Diff__singleton,axiom,
    ! [X: list_nat,A3: set_list_nat] :
      ( ( member_list_nat @ X @ A3 )
     => ( ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A3 @ ( insert_list_nat @ X @ bot_bot_set_list_nat ) ) )
        = ( minus_minus_nat @ ( finite_card_list_nat @ A3 ) @ one_one_nat ) ) ) ).

% card_Diff_singleton
thf(fact_8040_card__Diff__singleton,axiom,
    ! [X: product_unit,A3: set_Product_unit] :
      ( ( member_Product_unit @ X @ A3 )
     => ( ( finite410649719033368117t_unit @ ( minus_6452836326544984404t_unit @ A3 @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) )
        = ( minus_minus_nat @ ( finite410649719033368117t_unit @ A3 ) @ one_one_nat ) ) ) ).

% card_Diff_singleton
thf(fact_8041_card__Diff__singleton,axiom,
    ! [X: set_nat,A3: set_set_nat] :
      ( ( member_set_nat @ X @ A3 )
     => ( ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A3 @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) )
        = ( minus_minus_nat @ ( finite_card_set_nat @ A3 ) @ one_one_nat ) ) ) ).

% card_Diff_singleton
thf(fact_8042_card__Diff__singleton,axiom,
    ! [X: int,A3: set_int] :
      ( ( member_int @ X @ A3 )
     => ( ( finite_card_int @ ( minus_minus_set_int @ A3 @ ( insert_int @ X @ bot_bot_set_int ) ) )
        = ( minus_minus_nat @ ( finite_card_int @ A3 ) @ one_one_nat ) ) ) ).

% card_Diff_singleton
thf(fact_8043_card__Diff__singleton,axiom,
    ! [X: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ X @ A3 )
     => ( ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ A3 @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) )
        = ( minus_minus_nat @ ( finite711546835091564841at_nat @ A3 ) @ one_one_nat ) ) ) ).

% card_Diff_singleton
thf(fact_8044_card__Diff__singleton,axiom,
    ! [X: produc3843707927480180839at_nat,A3: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ X @ A3 )
     => ( ( finite3771342082235030671at_nat @ ( minus_3314409938677909166at_nat @ A3 @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) )
        = ( minus_minus_nat @ ( finite3771342082235030671at_nat @ A3 ) @ one_one_nat ) ) ) ).

% card_Diff_singleton
thf(fact_8045_card__Diff__singleton,axiom,
    ! [X: nat,A3: set_nat] :
      ( ( member_nat @ X @ A3 )
     => ( ( finite_card_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ X @ bot_bot_set_nat ) ) )
        = ( minus_minus_nat @ ( finite_card_nat @ A3 ) @ one_one_nat ) ) ) ).

% card_Diff_singleton
thf(fact_8046_card__atLeastAtMost,axiom,
    ! [L: nat,U: nat] :
      ( ( finite_card_nat @ ( set_or1269000886237332187st_nat @ L @ U ) )
      = ( minus_minus_nat @ ( suc @ U ) @ L ) ) ).

% card_atLeastAtMost
thf(fact_8047_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_8048_card__insert__le,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] : ( ord_less_eq_nat @ ( finite711546835091564841at_nat @ A3 ) @ ( finite711546835091564841at_nat @ ( insert8211810215607154385at_nat @ X @ A3 ) ) ) ).

% card_insert_le
thf(fact_8049_card__insert__le,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat] : ( ord_less_eq_nat @ ( finite3771342082235030671at_nat @ A3 ) @ ( finite3771342082235030671at_nat @ ( insert9069300056098147895at_nat @ X @ A3 ) ) ) ).

% card_insert_le
thf(fact_8050_card__insert__le,axiom,
    ! [A3: set_int,X: int] : ( ord_less_eq_nat @ ( finite_card_int @ A3 ) @ ( finite_card_int @ ( insert_int @ X @ A3 ) ) ) ).

% card_insert_le
thf(fact_8051_card__insert__le,axiom,
    ! [A3: set_nat,X: nat] : ( ord_less_eq_nat @ ( finite_card_nat @ A3 ) @ ( finite_card_nat @ ( insert_nat @ X @ A3 ) ) ) ).

% card_insert_le
thf(fact_8052_card__insert__le,axiom,
    ! [A3: set_list_nat,X: list_nat] : ( ord_less_eq_nat @ ( finite_card_list_nat @ A3 ) @ ( finite_card_list_nat @ ( insert_list_nat @ X @ A3 ) ) ) ).

% card_insert_le
thf(fact_8053_card__insert__le,axiom,
    ! [A3: set_Product_unit,X: product_unit] : ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ A3 ) @ ( finite410649719033368117t_unit @ ( insert_Product_unit @ X @ A3 ) ) ) ).

% card_insert_le
thf(fact_8054_card__insert__le,axiom,
    ! [A3: set_set_nat,X: set_nat] : ( ord_less_eq_nat @ ( finite_card_set_nat @ A3 ) @ ( finite_card_set_nat @ ( insert_set_nat @ X @ A3 ) ) ) ).

% card_insert_le
thf(fact_8055_card__1__singletonE,axiom,
    ! [A3: set_list_nat] :
      ( ( ( finite_card_list_nat @ A3 )
        = one_one_nat )
     => ~ ! [X3: list_nat] :
            ( A3
           != ( insert_list_nat @ X3 @ bot_bot_set_list_nat ) ) ) ).

% card_1_singletonE
thf(fact_8056_card__1__singletonE,axiom,
    ! [A3: set_Product_unit] :
      ( ( ( finite410649719033368117t_unit @ A3 )
        = one_one_nat )
     => ~ ! [X3: product_unit] :
            ( A3
           != ( insert_Product_unit @ X3 @ bot_bo3957492148770167129t_unit ) ) ) ).

% card_1_singletonE
thf(fact_8057_card__1__singletonE,axiom,
    ! [A3: set_set_nat] :
      ( ( ( finite_card_set_nat @ A3 )
        = one_one_nat )
     => ~ ! [X3: set_nat] :
            ( A3
           != ( insert_set_nat @ X3 @ bot_bot_set_set_nat ) ) ) ).

% card_1_singletonE
thf(fact_8058_card__1__singletonE,axiom,
    ! [A3: set_int] :
      ( ( ( finite_card_int @ A3 )
        = one_one_nat )
     => ~ ! [X3: int] :
            ( A3
           != ( insert_int @ X3 @ bot_bot_set_int ) ) ) ).

% card_1_singletonE
thf(fact_8059_card__1__singletonE,axiom,
    ! [A3: set_nat] :
      ( ( ( finite_card_nat @ A3 )
        = one_one_nat )
     => ~ ! [X3: nat] :
            ( A3
           != ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ).

% card_1_singletonE
thf(fact_8060_card__1__singletonE,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ( ( ( finite711546835091564841at_nat @ A3 )
        = one_one_nat )
     => ~ ! [X3: product_prod_nat_nat] :
            ( A3
           != ( insert8211810215607154385at_nat @ X3 @ bot_bo2099793752762293965at_nat ) ) ) ).

% card_1_singletonE
thf(fact_8061_card__1__singletonE,axiom,
    ! [A3: set_Pr4329608150637261639at_nat] :
      ( ( ( finite3771342082235030671at_nat @ A3 )
        = one_one_nat )
     => ~ ! [X3: produc3843707927480180839at_nat] :
            ( A3
           != ( insert9069300056098147895at_nat @ X3 @ bot_bo228742789529271731at_nat ) ) ) ).

% card_1_singletonE
thf(fact_8062_card__1__singleton__iff,axiom,
    ! [A3: set_list_nat] :
      ( ( ( finite_card_list_nat @ A3 )
        = ( suc @ zero_zero_nat ) )
      = ( ? [X2: list_nat] :
            ( A3
            = ( insert_list_nat @ X2 @ bot_bot_set_list_nat ) ) ) ) ).

% card_1_singleton_iff
thf(fact_8063_card__1__singleton__iff,axiom,
    ! [A3: set_Product_unit] :
      ( ( ( finite410649719033368117t_unit @ A3 )
        = ( suc @ zero_zero_nat ) )
      = ( ? [X2: product_unit] :
            ( A3
            = ( insert_Product_unit @ X2 @ bot_bo3957492148770167129t_unit ) ) ) ) ).

% card_1_singleton_iff
thf(fact_8064_card__1__singleton__iff,axiom,
    ! [A3: set_set_nat] :
      ( ( ( finite_card_set_nat @ A3 )
        = ( suc @ zero_zero_nat ) )
      = ( ? [X2: set_nat] :
            ( A3
            = ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) ) ) ) ).

% card_1_singleton_iff
thf(fact_8065_card__1__singleton__iff,axiom,
    ! [A3: set_int] :
      ( ( ( finite_card_int @ A3 )
        = ( suc @ zero_zero_nat ) )
      = ( ? [X2: int] :
            ( A3
            = ( insert_int @ X2 @ bot_bot_set_int ) ) ) ) ).

% card_1_singleton_iff
thf(fact_8066_card__1__singleton__iff,axiom,
    ! [A3: set_nat] :
      ( ( ( finite_card_nat @ A3 )
        = ( suc @ zero_zero_nat ) )
      = ( ? [X2: nat] :
            ( A3
            = ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ).

% card_1_singleton_iff
thf(fact_8067_card__1__singleton__iff,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ( ( ( finite711546835091564841at_nat @ A3 )
        = ( suc @ zero_zero_nat ) )
      = ( ? [X2: product_prod_nat_nat] :
            ( A3
            = ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) ) ) ).

% card_1_singleton_iff
thf(fact_8068_card__1__singleton__iff,axiom,
    ! [A3: set_Pr4329608150637261639at_nat] :
      ( ( ( finite3771342082235030671at_nat @ A3 )
        = ( suc @ zero_zero_nat ) )
      = ( ? [X2: produc3843707927480180839at_nat] :
            ( A3
            = ( insert9069300056098147895at_nat @ X2 @ bot_bo228742789529271731at_nat ) ) ) ) ).

% card_1_singleton_iff
thf(fact_8069_plus__integer__code_I1_J,axiom,
    ! [K: code_integer] :
      ( ( plus_p5714425477246183910nteger @ K @ zero_z3403309356797280102nteger )
      = K ) ).

% plus_integer_code(1)
thf(fact_8070_plus__integer__code_I2_J,axiom,
    ! [L: code_integer] :
      ( ( plus_p5714425477246183910nteger @ zero_z3403309356797280102nteger @ L )
      = L ) ).

% plus_integer_code(2)
thf(fact_8071_less__integer__code_I1_J,axiom,
    ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger ) ).

% less_integer_code(1)
thf(fact_8072_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_8073_uminus__integer__code_I1_J,axiom,
    ( ( uminus1351360451143612070nteger @ zero_z3403309356797280102nteger )
    = zero_z3403309356797280102nteger ) ).

% uminus_integer_code(1)
thf(fact_8074_minus__integer__code_I1_J,axiom,
    ! [K: code_integer] :
      ( ( minus_8373710615458151222nteger @ K @ zero_z3403309356797280102nteger )
      = K ) ).

% minus_integer_code(1)
thf(fact_8075_minus__integer__code_I2_J,axiom,
    ! [L: code_integer] :
      ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ L )
      = ( uminus1351360451143612070nteger @ L ) ) ).

% minus_integer_code(2)
thf(fact_8076_divmod__integer__def,axiom,
    ( code_divmod_integer
    = ( ^ [K3: code_integer,L3: code_integer] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ K3 @ L3 ) @ ( modulo364778990260209775nteger @ K3 @ L3 ) ) ) ) ).

% divmod_integer_def
thf(fact_8077_divmod__abs__def,axiom,
    ( code_divmod_abs
    = ( ^ [K3: code_integer,L3: code_integer] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( abs_abs_Code_integer @ K3 ) @ ( abs_abs_Code_integer @ L3 ) ) @ ( modulo364778990260209775nteger @ ( abs_abs_Code_integer @ K3 ) @ ( abs_abs_Code_integer @ L3 ) ) ) ) ) ).

% divmod_abs_def
thf(fact_8078_card__greaterThanLessThan__int,axiom,
    ! [L: int,U: int] :
      ( ( finite_card_int @ ( set_or5832277885323065728an_int @ L @ U ) )
      = ( nat2 @ ( minus_minus_int @ U @ ( plus_plus_int @ L @ one_one_int ) ) ) ) ).

% card_greaterThanLessThan_int
thf(fact_8079_divmod__abs__code_I6_J,axiom,
    ! [J: code_integer] :
      ( ( code_divmod_abs @ zero_z3403309356797280102nteger @ J )
      = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger ) ) ).

% divmod_abs_code(6)
thf(fact_8080_divmod__abs__code_I5_J,axiom,
    ! [J: code_integer] :
      ( ( code_divmod_abs @ J @ zero_z3403309356797280102nteger )
      = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ J ) ) ) ).

% divmod_abs_code(5)
thf(fact_8081_sqr_Osimps_I3_J,axiom,
    ! [N: num] :
      ( ( sqr @ ( bit1 @ N ) )
      = ( bit1 @ ( bit0 @ ( plus_plus_num @ ( sqr @ N ) @ N ) ) ) ) ).

% sqr.simps(3)
thf(fact_8082_quotient__of__number_I5_J,axiom,
    ! [K: num] :
      ( ( quotient_of @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) )
      = ( product_Pair_int_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) @ one_one_int ) ) ).

% quotient_of_number(5)
thf(fact_8083_prod__decode__aux_Oelims,axiom,
    ! [X: nat,Xa3: nat,Y3: product_prod_nat_nat] :
      ( ( ( nat_prod_decode_aux @ X @ Xa3 )
        = Y3 )
     => ( ( ( ord_less_eq_nat @ Xa3 @ X )
         => ( Y3
            = ( product_Pair_nat_nat @ Xa3 @ ( minus_minus_nat @ X @ Xa3 ) ) ) )
        & ( ~ ( ord_less_eq_nat @ Xa3 @ X )
         => ( Y3
            = ( nat_prod_decode_aux @ ( suc @ X ) @ ( minus_minus_nat @ Xa3 @ ( suc @ X ) ) ) ) ) ) ) ).

% prod_decode_aux.elims
thf(fact_8084_prod__decode__aux_Osimps,axiom,
    ( nat_prod_decode_aux
    = ( ^ [K3: nat,M2: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ M2 @ K3 ) @ ( product_Pair_nat_nat @ M2 @ ( minus_minus_nat @ K3 @ M2 ) ) @ ( nat_prod_decode_aux @ ( suc @ K3 ) @ ( minus_minus_nat @ M2 @ ( suc @ K3 ) ) ) ) ) ) ).

% prod_decode_aux.simps
thf(fact_8085_nat__of__integer__non__positive,axiom,
    ! [K: code_integer] :
      ( ( ord_le3102999989581377725nteger @ K @ zero_z3403309356797280102nteger )
     => ( ( code_nat_of_integer @ K )
        = zero_zero_nat ) ) ).

% nat_of_integer_non_positive
thf(fact_8086_nat__of__integer__of__nat,axiom,
    ! [N: nat] :
      ( ( code_nat_of_integer @ ( semiri4939895301339042750nteger @ N ) )
      = N ) ).

% nat_of_integer_of_nat
thf(fact_8087_of__nat__of__integer,axiom,
    ! [K: code_integer] :
      ( ( semiri4939895301339042750nteger @ ( code_nat_of_integer @ K ) )
      = ( ord_max_Code_integer @ zero_z3403309356797280102nteger @ K ) ) ).

% of_nat_of_integer
thf(fact_8088_card__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] :
      ( ( finite_card_nat @ ( set_or5834768355832116004an_nat @ L @ U ) )
      = ( minus_minus_nat @ U @ ( suc @ L ) ) ) ).

% card_greaterThanLessThan
thf(fact_8089_rat__one__code,axiom,
    ( ( quotient_of @ one_one_rat )
    = ( product_Pair_int_int @ one_one_int @ one_one_int ) ) ).

% rat_one_code
thf(fact_8090_rat__zero__code,axiom,
    ( ( quotient_of @ zero_zero_rat )
    = ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ).

% rat_zero_code
thf(fact_8091_quotient__of__number_I4_J,axiom,
    ( ( quotient_of @ ( uminus_uminus_rat @ one_one_rat ) )
    = ( product_Pair_int_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ) ) ).

% quotient_of_number(4)
thf(fact_8092_quotient__of__number_I3_J,axiom,
    ! [K: num] :
      ( ( quotient_of @ ( numeral_numeral_rat @ K ) )
      = ( product_Pair_int_int @ ( numeral_numeral_int @ K ) @ one_one_int ) ) ).

% quotient_of_number(3)
thf(fact_8093_prod__decode__aux_Ocases,axiom,
    ! [X: product_prod_nat_nat] :
      ~ ! [K2: nat,M6: nat] :
          ( X
         != ( product_Pair_nat_nat @ K2 @ M6 ) ) ).

% prod_decode_aux.cases
thf(fact_8094_sgn__rat__def,axiom,
    ( sgn_sgn_rat
    = ( ^ [A2: rat] : ( if_rat @ ( A2 = zero_zero_rat ) @ zero_zero_rat @ ( if_rat @ ( ord_less_rat @ zero_zero_rat @ A2 ) @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ) ) ) ).

% sgn_rat_def
thf(fact_8095_diff__rat__def,axiom,
    ( minus_minus_rat
    = ( ^ [Q3: rat,R5: rat] : ( plus_plus_rat @ Q3 @ ( uminus_uminus_rat @ R5 ) ) ) ) ).

% diff_rat_def
thf(fact_8096_abs__rat__def,axiom,
    ( abs_abs_rat
    = ( ^ [A2: rat] : ( if_rat @ ( ord_less_rat @ A2 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A2 ) @ A2 ) ) ) ).

% abs_rat_def
thf(fact_8097_sqr_Osimps_I2_J,axiom,
    ! [N: num] :
      ( ( sqr @ ( bit0 @ N ) )
      = ( bit0 @ ( bit0 @ ( sqr @ N ) ) ) ) ).

% sqr.simps(2)
thf(fact_8098_sqr_Osimps_I1_J,axiom,
    ( ( sqr @ one )
    = one ) ).

% sqr.simps(1)
thf(fact_8099_sqr__conv__mult,axiom,
    ( sqr
    = ( ^ [X2: num] : ( times_times_num @ X2 @ X2 ) ) ) ).

% sqr_conv_mult
thf(fact_8100_quotient__of__denom__pos,axiom,
    ! [R: rat,P5: int,Q4: int] :
      ( ( ( quotient_of @ R )
        = ( product_Pair_int_int @ P5 @ Q4 ) )
     => ( ord_less_int @ zero_zero_int @ Q4 ) ) ).

% quotient_of_denom_pos
thf(fact_8101_quotient__of__int,axiom,
    ! [A: int] :
      ( ( quotient_of @ ( of_int @ A ) )
      = ( product_Pair_int_int @ A @ one_one_int ) ) ).

% quotient_of_int
thf(fact_8102_pow_Osimps_I3_J,axiom,
    ! [X: num,Y3: num] :
      ( ( pow @ X @ ( bit1 @ Y3 ) )
      = ( times_times_num @ ( sqr @ ( pow @ X @ Y3 ) ) @ X ) ) ).

% pow.simps(3)
thf(fact_8103_Frct__code__post_I4_J,axiom,
    ! [K: num] :
      ( ( frct @ ( product_Pair_int_int @ ( numeral_numeral_int @ K ) @ one_one_int ) )
      = ( numeral_numeral_rat @ K ) ) ).

% Frct_code_post(4)
thf(fact_8104_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_8105_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_8106_prod__encode__eq,axiom,
    ! [X: product_prod_nat_nat,Y3: product_prod_nat_nat] :
      ( ( ( nat_prod_encode @ X )
        = ( nat_prod_encode @ Y3 ) )
      = ( X = Y3 ) ) ).

% prod_encode_eq
thf(fact_8107_less__eq__rat__def,axiom,
    ( ord_less_eq_rat
    = ( ^ [X2: rat,Y6: rat] :
          ( ( ord_less_rat @ X2 @ Y6 )
          | ( X2 = Y6 ) ) ) ) ).

% less_eq_rat_def
thf(fact_8108_Frct__code__post_I9_J,axiom,
    ! [Q4: product_prod_int_int] :
      ( ( uminus_uminus_rat @ ( uminus_uminus_rat @ ( frct @ Q4 ) ) )
      = ( frct @ Q4 ) ) ).

% Frct_code_post(9)
thf(fact_8109_pow_Osimps_I1_J,axiom,
    ! [X: num] :
      ( ( pow @ X @ one )
      = X ) ).

% pow.simps(1)
thf(fact_8110_le__prod__encode__2,axiom,
    ! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( nat_prod_encode @ ( product_Pair_nat_nat @ A @ B ) ) ) ).

% le_prod_encode_2
thf(fact_8111_le__prod__encode__1,axiom,
    ! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( nat_prod_encode @ ( product_Pair_nat_nat @ A @ B ) ) ) ).

% le_prod_encode_1
thf(fact_8112_Frct__code__post_I2_J,axiom,
    ! [A: int] :
      ( ( frct @ ( product_Pair_int_int @ A @ zero_zero_int ) )
      = zero_zero_rat ) ).

% Frct_code_post(2)
thf(fact_8113_Frct__code__post_I1_J,axiom,
    ! [A: int] :
      ( ( frct @ ( product_Pair_int_int @ zero_zero_int @ A ) )
      = zero_zero_rat ) ).

% Frct_code_post(1)
thf(fact_8114_Frct__code__post_I3_J,axiom,
    ( ( frct @ ( product_Pair_int_int @ one_one_int @ one_one_int ) )
    = one_one_rat ) ).

% Frct_code_post(3)
thf(fact_8115_pow_Osimps_I2_J,axiom,
    ! [X: num,Y3: num] :
      ( ( pow @ X @ ( bit0 @ Y3 ) )
      = ( sqr @ ( pow @ X @ Y3 ) ) ) ).

% pow.simps(2)
thf(fact_8116_Frct__code__post_I8_J,axiom,
    ! [A: int,B: int] :
      ( ( frct @ ( product_Pair_int_int @ A @ ( uminus_uminus_int @ B ) ) )
      = ( uminus_uminus_rat @ ( frct @ ( product_Pair_int_int @ A @ B ) ) ) ) ).

% Frct_code_post(8)
thf(fact_8117_Frct__code__post_I7_J,axiom,
    ! [A: int,B: int] :
      ( ( frct @ ( product_Pair_int_int @ ( uminus_uminus_int @ A ) @ B ) )
      = ( uminus_uminus_rat @ ( frct @ ( product_Pair_int_int @ A @ B ) ) ) ) ).

% Frct_code_post(7)
thf(fact_8118_nth__sorted__list__of__set__greaterThanLessThan,axiom,
    ! [N: nat,J: nat,I2: nat] :
      ( ( ord_less_nat @ N @ ( minus_minus_nat @ J @ ( suc @ I2 ) ) )
     => ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I2 @ J ) ) @ N )
        = ( suc @ ( plus_plus_nat @ I2 @ N ) ) ) ) ).

% nth_sorted_list_of_set_greaterThanLessThan
thf(fact_8119_rat__sgn__code,axiom,
    ! [P5: rat] :
      ( ( quotient_of @ ( sgn_sgn_rat @ P5 ) )
      = ( product_Pair_int_int @ ( sgn_sgn_int @ ( product_fst_int_int @ ( quotient_of @ P5 ) ) ) @ one_one_int ) ) ).

% rat_sgn_code
thf(fact_8120_fst__divmod__integer,axiom,
    ! [K: code_integer,L: code_integer] :
      ( ( produc8508995932063986495nteger @ ( code_divmod_integer @ K @ L ) )
      = ( divide6298287555418463151nteger @ K @ L ) ) ).

% fst_divmod_integer
thf(fact_8121_fst__divmod__abs,axiom,
    ! [K: code_integer,L: code_integer] :
      ( ( produc8508995932063986495nteger @ ( code_divmod_abs @ K @ L ) )
      = ( divide6298287555418463151nteger @ ( abs_abs_Code_integer @ K ) @ ( abs_abs_Code_integer @ L ) ) ) ).

% fst_divmod_abs
thf(fact_8122_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_8123_sorted__list__of__set__greaterThanLessThan,axiom,
    ! [I2: nat,J: nat] :
      ( ( ord_less_nat @ ( suc @ I2 ) @ J )
     => ( ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I2 @ J ) )
        = ( cons_nat @ ( suc @ I2 ) @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ ( suc @ I2 ) @ J ) ) ) ) ) ).

% sorted_list_of_set_greaterThanLessThan
thf(fact_8124_bezw_Osimps,axiom,
    ( bezw
    = ( ^ [X2: nat,Y6: nat] : ( if_Pro3027730157355071871nt_int @ ( Y6 = zero_zero_nat ) @ ( product_Pair_int_int @ one_one_int @ zero_zero_int ) @ ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Y6 @ ( modulo_modulo_nat @ X2 @ Y6 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Y6 @ ( modulo_modulo_nat @ X2 @ Y6 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Y6 @ ( modulo_modulo_nat @ X2 @ Y6 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X2 @ Y6 ) ) ) ) ) ) ) ) ).

% bezw.simps
thf(fact_8125_bezw_Oelims,axiom,
    ! [X: nat,Xa3: nat,Y3: product_prod_int_int] :
      ( ( ( bezw @ X @ Xa3 )
        = Y3 )
     => ( ( ( Xa3 = zero_zero_nat )
         => ( Y3
            = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
        & ( ( Xa3 != zero_zero_nat )
         => ( Y3
            = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Xa3 @ ( modulo_modulo_nat @ X @ Xa3 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Xa3 @ ( modulo_modulo_nat @ X @ Xa3 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Xa3 @ ( modulo_modulo_nat @ X @ Xa3 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X @ Xa3 ) ) ) ) ) ) ) ) ) ).

% bezw.elims
thf(fact_8126_bezw__non__0,axiom,
    ! [Y3: nat,X: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ Y3 )
     => ( ( bezw @ X @ Y3 )
        = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Y3 @ ( modulo_modulo_nat @ X @ Y3 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Y3 @ ( modulo_modulo_nat @ X @ Y3 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Y3 @ ( modulo_modulo_nat @ X @ Y3 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X @ Y3 ) ) ) ) ) ) ) ).

% bezw_non_0
thf(fact_8127_nth__sorted__list__of__set__greaterThanAtMost,axiom,
    ! [N: nat,J: nat,I2: nat] :
      ( ( ord_less_nat @ N @ ( minus_minus_nat @ J @ I2 ) )
     => ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I2 @ J ) ) @ N )
        = ( suc @ ( plus_plus_nat @ I2 @ N ) ) ) ) ).

% nth_sorted_list_of_set_greaterThanAtMost
thf(fact_8128_atLeastSucAtMost__greaterThanAtMost,axiom,
    ! [L: nat,U: nat] :
      ( ( set_or1269000886237332187st_nat @ ( suc @ L ) @ U )
      = ( set_or6659071591806873216st_nat @ L @ U ) ) ).

% atLeastSucAtMost_greaterThanAtMost
thf(fact_8129_quotient__of__denom__pos_H,axiom,
    ! [R: rat] : ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ ( quotient_of @ R ) ) ) ).

% quotient_of_denom_pos'
thf(fact_8130_sorted__list__of__set__greaterThanAtMost,axiom,
    ! [I2: nat,J: nat] :
      ( ( ord_less_eq_nat @ ( suc @ I2 ) @ J )
     => ( ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I2 @ J ) )
        = ( cons_nat @ ( suc @ I2 ) @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ ( suc @ I2 ) @ J ) ) ) ) ) ).

% sorted_list_of_set_greaterThanAtMost
thf(fact_8131_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_8132_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_8133_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_8134_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_8135_snd__divmod__integer,axiom,
    ! [K: code_integer,L: code_integer] :
      ( ( produc6174133586879617921nteger @ ( code_divmod_integer @ K @ L ) )
      = ( modulo364778990260209775nteger @ K @ L ) ) ).

% snd_divmod_integer
thf(fact_8136_snd__divmod__abs,axiom,
    ! [K: code_integer,L: code_integer] :
      ( ( produc6174133586879617921nteger @ ( code_divmod_abs @ K @ L ) )
      = ( modulo364778990260209775nteger @ ( abs_abs_Code_integer @ K ) @ ( abs_abs_Code_integer @ L ) ) ) ).

% snd_divmod_abs
thf(fact_8137_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_8138_atLeastPlusOneAtMost__greaterThanAtMost__int,axiom,
    ! [L: int,U: int] :
      ( ( set_or1266510415728281911st_int @ ( plus_plus_int @ L @ one_one_int ) @ U )
      = ( set_or6656581121297822940st_int @ L @ U ) ) ).

% atLeastPlusOneAtMost_greaterThanAtMost_int
thf(fact_8139_Divides_Oadjust__mod__def,axiom,
    ( adjust_mod
    = ( ^ [L3: int,R5: int] : ( if_int @ ( R5 = zero_zero_int ) @ zero_zero_int @ ( minus_minus_int @ L3 @ R5 ) ) ) ) ).

% Divides.adjust_mod_def
thf(fact_8140_bezw_Opelims,axiom,
    ! [X: nat,Xa3: nat,Y3: product_prod_int_int] :
      ( ( ( bezw @ X @ Xa3 )
        = Y3 )
     => ( ( accp_P4275260045618599050at_nat @ bezw_rel @ ( product_Pair_nat_nat @ X @ Xa3 ) )
       => ~ ( ( ( ( Xa3 = zero_zero_nat )
               => ( Y3
                  = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
              & ( ( Xa3 != zero_zero_nat )
               => ( Y3
                  = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Xa3 @ ( modulo_modulo_nat @ X @ Xa3 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Xa3 @ ( modulo_modulo_nat @ X @ Xa3 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Xa3 @ ( modulo_modulo_nat @ X @ Xa3 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X @ Xa3 ) ) ) ) ) ) ) )
           => ~ ( accp_P4275260045618599050at_nat @ bezw_rel @ ( product_Pair_nat_nat @ X @ Xa3 ) ) ) ) ) ).

% bezw.pelims
thf(fact_8141_list__encode_Osimps_I2_J,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( nat_list_encode @ ( cons_nat @ X @ Xs ) )
      = ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X @ ( nat_list_encode @ Xs ) ) ) ) ) ).

% list_encode.simps(2)
thf(fact_8142_bezw__aux,axiom,
    ! [X: nat,Y3: nat] :
      ( ( semiri1314217659103216013at_int @ ( gcd_gcd_nat @ X @ Y3 ) )
      = ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ ( bezw @ X @ Y3 ) ) @ ( semiri1314217659103216013at_int @ X ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ X @ Y3 ) ) @ ( semiri1314217659103216013at_int @ Y3 ) ) ) ) ).

% bezw_aux
thf(fact_8143_gcd__0__left__nat,axiom,
    ! [X: nat] :
      ( ( gcd_gcd_nat @ zero_zero_nat @ X )
      = X ) ).

% gcd_0_left_nat
thf(fact_8144_gcd__0__nat,axiom,
    ! [X: nat] :
      ( ( gcd_gcd_nat @ X @ zero_zero_nat )
      = X ) ).

% gcd_0_nat
thf(fact_8145_gcd__nat_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( gcd_gcd_nat @ A @ zero_zero_nat )
      = A ) ).

% gcd_nat.right_neutral
thf(fact_8146_gcd__nat_Oneutr__eq__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( zero_zero_nat
        = ( gcd_gcd_nat @ A @ B ) )
      = ( ( A = zero_zero_nat )
        & ( B = zero_zero_nat ) ) ) ).

% gcd_nat.neutr_eq_iff
thf(fact_8147_gcd__nat_Oleft__neutral,axiom,
    ! [A: nat] :
      ( ( gcd_gcd_nat @ zero_zero_nat @ A )
      = A ) ).

% gcd_nat.left_neutral
thf(fact_8148_gcd__nat_Oeq__neutr__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( ( gcd_gcd_nat @ A @ B )
        = zero_zero_nat )
      = ( ( A = zero_zero_nat )
        & ( B = zero_zero_nat ) ) ) ).

% gcd_nat.eq_neutr_iff
thf(fact_8149_gcd__1__nat,axiom,
    ! [M: nat] :
      ( ( gcd_gcd_nat @ M @ one_one_nat )
      = one_one_nat ) ).

% gcd_1_nat
thf(fact_8150_gcd__Suc__0,axiom,
    ! [M: nat] :
      ( ( gcd_gcd_nat @ M @ ( suc @ zero_zero_nat ) )
      = ( suc @ zero_zero_nat ) ) ).

% gcd_Suc_0
thf(fact_8151_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_8152_list__encode__eq,axiom,
    ! [X: list_nat,Y3: list_nat] :
      ( ( ( nat_list_encode @ X )
        = ( nat_list_encode @ Y3 ) )
      = ( X = Y3 ) ) ).

% list_encode_eq
thf(fact_8153_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_8154_gcd__le1__nat,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( ord_less_eq_nat @ ( gcd_gcd_nat @ A @ B ) @ A ) ) ).

% gcd_le1_nat
thf(fact_8155_gcd__le2__nat,axiom,
    ! [B: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( ord_less_eq_nat @ ( gcd_gcd_nat @ A @ B ) @ B ) ) ).

% gcd_le2_nat
thf(fact_8156_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_8157_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_8158_gcd__non__0__nat,axiom,
    ! [Y3: nat,X: nat] :
      ( ( Y3 != zero_zero_nat )
     => ( ( gcd_gcd_nat @ X @ Y3 )
        = ( gcd_gcd_nat @ Y3 @ ( modulo_modulo_nat @ X @ Y3 ) ) ) ) ).

% gcd_non_0_nat
thf(fact_8159_gcd__nat_Osimps,axiom,
    ( gcd_gcd_nat
    = ( ^ [X2: nat,Y6: nat] : ( if_nat @ ( Y6 = zero_zero_nat ) @ X2 @ ( gcd_gcd_nat @ Y6 @ ( modulo_modulo_nat @ X2 @ Y6 ) ) ) ) ) ).

% gcd_nat.simps
thf(fact_8160_gcd__nat_Oelims,axiom,
    ! [X: nat,Xa3: nat,Y3: nat] :
      ( ( ( gcd_gcd_nat @ X @ Xa3 )
        = Y3 )
     => ( ( ( Xa3 = zero_zero_nat )
         => ( Y3 = X ) )
        & ( ( Xa3 != zero_zero_nat )
         => ( Y3
            = ( gcd_gcd_nat @ Xa3 @ ( modulo_modulo_nat @ X @ Xa3 ) ) ) ) ) ) ).

% gcd_nat.elims
thf(fact_8161_bezout__nat,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ? [X3: nat,Y4: nat] :
          ( ( times_times_nat @ A @ X3 )
          = ( plus_plus_nat @ ( times_times_nat @ B @ Y4 ) @ ( gcd_gcd_nat @ A @ B ) ) ) ) ).

% bezout_nat
thf(fact_8162_bezout__gcd__nat_H,axiom,
    ! [B: nat,A: nat] :
    ? [X3: nat,Y4: nat] :
      ( ( ( ord_less_eq_nat @ ( times_times_nat @ B @ Y4 ) @ ( times_times_nat @ A @ X3 ) )
        & ( ( minus_minus_nat @ ( times_times_nat @ A @ X3 ) @ ( times_times_nat @ B @ Y4 ) )
          = ( gcd_gcd_nat @ A @ B ) ) )
      | ( ( ord_less_eq_nat @ ( times_times_nat @ A @ Y4 ) @ ( times_times_nat @ B @ X3 ) )
        & ( ( minus_minus_nat @ ( times_times_nat @ B @ X3 ) @ ( times_times_nat @ A @ Y4 ) )
          = ( gcd_gcd_nat @ A @ B ) ) ) ) ).

% bezout_gcd_nat'
thf(fact_8163_prod__decode__aux_Opelims,axiom,
    ! [X: nat,Xa3: nat,Y3: product_prod_nat_nat] :
      ( ( ( nat_prod_decode_aux @ X @ Xa3 )
        = Y3 )
     => ( ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X @ Xa3 ) )
       => ~ ( ( ( ( ord_less_eq_nat @ Xa3 @ X )
               => ( Y3
                  = ( product_Pair_nat_nat @ Xa3 @ ( minus_minus_nat @ X @ Xa3 ) ) ) )
              & ( ~ ( ord_less_eq_nat @ Xa3 @ X )
               => ( Y3
                  = ( nat_prod_decode_aux @ ( suc @ X ) @ ( minus_minus_nat @ Xa3 @ ( suc @ X ) ) ) ) ) )
           => ~ ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X @ Xa3 ) ) ) ) ) ).

% prod_decode_aux.pelims
thf(fact_8164_gcd__nat_Opelims,axiom,
    ! [X: nat,Xa3: nat,Y3: nat] :
      ( ( ( gcd_gcd_nat @ X @ Xa3 )
        = Y3 )
     => ( ( accp_P4275260045618599050at_nat @ gcd_nat_rel @ ( product_Pair_nat_nat @ X @ Xa3 ) )
       => ~ ( ( ( ( Xa3 = zero_zero_nat )
               => ( Y3 = X ) )
              & ( ( Xa3 != zero_zero_nat )
               => ( Y3
                  = ( gcd_gcd_nat @ Xa3 @ ( modulo_modulo_nat @ X @ Xa3 ) ) ) ) )
           => ~ ( accp_P4275260045618599050at_nat @ gcd_nat_rel @ ( product_Pair_nat_nat @ X @ Xa3 ) ) ) ) ) ).

% gcd_nat.pelims
thf(fact_8165_list__encode_Oelims,axiom,
    ! [X: list_nat,Y3: nat] :
      ( ( ( nat_list_encode @ X )
        = Y3 )
     => ( ( ( X = nil_nat )
         => ( Y3 != zero_zero_nat ) )
       => ~ ! [X3: nat,Xs2: list_nat] :
              ( ( X
                = ( cons_nat @ X3 @ Xs2 ) )
             => ( Y3
               != ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X3 @ ( nat_list_encode @ Xs2 ) ) ) ) ) ) ) ) ).

% list_encode.elims
thf(fact_8166_zero__rat__def,axiom,
    ( zero_zero_rat
    = ( abs_Rat @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ) ).

% zero_rat_def
thf(fact_8167_pair__lessI2,axiom,
    ! [A: nat,B: nat,S: nat,T: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ S @ T )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B @ T ) ) @ fun_pair_less ) ) ) ).

% pair_lessI2
thf(fact_8168_gcd__1__int,axiom,
    ! [M: int] :
      ( ( gcd_gcd_int @ M @ one_one_int )
      = one_one_int ) ).

% gcd_1_int
thf(fact_8169_gcd__neg2__int,axiom,
    ! [X: int,Y3: int] :
      ( ( gcd_gcd_int @ X @ ( uminus_uminus_int @ Y3 ) )
      = ( gcd_gcd_int @ X @ Y3 ) ) ).

% gcd_neg2_int
thf(fact_8170_gcd__neg1__int,axiom,
    ! [X: int,Y3: int] :
      ( ( gcd_gcd_int @ ( uminus_uminus_int @ X ) @ Y3 )
      = ( gcd_gcd_int @ X @ Y3 ) ) ).

% gcd_neg1_int
thf(fact_8171_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_8172_gcd__neg__numeral__1__int,axiom,
    ! [N: num,X: int] :
      ( ( gcd_gcd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ X )
      = ( gcd_gcd_int @ ( numeral_numeral_int @ N ) @ X ) ) ).

% gcd_neg_numeral_1_int
thf(fact_8173_gcd__neg__numeral__2__int,axiom,
    ! [X: int,N: num] :
      ( ( gcd_gcd_int @ X @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( gcd_gcd_int @ X @ ( numeral_numeral_int @ N ) ) ) ).

% gcd_neg_numeral_2_int
thf(fact_8174_gcd__0__left__int,axiom,
    ! [X: int] :
      ( ( gcd_gcd_int @ zero_zero_int @ X )
      = ( abs_abs_int @ X ) ) ).

% gcd_0_left_int
thf(fact_8175_gcd__0__int,axiom,
    ! [X: int] :
      ( ( gcd_gcd_int @ X @ zero_zero_int )
      = ( abs_abs_int @ X ) ) ).

% gcd_0_int
thf(fact_8176_gcd__int__int__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( gcd_gcd_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
      = ( semiri1314217659103216013at_int @ ( gcd_gcd_nat @ M @ N ) ) ) ).

% gcd_int_int_eq
thf(fact_8177_gcd__nat__abs__right__eq,axiom,
    ! [N: nat,K: int] :
      ( ( gcd_gcd_nat @ N @ ( nat2 @ ( abs_abs_int @ K ) ) )
      = ( nat2 @ ( gcd_gcd_int @ ( semiri1314217659103216013at_int @ N ) @ K ) ) ) ).

% gcd_nat_abs_right_eq
thf(fact_8178_gcd__nat__abs__left__eq,axiom,
    ! [K: int,N: nat] :
      ( ( gcd_gcd_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ N )
      = ( nat2 @ ( gcd_gcd_int @ K @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).

% gcd_nat_abs_left_eq
thf(fact_8179_pair__less__iff1,axiom,
    ! [X: nat,Y3: nat,Z2: nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ ( product_Pair_nat_nat @ X @ Z2 ) ) @ fun_pair_less )
      = ( ord_less_nat @ Y3 @ Z2 ) ) ).

% pair_less_iff1
thf(fact_8180_list__encode_Ocases,axiom,
    ! [X: list_nat] :
      ( ( X != nil_nat )
     => ~ ! [X3: nat,Xs2: list_nat] :
            ( X
           != ( cons_nat @ X3 @ Xs2 ) ) ) ).

% list_encode.cases
thf(fact_8181_gcd__ge__0__int,axiom,
    ! [X: int,Y3: int] : ( ord_less_eq_int @ zero_zero_int @ ( gcd_gcd_int @ X @ Y3 ) ) ).

% gcd_ge_0_int
thf(fact_8182_bezout__int,axiom,
    ! [X: int,Y3: int] :
    ? [U2: int,V3: int] :
      ( ( plus_plus_int @ ( times_times_int @ U2 @ X ) @ ( times_times_int @ V3 @ Y3 ) )
      = ( gcd_gcd_int @ X @ Y3 ) ) ).

% bezout_int
thf(fact_8183_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_8184_list__encode_Osimps_I1_J,axiom,
    ( ( nat_list_encode @ nil_nat )
    = zero_zero_nat ) ).

% list_encode.simps(1)
thf(fact_8185_gcd__le1__int,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ord_less_eq_int @ ( gcd_gcd_int @ A @ B ) @ A ) ) ).

% gcd_le1_int
thf(fact_8186_gcd__le2__int,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ord_less_eq_int @ ( gcd_gcd_int @ A @ B ) @ B ) ) ).

% gcd_le2_int
thf(fact_8187_gcd__cases__int,axiom,
    ! [X: int,Y3: int,P: int > $o] :
      ( ( ( ord_less_eq_int @ zero_zero_int @ X )
       => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
         => ( P @ ( gcd_gcd_int @ X @ Y3 ) ) ) )
     => ( ( ( ord_less_eq_int @ zero_zero_int @ X )
         => ( ( ord_less_eq_int @ Y3 @ zero_zero_int )
           => ( P @ ( gcd_gcd_int @ X @ ( uminus_uminus_int @ Y3 ) ) ) ) )
       => ( ( ( ord_less_eq_int @ X @ zero_zero_int )
           => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
             => ( P @ ( gcd_gcd_int @ ( uminus_uminus_int @ X ) @ Y3 ) ) ) )
         => ( ( ( ord_less_eq_int @ X @ zero_zero_int )
             => ( ( ord_less_eq_int @ Y3 @ zero_zero_int )
               => ( P @ ( gcd_gcd_int @ ( uminus_uminus_int @ X ) @ ( uminus_uminus_int @ Y3 ) ) ) ) )
           => ( P @ ( gcd_gcd_int @ X @ Y3 ) ) ) ) ) ) ).

% gcd_cases_int
thf(fact_8188_gcd__unique__int,axiom,
    ! [D: int,A: int,B: int] :
      ( ( ( ord_less_eq_int @ zero_zero_int @ D )
        & ( dvd_dvd_int @ D @ A )
        & ( dvd_dvd_int @ D @ B )
        & ! [E3: int] :
            ( ( ( dvd_dvd_int @ E3 @ A )
              & ( dvd_dvd_int @ E3 @ B ) )
           => ( dvd_dvd_int @ E3 @ D ) ) )
      = ( D
        = ( gcd_gcd_int @ A @ B ) ) ) ).

% gcd_unique_int
thf(fact_8189_gcd__non__0__int,axiom,
    ! [Y3: int,X: int] :
      ( ( ord_less_int @ zero_zero_int @ Y3 )
     => ( ( gcd_gcd_int @ X @ Y3 )
        = ( gcd_gcd_int @ Y3 @ ( modulo_modulo_int @ X @ Y3 ) ) ) ) ).

% gcd_non_0_int
thf(fact_8190_gcd__code__int,axiom,
    ( gcd_gcd_int
    = ( ^ [K3: int,L3: int] : ( abs_abs_int @ ( if_int @ ( L3 = zero_zero_int ) @ K3 @ ( gcd_gcd_int @ L3 @ ( modulo_modulo_int @ ( abs_abs_int @ K3 ) @ ( abs_abs_int @ L3 ) ) ) ) ) ) ) ).

% gcd_code_int
thf(fact_8191_pair__lessI1,axiom,
    ! [A: nat,B: nat,S: nat,T: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B @ T ) ) @ fun_pair_less ) ) ).

% pair_lessI1
thf(fact_8192_gcd__int__def,axiom,
    ( gcd_gcd_int
    = ( ^ [X2: int,Y6: int] : ( semiri1314217659103216013at_int @ ( gcd_gcd_nat @ ( nat2 @ ( abs_abs_int @ X2 ) ) @ ( nat2 @ ( abs_abs_int @ Y6 ) ) ) ) ) ) ).

% gcd_int_def
thf(fact_8193_one__rat__def,axiom,
    ( one_one_rat
    = ( abs_Rat @ ( product_Pair_int_int @ one_one_int @ one_one_int ) ) ) ).

% one_rat_def
thf(fact_8194_plus__rat_Oabs__eq,axiom,
    ! [Xa3: product_prod_int_int,X: product_prod_int_int] :
      ( ( ratrel @ Xa3 @ Xa3 )
     => ( ( ratrel @ X @ X )
       => ( ( plus_plus_rat @ ( abs_Rat @ Xa3 ) @ ( abs_Rat @ X ) )
          = ( abs_Rat @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ Xa3 ) @ ( product_snd_int_int @ X ) ) @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ Xa3 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ Xa3 ) @ ( product_snd_int_int @ X ) ) ) ) ) ) ) ).

% plus_rat.abs_eq
thf(fact_8195_pair__leqI2,axiom,
    ! [A: nat,B: nat,S: nat,T: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ S @ T )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B @ T ) ) @ fun_pair_leq ) ) ) ).

% pair_leqI2
thf(fact_8196_pair__leqI1,axiom,
    ! [A: nat,B: nat,S: nat,T: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B @ T ) ) @ fun_pair_leq ) ) ).

% pair_leqI1
thf(fact_8197_smax__insertI,axiom,
    ! [Y3: product_prod_nat_nat,Y7: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,X8: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ Y3 @ Y7 )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y3 ) @ fun_pair_less )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X8 @ Y7 ) @ fun_max_strict )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( insert8211810215607154385at_nat @ X @ X8 ) @ Y7 ) @ fun_max_strict ) ) ) ) ).

% smax_insertI
thf(fact_8198_smin__insertI,axiom,
    ! [X: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat,Y3: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ X @ XS )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y3 ) @ fun_pair_less )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ YS ) @ fun_min_strict )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ ( insert8211810215607154385at_nat @ Y3 @ YS ) ) @ fun_min_strict ) ) ) ) ).

% smin_insertI
thf(fact_8199_ratrel__iff,axiom,
    ( ratrel
    = ( ^ [X2: product_prod_int_int,Y6: product_prod_int_int] :
          ( ( ( product_snd_int_int @ X2 )
           != zero_zero_int )
          & ( ( product_snd_int_int @ Y6 )
           != zero_zero_int )
          & ( ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_snd_int_int @ Y6 ) )
            = ( times_times_int @ ( product_fst_int_int @ Y6 ) @ ( product_snd_int_int @ X2 ) ) ) ) ) ) ).

% ratrel_iff
thf(fact_8200_smin__emptyI,axiom,
    ! [X8: set_Pr1261947904930325089at_nat] :
      ( ( X8 != bot_bo2099793752762293965at_nat )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X8 @ bot_bo2099793752762293965at_nat ) @ fun_min_strict ) ) ).

% smin_emptyI
thf(fact_8201_one__rat_Orsp,axiom,
    ratrel @ ( product_Pair_int_int @ one_one_int @ one_one_int ) @ ( product_Pair_int_int @ one_one_int @ one_one_int ) ).

% one_rat.rsp
thf(fact_8202_zero__rat_Orsp,axiom,
    ratrel @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ).

% zero_rat.rsp
thf(fact_8203_times__rat_Oabs__eq,axiom,
    ! [Xa3: product_prod_int_int,X: product_prod_int_int] :
      ( ( ratrel @ Xa3 @ Xa3 )
     => ( ( ratrel @ X @ X )
       => ( ( times_times_rat @ ( abs_Rat @ Xa3 ) @ ( abs_Rat @ X ) )
          = ( abs_Rat @ ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ Xa3 ) @ ( product_fst_int_int @ X ) ) @ ( times_times_int @ ( product_snd_int_int @ Xa3 ) @ ( product_snd_int_int @ X ) ) ) ) ) ) ) ).

% times_rat.abs_eq
thf(fact_8204_uminus__rat_Oabs__eq,axiom,
    ! [X: product_prod_int_int] :
      ( ( ratrel @ X @ X )
     => ( ( uminus_uminus_rat @ ( abs_Rat @ X ) )
        = ( abs_Rat @ ( product_Pair_int_int @ ( uminus_uminus_int @ ( product_fst_int_int @ X ) ) @ ( product_snd_int_int @ X ) ) ) ) ) ).

% uminus_rat.abs_eq
thf(fact_8205_wmin__insertI,axiom,
    ! [X: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat,Y3: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ X @ XS )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y3 ) @ fun_pair_leq )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ YS ) @ fun_min_weak )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ ( insert8211810215607154385at_nat @ Y3 @ YS ) ) @ fun_min_weak ) ) ) ) ).

% wmin_insertI
thf(fact_8206_wmax__insertI,axiom,
    ! [Y3: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ Y3 @ YS )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y3 ) @ fun_pair_leq )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ YS ) @ fun_max_weak )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( insert8211810215607154385at_nat @ X @ XS ) @ YS ) @ fun_max_weak ) ) ) ) ).

% wmax_insertI
thf(fact_8207_positive_Oabs__eq,axiom,
    ! [X: product_prod_int_int] :
      ( ( ratrel @ X @ X )
     => ( ( positive @ ( abs_Rat @ X ) )
        = ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ X ) ) ) ) ) ).

% positive.abs_eq
thf(fact_8208_upto_Opelims,axiom,
    ! [X: int,Xa3: int,Y3: list_int] :
      ( ( ( upto @ X @ Xa3 )
        = Y3 )
     => ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X @ Xa3 ) )
       => ~ ( ( ( ( ord_less_eq_int @ X @ Xa3 )
               => ( Y3
                  = ( cons_int @ X @ ( upto @ ( plus_plus_int @ X @ one_one_int ) @ Xa3 ) ) ) )
              & ( ~ ( ord_less_eq_int @ X @ Xa3 )
               => ( Y3 = nil_int ) ) )
           => ~ ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X @ Xa3 ) ) ) ) ) ).

% upto.pelims
thf(fact_8209_upto_Opsimps,axiom,
    ! [I2: int,J: int] :
      ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I2 @ J ) )
     => ( ( ( ord_less_eq_int @ I2 @ J )
         => ( ( upto @ I2 @ J )
            = ( cons_int @ I2 @ ( upto @ ( plus_plus_int @ I2 @ one_one_int ) @ J ) ) ) )
        & ( ~ ( ord_less_eq_int @ I2 @ J )
         => ( ( upto @ I2 @ J )
            = nil_int ) ) ) ) ).

% upto.psimps
thf(fact_8210_upto__Nil,axiom,
    ! [I2: int,J: int] :
      ( ( ( upto @ I2 @ J )
        = nil_int )
      = ( ord_less_int @ J @ I2 ) ) ).

% upto_Nil
thf(fact_8211_upto__Nil2,axiom,
    ! [I2: int,J: int] :
      ( ( nil_int
        = ( upto @ I2 @ J ) )
      = ( ord_less_int @ J @ I2 ) ) ).

% upto_Nil2
thf(fact_8212_upto__empty,axiom,
    ! [J: int,I2: int] :
      ( ( ord_less_int @ J @ I2 )
     => ( ( upto @ I2 @ J )
        = nil_int ) ) ).

% upto_empty
thf(fact_8213_nth__upto,axiom,
    ! [I2: int,K: nat,J: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ I2 @ ( semiri1314217659103216013at_int @ K ) ) @ J )
     => ( ( nth_int @ ( upto @ I2 @ J ) @ K )
        = ( plus_plus_int @ I2 @ ( semiri1314217659103216013at_int @ K ) ) ) ) ).

% nth_upto
thf(fact_8214_length__upto,axiom,
    ! [I2: int,J: int] :
      ( ( size_size_list_int @ ( upto @ I2 @ J ) )
      = ( nat2 @ ( plus_plus_int @ ( minus_minus_int @ J @ I2 ) @ one_one_int ) ) ) ).

% length_upto
thf(fact_8215_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_8216_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_8217_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_8218_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_8219_wmin__emptyI,axiom,
    ! [X8: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X8 @ bot_bo2099793752762293965at_nat ) @ fun_min_weak ) ).

% wmin_emptyI
thf(fact_8220_positive__minus,axiom,
    ! [X: rat] :
      ( ~ ( positive @ X )
     => ( ( X != zero_zero_rat )
       => ( positive @ ( uminus_uminus_rat @ X ) ) ) ) ).

% positive_minus
thf(fact_8221_greaterThanAtMost__upto,axiom,
    ( set_or6656581121297822940st_int
    = ( ^ [I4: int,J3: int] : ( set_int2 @ ( upto @ ( plus_plus_int @ I4 @ one_one_int ) @ J3 ) ) ) ) ).

% greaterThanAtMost_upto
thf(fact_8222_less__rat__def,axiom,
    ( ord_less_rat
    = ( ^ [X2: rat,Y6: rat] : ( positive @ ( minus_minus_rat @ Y6 @ X2 ) ) ) ) ).

% less_rat_def
thf(fact_8223_upto__rec1,axiom,
    ! [I2: int,J: int] :
      ( ( ord_less_eq_int @ I2 @ J )
     => ( ( upto @ I2 @ J )
        = ( cons_int @ I2 @ ( upto @ ( plus_plus_int @ I2 @ one_one_int ) @ J ) ) ) ) ).

% upto_rec1
thf(fact_8224_upto_Oelims,axiom,
    ! [X: int,Xa3: int,Y3: list_int] :
      ( ( ( upto @ X @ Xa3 )
        = Y3 )
     => ( ( ( ord_less_eq_int @ X @ Xa3 )
         => ( Y3
            = ( cons_int @ X @ ( upto @ ( plus_plus_int @ X @ one_one_int ) @ Xa3 ) ) ) )
        & ( ~ ( ord_less_eq_int @ X @ Xa3 )
         => ( Y3 = nil_int ) ) ) ) ).

% upto.elims
thf(fact_8225_upto_Osimps,axiom,
    ( upto
    = ( ^ [I4: int,J3: int] : ( if_list_int @ ( ord_less_eq_int @ I4 @ J3 ) @ ( cons_int @ I4 @ ( upto @ ( plus_plus_int @ I4 @ one_one_int ) @ J3 ) ) @ nil_int ) ) ) ).

% upto.simps
thf(fact_8226_greaterThanLessThan__upto,axiom,
    ( set_or5832277885323065728an_int
    = ( ^ [I4: int,J3: int] : ( set_int2 @ ( upto @ ( plus_plus_int @ I4 @ one_one_int ) @ ( minus_minus_int @ J3 @ one_one_int ) ) ) ) ) ).

% greaterThanLessThan_upto
thf(fact_8227_positive_Orep__eq,axiom,
    ( positive
    = ( ^ [X2: rat] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ ( rep_Rat @ X2 ) ) @ ( product_snd_int_int @ ( rep_Rat @ X2 ) ) ) ) ) ) ).

% positive.rep_eq
thf(fact_8228_inverse__rat_Oabs__eq,axiom,
    ! [X: product_prod_int_int] :
      ( ( ratrel @ X @ X )
     => ( ( inverse_inverse_rat @ ( abs_Rat @ X ) )
        = ( abs_Rat
          @ ( if_Pro3027730157355071871nt_int
            @ ( ( product_fst_int_int @ X )
              = zero_zero_int )
            @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
            @ ( product_Pair_int_int @ ( product_snd_int_int @ X ) @ ( product_fst_int_int @ X ) ) ) ) ) ) ).

% inverse_rat.abs_eq
thf(fact_8229_min__rpair__set,axiom,
    fun_re2478310338295953701at_nat @ ( produc9060074326276436823at_nat @ fun_min_strict @ fun_min_weak ) ).

% min_rpair_set
thf(fact_8230_max__rpair__set,axiom,
    fun_re2478310338295953701at_nat @ ( produc9060074326276436823at_nat @ fun_max_strict @ fun_max_weak ) ).

% max_rpair_set
thf(fact_8231_upto__rec2,axiom,
    ! [I2: int,J: int] :
      ( ( ord_less_eq_int @ I2 @ J )
     => ( ( upto @ I2 @ J )
        = ( append_int @ ( upto @ I2 @ ( minus_minus_int @ J @ one_one_int ) ) @ ( cons_int @ J @ nil_int ) ) ) ) ).

% upto_rec2
thf(fact_8232_upto__split2,axiom,
    ! [I2: int,J: int,K: int] :
      ( ( ord_less_eq_int @ I2 @ J )
     => ( ( ord_less_eq_int @ J @ K )
       => ( ( upto @ I2 @ K )
          = ( append_int @ ( upto @ I2 @ J ) @ ( upto @ ( plus_plus_int @ J @ one_one_int ) @ K ) ) ) ) ) ).

% upto_split2
thf(fact_8233_upto__split1,axiom,
    ! [I2: int,J: int,K: int] :
      ( ( ord_less_eq_int @ I2 @ J )
     => ( ( ord_less_eq_int @ J @ K )
       => ( ( upto @ I2 @ K )
          = ( append_int @ ( upto @ I2 @ ( minus_minus_int @ J @ one_one_int ) ) @ ( upto @ J @ K ) ) ) ) ) ).

% upto_split1
thf(fact_8234_upto__split3,axiom,
    ! [I2: int,J: int,K: int] :
      ( ( ord_less_eq_int @ I2 @ J )
     => ( ( ord_less_eq_int @ J @ K )
       => ( ( upto @ I2 @ K )
          = ( append_int @ ( upto @ I2 @ ( minus_minus_int @ J @ one_one_int ) ) @ ( cons_int @ J @ ( upto @ ( plus_plus_int @ J @ one_one_int ) @ K ) ) ) ) ) ) ).

% upto_split3
thf(fact_8235_list__encode_Opelims,axiom,
    ! [X: list_nat,Y3: nat] :
      ( ( ( nat_list_encode @ X )
        = Y3 )
     => ( ( accp_list_nat @ nat_list_encode_rel @ X )
       => ( ( ( X = nil_nat )
           => ( ( Y3 = zero_zero_nat )
             => ~ ( accp_list_nat @ nat_list_encode_rel @ nil_nat ) ) )
         => ~ ! [X3: nat,Xs2: list_nat] :
                ( ( X
                  = ( cons_nat @ X3 @ Xs2 ) )
               => ( ( Y3
                    = ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X3 @ ( nat_list_encode @ Xs2 ) ) ) ) )
                 => ~ ( accp_list_nat @ nat_list_encode_rel @ ( cons_nat @ X3 @ Xs2 ) ) ) ) ) ) ) ).

% list_encode.pelims
thf(fact_8236_min__strict__def,axiom,
    ( fun_min_strict
    = ( min_ex6901939911449802026at_nat @ fun_pair_less ) ) ).

% min_strict_def
thf(fact_8237_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_8238_take__upt,axiom,
    ! [I2: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ M ) @ N )
     => ( ( take_nat @ M @ ( upt @ I2 @ N ) )
        = ( upt @ I2 @ ( plus_plus_nat @ I2 @ M ) ) ) ) ).

% take_upt
thf(fact_8239_upt__conv__Nil,axiom,
    ! [J: nat,I2: nat] :
      ( ( ord_less_eq_nat @ J @ I2 )
     => ( ( upt @ I2 @ J )
        = nil_nat ) ) ).

% upt_conv_Nil
thf(fact_8240_upt__eq__Nil__conv,axiom,
    ! [I2: nat,J: nat] :
      ( ( ( upt @ I2 @ J )
        = nil_nat )
      = ( ( J = zero_zero_nat )
        | ( ord_less_eq_nat @ J @ I2 ) ) ) ).

% upt_eq_Nil_conv
thf(fact_8241_nth__upt,axiom,
    ! [I2: nat,K: nat,J: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ J )
     => ( ( nth_nat @ ( upt @ I2 @ J ) @ K )
        = ( plus_plus_nat @ I2 @ K ) ) ) ).

% nth_upt
thf(fact_8242_upt__add__eq__append,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I2 @ J )
     => ( ( upt @ I2 @ ( plus_plus_nat @ J @ K ) )
        = ( append_nat @ ( upt @ I2 @ J ) @ ( upt @ J @ ( plus_plus_nat @ J @ K ) ) ) ) ) ).

% upt_add_eq_append
thf(fact_8243_atLeastAtMost__upt,axiom,
    ( set_or1269000886237332187st_nat
    = ( ^ [N3: nat,M2: nat] : ( set_nat2 @ ( upt @ N3 @ ( suc @ M2 ) ) ) ) ) ).

% atLeastAtMost_upt
thf(fact_8244_upt__Suc,axiom,
    ! [I2: nat,J: nat] :
      ( ( ( ord_less_eq_nat @ I2 @ J )
       => ( ( upt @ I2 @ ( suc @ J ) )
          = ( append_nat @ ( upt @ I2 @ J ) @ ( cons_nat @ J @ nil_nat ) ) ) )
      & ( ~ ( ord_less_eq_nat @ I2 @ J )
       => ( ( upt @ I2 @ ( suc @ J ) )
          = nil_nat ) ) ) ).

% upt_Suc
thf(fact_8245_upt__Suc__append,axiom,
    ! [I2: nat,J: nat] :
      ( ( ord_less_eq_nat @ I2 @ J )
     => ( ( upt @ I2 @ ( suc @ J ) )
        = ( append_nat @ ( upt @ I2 @ J ) @ ( cons_nat @ J @ nil_nat ) ) ) ) ).

% upt_Suc_append
thf(fact_8246_upt__conv__Cons__Cons,axiom,
    ! [M: nat,N: nat,Ns: list_nat,Q4: nat] :
      ( ( ( cons_nat @ M @ ( cons_nat @ N @ Ns ) )
        = ( upt @ M @ Q4 ) )
      = ( ( cons_nat @ N @ Ns )
        = ( upt @ ( suc @ M ) @ Q4 ) ) ) ).

% upt_conv_Cons_Cons
thf(fact_8247_upt__0,axiom,
    ! [I2: nat] :
      ( ( upt @ I2 @ zero_zero_nat )
      = nil_nat ) ).

% upt_0
thf(fact_8248_greaterThanAtMost__upt,axiom,
    ( set_or6659071591806873216st_nat
    = ( ^ [N3: nat,M2: nat] : ( set_nat2 @ ( upt @ ( suc @ N3 ) @ ( suc @ M2 ) ) ) ) ) ).

% greaterThanAtMost_upt
thf(fact_8249_greaterThanLessThan__upt,axiom,
    ( set_or5834768355832116004an_nat
    = ( ^ [N3: nat,M2: nat] : ( set_nat2 @ ( upt @ ( suc @ N3 ) @ M2 ) ) ) ) ).

% greaterThanLessThan_upt
thf(fact_8250_upt__conv__Cons,axiom,
    ! [I2: nat,J: nat] :
      ( ( ord_less_nat @ I2 @ J )
     => ( ( upt @ I2 @ J )
        = ( cons_nat @ I2 @ ( upt @ ( suc @ I2 ) @ J ) ) ) ) ).

% upt_conv_Cons
thf(fact_8251_upt__eq__Cons__conv,axiom,
    ! [I2: nat,J: nat,X: nat,Xs: list_nat] :
      ( ( ( upt @ I2 @ J )
        = ( cons_nat @ X @ Xs ) )
      = ( ( ord_less_nat @ I2 @ J )
        & ( I2 = X )
        & ( ( upt @ ( plus_plus_nat @ I2 @ one_one_nat ) @ J )
          = Xs ) ) ) ).

% upt_eq_Cons_conv
thf(fact_8252_upt__rec,axiom,
    ( upt
    = ( ^ [I4: nat,J3: nat] : ( if_list_nat @ ( ord_less_nat @ I4 @ J3 ) @ ( cons_nat @ I4 @ ( upt @ ( suc @ I4 ) @ J3 ) ) @ nil_nat ) ) ) ).

% upt_rec
thf(fact_8253_min__weak__def,axiom,
    ( fun_min_weak
    = ( sup_su5525570899277871387at_nat @ ( min_ex6901939911449802026at_nat @ fun_pair_leq ) @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ bot_bo2099793752762293965at_nat @ bot_bo2099793752762293965at_nat ) @ bot_bo228742789529271731at_nat ) ) ) ).

% min_weak_def
thf(fact_8254_max__weak__def,axiom,
    ( fun_max_weak
    = ( sup_su5525570899277871387at_nat @ ( max_ex8135407076693332796at_nat @ fun_pair_leq ) @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ bot_bo2099793752762293965at_nat @ bot_bo2099793752762293965at_nat ) @ bot_bo228742789529271731at_nat ) ) ) ).

% max_weak_def
thf(fact_8255_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_8256_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_8257_lessThan__0,axiom,
    ( ( set_ord_lessThan_nat @ zero_zero_nat )
    = bot_bot_set_nat ) ).

% lessThan_0
thf(fact_8258_card__atMost,axiom,
    ! [U: nat] :
      ( ( finite_card_nat @ ( set_ord_atMost_nat @ U ) )
      = ( suc @ U ) ) ).

% card_atMost
thf(fact_8259_atMost__0,axiom,
    ( ( set_ord_atMost_nat @ zero_zero_nat )
    = ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ).

% atMost_0
thf(fact_8260_sup__nat__def,axiom,
    sup_sup_nat = ord_max_nat ).

% sup_nat_def
thf(fact_8261_lessThan__Suc__atMost,axiom,
    ! [K: nat] :
      ( ( set_ord_lessThan_nat @ ( suc @ K ) )
      = ( set_ord_atMost_nat @ K ) ) ).

% lessThan_Suc_atMost
thf(fact_8262_sup__int__def,axiom,
    sup_sup_int = ord_max_int ).

% sup_int_def
thf(fact_8263_atMost__atLeast0,axiom,
    ( set_ord_atMost_nat
    = ( set_or1269000886237332187st_nat @ zero_zero_nat ) ) ).

% atMost_atLeast0
thf(fact_8264_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_8265_lessThan__Suc,axiom,
    ! [K: nat] :
      ( ( set_ord_lessThan_nat @ ( suc @ K ) )
      = ( insert_nat @ K @ ( set_ord_lessThan_nat @ K ) ) ) ).

% lessThan_Suc
thf(fact_8266_lessThan__empty__iff,axiom,
    ! [N: nat] :
      ( ( ( set_ord_lessThan_nat @ N )
        = bot_bot_set_nat )
      = ( N = zero_zero_nat ) ) ).

% lessThan_empty_iff
thf(fact_8267_atMost__Suc,axiom,
    ! [K: nat] :
      ( ( set_ord_atMost_nat @ ( suc @ K ) )
      = ( insert_nat @ ( suc @ K ) @ ( set_ord_atMost_nat @ K ) ) ) ).

% atMost_Suc
thf(fact_8268_atLeast__upt,axiom,
    ( set_ord_lessThan_nat
    = ( ^ [N3: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ N3 ) ) ) ) ).

% atLeast_upt
thf(fact_8269_atMost__upto,axiom,
    ( set_ord_atMost_nat
    = ( ^ [N3: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ ( suc @ N3 ) ) ) ) ) ).

% atMost_upto
thf(fact_8270_max__strict__def,axiom,
    ( fun_max_strict
    = ( max_ex8135407076693332796at_nat @ fun_pair_less ) ) ).

% max_strict_def
thf(fact_8271_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_8272_hd__upt,axiom,
    ! [I2: nat,J: nat] :
      ( ( ord_less_nat @ I2 @ J )
     => ( ( hd_nat @ ( upt @ I2 @ J ) )
        = I2 ) ) ).

% hd_upt
thf(fact_8273_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_8274_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_8275_atLeastLessThan__singleton,axiom,
    ! [M: nat] :
      ( ( set_or4665077453230672383an_nat @ M @ ( suc @ M ) )
      = ( insert_nat @ M @ bot_bot_set_nat ) ) ).

% atLeastLessThan_singleton
thf(fact_8276_finite__nat__set__iff__bounded,axiom,
    ( finite_finite_nat
    = ( ^ [N6: set_nat] :
        ? [M2: nat] :
        ! [X2: nat] :
          ( ( member_nat @ X2 @ N6 )
         => ( ord_less_nat @ X2 @ M2 ) ) ) ) ).

% finite_nat_set_iff_bounded
thf(fact_8277_bounded__nat__set__is__finite,axiom,
    ! [N5: set_nat,N: nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ N5 )
         => ( ord_less_nat @ X3 @ N ) )
     => ( finite_finite_nat @ N5 ) ) ).

% bounded_nat_set_is_finite
thf(fact_8278_finite__nat__set__iff__bounded__le,axiom,
    ( finite_finite_nat
    = ( ^ [N6: set_nat] :
        ? [M2: nat] :
        ! [X2: nat] :
          ( ( member_nat @ X2 @ N6 )
         => ( ord_less_eq_nat @ X2 @ M2 ) ) ) ) ).

% finite_nat_set_iff_bounded_le
thf(fact_8279_subset__eq__atLeast0__lessThan__finite,axiom,
    ! [N5: set_nat,N: nat] :
      ( ( ord_less_eq_set_nat @ N5 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
     => ( finite_finite_nat @ N5 ) ) ).

% subset_eq_atLeast0_lessThan_finite
thf(fact_8280_finite__set__decode,axiom,
    ! [N: nat] : ( finite_finite_nat @ ( nat_set_decode @ N ) ) ).

% finite_set_decode
thf(fact_8281_ex__nat__less__eq,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ? [M2: nat] :
            ( ( ord_less_nat @ M2 @ N )
            & ( P @ M2 ) ) )
      = ( ? [X2: nat] :
            ( ( member_nat @ X2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
            & ( P @ X2 ) ) ) ) ).

% ex_nat_less_eq
thf(fact_8282_all__nat__less__eq,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ! [M2: nat] :
            ( ( ord_less_nat @ M2 @ N )
           => ( P @ M2 ) ) )
      = ( ! [X2: nat] :
            ( ( member_nat @ X2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
           => ( P @ X2 ) ) ) ) ).

% all_nat_less_eq
thf(fact_8283_atLeastLessThan__add__Un,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I2 @ J )
     => ( ( set_or4665077453230672383an_nat @ I2 @ ( plus_plus_nat @ J @ K ) )
        = ( sup_sup_set_nat @ ( set_or4665077453230672383an_nat @ I2 @ J ) @ ( set_or4665077453230672383an_nat @ J @ ( plus_plus_nat @ J @ K ) ) ) ) ) ).

% atLeastLessThan_add_Un
thf(fact_8284_atLeastLessThanSuc__atLeastAtMost,axiom,
    ! [L: nat,U: nat] :
      ( ( set_or4665077453230672383an_nat @ L @ ( suc @ U ) )
      = ( set_or1269000886237332187st_nat @ L @ U ) ) ).

% atLeastLessThanSuc_atLeastAtMost
thf(fact_8285_lessThan__atLeast0,axiom,
    ( set_ord_lessThan_nat
    = ( set_or4665077453230672383an_nat @ zero_zero_nat ) ) ).

% lessThan_atLeast0
thf(fact_8286_atLeastLessThan0,axiom,
    ! [M: nat] :
      ( ( set_or4665077453230672383an_nat @ M @ zero_zero_nat )
      = bot_bot_set_nat ) ).

% atLeastLessThan0
thf(fact_8287_atLeastSucLessThan__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] :
      ( ( set_or4665077453230672383an_nat @ ( suc @ L ) @ U )
      = ( set_or5834768355832116004an_nat @ L @ U ) ) ).

% atLeastSucLessThan_greaterThanLessThan
thf(fact_8288_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_8289_subset__eq__atLeast0__atMost__finite,axiom,
    ! [N5: set_nat,N: nat] :
      ( ( ord_less_eq_set_nat @ N5 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
     => ( finite_finite_nat @ N5 ) ) ).

% subset_eq_atLeast0_atMost_finite
thf(fact_8290_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_8291_subset__eq__atLeast0__lessThan__card,axiom,
    ! [N5: set_nat,N: nat] :
      ( ( ord_less_eq_set_nat @ N5 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
     => ( ord_less_eq_nat @ ( finite_card_nat @ N5 ) @ N ) ) ).

% subset_eq_atLeast0_lessThan_card
thf(fact_8292_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_8293_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_8294_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_8295_set__decode__inverse,axiom,
    ! [N: nat] :
      ( ( nat_set_encode @ ( nat_set_decode @ N ) )
      = N ) ).

% set_decode_inverse
thf(fact_8296_set__encode__empty,axiom,
    ( ( nat_set_encode @ bot_bot_set_nat )
    = zero_zero_nat ) ).

% set_encode_empty
thf(fact_8297_set__encode__inverse,axiom,
    ! [A3: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( nat_set_decode @ ( nat_set_encode @ A3 ) )
        = A3 ) ) ).

% set_encode_inverse
thf(fact_8298_finite__atLeastZeroLessThan__int,axiom,
    ! [U: int] : ( finite_finite_int @ ( set_or4662586982721622107an_int @ zero_zero_int @ U ) ) ).

% finite_atLeastZeroLessThan_int
thf(fact_8299_set__encode__eq,axiom,
    ! [A3: set_nat,B5: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( finite_finite_nat @ B5 )
       => ( ( ( nat_set_encode @ A3 )
            = ( nat_set_encode @ B5 ) )
          = ( A3 = B5 ) ) ) ) ).

% set_encode_eq
thf(fact_8300_atLeastLessThanPlusOne__atLeastAtMost__int,axiom,
    ! [L: int,U: int] :
      ( ( set_or4662586982721622107an_int @ L @ ( plus_plus_int @ U @ one_one_int ) )
      = ( set_or1266510415728281911st_int @ L @ U ) ) ).

% atLeastLessThanPlusOne_atLeastAtMost_int
thf(fact_8301_card__atLeastZeroLessThan__int,axiom,
    ! [U: int] :
      ( ( finite_card_int @ ( set_or4662586982721622107an_int @ zero_zero_int @ U ) )
      = ( nat2 @ U ) ) ).

% card_atLeastZeroLessThan_int
thf(fact_8302_atLeastPlusOneLessThan__greaterThanLessThan__int,axiom,
    ! [L: int,U: int] :
      ( ( set_or4662586982721622107an_int @ ( plus_plus_int @ L @ one_one_int ) @ U )
      = ( set_or5832277885323065728an_int @ L @ U ) ) ).

% atLeastPlusOneLessThan_greaterThanLessThan_int
thf(fact_8303_set__encode__inf,axiom,
    ! [A3: set_nat] :
      ( ~ ( finite_finite_nat @ A3 )
     => ( ( nat_set_encode @ A3 )
        = zero_zero_nat ) ) ).

% set_encode_inf
thf(fact_8304_atLeastLessThan__upto,axiom,
    ( set_or4662586982721622107an_int
    = ( ^ [I4: int,J3: int] : ( set_int2 @ ( upto @ I4 @ ( minus_minus_int @ J3 @ one_one_int ) ) ) ) ) ).

% atLeastLessThan_upto
thf(fact_8305_wmax__emptyI,axiom,
    ! [X8: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ X8 )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ bot_bo2099793752762293965at_nat @ X8 ) @ fun_max_weak ) ) ).

% wmax_emptyI
thf(fact_8306_smax__emptyI,axiom,
    ! [Y7: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ Y7 )
     => ( ( Y7 != bot_bo2099793752762293965at_nat )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ bot_bo2099793752762293965at_nat @ Y7 ) @ fun_max_strict ) ) ) ).

% smax_emptyI
thf(fact_8307_image__Suc__atLeastAtMost,axiom,
    ! [I2: nat,J: nat] :
      ( ( image_nat_nat @ suc @ ( set_or1269000886237332187st_nat @ I2 @ J ) )
      = ( set_or1269000886237332187st_nat @ ( suc @ I2 ) @ ( suc @ J ) ) ) ).

% image_Suc_atLeastAtMost
thf(fact_8308_image__Suc__atLeastLessThan,axiom,
    ! [I2: nat,J: nat] :
      ( ( image_nat_nat @ suc @ ( set_or4665077453230672383an_nat @ I2 @ J ) )
      = ( set_or4665077453230672383an_nat @ ( suc @ I2 ) @ ( suc @ J ) ) ) ).

% image_Suc_atLeastLessThan
thf(fact_8309_zero__notin__Suc__image,axiom,
    ! [A3: set_nat] :
      ~ ( member_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ A3 ) ) ).

% zero_notin_Suc_image
thf(fact_8310_image__int__atLeastLessThan,axiom,
    ! [A: nat,B: nat] :
      ( ( image_nat_int @ semiri1314217659103216013at_int @ ( set_or4665077453230672383an_nat @ A @ B ) )
      = ( set_or4662586982721622107an_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).

% image_int_atLeastLessThan
thf(fact_8311_image__int__atLeastAtMost,axiom,
    ! [A: nat,B: nat] :
      ( ( image_nat_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).

% image_int_atLeastAtMost
thf(fact_8312_prod__Suc__fact,axiom,
    ! [N: nat] :
      ( ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
      = ( semiri1408675320244567234ct_nat @ N ) ) ).

% prod_Suc_fact
thf(fact_8313_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_8314_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_8315_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_8316_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_8317_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_8318_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_8319_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_8320_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_8321_inf__nat__def,axiom,
    inf_inf_nat = ord_min_nat ).

% inf_nat_def
thf(fact_8322_inf__int__def,axiom,
    inf_inf_int = ord_min_int ).

% inf_int_def
thf(fact_8323_finite__Collect__less__nat,axiom,
    ! [K: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [N3: nat] : ( ord_less_nat @ N3 @ K ) ) ) ).

% finite_Collect_less_nat
thf(fact_8324_finite__Collect__le__nat,axiom,
    ! [K: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [N3: nat] : ( ord_less_eq_nat @ N3 @ K ) ) ) ).

% finite_Collect_le_nat
thf(fact_8325_card__Collect__less__nat,axiom,
    ! [N: nat] :
      ( ( finite_card_nat
        @ ( collect_nat
          @ ^ [I4: nat] : ( ord_less_nat @ I4 @ N ) ) )
      = N ) ).

% card_Collect_less_nat
thf(fact_8326_finite__interval__int1,axiom,
    ! [A: int,B: int] :
      ( finite_finite_int
      @ ( collect_int
        @ ^ [I4: int] :
            ( ( ord_less_eq_int @ A @ I4 )
            & ( ord_less_eq_int @ I4 @ B ) ) ) ) ).

% finite_interval_int1
thf(fact_8327_finite__interval__int4,axiom,
    ! [A: int,B: int] :
      ( finite_finite_int
      @ ( collect_int
        @ ^ [I4: int] :
            ( ( ord_less_int @ A @ I4 )
            & ( ord_less_int @ I4 @ B ) ) ) ) ).

% finite_interval_int4
thf(fact_8328_card__Collect__le__nat,axiom,
    ! [N: nat] :
      ( ( finite_card_nat
        @ ( collect_nat
          @ ^ [I4: nat] : ( ord_less_eq_nat @ I4 @ N ) ) )
      = ( suc @ N ) ) ).

% card_Collect_le_nat
thf(fact_8329_finite__interval__int2,axiom,
    ! [A: int,B: int] :
      ( finite_finite_int
      @ ( collect_int
        @ ^ [I4: int] :
            ( ( ord_less_eq_int @ A @ I4 )
            & ( ord_less_int @ I4 @ B ) ) ) ) ).

% finite_interval_int2
thf(fact_8330_finite__interval__int3,axiom,
    ! [A: int,B: int] :
      ( finite_finite_int
      @ ( collect_int
        @ ^ [I4: int] :
            ( ( ord_less_int @ A @ I4 )
            & ( ord_less_eq_int @ I4 @ B ) ) ) ) ).

% finite_interval_int3
thf(fact_8331_Gcd__int__eq,axiom,
    ! [N5: set_nat] :
      ( ( gcd_Gcd_int @ ( image_nat_int @ semiri1314217659103216013at_int @ N5 ) )
      = ( semiri1314217659103216013at_int @ ( gcd_Gcd_nat @ N5 ) ) ) ).

% Gcd_int_eq
thf(fact_8332_prod__int__eq,axiom,
    ! [I2: nat,J: nat] :
      ( ( groups705719431365010083at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ I2 @ J ) )
      = ( groups1705073143266064639nt_int
        @ ^ [X2: int] : X2
        @ ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ I2 ) @ ( semiri1314217659103216013at_int @ J ) ) ) ) ).

% prod_int_eq
thf(fact_8333_nat__less__as__int,axiom,
    ( ord_less_nat
    = ( ^ [A2: nat,B2: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).

% nat_less_as_int
thf(fact_8334_mono__Suc,axiom,
    order_mono_nat_nat @ suc ).

% mono_Suc
thf(fact_8335_nat__leq__as__int,axiom,
    ( ord_less_eq_nat
    = ( ^ [A2: nat,B2: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).

% nat_leq_as_int
thf(fact_8336_finite__less__ub,axiom,
    ! [F: nat > nat,U: nat] :
      ( ! [N4: nat] : ( ord_less_eq_nat @ N4 @ ( F @ N4 ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ U ) ) ) ) ).

% finite_less_ub
thf(fact_8337_finite__M__bounded__by__nat,axiom,
    ! [P: nat > $o,I2: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [K3: nat] :
            ( ( P @ K3 )
            & ( ord_less_nat @ K3 @ I2 ) ) ) ) ).

% finite_M_bounded_by_nat
thf(fact_8338_finite__divisors__int,axiom,
    ! [I2: int] :
      ( ( I2 != zero_zero_int )
     => ( finite_finite_int
        @ ( collect_int
          @ ^ [D4: int] : ( dvd_dvd_int @ D4 @ I2 ) ) ) ) ).

% finite_divisors_int
thf(fact_8339_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_8340_mono__ge2__power__minus__self,axiom,
    ! [K: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
     => ( order_mono_nat_nat
        @ ^ [M2: nat] : ( minus_minus_nat @ ( power_power_nat @ K @ M2 ) @ M2 ) ) ) ).

% mono_ge2_power_minus_self
thf(fact_8341_card__length__sum__list__rec,axiom,
    ! [M: nat,N5: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ M )
     => ( ( finite_card_list_nat
          @ ( collect_list_nat
            @ ^ [L3: list_nat] :
                ( ( ( size_size_list_nat @ L3 )
                  = M )
                & ( ( groups4561878855575611511st_nat @ L3 )
                  = N5 ) ) ) )
        = ( plus_plus_nat
          @ ( finite_card_list_nat
            @ ( collect_list_nat
              @ ^ [L3: list_nat] :
                  ( ( ( size_size_list_nat @ L3 )
                    = ( minus_minus_nat @ M @ one_one_nat ) )
                  & ( ( groups4561878855575611511st_nat @ L3 )
                    = N5 ) ) ) )
          @ ( finite_card_list_nat
            @ ( collect_list_nat
              @ ^ [L3: list_nat] :
                  ( ( ( size_size_list_nat @ L3 )
                    = M )
                  & ( ( plus_plus_nat @ ( groups4561878855575611511st_nat @ L3 ) @ one_one_nat )
                    = N5 ) ) ) ) ) ) ) ).

% card_length_sum_list_rec
thf(fact_8342_nat__plus__as__int,axiom,
    ( plus_plus_nat
    = ( ^ [A2: nat,B2: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ) ).

% nat_plus_as_int
thf(fact_8343_nat__times__as__int,axiom,
    ( times_times_nat
    = ( ^ [A2: nat,B2: nat] : ( nat2 @ ( times_times_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ) ).

% nat_times_as_int
thf(fact_8344_card__length__sum__list,axiom,
    ! [M: nat,N5: nat] :
      ( ( finite_card_list_nat
        @ ( collect_list_nat
          @ ^ [L3: list_nat] :
              ( ( ( size_size_list_nat @ L3 )
                = M )
              & ( ( groups4561878855575611511st_nat @ L3 )
                = N5 ) ) ) )
      = ( binomial @ ( minus_minus_nat @ ( plus_plus_nat @ N5 @ M ) @ one_one_nat ) @ N5 ) ) ).

% card_length_sum_list
thf(fact_8345_nat__minus__as__int,axiom,
    ( minus_minus_nat
    = ( ^ [A2: nat,B2: nat] : ( nat2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ) ).

% nat_minus_as_int
thf(fact_8346_finite__divisors__nat,axiom,
    ! [M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [D4: nat] : ( dvd_dvd_nat @ D4 @ M ) ) ) ) ).

% finite_divisors_nat
thf(fact_8347_card__less,axiom,
    ! [M7: set_nat,I2: nat] :
      ( ( member_nat @ zero_zero_nat @ M7 )
     => ( ( finite_card_nat
          @ ( collect_nat
            @ ^ [K3: nat] :
                ( ( member_nat @ K3 @ M7 )
                & ( ord_less_nat @ K3 @ ( suc @ I2 ) ) ) ) )
       != zero_zero_nat ) ) ).

% card_less
thf(fact_8348_card__less__Suc,axiom,
    ! [M7: set_nat,I2: nat] :
      ( ( member_nat @ zero_zero_nat @ M7 )
     => ( ( suc
          @ ( finite_card_nat
            @ ( collect_nat
              @ ^ [K3: nat] :
                  ( ( member_nat @ ( suc @ K3 ) @ M7 )
                  & ( ord_less_nat @ K3 @ I2 ) ) ) ) )
        = ( finite_card_nat
          @ ( collect_nat
            @ ^ [K3: nat] :
                ( ( member_nat @ K3 @ M7 )
                & ( ord_less_nat @ K3 @ ( suc @ I2 ) ) ) ) ) ) ) ).

% card_less_Suc
thf(fact_8349_card__less__Suc2,axiom,
    ! [M7: set_nat,I2: nat] :
      ( ~ ( member_nat @ zero_zero_nat @ M7 )
     => ( ( finite_card_nat
          @ ( collect_nat
            @ ^ [K3: nat] :
                ( ( member_nat @ ( suc @ K3 ) @ M7 )
                & ( ord_less_nat @ K3 @ I2 ) ) ) )
        = ( finite_card_nat
          @ ( collect_nat
            @ ^ [K3: nat] :
                ( ( member_nat @ K3 @ M7 )
                & ( ord_less_nat @ K3 @ ( suc @ I2 ) ) ) ) ) ) ) ).

% card_less_Suc2
thf(fact_8350_prod__int__plus__eq,axiom,
    ! [I2: nat,J: nat] :
      ( ( groups705719431365010083at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ I2 @ ( plus_plus_nat @ I2 @ J ) ) )
      = ( groups1705073143266064639nt_int
        @ ^ [X2: int] : X2
        @ ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ I2 ) @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ I2 @ J ) ) ) ) ) ).

% prod_int_plus_eq
thf(fact_8351_nat__div__as__int,axiom,
    ( divide_divide_nat
    = ( ^ [A2: nat,B2: nat] : ( nat2 @ ( divide_divide_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ) ).

% nat_div_as_int
thf(fact_8352_nat__mod__as__int,axiom,
    ( modulo_modulo_nat
    = ( ^ [A2: nat,B2: nat] : ( nat2 @ ( modulo_modulo_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ) ).

% nat_mod_as_int
thf(fact_8353_image__add__int__atLeastLessThan,axiom,
    ! [L: int,U: int] :
      ( ( image_int_int
        @ ^ [X2: int] : ( plus_plus_int @ X2 @ L )
        @ ( set_or4662586982721622107an_int @ zero_zero_int @ ( minus_minus_int @ U @ L ) ) )
      = ( set_or4662586982721622107an_int @ L @ U ) ) ).

% image_add_int_atLeastLessThan
thf(fact_8354_map__add__upt,axiom,
    ! [N: nat,M: nat] :
      ( ( map_nat_nat
        @ ^ [I4: nat] : ( plus_plus_nat @ I4 @ N )
        @ ( upt @ zero_zero_nat @ M ) )
      = ( upt @ N @ ( plus_plus_nat @ M @ N ) ) ) ).

% map_add_upt
thf(fact_8355_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_8356_ratrel__def,axiom,
    ( ratrel
    = ( ^ [X2: product_prod_int_int,Y6: product_prod_int_int] :
          ( ( ( product_snd_int_int @ X2 )
           != zero_zero_int )
          & ( ( product_snd_int_int @ Y6 )
           != zero_zero_int )
          & ( ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_snd_int_int @ Y6 ) )
            = ( times_times_int @ ( product_fst_int_int @ Y6 ) @ ( product_snd_int_int @ X2 ) ) ) ) ) ) ).

% ratrel_def
thf(fact_8357_map__decr__upt,axiom,
    ! [M: nat,N: nat] :
      ( ( map_nat_nat
        @ ^ [N3: nat] : ( minus_minus_nat @ N3 @ ( suc @ zero_zero_nat ) )
        @ ( upt @ ( suc @ M ) @ ( suc @ N ) ) )
      = ( upt @ M @ N ) ) ).

% map_decr_upt
thf(fact_8358_diff__nat__eq__if,axiom,
    ! [Z5: int,Z2: int] :
      ( ( ( ord_less_int @ Z5 @ zero_zero_int )
       => ( ( minus_minus_nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z5 ) )
          = ( nat2 @ Z2 ) ) )
      & ( ~ ( ord_less_int @ Z5 @ zero_zero_int )
       => ( ( minus_minus_nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z5 ) )
          = ( if_nat @ ( ord_less_int @ ( minus_minus_int @ Z2 @ Z5 ) @ zero_zero_int ) @ zero_zero_nat @ ( nat2 @ ( minus_minus_int @ Z2 @ Z5 ) ) ) ) ) ) ).

% diff_nat_eq_if
thf(fact_8359_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_8360_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
            @ ^ [X2: nat] : X2
            @ ( set_or1269000886237332187st_nat @ ( suc @ N ) @ M ) ) ) ) ) ).

% fact_eq_fact_times
thf(fact_8361_set__decode__def,axiom,
    ( nat_set_decode
    = ( ^ [X2: nat] :
          ( collect_nat
          @ ^ [N3: nat] :
              ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ) ) ).

% set_decode_def
thf(fact_8362_image__minus__const__atLeastLessThan__nat,axiom,
    ! [C: nat,Y3: nat,X: nat] :
      ( ( ( ord_less_nat @ C @ Y3 )
       => ( ( image_nat_nat
            @ ^ [I4: nat] : ( minus_minus_nat @ I4 @ C )
            @ ( set_or4665077453230672383an_nat @ X @ Y3 ) )
          = ( set_or4665077453230672383an_nat @ ( minus_minus_nat @ X @ C ) @ ( minus_minus_nat @ Y3 @ C ) ) ) )
      & ( ~ ( ord_less_nat @ C @ Y3 )
       => ( ( ( ord_less_nat @ X @ Y3 )
           => ( ( image_nat_nat
                @ ^ [I4: nat] : ( minus_minus_nat @ I4 @ C )
                @ ( set_or4665077453230672383an_nat @ X @ Y3 ) )
              = ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) )
          & ( ~ ( ord_less_nat @ X @ Y3 )
           => ( ( image_nat_nat
                @ ^ [I4: nat] : ( minus_minus_nat @ I4 @ C )
                @ ( set_or4665077453230672383an_nat @ X @ Y3 ) )
              = bot_bot_set_nat ) ) ) ) ) ).

% image_minus_const_atLeastLessThan_nat
thf(fact_8363_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
          @ ^ [X2: nat] : X2
          @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M ) ) ) ) ).

% fact_div_fact
thf(fact_8364_normalize__def,axiom,
    ( normalize
    = ( ^ [P6: product_prod_int_int] :
          ( if_Pro3027730157355071871nt_int @ ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ P6 ) ) @ ( product_Pair_int_int @ ( divide_divide_int @ ( product_fst_int_int @ P6 ) @ ( gcd_gcd_int @ ( product_fst_int_int @ P6 ) @ ( product_snd_int_int @ P6 ) ) ) @ ( divide_divide_int @ ( product_snd_int_int @ P6 ) @ ( gcd_gcd_int @ ( product_fst_int_int @ P6 ) @ ( product_snd_int_int @ P6 ) ) ) )
          @ ( if_Pro3027730157355071871nt_int
            @ ( ( product_snd_int_int @ P6 )
              = zero_zero_int )
            @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
            @ ( product_Pair_int_int @ ( divide_divide_int @ ( product_fst_int_int @ P6 ) @ ( uminus_uminus_int @ ( gcd_gcd_int @ ( product_fst_int_int @ P6 ) @ ( product_snd_int_int @ P6 ) ) ) ) @ ( divide_divide_int @ ( product_snd_int_int @ P6 ) @ ( uminus_uminus_int @ ( gcd_gcd_int @ ( product_fst_int_int @ P6 ) @ ( product_snd_int_int @ P6 ) ) ) ) ) ) ) ) ) ).

% normalize_def
thf(fact_8365_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,S3: code_integer] : ( produc6677183202524767010eger_o @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ K3 ) @ R5 @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ S3 ) ) @ ( S3 = one_one_Code_integer ) )
            @ ( code_divmod_abs @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% bit_cut_integer_code
thf(fact_8366_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_8367_integer__of__int__eq__of__int,axiom,
    code_integer_of_int = ring_18347121197199848620nteger ).

% integer_of_int_eq_of_int
thf(fact_8368_sum__list__upt,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( groups4561878855575611511st_nat @ ( upt @ M @ N ) )
        = ( groups3542108847815614940at_nat
          @ ^ [X2: nat] : X2
          @ ( set_or4665077453230672383an_nat @ M @ N ) ) ) ) ).

% sum_list_upt
thf(fact_8369_unset__bit__integer_Oabs__eq,axiom,
    ! [Xa3: nat,X: int] :
      ( ( bit_se8260200283734997820nteger @ Xa3 @ ( code_integer_of_int @ X ) )
      = ( code_integer_of_int @ ( bit_se4203085406695923979it_int @ Xa3 @ X ) ) ) ).

% unset_bit_integer.abs_eq
thf(fact_8370_bit__integer_Oabs__eq,axiom,
    ! [X: int] :
      ( ( bit_se9216721137139052372nteger @ ( code_integer_of_int @ X ) )
      = ( bit_se1146084159140164899it_int @ X ) ) ).

% bit_integer.abs_eq
thf(fact_8371_or__integer_Oabs__eq,axiom,
    ! [Xa3: int,X: int] :
      ( ( bit_se1080825931792720795nteger @ ( code_integer_of_int @ Xa3 ) @ ( code_integer_of_int @ X ) )
      = ( code_integer_of_int @ ( bit_se1409905431419307370or_int @ Xa3 @ X ) ) ) ).

% or_integer.abs_eq
thf(fact_8372_and__integer_Oabs__eq,axiom,
    ! [Xa3: int,X: int] :
      ( ( bit_se3949692690581998587nteger @ ( code_integer_of_int @ Xa3 ) @ ( code_integer_of_int @ X ) )
      = ( code_integer_of_int @ ( bit_se725231765392027082nd_int @ Xa3 @ X ) ) ) ).

% and_integer.abs_eq
thf(fact_8373_euclidean__size__integer_Oabs__eq,axiom,
    ! [X: int] :
      ( ( euclid6377331345833325938nteger @ ( code_integer_of_int @ X ) )
      = ( euclid4774559944035922753ze_int @ X ) ) ).

% euclidean_size_integer.abs_eq
thf(fact_8374_not__integer_Oabs__eq,axiom,
    ! [X: int] :
      ( ( bit_ri7632146776885996613nteger @ ( code_integer_of_int @ X ) )
      = ( code_integer_of_int @ ( bit_ri7919022796975470100ot_int @ X ) ) ) ).

% not_integer.abs_eq
thf(fact_8375_take__bit__integer_Oabs__eq,axiom,
    ! [Xa3: nat,X: int] :
      ( ( bit_se1745604003318907178nteger @ Xa3 @ ( code_integer_of_int @ X ) )
      = ( code_integer_of_int @ ( bit_se2923211474154528505it_int @ Xa3 @ X ) ) ) ).

% take_bit_integer.abs_eq
thf(fact_8376_xor__integer_Oabs__eq,axiom,
    ! [Xa3: int,X: int] :
      ( ( bit_se3222712562003087583nteger @ ( code_integer_of_int @ Xa3 ) @ ( code_integer_of_int @ X ) )
      = ( code_integer_of_int @ ( bit_se6526347334894502574or_int @ Xa3 @ X ) ) ) ).

% xor_integer.abs_eq
thf(fact_8377_times__integer_Oabs__eq,axiom,
    ! [Xa3: int,X: int] :
      ( ( times_3573771949741848930nteger @ ( code_integer_of_int @ Xa3 ) @ ( code_integer_of_int @ X ) )
      = ( code_integer_of_int @ ( times_times_int @ Xa3 @ X ) ) ) ).

% times_integer.abs_eq
thf(fact_8378_push__bit__integer_Oabs__eq,axiom,
    ! [Xa3: nat,X: int] :
      ( ( bit_se7788150548672797655nteger @ Xa3 @ ( code_integer_of_int @ X ) )
      = ( code_integer_of_int @ ( bit_se545348938243370406it_int @ Xa3 @ X ) ) ) ).

% push_bit_integer.abs_eq
thf(fact_8379_drop__bit__integer_Oabs__eq,axiom,
    ! [Xa3: nat,X: int] :
      ( ( bit_se3928097537394005634nteger @ Xa3 @ ( code_integer_of_int @ X ) )
      = ( code_integer_of_int @ ( bit_se8568078237143864401it_int @ Xa3 @ X ) ) ) ).

% drop_bit_integer.abs_eq
thf(fact_8380_mask__integer_Oabs__eq,axiom,
    ( bit_se2119862282449309892nteger
    = ( ^ [X2: nat] : ( code_integer_of_int @ ( bit_se2000444600071755411sk_int @ X2 ) ) ) ) ).

% mask_integer.abs_eq
thf(fact_8381_set__bit__integer_Oabs__eq,axiom,
    ! [Xa3: nat,X: int] :
      ( ( bit_se2793503036327961859nteger @ Xa3 @ ( code_integer_of_int @ X ) )
      = ( code_integer_of_int @ ( bit_se7879613467334960850it_int @ Xa3 @ X ) ) ) ).

% set_bit_integer.abs_eq
thf(fact_8382_flip__bit__integer_Oabs__eq,axiom,
    ! [Xa3: nat,X: int] :
      ( ( bit_se1345352211410354436nteger @ Xa3 @ ( code_integer_of_int @ X ) )
      = ( code_integer_of_int @ ( bit_se2159334234014336723it_int @ Xa3 @ X ) ) ) ).

% flip_bit_integer.abs_eq
thf(fact_8383_division__segment__integer_Oabs__eq,axiom,
    ! [X: int] :
      ( ( euclid6289375185220004616nteger @ ( code_integer_of_int @ X ) )
      = ( code_integer_of_int @ ( euclid3395696857347342551nt_int @ X ) ) ) ).

% division_segment_integer.abs_eq
thf(fact_8384_zero__integer__def,axiom,
    ( zero_z3403309356797280102nteger
    = ( code_integer_of_int @ zero_zero_int ) ) ).

% zero_integer_def
thf(fact_8385_less__integer_Oabs__eq,axiom,
    ! [Xa3: int,X: int] :
      ( ( ord_le6747313008572928689nteger @ ( code_integer_of_int @ Xa3 ) @ ( code_integer_of_int @ X ) )
      = ( ord_less_int @ Xa3 @ X ) ) ).

% less_integer.abs_eq
thf(fact_8386_plus__integer_Oabs__eq,axiom,
    ! [Xa3: int,X: int] :
      ( ( plus_p5714425477246183910nteger @ ( code_integer_of_int @ Xa3 ) @ ( code_integer_of_int @ X ) )
      = ( code_integer_of_int @ ( plus_plus_int @ Xa3 @ X ) ) ) ).

% plus_integer.abs_eq
thf(fact_8387_uminus__integer_Oabs__eq,axiom,
    ! [X: int] :
      ( ( uminus1351360451143612070nteger @ ( code_integer_of_int @ X ) )
      = ( code_integer_of_int @ ( uminus_uminus_int @ X ) ) ) ).

% uminus_integer.abs_eq
thf(fact_8388_one__integer__def,axiom,
    ( one_one_Code_integer
    = ( code_integer_of_int @ one_one_int ) ) ).

% one_integer_def
thf(fact_8389_less__eq__integer_Oabs__eq,axiom,
    ! [Xa3: int,X: int] :
      ( ( ord_le3102999989581377725nteger @ ( code_integer_of_int @ Xa3 ) @ ( code_integer_of_int @ X ) )
      = ( ord_less_eq_int @ Xa3 @ X ) ) ).

% less_eq_integer.abs_eq
thf(fact_8390_minus__integer_Oabs__eq,axiom,
    ! [Xa3: int,X: int] :
      ( ( minus_8373710615458151222nteger @ ( code_integer_of_int @ Xa3 ) @ ( code_integer_of_int @ X ) )
      = ( code_integer_of_int @ ( minus_minus_int @ Xa3 @ X ) ) ) ).

% minus_integer.abs_eq
thf(fact_8391_abs__integer_Oabs__eq,axiom,
    ! [X: int] :
      ( ( abs_abs_Code_integer @ ( code_integer_of_int @ X ) )
      = ( code_integer_of_int @ ( abs_abs_int @ X ) ) ) ).

% abs_integer.abs_eq
thf(fact_8392_divide__integer_Oabs__eq,axiom,
    ! [Xa3: int,X: int] :
      ( ( divide6298287555418463151nteger @ ( code_integer_of_int @ Xa3 ) @ ( code_integer_of_int @ X ) )
      = ( code_integer_of_int @ ( divide_divide_int @ Xa3 @ X ) ) ) ).

% divide_integer.abs_eq
thf(fact_8393_nat__of__integer_Oabs__eq,axiom,
    ! [X: int] :
      ( ( code_nat_of_integer @ ( code_integer_of_int @ X ) )
      = ( nat2 @ X ) ) ).

% nat_of_integer.abs_eq
thf(fact_8394_modulo__integer_Oabs__eq,axiom,
    ! [Xa3: int,X: int] :
      ( ( modulo364778990260209775nteger @ ( code_integer_of_int @ Xa3 ) @ ( code_integer_of_int @ X ) )
      = ( code_integer_of_int @ ( modulo_modulo_int @ Xa3 @ X ) ) ) ).

% modulo_integer.abs_eq
thf(fact_8395_sgn__integer_Oabs__eq,axiom,
    ! [X: int] :
      ( ( sgn_sgn_Code_integer @ ( code_integer_of_int @ X ) )
      = ( code_integer_of_int @ ( sgn_sgn_int @ X ) ) ) ).

% sgn_integer.abs_eq
thf(fact_8396_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_8397_sum__choose__lower,axiom,
    ! [R: nat,N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [K3: nat] : ( binomial @ ( plus_plus_nat @ R @ K3 ) @ K3 )
        @ ( set_ord_atMost_nat @ N ) )
      = ( binomial @ ( suc @ ( plus_plus_nat @ R @ N ) ) @ N ) ) ).

% sum_choose_lower
thf(fact_8398_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_8399_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_8400_card__sum__le__nat__sum,axiom,
    ! [S4: set_nat] :
      ( ord_less_eq_nat
      @ ( groups3542108847815614940at_nat
        @ ^ [X2: nat] : X2
        @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( finite_card_nat @ S4 ) ) )
      @ ( groups3542108847815614940at_nat
        @ ^ [X2: nat] : X2
        @ S4 ) ) ).

% card_sum_le_nat_sum
thf(fact_8401_set__encode__def,axiom,
    ( nat_set_encode
    = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% set_encode_def
thf(fact_8402_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_8403_vandermonde,axiom,
    ! [M: nat,N: nat,R: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [K3: nat] : ( times_times_nat @ ( binomial @ M @ K3 ) @ ( binomial @ N @ ( minus_minus_nat @ R @ K3 ) ) )
        @ ( set_ord_atMost_nat @ R ) )
      = ( binomial @ ( plus_plus_nat @ M @ N ) @ R ) ) ).

% vandermonde
thf(fact_8404_binomial,axiom,
    ! [A: nat,B: nat,N: nat] :
      ( ( power_power_nat @ ( plus_plus_nat @ A @ B ) @ N )
      = ( groups3542108847815614940at_nat
        @ ^ [K3: nat] : ( times_times_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( binomial @ N @ K3 ) ) @ ( power_power_nat @ A @ K3 ) ) @ ( power_power_nat @ B @ ( minus_minus_nat @ N @ K3 ) ) )
        @ ( set_ord_atMost_nat @ N ) ) ) ).

% binomial
thf(fact_8405_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_8406_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
          @ ^ [Q3: nat] : ( ord_less_nat @ Q3 @ N ) ) ) ) ).

% mask_eq_sum_exp_nat
thf(fact_8407_gauss__sum__nat,axiom,
    ! [N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [X2: nat] : X2
        @ ( 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_8408_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_8409_Sum__Ico__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [X2: nat] : X2
        @ ( 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_8410_arith__series__nat,axiom,
    ! [A: nat,D: nat,N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [I4: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ I4 @ 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_8411_Sum__Icc__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [X2: nat] : X2
        @ ( 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_8412_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_8413_choose__linear__sum,axiom,
    ! [N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [I4: nat] : ( times_times_nat @ I4 @ ( binomial @ N @ I4 ) )
        @ ( 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_8414_nat__of__integer__code,axiom,
    ( code_nat_of_integer
    = ( ^ [K3: code_integer] :
          ( if_nat @ ( ord_le3102999989581377725nteger @ K3 @ zero_z3403309356797280102nteger ) @ zero_zero_nat
          @ ( produc1555791787009142072er_nat
            @ ^ [L3: code_integer,J3: code_integer] : ( if_nat @ ( J3 = zero_z3403309356797280102nteger ) @ ( plus_plus_nat @ ( code_nat_of_integer @ L3 ) @ ( code_nat_of_integer @ L3 ) ) @ ( plus_plus_nat @ ( plus_plus_nat @ ( code_nat_of_integer @ L3 ) @ ( code_nat_of_integer @ L3 ) ) @ one_one_nat ) )
            @ ( code_divmod_integer @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% nat_of_integer_code
thf(fact_8415_prod__encode__def,axiom,
    ( nat_prod_encode
    = ( produc6842872674320459806at_nat
      @ ^ [M2: nat,N3: nat] : ( plus_plus_nat @ ( nat_triangle @ ( plus_plus_nat @ M2 @ N3 ) ) @ M2 ) ) ) ).

% prod_encode_def
thf(fact_8416_rat__less__code,axiom,
    ( ord_less_rat
    = ( ^ [P6: rat,Q3: rat] :
          ( produc4947309494688390418_int_o
          @ ^ [A2: int,C3: int] :
              ( produc4947309494688390418_int_o
              @ ^ [B2: int,D4: int] : ( ord_less_int @ ( times_times_int @ A2 @ D4 ) @ ( times_times_int @ C3 @ B2 ) )
              @ ( quotient_of @ Q3 ) )
          @ ( quotient_of @ P6 ) ) ) ) ).

% rat_less_code
thf(fact_8417_rat__less__eq__code,axiom,
    ( ord_less_eq_rat
    = ( ^ [P6: rat,Q3: rat] :
          ( produc4947309494688390418_int_o
          @ ^ [A2: int,C3: int] :
              ( produc4947309494688390418_int_o
              @ ^ [B2: int,D4: int] : ( ord_less_eq_int @ ( times_times_int @ A2 @ D4 ) @ ( times_times_int @ C3 @ B2 ) )
              @ ( quotient_of @ Q3 ) )
          @ ( quotient_of @ P6 ) ) ) ) ).

% rat_less_eq_code
thf(fact_8418_rat__uminus__code,axiom,
    ! [P5: rat] :
      ( ( quotient_of @ ( uminus_uminus_rat @ P5 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A2: int] : ( product_Pair_int_int @ ( uminus_uminus_int @ A2 ) )
        @ ( quotient_of @ P5 ) ) ) ).

% rat_uminus_code
thf(fact_8419_Divides_Oadjust__div__def,axiom,
    ( adjust_div
    = ( produc8211389475949308722nt_int
      @ ^ [Q3: int,R5: int] : ( plus_plus_int @ Q3 @ ( zero_n2684676970156552555ol_int @ ( R5 != zero_zero_int ) ) ) ) ) ).

% Divides.adjust_div_def
thf(fact_8420_rat__divide__code,axiom,
    ! [P5: rat,Q4: rat] :
      ( ( quotient_of @ ( divide_divide_rat @ P5 @ Q4 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A2: int,C3: int] :
            ( produc4245557441103728435nt_int
            @ ^ [B2: int,D4: int] : ( normalize @ ( product_Pair_int_int @ ( times_times_int @ A2 @ D4 ) @ ( times_times_int @ C3 @ B2 ) ) )
            @ ( quotient_of @ Q4 ) )
        @ ( quotient_of @ P5 ) ) ) ).

% rat_divide_code
thf(fact_8421_rat__times__code,axiom,
    ! [P5: rat,Q4: rat] :
      ( ( quotient_of @ ( times_times_rat @ P5 @ Q4 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A2: int,C3: int] :
            ( produc4245557441103728435nt_int
            @ ^ [B2: int,D4: int] : ( normalize @ ( product_Pair_int_int @ ( times_times_int @ A2 @ B2 ) @ ( times_times_int @ C3 @ D4 ) ) )
            @ ( quotient_of @ Q4 ) )
        @ ( quotient_of @ P5 ) ) ) ).

% rat_times_code
thf(fact_8422_rat__plus__code,axiom,
    ! [P5: rat,Q4: rat] :
      ( ( quotient_of @ ( plus_plus_rat @ P5 @ Q4 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A2: int,C3: int] :
            ( produc4245557441103728435nt_int
            @ ^ [B2: int,D4: int] : ( normalize @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ A2 @ D4 ) @ ( times_times_int @ B2 @ C3 ) ) @ ( times_times_int @ C3 @ D4 ) ) )
            @ ( quotient_of @ Q4 ) )
        @ ( quotient_of @ P5 ) ) ) ).

% rat_plus_code
thf(fact_8423_rat__minus__code,axiom,
    ! [P5: rat,Q4: rat] :
      ( ( quotient_of @ ( minus_minus_rat @ P5 @ Q4 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A2: int,C3: int] :
            ( produc4245557441103728435nt_int
            @ ^ [B2: int,D4: int] : ( normalize @ ( product_Pair_int_int @ ( minus_minus_int @ ( times_times_int @ A2 @ D4 ) @ ( times_times_int @ B2 @ C3 ) ) @ ( times_times_int @ C3 @ D4 ) ) )
            @ ( quotient_of @ Q4 ) )
        @ ( quotient_of @ P5 ) ) ) ).

% rat_minus_code
thf(fact_8424_divmod__step__nat__def,axiom,
    ( unique5026877609467782581ep_nat
    = ( ^ [L3: num] :
          ( produc2626176000494625587at_nat
          @ ^ [Q3: nat,R5: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L3 ) @ R5 ) @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q3 ) @ one_one_nat ) @ ( minus_minus_nat @ R5 @ ( numeral_numeral_nat @ L3 ) ) ) @ ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q3 ) @ R5 ) ) ) ) ) ).

% divmod_step_nat_def
thf(fact_8425_rat__inverse__code,axiom,
    ! [P5: rat] :
      ( ( quotient_of @ ( inverse_inverse_rat @ P5 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A2: int,B2: int] : ( if_Pro3027730157355071871nt_int @ ( A2 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ ( times_times_int @ ( sgn_sgn_int @ A2 ) @ B2 ) @ ( abs_abs_int @ A2 ) ) )
        @ ( quotient_of @ P5 ) ) ) ).

% rat_inverse_code
thf(fact_8426_divmod__step__int__def,axiom,
    ( unique5024387138958732305ep_int
    = ( ^ [L3: num] :
          ( produc4245557441103728435nt_int
          @ ^ [Q3: int,R5: int] : ( if_Pro3027730157355071871nt_int @ ( ord_less_eq_int @ ( numeral_numeral_int @ L3 ) @ R5 ) @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q3 ) @ one_one_int ) @ ( minus_minus_int @ R5 @ ( numeral_numeral_int @ L3 ) ) ) @ ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q3 ) @ R5 ) ) ) ) ) ).

% divmod_step_int_def
thf(fact_8427_Sum__Icc__int,axiom,
    ! [M: int,N: int] :
      ( ( ord_less_eq_int @ M @ N )
     => ( ( groups4538972089207619220nt_int
          @ ^ [X2: int] : X2
          @ ( 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_8428_divmod__step__integer__def,axiom,
    ( unique4921790084139445826nteger
    = ( ^ [L3: num] :
          ( produc6916734918728496179nteger
          @ ^ [Q3: code_integer,R5: code_integer] : ( if_Pro6119634080678213985nteger @ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L3 ) @ R5 ) @ ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q3 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R5 @ ( numera6620942414471956472nteger @ L3 ) ) ) @ ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q3 ) @ R5 ) ) ) ) ) ).

% divmod_step_integer_def
thf(fact_8429_num__of__integer__code,axiom,
    ( code_num_of_integer
    = ( ^ [K3: code_integer] :
          ( if_num @ ( ord_le3102999989581377725nteger @ K3 @ one_one_Code_integer ) @ one
          @ ( produc7336495610019696514er_num
            @ ^ [L3: code_integer,J3: code_integer] : ( if_num @ ( J3 = zero_z3403309356797280102nteger ) @ ( plus_plus_num @ ( code_num_of_integer @ L3 ) @ ( code_num_of_integer @ L3 ) ) @ ( plus_plus_num @ ( plus_plus_num @ ( code_num_of_integer @ L3 ) @ ( code_num_of_integer @ L3 ) ) @ one ) )
            @ ( code_divmod_integer @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% num_of_integer_code
thf(fact_8430_num__of__integer_Oabs__eq,axiom,
    ! [X: int] :
      ( ( code_num_of_integer @ ( code_integer_of_int @ X ) )
      = ( num_of_nat @ ( nat2 @ X ) ) ) ).

% num_of_integer.abs_eq
thf(fact_8431_divmod__nat__if,axiom,
    ( divmod_nat
    = ( ^ [M2: nat,N3: nat] :
          ( if_Pro6206227464963214023at_nat
          @ ( ( N3 = zero_zero_nat )
            | ( ord_less_nat @ M2 @ N3 ) )
          @ ( product_Pair_nat_nat @ zero_zero_nat @ M2 )
          @ ( produc2626176000494625587at_nat
            @ ^ [Q3: nat] : ( product_Pair_nat_nat @ ( suc @ Q3 ) )
            @ ( divmod_nat @ ( minus_minus_nat @ M2 @ N3 ) @ N3 ) ) ) ) ) ).

% divmod_nat_if
thf(fact_8432_int__ge__less__than__def,axiom,
    ( int_ge_less_than
    = ( ^ [D4: int] :
          ( collec213857154873943460nt_int
          @ ( produc4947309494688390418_int_o
            @ ^ [Z7: int,Z6: int] :
                ( ( ord_less_eq_int @ D4 @ Z7 )
                & ( ord_less_int @ Z7 @ Z6 ) ) ) ) ) ) ).

% int_ge_less_than_def
thf(fact_8433_int__ge__less__than2__def,axiom,
    ( int_ge_less_than2
    = ( ^ [D4: int] :
          ( collec213857154873943460nt_int
          @ ( produc4947309494688390418_int_o
            @ ^ [Z7: int,Z6: int] :
                ( ( ord_less_eq_int @ D4 @ Z6 )
                & ( ord_less_int @ Z7 @ Z6 ) ) ) ) ) ) ).

% int_ge_less_than2_def
thf(fact_8434_divmod__integer__code,axiom,
    ( code_divmod_integer
    = ( ^ [K3: code_integer,L3: code_integer] :
          ( if_Pro6119634080678213985nteger @ ( K3 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
          @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ L3 )
            @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ K3 ) @ ( code_divmod_abs @ K3 @ L3 )
              @ ( produc6916734918728496179nteger
                @ ^ [R5: code_integer,S3: code_integer] : ( if_Pro6119634080678213985nteger @ ( S3 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ L3 @ S3 ) ) )
                @ ( code_divmod_abs @ K3 @ L3 ) ) )
            @ ( if_Pro6119634080678213985nteger @ ( L3 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ K3 )
              @ ( produc6499014454317279255nteger @ uminus1351360451143612070nteger
                @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ K3 @ zero_z3403309356797280102nteger ) @ ( code_divmod_abs @ K3 @ L3 )
                  @ ( produc6916734918728496179nteger
                    @ ^ [R5: code_integer,S3: code_integer] : ( if_Pro6119634080678213985nteger @ ( S3 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ L3 ) @ S3 ) ) )
                    @ ( code_divmod_abs @ K3 @ L3 ) ) ) ) ) ) ) ) ) ).

% divmod_integer_code
thf(fact_8435_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
              @ ^ [L3: code_integer,J3: code_integer] : ( if_int @ ( J3 = zero_z3403309356797280102nteger ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( code_int_of_integer @ L3 ) ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( code_int_of_integer @ L3 ) ) @ one_one_int ) )
              @ ( code_divmod_integer @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% int_of_integer_code
thf(fact_8436_inverse__rat__def,axiom,
    ( inverse_inverse_rat
    = ( map_fu5673905371560938248nt_rat @ rep_Rat @ abs_Rat
      @ ^ [X2: product_prod_int_int] :
          ( if_Pro3027730157355071871nt_int
          @ ( ( product_fst_int_int @ X2 )
            = zero_zero_int )
          @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
          @ ( product_Pair_int_int @ ( product_snd_int_int @ X2 ) @ ( product_fst_int_int @ X2 ) ) ) ) ) ).

% inverse_rat_def
thf(fact_8437_int__of__integer__inverse,axiom,
    ! [X: code_integer] :
      ( ( code_integer_of_int @ ( code_int_of_integer @ X ) )
      = X ) ).

% int_of_integer_inverse
thf(fact_8438_int__of__integer__integer__of__int,axiom,
    ! [K: int] :
      ( ( code_int_of_integer @ ( code_integer_of_int @ K ) )
      = K ) ).

% int_of_integer_integer_of_int
thf(fact_8439_integer__of__int__int__of__integer,axiom,
    ! [K: code_integer] :
      ( ( code_integer_of_int @ ( code_int_of_integer @ K ) )
      = K ) ).

% integer_of_int_int_of_integer
thf(fact_8440_int__of__integer__sub,axiom,
    ! [K: num,L: num] :
      ( ( code_int_of_integer @ ( neg_nu5755505904847501662nteger @ K @ L ) )
      = ( neg_numeral_sub_int @ K @ L ) ) ).

% int_of_integer_sub
thf(fact_8441_int__of__integer__min,axiom,
    ! [K: code_integer,L: code_integer] :
      ( ( code_int_of_integer @ ( ord_min_Code_integer @ K @ L ) )
      = ( ord_min_int @ ( code_int_of_integer @ K ) @ ( code_int_of_integer @ L ) ) ) ).

% int_of_integer_min
thf(fact_8442_of__int__integer__of,axiom,
    ! [K: code_integer] :
      ( ( ring_18347121197199848620nteger @ ( code_int_of_integer @ K ) )
      = K ) ).

% of_int_integer_of
thf(fact_8443_int__of__integer__of__int,axiom,
    ! [K: int] :
      ( ( code_int_of_integer @ ( ring_18347121197199848620nteger @ K ) )
      = K ) ).

% int_of_integer_of_int
thf(fact_8444_zero__integer_Orep__eq,axiom,
    ( ( code_int_of_integer @ zero_z3403309356797280102nteger )
    = zero_zero_int ) ).

% zero_integer.rep_eq
thf(fact_8445_int__of__integer__numeral,axiom,
    ! [K: num] :
      ( ( code_int_of_integer @ ( numera6620942414471956472nteger @ K ) )
      = ( numeral_numeral_int @ K ) ) ).

% int_of_integer_numeral
thf(fact_8446_plus__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa3: code_integer] :
      ( ( code_int_of_integer @ ( plus_p5714425477246183910nteger @ X @ Xa3 ) )
      = ( plus_plus_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa3 ) ) ) ).

% plus_integer.rep_eq
thf(fact_8447_uminus__integer_Orep__eq,axiom,
    ! [X: code_integer] :
      ( ( code_int_of_integer @ ( uminus1351360451143612070nteger @ X ) )
      = ( uminus_uminus_int @ ( code_int_of_integer @ X ) ) ) ).

% uminus_integer.rep_eq
thf(fact_8448_times__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa3: code_integer] :
      ( ( code_int_of_integer @ ( times_3573771949741848930nteger @ X @ Xa3 ) )
      = ( times_times_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa3 ) ) ) ).

% times_integer.rep_eq
thf(fact_8449_one__integer_Orep__eq,axiom,
    ( ( code_int_of_integer @ one_one_Code_integer )
    = one_one_int ) ).

% one_integer.rep_eq
thf(fact_8450_minus__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa3: code_integer] :
      ( ( code_int_of_integer @ ( minus_8373710615458151222nteger @ X @ Xa3 ) )
      = ( minus_minus_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa3 ) ) ) ).

% minus_integer.rep_eq
thf(fact_8451_abs__integer_Orep__eq,axiom,
    ! [X: code_integer] :
      ( ( code_int_of_integer @ ( abs_abs_Code_integer @ X ) )
      = ( abs_abs_int @ ( code_int_of_integer @ X ) ) ) ).

% abs_integer.rep_eq
thf(fact_8452_divide__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa3: code_integer] :
      ( ( code_int_of_integer @ ( divide6298287555418463151nteger @ X @ Xa3 ) )
      = ( divide_divide_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa3 ) ) ) ).

% divide_integer.rep_eq
thf(fact_8453_modulo__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa3: code_integer] :
      ( ( code_int_of_integer @ ( modulo364778990260209775nteger @ X @ Xa3 ) )
      = ( modulo_modulo_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa3 ) ) ) ).

% modulo_integer.rep_eq
thf(fact_8454_sgn__integer_Orep__eq,axiom,
    ! [X: code_integer] :
      ( ( code_int_of_integer @ ( sgn_sgn_Code_integer @ X ) )
      = ( sgn_sgn_int @ ( code_int_of_integer @ X ) ) ) ).

% sgn_integer.rep_eq
thf(fact_8455_int__of__integer__of__nat,axiom,
    ! [N: nat] :
      ( ( code_int_of_integer @ ( semiri4939895301339042750nteger @ N ) )
      = ( semiri1314217659103216013at_int @ N ) ) ).

% int_of_integer_of_nat
thf(fact_8456_division__segment__integer_Orep__eq,axiom,
    ! [X: code_integer] :
      ( ( code_int_of_integer @ ( euclid6289375185220004616nteger @ X ) )
      = ( euclid3395696857347342551nt_int @ ( code_int_of_integer @ X ) ) ) ).

% division_segment_integer.rep_eq
thf(fact_8457_euclidean__size__integer_Orep__eq,axiom,
    ( euclid6377331345833325938nteger
    = ( ^ [X2: code_integer] : ( euclid4774559944035922753ze_int @ ( code_int_of_integer @ X2 ) ) ) ) ).

% euclidean_size_integer.rep_eq
thf(fact_8458_int__of__integer__max,axiom,
    ! [K: code_integer,L: code_integer] :
      ( ( code_int_of_integer @ ( ord_max_Code_integer @ K @ L ) )
      = ( ord_max_int @ ( code_int_of_integer @ K ) @ ( code_int_of_integer @ L ) ) ) ).

% int_of_integer_max
thf(fact_8459_integer__eq__iff,axiom,
    ( ( ^ [Y: code_integer,Z: code_integer] : ( Y = Z ) )
    = ( ^ [K3: code_integer,L3: code_integer] :
          ( ( code_int_of_integer @ K3 )
          = ( code_int_of_integer @ L3 ) ) ) ) ).

% integer_eq_iff
thf(fact_8460_integer__eqI,axiom,
    ! [K: code_integer,L: code_integer] :
      ( ( ( code_int_of_integer @ K )
        = ( code_int_of_integer @ L ) )
     => ( K = L ) ) ).

% integer_eqI
thf(fact_8461_int__of__integer__inject,axiom,
    ! [X: code_integer,Y3: code_integer] :
      ( ( ( code_int_of_integer @ X )
        = ( code_int_of_integer @ Y3 ) )
      = ( X = Y3 ) ) ).

% int_of_integer_inject
thf(fact_8462_less__integer_Orep__eq,axiom,
    ( ord_le6747313008572928689nteger
    = ( ^ [X2: code_integer,Xa4: code_integer] : ( ord_less_int @ ( code_int_of_integer @ X2 ) @ ( code_int_of_integer @ Xa4 ) ) ) ) ).

% less_integer.rep_eq
thf(fact_8463_integer__less__iff,axiom,
    ( ord_le6747313008572928689nteger
    = ( ^ [K3: code_integer,L3: code_integer] : ( ord_less_int @ ( code_int_of_integer @ K3 ) @ ( code_int_of_integer @ L3 ) ) ) ) ).

% integer_less_iff
thf(fact_8464_integer__less__eq__iff,axiom,
    ( ord_le3102999989581377725nteger
    = ( ^ [K3: code_integer,L3: code_integer] : ( ord_less_eq_int @ ( code_int_of_integer @ K3 ) @ ( code_int_of_integer @ L3 ) ) ) ) ).

% integer_less_eq_iff
thf(fact_8465_less__eq__integer_Orep__eq,axiom,
    ( ord_le3102999989581377725nteger
    = ( ^ [X2: code_integer,Xa4: code_integer] : ( ord_less_eq_int @ ( code_int_of_integer @ X2 ) @ ( code_int_of_integer @ Xa4 ) ) ) ) ).

% less_eq_integer.rep_eq
thf(fact_8466_nat__of__integer_Orep__eq,axiom,
    ( code_nat_of_integer
    = ( ^ [X2: code_integer] : ( nat2 @ ( code_int_of_integer @ X2 ) ) ) ) ).

% nat_of_integer.rep_eq
thf(fact_8467_take__bit__integer_Orep__eq,axiom,
    ! [X: nat,Xa3: code_integer] :
      ( ( code_int_of_integer @ ( bit_se1745604003318907178nteger @ X @ Xa3 ) )
      = ( bit_se2923211474154528505it_int @ X @ ( code_int_of_integer @ Xa3 ) ) ) ).

% take_bit_integer.rep_eq
thf(fact_8468_not__integer_Orep__eq,axiom,
    ! [X: code_integer] :
      ( ( code_int_of_integer @ ( bit_ri7632146776885996613nteger @ X ) )
      = ( bit_ri7919022796975470100ot_int @ ( code_int_of_integer @ X ) ) ) ).

% not_integer.rep_eq
thf(fact_8469_and__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa3: code_integer] :
      ( ( code_int_of_integer @ ( bit_se3949692690581998587nteger @ X @ Xa3 ) )
      = ( bit_se725231765392027082nd_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa3 ) ) ) ).

% and_integer.rep_eq
thf(fact_8470_bit__integer_Orep__eq,axiom,
    ( bit_se9216721137139052372nteger
    = ( ^ [X2: code_integer] : ( bit_se1146084159140164899it_int @ ( code_int_of_integer @ X2 ) ) ) ) ).

% bit_integer.rep_eq
thf(fact_8471_or__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa3: code_integer] :
      ( ( code_int_of_integer @ ( bit_se1080825931792720795nteger @ X @ Xa3 ) )
      = ( bit_se1409905431419307370or_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa3 ) ) ) ).

% or_integer.rep_eq
thf(fact_8472_xor__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa3: code_integer] :
      ( ( code_int_of_integer @ ( bit_se3222712562003087583nteger @ X @ Xa3 ) )
      = ( bit_se6526347334894502574or_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa3 ) ) ) ).

% xor_integer.rep_eq
thf(fact_8473_push__bit__integer_Orep__eq,axiom,
    ! [X: nat,Xa3: code_integer] :
      ( ( code_int_of_integer @ ( bit_se7788150548672797655nteger @ X @ Xa3 ) )
      = ( bit_se545348938243370406it_int @ X @ ( code_int_of_integer @ Xa3 ) ) ) ).

% push_bit_integer.rep_eq
thf(fact_8474_drop__bit__integer_Orep__eq,axiom,
    ! [X: nat,Xa3: code_integer] :
      ( ( code_int_of_integer @ ( bit_se3928097537394005634nteger @ X @ Xa3 ) )
      = ( bit_se8568078237143864401it_int @ X @ ( code_int_of_integer @ Xa3 ) ) ) ).

% drop_bit_integer.rep_eq
thf(fact_8475_mask__integer_Orep__eq,axiom,
    ! [X: nat] :
      ( ( code_int_of_integer @ ( bit_se2119862282449309892nteger @ X ) )
      = ( bit_se2000444600071755411sk_int @ X ) ) ).

% mask_integer.rep_eq
thf(fact_8476_unset__bit__integer_Orep__eq,axiom,
    ! [X: nat,Xa3: code_integer] :
      ( ( code_int_of_integer @ ( bit_se8260200283734997820nteger @ X @ Xa3 ) )
      = ( bit_se4203085406695923979it_int @ X @ ( code_int_of_integer @ Xa3 ) ) ) ).

% unset_bit_integer.rep_eq
thf(fact_8477_set__bit__integer_Orep__eq,axiom,
    ! [X: nat,Xa3: code_integer] :
      ( ( code_int_of_integer @ ( bit_se2793503036327961859nteger @ X @ Xa3 ) )
      = ( bit_se7879613467334960850it_int @ X @ ( code_int_of_integer @ Xa3 ) ) ) ).

% set_bit_integer.rep_eq
thf(fact_8478_flip__bit__integer_Orep__eq,axiom,
    ! [X: nat,Xa3: code_integer] :
      ( ( code_int_of_integer @ ( bit_se1345352211410354436nteger @ X @ Xa3 ) )
      = ( bit_se2159334234014336723it_int @ X @ ( code_int_of_integer @ Xa3 ) ) ) ).

% flip_bit_integer.rep_eq
thf(fact_8479_num__of__integer_Orep__eq,axiom,
    ( code_num_of_integer
    = ( ^ [X2: code_integer] : ( num_of_nat @ ( nat2 @ ( code_int_of_integer @ X2 ) ) ) ) ) ).

% num_of_integer.rep_eq
thf(fact_8480_uminus__rat__def,axiom,
    ( uminus_uminus_rat
    = ( map_fu5673905371560938248nt_rat @ rep_Rat @ abs_Rat
      @ ^ [X2: product_prod_int_int] : ( product_Pair_int_int @ ( uminus_uminus_int @ ( product_fst_int_int @ X2 ) ) @ ( product_snd_int_int @ X2 ) ) ) ) ).

% uminus_rat_def
thf(fact_8481_plus__rat__def,axiom,
    ( plus_plus_rat
    = ( map_fu4333342158222067775at_rat @ rep_Rat @ ( map_fu5673905371560938248nt_rat @ rep_Rat @ abs_Rat )
      @ ^ [X2: product_prod_int_int,Y6: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_snd_int_int @ Y6 ) ) @ ( times_times_int @ ( product_fst_int_int @ Y6 ) @ ( product_snd_int_int @ X2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X2 ) @ ( product_snd_int_int @ Y6 ) ) ) ) ) ).

% plus_rat_def
thf(fact_8482_times__rat__def,axiom,
    ( times_times_rat
    = ( map_fu4333342158222067775at_rat @ rep_Rat @ ( map_fu5673905371560938248nt_rat @ rep_Rat @ abs_Rat )
      @ ^ [X2: product_prod_int_int,Y6: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_fst_int_int @ Y6 ) ) @ ( times_times_int @ ( product_snd_int_int @ X2 ) @ ( product_snd_int_int @ Y6 ) ) ) ) ) ).

% times_rat_def
thf(fact_8483_sorted__wrt__upt,axiom,
    ! [M: nat,N: nat] : ( sorted_wrt_nat @ ord_less_nat @ ( upt @ M @ N ) ) ).

% sorted_wrt_upt
thf(fact_8484_sorted__upt,axiom,
    ! [M: nat,N: nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( upt @ M @ N ) ) ).

% sorted_upt
thf(fact_8485_sorted__wrt__less__idx,axiom,
    ! [Ns: list_nat,I2: nat] :
      ( ( sorted_wrt_nat @ ord_less_nat @ Ns )
     => ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ns ) )
       => ( ord_less_eq_nat @ I2 @ ( nth_nat @ Ns @ I2 ) ) ) ) ).

% sorted_wrt_less_idx
thf(fact_8486_sorted__upto,axiom,
    ! [M: int,N: int] : ( sorted_wrt_int @ ord_less_eq_int @ ( upto @ M @ N ) ) ).

% sorted_upto
thf(fact_8487_sorted__wrt__upto,axiom,
    ! [I2: int,J: int] : ( sorted_wrt_int @ ord_less_int @ ( upto @ I2 @ J ) ) ).

% sorted_wrt_upto
thf(fact_8488_range__mod,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( image_nat_nat
          @ ^ [M2: nat] : ( modulo_modulo_nat @ M2 @ N )
          @ top_top_set_nat )
        = ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ).

% range_mod
thf(fact_8489_card__UNIV__unit,axiom,
    ( ( finite410649719033368117t_unit @ top_to1996260823553986621t_unit )
    = one_one_nat ) ).

% card_UNIV_unit
thf(fact_8490_infinite__UNIV__int,axiom,
    ~ ( finite_finite_int @ top_top_set_int ) ).

% infinite_UNIV_int
thf(fact_8491_integer__of__int__cases,axiom,
    ! [X: code_integer] :
      ~ ! [Y4: int] :
          ( ( X
            = ( code_integer_of_int @ Y4 ) )
         => ~ ( member_int @ Y4 @ top_top_set_int ) ) ).

% integer_of_int_cases
thf(fact_8492_integer__of__int__induct,axiom,
    ! [P: code_integer > $o,X: code_integer] :
      ( ! [Y4: int] :
          ( ( member_int @ Y4 @ top_top_set_int )
         => ( P @ ( code_integer_of_int @ Y4 ) ) )
     => ( P @ X ) ) ).

% integer_of_int_induct
thf(fact_8493_integer__of__int__inject,axiom,
    ! [X: int,Y3: int] :
      ( ( member_int @ X @ top_top_set_int )
     => ( ( member_int @ Y3 @ top_top_set_int )
       => ( ( ( code_integer_of_int @ X )
            = ( code_integer_of_int @ Y3 ) )
          = ( X = Y3 ) ) ) ) ).

% integer_of_int_inject
thf(fact_8494_int__of__integer,axiom,
    ! [X: code_integer] : ( member_int @ ( code_int_of_integer @ X ) @ top_top_set_int ) ).

% int_of_integer
thf(fact_8495_int__of__integer__cases,axiom,
    ! [Y3: int] :
      ( ( member_int @ Y3 @ top_top_set_int )
     => ~ ! [X3: code_integer] :
            ( Y3
           != ( code_int_of_integer @ X3 ) ) ) ).

% int_of_integer_cases
thf(fact_8496_int__of__integer__induct,axiom,
    ! [Y3: int,P: int > $o] :
      ( ( member_int @ Y3 @ top_top_set_int )
     => ( ! [X3: code_integer] : ( P @ ( code_int_of_integer @ X3 ) )
       => ( P @ Y3 ) ) ) ).

% int_of_integer_induct
thf(fact_8497_integer__of__int__inverse,axiom,
    ! [Y3: int] :
      ( ( member_int @ Y3 @ top_top_set_int )
     => ( ( code_int_of_integer @ ( code_integer_of_int @ Y3 ) )
        = Y3 ) ) ).

% integer_of_int_inverse
thf(fact_8498_surj__list__encode,axiom,
    ( ( image_list_nat_nat @ nat_list_encode @ top_top_set_list_nat )
    = top_top_set_nat ) ).

% surj_list_encode
thf(fact_8499_surj__prod__encode,axiom,
    ( ( image_2486076414777270412at_nat @ nat_prod_encode @ top_to4669805908274784177at_nat )
    = top_top_set_nat ) ).

% surj_prod_encode
thf(fact_8500_int__in__range__abs,axiom,
    ! [N: nat] : ( member_int @ ( semiri1314217659103216013at_int @ N ) @ ( image_int_int @ abs_abs_int @ top_top_set_int ) ) ).

% int_in_range_abs
thf(fact_8501_divmod__integer__eq__cases,axiom,
    ( code_divmod_integer
    = ( ^ [K3: code_integer,L3: code_integer] :
          ( if_Pro6119634080678213985nteger @ ( K3 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
          @ ( if_Pro6119634080678213985nteger @ ( L3 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ K3 )
            @ ( comp_C1593894019821074884nteger @ ( comp_C8797469213163452608nteger @ produc6499014454317279255nteger @ times_3573771949741848930nteger ) @ sgn_sgn_Code_integer @ L3
              @ ( if_Pro6119634080678213985nteger
                @ ( ( sgn_sgn_Code_integer @ K3 )
                  = ( sgn_sgn_Code_integer @ L3 ) )
                @ ( code_divmod_abs @ K3 @ L3 )
                @ ( produc6916734918728496179nteger
                  @ ^ [R5: code_integer,S3: code_integer] : ( if_Pro6119634080678213985nteger @ ( S3 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ L3 ) @ S3 ) ) )
                  @ ( code_divmod_abs @ K3 @ L3 ) ) ) ) ) ) ) ) ).

% divmod_integer_eq_cases
thf(fact_8502_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_8503_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_8504_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_8505_max__Suc1,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_max_nat @ ( suc @ N ) @ M )
      = ( case_nat_nat @ ( suc @ N )
        @ ^ [M4: nat] : ( suc @ ( ord_max_nat @ N @ M4 ) )
        @ M ) ) ).

% max_Suc1
thf(fact_8506_max__Suc2,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_max_nat @ M @ ( suc @ N ) )
      = ( case_nat_nat @ ( suc @ N )
        @ ^ [M4: nat] : ( suc @ ( ord_max_nat @ M4 @ N ) )
        @ M ) ) ).

% max_Suc2
thf(fact_8507_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_8508_min__Suc1,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_min_nat @ ( suc @ N ) @ M )
      = ( case_nat_nat @ zero_zero_nat
        @ ^ [M4: nat] : ( suc @ ( ord_min_nat @ N @ M4 ) )
        @ M ) ) ).

% min_Suc1
thf(fact_8509_min__Suc2,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_min_nat @ M @ ( suc @ N ) )
      = ( case_nat_nat @ zero_zero_nat
        @ ^ [M4: nat] : ( suc @ ( ord_min_nat @ M4 @ N ) )
        @ M ) ) ).

% min_Suc2
thf(fact_8510_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_8511_euclidean__size__int__def,axiom,
    ( euclid4774559944035922753ze_int
    = ( comp_int_nat_int @ nat2 @ abs_abs_int ) ) ).

% euclidean_size_int_def
thf(fact_8512_pred__def,axiom,
    ( pred
    = ( case_nat_nat @ zero_zero_nat
      @ ^ [X24: nat] : X24 ) ) ).

% pred_def
thf(fact_8513_Gcd__int__def,axiom,
    ( gcd_Gcd_int
    = ( ^ [K6: set_int] : ( semiri1314217659103216013at_int @ ( gcd_Gcd_nat @ ( image_int_nat @ ( comp_int_nat_int @ nat2 @ abs_abs_int ) @ K6 ) ) ) ) ) ).

% Gcd_int_def
thf(fact_8514_Code__Numeral_Onegative__def,axiom,
    ( code_negative
    = ( comp_C3531382070062128313er_num @ uminus1351360451143612070nteger @ numera6620942414471956472nteger ) ) ).

% Code_Numeral.negative_def
thf(fact_8515_range__abs__Nats,axiom,
    ( ( image_int_int @ abs_abs_int @ top_top_set_int )
    = semiring_1_Nats_int ) ).

% range_abs_Nats
thf(fact_8516_Sup__nat__empty,axiom,
    ( ( complete_Sup_Sup_nat @ bot_bot_set_nat )
    = zero_zero_nat ) ).

% Sup_nat_empty
thf(fact_8517_surj__int__encode,axiom,
    ( ( image_int_nat @ nat_int_encode @ top_top_set_int )
    = top_top_set_nat ) ).

% surj_int_encode
thf(fact_8518_Suc__funpow,axiom,
    ! [N: nat] :
      ( ( compow_nat_nat @ N @ suc )
      = ( plus_plus_nat @ N ) ) ).

% Suc_funpow
thf(fact_8519_int__encode__eq,axiom,
    ! [X: int,Y3: int] :
      ( ( ( nat_int_encode @ X )
        = ( nat_int_encode @ Y3 ) )
      = ( X = Y3 ) ) ).

% int_encode_eq
thf(fact_8520_Inf__int__def,axiom,
    ( complete_Inf_Inf_int
    = ( ^ [X7: set_int] : ( uminus_uminus_int @ ( complete_Sup_Sup_int @ ( image_int_int @ uminus_uminus_int @ X7 ) ) ) ) ) ).

% Inf_int_def
thf(fact_8521_measure__function__int,axiom,
    fun_is_measure_int @ ( comp_int_nat_int @ nat2 @ abs_abs_int ) ).

% measure_function_int
thf(fact_8522_surj__int__decode,axiom,
    ( ( image_nat_int @ nat_int_decode @ top_top_set_nat )
    = top_top_set_int ) ).

% surj_int_decode
thf(fact_8523_int__decode__inverse,axiom,
    ! [N: nat] :
      ( ( nat_int_encode @ ( nat_int_decode @ N ) )
      = N ) ).

% int_decode_inverse
thf(fact_8524_int__encode__inverse,axiom,
    ! [X: int] :
      ( ( nat_int_decode @ ( nat_int_encode @ X ) )
      = X ) ).

% int_encode_inverse
thf(fact_8525_int__decode__eq,axiom,
    ! [X: nat,Y3: nat] :
      ( ( ( nat_int_decode @ X )
        = ( nat_int_decode @ Y3 ) )
      = ( X = Y3 ) ) ).

% int_decode_eq
thf(fact_8526_max__nat_Osemilattice__neutr__order__axioms,axiom,
    ( semila1623282765462674594er_nat @ ord_max_nat @ zero_zero_nat
    @ ^ [X2: nat,Y6: nat] : ( ord_less_eq_nat @ Y6 @ X2 )
    @ ^ [X2: nat,Y6: nat] : ( ord_less_nat @ Y6 @ X2 ) ) ).

% max_nat.semilattice_neutr_order_axioms
thf(fact_8527_times__int_Oabs__eq,axiom,
    ! [Xa3: product_prod_nat_nat,X: product_prod_nat_nat] :
      ( ( times_times_int @ ( abs_Integ @ Xa3 ) @ ( abs_Integ @ X ) )
      = ( abs_Integ
        @ ( produc27273713700761075at_nat
          @ ^ [X2: nat,Y6: nat] :
              ( produc2626176000494625587at_nat
              @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X2 @ U3 ) @ ( times_times_nat @ Y6 @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X2 @ V4 ) @ ( times_times_nat @ Y6 @ U3 ) ) ) )
          @ Xa3
          @ X ) ) ) ).

% times_int.abs_eq
thf(fact_8528_int_Oabs__induct,axiom,
    ! [P: int > $o,X: int] :
      ( ! [Y4: product_prod_nat_nat] : ( P @ ( abs_Integ @ Y4 ) )
     => ( P @ X ) ) ).

% int.abs_induct
thf(fact_8529_eq__Abs__Integ,axiom,
    ! [Z2: int] :
      ~ ! [X3: nat,Y4: nat] :
          ( Z2
         != ( abs_Integ @ ( product_Pair_nat_nat @ X3 @ Y4 ) ) ) ).

% eq_Abs_Integ
thf(fact_8530_zero__int__def,axiom,
    ( zero_zero_int
    = ( abs_Integ @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) ) ) ).

% zero_int_def
thf(fact_8531_int__def,axiom,
    ( semiri1314217659103216013at_int
    = ( ^ [N3: nat] : ( abs_Integ @ ( product_Pair_nat_nat @ N3 @ zero_zero_nat ) ) ) ) ).

% int_def
thf(fact_8532_nat_Oabs__eq,axiom,
    ! [X: product_prod_nat_nat] :
      ( ( nat2 @ ( abs_Integ @ X ) )
      = ( produc6842872674320459806at_nat @ minus_minus_nat @ X ) ) ).

% nat.abs_eq
thf(fact_8533_uminus__int_Oabs__eq,axiom,
    ! [X: product_prod_nat_nat] :
      ( ( uminus_uminus_int @ ( abs_Integ @ X ) )
      = ( abs_Integ
        @ ( produc2626176000494625587at_nat
          @ ^ [X2: nat,Y6: nat] : ( product_Pair_nat_nat @ Y6 @ X2 )
          @ X ) ) ) ).

% uminus_int.abs_eq
thf(fact_8534_one__int__def,axiom,
    ( one_one_int
    = ( abs_Integ @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) ) ) ).

% one_int_def
thf(fact_8535_gcd__nat_Osemilattice__neutr__order__axioms,axiom,
    ( semila1623282765462674594er_nat @ gcd_gcd_nat @ zero_zero_nat @ dvd_dvd_nat
    @ ^ [M2: nat,N3: nat] :
        ( ( dvd_dvd_nat @ M2 @ N3 )
        & ( M2 != N3 ) ) ) ).

% gcd_nat.semilattice_neutr_order_axioms
thf(fact_8536_less__int_Oabs__eq,axiom,
    ! [Xa3: product_prod_nat_nat,X: product_prod_nat_nat] :
      ( ( ord_less_int @ ( abs_Integ @ Xa3 ) @ ( abs_Integ @ X ) )
      = ( produc8739625826339149834_nat_o
        @ ^ [X2: nat,Y6: nat] :
            ( produc6081775807080527818_nat_o
            @ ^ [U3: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X2 @ V4 ) @ ( plus_plus_nat @ U3 @ Y6 ) ) )
        @ Xa3
        @ X ) ) ).

% less_int.abs_eq
thf(fact_8537_less__eq__int_Oabs__eq,axiom,
    ! [Xa3: product_prod_nat_nat,X: product_prod_nat_nat] :
      ( ( ord_less_eq_int @ ( abs_Integ @ Xa3 ) @ ( abs_Integ @ X ) )
      = ( produc8739625826339149834_nat_o
        @ ^ [X2: nat,Y6: nat] :
            ( produc6081775807080527818_nat_o
            @ ^ [U3: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X2 @ V4 ) @ ( plus_plus_nat @ U3 @ Y6 ) ) )
        @ Xa3
        @ X ) ) ).

% less_eq_int.abs_eq
thf(fact_8538_plus__int_Oabs__eq,axiom,
    ! [Xa3: product_prod_nat_nat,X: product_prod_nat_nat] :
      ( ( plus_plus_int @ ( abs_Integ @ Xa3 ) @ ( abs_Integ @ X ) )
      = ( abs_Integ
        @ ( produc27273713700761075at_nat
          @ ^ [X2: nat,Y6: nat] :
              ( produc2626176000494625587at_nat
              @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X2 @ U3 ) @ ( plus_plus_nat @ Y6 @ V4 ) ) )
          @ Xa3
          @ X ) ) ) ).

% plus_int.abs_eq
thf(fact_8539_minus__int_Oabs__eq,axiom,
    ! [Xa3: product_prod_nat_nat,X: product_prod_nat_nat] :
      ( ( minus_minus_int @ ( abs_Integ @ Xa3 ) @ ( abs_Integ @ X ) )
      = ( abs_Integ
        @ ( produc27273713700761075at_nat
          @ ^ [X2: nat,Y6: nat] :
              ( produc2626176000494625587at_nat
              @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X2 @ V4 ) @ ( plus_plus_nat @ Y6 @ U3 ) ) )
          @ Xa3
          @ X ) ) ) ).

% minus_int.abs_eq
thf(fact_8540_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
        @ ^ [Q3: 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 @ Q3 ) ) )
        @ ( bit_take_bit_num @ ( numeral_numeral_nat @ M ) @ N ) ) ) ).

% take_bit_numeral_minus_numeral_int
thf(fact_8541_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_8542_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_8543_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_8544_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_8545_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_8546_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_8547_less__eq__int_Orep__eq,axiom,
    ( ord_less_eq_int
    = ( ^ [X2: int,Xa4: int] :
          ( produc8739625826339149834_nat_o
          @ ^ [Y6: nat,Z6: nat] :
              ( produc6081775807080527818_nat_o
              @ ^ [U3: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ Y6 @ V4 ) @ ( plus_plus_nat @ U3 @ Z6 ) ) )
          @ ( rep_Integ @ X2 )
          @ ( rep_Integ @ Xa4 ) ) ) ) ).

% less_eq_int.rep_eq
thf(fact_8548_less__int_Orep__eq,axiom,
    ( ord_less_int
    = ( ^ [X2: int,Xa4: int] :
          ( produc8739625826339149834_nat_o
          @ ^ [Y6: nat,Z6: nat] :
              ( produc6081775807080527818_nat_o
              @ ^ [U3: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ Y6 @ V4 ) @ ( plus_plus_nat @ U3 @ Z6 ) ) )
          @ ( rep_Integ @ X2 )
          @ ( rep_Integ @ Xa4 ) ) ) ) ).

% less_int.rep_eq
thf(fact_8549_nat_Orep__eq,axiom,
    ( nat2
    = ( ^ [X2: int] : ( produc6842872674320459806at_nat @ minus_minus_nat @ ( rep_Integ @ X2 ) ) ) ) ).

% nat.rep_eq
thf(fact_8550_uminus__int__def,axiom,
    ( uminus_uminus_int
    = ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ
      @ ( produc2626176000494625587at_nat
        @ ^ [X2: nat,Y6: nat] : ( product_Pair_nat_nat @ Y6 @ X2 ) ) ) ) ).

% uminus_int_def
thf(fact_8551_times__int__def,axiom,
    ( times_times_int
    = ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
      @ ( produc27273713700761075at_nat
        @ ^ [X2: nat,Y6: nat] :
            ( produc2626176000494625587at_nat
            @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X2 @ U3 ) @ ( times_times_nat @ Y6 @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X2 @ V4 ) @ ( times_times_nat @ Y6 @ U3 ) ) ) ) ) ) ) ).

% times_int_def
thf(fact_8552_minus__int__def,axiom,
    ( minus_minus_int
    = ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
      @ ( produc27273713700761075at_nat
        @ ^ [X2: nat,Y6: nat] :
            ( produc2626176000494625587at_nat
            @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X2 @ V4 ) @ ( plus_plus_nat @ Y6 @ U3 ) ) ) ) ) ) ).

% minus_int_def
thf(fact_8553_plus__int__def,axiom,
    ( plus_plus_int
    = ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
      @ ( produc27273713700761075at_nat
        @ ^ [X2: nat,Y6: nat] :
            ( produc2626176000494625587at_nat
            @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X2 @ U3 ) @ ( plus_plus_nat @ Y6 @ V4 ) ) ) ) ) ) ).

% plus_int_def
thf(fact_8554_of__nat__eq__id,axiom,
    semiri1316708129612266289at_nat = id_nat ).

% of_nat_eq_id
thf(fact_8555_less__int__def,axiom,
    ( ord_less_int
    = ( map_fu434086159418415080_int_o @ rep_Integ @ ( map_fu4826362097070443709at_o_o @ rep_Integ @ id_o )
      @ ( produc8739625826339149834_nat_o
        @ ^ [X2: nat,Y6: nat] :
            ( produc6081775807080527818_nat_o
            @ ^ [U3: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X2 @ V4 ) @ ( plus_plus_nat @ U3 @ Y6 ) ) ) ) ) ) ).

% less_int_def
thf(fact_8556_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
        @ ^ [X2: nat,Y6: nat] :
            ( produc6081775807080527818_nat_o
            @ ^ [U3: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X2 @ V4 ) @ ( plus_plus_nat @ U3 @ Y6 ) ) ) ) ) ) ).

% less_eq_int_def
thf(fact_8557_nat__def,axiom,
    ( nat2
    = ( map_fu2345160673673942751at_nat @ rep_Integ @ id_nat @ ( produc6842872674320459806at_nat @ minus_minus_nat ) ) ) ).

% nat_def
thf(fact_8558_positive__def,axiom,
    ( positive
    = ( map_fu898904425404107465nt_o_o @ rep_Rat @ id_o
      @ ^ [X2: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_snd_int_int @ X2 ) ) ) ) ) ).

% positive_def
thf(fact_8559_euclidean__size__nat__def,axiom,
    euclid4777050414544973029ze_nat = id_nat ).

% euclidean_size_nat_def
thf(fact_8560_euclidean__size__integer__def,axiom,
    ( euclid6377331345833325938nteger
    = ( map_fu6539832666145259331at_nat @ code_int_of_integer @ id_nat @ euclid4774559944035922753ze_int ) ) ).

% euclidean_size_integer_def
thf(fact_8561_Max__divisors__self__nat,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( ( lattic8265883725875713057ax_nat
          @ ( collect_nat
            @ ^ [D4: nat] : ( dvd_dvd_nat @ D4 @ N ) ) )
        = N ) ) ).

% Max_divisors_self_nat
thf(fact_8562_Sup__nat__def,axiom,
    ( complete_Sup_Sup_nat
    = ( ^ [X7: set_nat] : ( if_nat @ ( X7 = bot_bot_set_nat ) @ zero_zero_nat @ ( lattic8265883725875713057ax_nat @ X7 ) ) ) ) ).

% Sup_nat_def
thf(fact_8563_card__le__Suc__Max,axiom,
    ! [S4: set_nat] :
      ( ( finite_finite_nat @ S4 )
     => ( ord_less_eq_nat @ ( finite_card_nat @ S4 ) @ ( suc @ ( lattic8265883725875713057ax_nat @ S4 ) ) ) ) ).

% card_le_Suc_Max
thf(fact_8564_divide__nat__def,axiom,
    ( divide_divide_nat
    = ( ^ [M2: nat,N3: nat] :
          ( if_nat @ ( N3 = zero_zero_nat ) @ zero_zero_nat
          @ ( lattic8265883725875713057ax_nat
            @ ( collect_nat
              @ ^ [K3: nat] : ( ord_less_eq_nat @ ( times_times_nat @ K3 @ N3 ) @ M2 ) ) ) ) ) ) ).

% divide_nat_def
thf(fact_8565_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
            @ ^ [D4: nat] :
                ( ( dvd_dvd_nat @ D4 @ M )
                & ( dvd_dvd_nat @ D4 @ N ) ) ) ) ) ) ).

% gcd_is_Max_divisors_nat
thf(fact_8566_nat__of__integer__def,axiom,
    ( code_nat_of_integer
    = ( map_fu6539832666145259331at_nat @ code_int_of_integer @ id_nat @ nat2 ) ) ).

% nat_of_integer_def
thf(fact_8567_Gcd__eq__Max,axiom,
    ! [M7: set_nat] :
      ( ( finite_finite_nat @ M7 )
     => ( ( M7 != bot_bot_set_nat )
       => ( ~ ( member_nat @ zero_zero_nat @ M7 )
         => ( ( gcd_Gcd_nat @ M7 )
            = ( lattic8265883725875713057ax_nat
              @ ( comple7806235888213564991et_nat
                @ ( image_nat_set_nat
                  @ ^ [M2: nat] :
                      ( collect_nat
                      @ ^ [D4: nat] : ( dvd_dvd_nat @ D4 @ M2 ) )
                  @ M7 ) ) ) ) ) ) ) ).

% Gcd_eq_Max
thf(fact_8568_mask__integer__def,axiom,
    ( bit_se2119862282449309892nteger
    = ( map_fu6290471996055670595nteger @ id_nat @ code_integer_of_int @ bit_se2000444600071755411sk_int ) ) ).

% mask_integer_def
thf(fact_8569_positive_Orsp,axiom,
    ( bNF_re8699439704749558557nt_o_o @ ratrel
    @ ^ [Y: $o,Z: $o] : ( Y = Z )
    @ ^ [X2: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_snd_int_int @ X2 ) ) )
    @ ^ [X2: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_snd_int_int @ X2 ) ) ) ) ).

% positive.rsp
thf(fact_8570_Max__divisors__self__int,axiom,
    ! [N: int] :
      ( ( N != zero_zero_int )
     => ( ( lattic8263393255366662781ax_int
          @ ( collect_int
            @ ^ [D4: int] : ( dvd_dvd_int @ D4 @ N ) ) )
        = ( abs_abs_int @ N ) ) ) ).

% Max_divisors_self_int
thf(fact_8571_gcd__is__Max__divisors__int,axiom,
    ! [N: int,M: int] :
      ( ( N != zero_zero_int )
     => ( ( gcd_gcd_int @ M @ N )
        = ( lattic8263393255366662781ax_int
          @ ( collect_int
            @ ^ [D4: int] :
                ( ( dvd_dvd_int @ D4 @ M )
                & ( dvd_dvd_int @ D4 @ N ) ) ) ) ) ) ).

% gcd_is_Max_divisors_int
thf(fact_8572_integer__of__nat__def,axiom,
    ( code_integer_of_nat
    = ( map_fu6290471996055670595nteger @ id_nat @ code_integer_of_int @ semiri1314217659103216013at_int ) ) ).

% integer_of_nat_def
thf(fact_8573_nat__of__integer__integer__of__nat,axiom,
    ! [N: nat] :
      ( ( code_nat_of_integer @ ( code_integer_of_nat @ N ) )
      = N ) ).

% nat_of_integer_integer_of_nat
thf(fact_8574_integer__of__nat_Orep__eq,axiom,
    ! [X: nat] :
      ( ( code_int_of_integer @ ( code_integer_of_nat @ X ) )
      = ( semiri1314217659103216013at_int @ X ) ) ).

% integer_of_nat.rep_eq
thf(fact_8575_int__of__integer__integer__of__nat,axiom,
    ! [N: nat] :
      ( ( code_int_of_integer @ ( code_integer_of_nat @ N ) )
      = ( semiri1314217659103216013at_int @ N ) ) ).

% int_of_integer_integer_of_nat
thf(fact_8576_euclidean__size__integer_Orsp,axiom,
    ( bNF_re3715656647883201625at_nat
    @ ^ [Y: int,Z: int] : ( Y = Z )
    @ ^ [Y: nat,Z: nat] : ( Y = Z )
    @ euclid4774559944035922753ze_int
    @ euclid4774559944035922753ze_int ) ).

% euclidean_size_integer.rsp
thf(fact_8577_euclidean__size__natural_Orsp,axiom,
    ( bNF_re5653821019739307937at_nat
    @ ^ [Y: nat,Z: nat] : ( Y = Z )
    @ ^ [Y: nat,Z: nat] : ( Y = Z )
    @ euclid4777050414544973029ze_nat
    @ euclid4777050414544973029ze_nat ) ).

% euclidean_size_natural.rsp
thf(fact_8578_dup_Orsp,axiom,
    ( bNF_re4712519889275205905nt_int
    @ ^ [Y: int,Z: int] : ( Y = Z )
    @ ^ [Y: int,Z: int] : ( Y = Z )
    @ ^ [K3: int] : ( plus_plus_int @ K3 @ K3 )
    @ ^ [K3: int] : ( plus_plus_int @ K3 @ K3 ) ) ).

% dup.rsp
thf(fact_8579_sub_Orsp,axiom,
    ( bNF_re8402795839162346335um_int
    @ ^ [Y: num,Z: num] : ( Y = Z )
    @ ( bNF_re1822329894187522285nt_int
      @ ^ [Y: num,Z: num] : ( Y = Z )
      @ ^ [Y: int,Z: int] : ( Y = Z ) )
    @ ^ [M2: num,N3: num] : ( minus_minus_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N3 ) )
    @ ^ [M2: num,N3: num] : ( minus_minus_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N3 ) ) ) ).

% sub.rsp
thf(fact_8580_less__eq__integer_Orsp,axiom,
    ( bNF_re3403563459893282935_int_o
    @ ^ [Y: int,Z: int] : ( Y = Z )
    @ ( bNF_re5089333283451836215nt_o_o
      @ ^ [Y: int,Z: int] : ( Y = Z )
      @ ^ [Y: $o,Z: $o] : ( Y = Z ) )
    @ ord_less_eq_int
    @ ord_less_eq_int ) ).

% less_eq_integer.rsp
thf(fact_8581_less__eq__natural_Orsp,axiom,
    ( bNF_re578469030762574527_nat_o
    @ ^ [Y: nat,Z: nat] : ( Y = Z )
    @ ( bNF_re4705727531993890431at_o_o
      @ ^ [Y: nat,Z: nat] : ( Y = Z )
      @ ^ [Y: $o,Z: $o] : ( Y = Z ) )
    @ ord_less_eq_nat
    @ ord_less_eq_nat ) ).

% less_eq_natural.rsp
thf(fact_8582_less__integer_Orsp,axiom,
    ( bNF_re3403563459893282935_int_o
    @ ^ [Y: int,Z: int] : ( Y = Z )
    @ ( bNF_re5089333283451836215nt_o_o
      @ ^ [Y: int,Z: int] : ( Y = Z )
      @ ^ [Y: $o,Z: $o] : ( Y = Z ) )
    @ ord_less_int
    @ ord_less_int ) ).

% less_integer.rsp
thf(fact_8583_less__natural_Orsp,axiom,
    ( bNF_re578469030762574527_nat_o
    @ ^ [Y: nat,Z: nat] : ( Y = Z )
    @ ( bNF_re4705727531993890431at_o_o
      @ ^ [Y: nat,Z: nat] : ( Y = Z )
      @ ^ [Y: $o,Z: $o] : ( Y = Z ) )
    @ ord_less_nat
    @ ord_less_nat ) ).

% less_natural.rsp
thf(fact_8584_uminus__integer_Orsp,axiom,
    ( bNF_re4712519889275205905nt_int
    @ ^ [Y: int,Z: int] : ( Y = Z )
    @ ^ [Y: int,Z: int] : ( Y = Z )
    @ uminus_uminus_int
    @ uminus_uminus_int ) ).

% uminus_integer.rsp
thf(fact_8585_times__natural_Orsp,axiom,
    ( bNF_re1345281282404953727at_nat
    @ ^ [Y: nat,Z: nat] : ( Y = Z )
    @ ( bNF_re5653821019739307937at_nat
      @ ^ [Y: nat,Z: nat] : ( Y = Z )
      @ ^ [Y: nat,Z: nat] : ( Y = Z ) )
    @ times_times_nat
    @ times_times_nat ) ).

% times_natural.rsp
thf(fact_8586_times__integer_Orsp,axiom,
    ( bNF_re711492959462206631nt_int
    @ ^ [Y: int,Z: int] : ( Y = Z )
    @ ( bNF_re4712519889275205905nt_int
      @ ^ [Y: int,Z: int] : ( Y = Z )
      @ ^ [Y: int,Z: int] : ( Y = Z ) )
    @ times_times_int
    @ times_times_int ) ).

% times_integer.rsp
thf(fact_8587_natural__of__integer_Orsp,axiom,
    ( bNF_re3715656647883201625at_nat
    @ ^ [Y: int,Z: int] : ( Y = Z )
    @ ^ [Y: nat,Z: nat] : ( Y = Z )
    @ nat2
    @ nat2 ) ).

% natural_of_integer.rsp
thf(fact_8588_Suc_Orsp,axiom,
    ( bNF_re5653821019739307937at_nat
    @ ^ [Y: nat,Z: nat] : ( Y = Z )
    @ ^ [Y: nat,Z: nat] : ( Y = Z )
    @ suc
    @ suc ) ).

% Suc.rsp
thf(fact_8589_modulo__integer_Orsp,axiom,
    ( bNF_re711492959462206631nt_int
    @ ^ [Y: int,Z: int] : ( Y = Z )
    @ ( bNF_re4712519889275205905nt_int
      @ ^ [Y: int,Z: int] : ( Y = Z )
      @ ^ [Y: int,Z: int] : ( Y = Z ) )
    @ modulo_modulo_int
    @ modulo_modulo_int ) ).

% modulo_integer.rsp
thf(fact_8590_modulo__natural_Orsp,axiom,
    ( bNF_re1345281282404953727at_nat
    @ ^ [Y: nat,Z: nat] : ( Y = Z )
    @ ( bNF_re5653821019739307937at_nat
      @ ^ [Y: nat,Z: nat] : ( Y = Z )
      @ ^ [Y: nat,Z: nat] : ( Y = Z ) )
    @ modulo_modulo_nat
    @ modulo_modulo_nat ) ).

% modulo_natural.rsp
thf(fact_8591_sgn__integer_Orsp,axiom,
    ( bNF_re4712519889275205905nt_int
    @ ^ [Y: int,Z: int] : ( Y = Z )
    @ ^ [Y: int,Z: int] : ( Y = Z )
    @ sgn_sgn_int
    @ sgn_sgn_int ) ).

% sgn_integer.rsp
thf(fact_8592_plus__natural_Orsp,axiom,
    ( bNF_re1345281282404953727at_nat
    @ ^ [Y: nat,Z: nat] : ( Y = Z )
    @ ( bNF_re5653821019739307937at_nat
      @ ^ [Y: nat,Z: nat] : ( Y = Z )
      @ ^ [Y: nat,Z: nat] : ( Y = Z ) )
    @ plus_plus_nat
    @ plus_plus_nat ) ).

% plus_natural.rsp
thf(fact_8593_plus__integer_Orsp,axiom,
    ( bNF_re711492959462206631nt_int
    @ ^ [Y: int,Z: int] : ( Y = Z )
    @ ( bNF_re4712519889275205905nt_int
      @ ^ [Y: int,Z: int] : ( Y = Z )
      @ ^ [Y: int,Z: int] : ( Y = Z ) )
    @ plus_plus_int
    @ plus_plus_int ) ).

% plus_integer.rsp
thf(fact_8594_divide__natural_Orsp,axiom,
    ( bNF_re1345281282404953727at_nat
    @ ^ [Y: nat,Z: nat] : ( Y = Z )
    @ ( bNF_re5653821019739307937at_nat
      @ ^ [Y: nat,Z: nat] : ( Y = Z )
      @ ^ [Y: nat,Z: nat] : ( Y = Z ) )
    @ divide_divide_nat
    @ divide_divide_nat ) ).

% divide_natural.rsp
thf(fact_8595_divide__integer_Orsp,axiom,
    ( bNF_re711492959462206631nt_int
    @ ^ [Y: int,Z: int] : ( Y = Z )
    @ ( bNF_re4712519889275205905nt_int
      @ ^ [Y: int,Z: int] : ( Y = Z )
      @ ^ [Y: int,Z: int] : ( Y = Z ) )
    @ divide_divide_int
    @ divide_divide_int ) ).

% divide_integer.rsp
thf(fact_8596_abs__integer_Orsp,axiom,
    ( bNF_re4712519889275205905nt_int
    @ ^ [Y: int,Z: int] : ( Y = Z )
    @ ^ [Y: int,Z: int] : ( Y = Z )
    @ abs_abs_int
    @ abs_abs_int ) ).

% abs_integer.rsp
thf(fact_8597_minus__integer_Orsp,axiom,
    ( bNF_re711492959462206631nt_int
    @ ^ [Y: int,Z: int] : ( Y = Z )
    @ ( bNF_re4712519889275205905nt_int
      @ ^ [Y: int,Z: int] : ( Y = Z )
      @ ^ [Y: int,Z: int] : ( Y = Z ) )
    @ minus_minus_int
    @ minus_minus_int ) ).

% minus_integer.rsp
thf(fact_8598_minus__natural_Orsp,axiom,
    ( bNF_re1345281282404953727at_nat
    @ ^ [Y: nat,Z: nat] : ( Y = Z )
    @ ( bNF_re5653821019739307937at_nat
      @ ^ [Y: nat,Z: nat] : ( Y = Z )
      @ ^ [Y: nat,Z: nat] : ( Y = Z ) )
    @ minus_minus_nat
    @ minus_minus_nat ) ).

% minus_natural.rsp
thf(fact_8599_integer__of__natural_Orsp,axiom,
    ( bNF_re6650684261131312217nt_int
    @ ^ [Y: nat,Z: nat] : ( Y = Z )
    @ ^ [Y: int,Z: int] : ( Y = Z )
    @ semiri1314217659103216013at_int
    @ semiri1314217659103216013at_int ) ).

% integer_of_natural.rsp
thf(fact_8600_push__bit__natural_Orsp,axiom,
    ( bNF_re1345281282404953727at_nat
    @ ^ [Y: nat,Z: nat] : ( Y = Z )
    @ ( bNF_re5653821019739307937at_nat
      @ ^ [Y: nat,Z: nat] : ( Y = Z )
      @ ^ [Y: nat,Z: nat] : ( Y = Z ) )
    @ bit_se547839408752420682it_nat
    @ bit_se547839408752420682it_nat ) ).

% push_bit_natural.rsp
thf(fact_8601_push__bit__integer_Orsp,axiom,
    ( bNF_re4785983289428654063nt_int
    @ ^ [Y: nat,Z: nat] : ( Y = Z )
    @ ( bNF_re4712519889275205905nt_int
      @ ^ [Y: int,Z: int] : ( Y = Z )
      @ ^ [Y: int,Z: int] : ( Y = Z ) )
    @ bit_se545348938243370406it_int
    @ bit_se545348938243370406it_int ) ).

% push_bit_integer.rsp
thf(fact_8602_flip__bit__integer_Orsp,axiom,
    ( bNF_re4785983289428654063nt_int
    @ ^ [Y: nat,Z: nat] : ( Y = Z )
    @ ( bNF_re4712519889275205905nt_int
      @ ^ [Y: int,Z: int] : ( Y = Z )
      @ ^ [Y: int,Z: int] : ( Y = Z ) )
    @ bit_se2159334234014336723it_int
    @ bit_se2159334234014336723it_int ) ).

% flip_bit_integer.rsp
thf(fact_8603_flip__bit__natural_Orsp,axiom,
    ( bNF_re1345281282404953727at_nat
    @ ^ [Y: nat,Z: nat] : ( Y = Z )
    @ ( bNF_re5653821019739307937at_nat
      @ ^ [Y: nat,Z: nat] : ( Y = Z )
      @ ^ [Y: nat,Z: nat] : ( Y = Z ) )
    @ bit_se2161824704523386999it_nat
    @ bit_se2161824704523386999it_nat ) ).

% flip_bit_natural.rsp
thf(fact_8604_take__bit__natural_Orsp,axiom,
    ( bNF_re1345281282404953727at_nat
    @ ^ [Y: nat,Z: nat] : ( Y = Z )
    @ ( bNF_re5653821019739307937at_nat
      @ ^ [Y: nat,Z: nat] : ( Y = Z )
      @ ^ [Y: nat,Z: nat] : ( Y = Z ) )
    @ bit_se2925701944663578781it_nat
    @ bit_se2925701944663578781it_nat ) ).

% take_bit_natural.rsp
thf(fact_8605_take__bit__integer_Orsp,axiom,
    ( bNF_re4785983289428654063nt_int
    @ ^ [Y: nat,Z: nat] : ( Y = Z )
    @ ( bNF_re4712519889275205905nt_int
      @ ^ [Y: int,Z: int] : ( Y = Z )
      @ ^ [Y: int,Z: int] : ( Y = Z ) )
    @ bit_se2923211474154528505it_int
    @ bit_se2923211474154528505it_int ) ).

% take_bit_integer.rsp
thf(fact_8606_and__natural_Orsp,axiom,
    ( bNF_re1345281282404953727at_nat
    @ ^ [Y: nat,Z: nat] : ( Y = Z )
    @ ( bNF_re5653821019739307937at_nat
      @ ^ [Y: nat,Z: nat] : ( Y = Z )
      @ ^ [Y: nat,Z: nat] : ( Y = Z ) )
    @ bit_se727722235901077358nd_nat
    @ bit_se727722235901077358nd_nat ) ).

% and_natural.rsp
thf(fact_8607_and__integer_Orsp,axiom,
    ( bNF_re711492959462206631nt_int
    @ ^ [Y: int,Z: int] : ( Y = Z )
    @ ( bNF_re4712519889275205905nt_int
      @ ^ [Y: int,Z: int] : ( Y = Z )
      @ ^ [Y: int,Z: int] : ( Y = Z ) )
    @ bit_se725231765392027082nd_int
    @ bit_se725231765392027082nd_int ) ).

% and_integer.rsp
thf(fact_8608_unset__bit__natural_Orsp,axiom,
    ( bNF_re1345281282404953727at_nat
    @ ^ [Y: nat,Z: nat] : ( Y = Z )
    @ ( bNF_re5653821019739307937at_nat
      @ ^ [Y: nat,Z: nat] : ( Y = Z )
      @ ^ [Y: nat,Z: nat] : ( Y = Z ) )
    @ bit_se4205575877204974255it_nat
    @ bit_se4205575877204974255it_nat ) ).

% unset_bit_natural.rsp
thf(fact_8609_unset__bit__integer_Orsp,axiom,
    ( bNF_re4785983289428654063nt_int
    @ ^ [Y: nat,Z: nat] : ( Y = Z )
    @ ( bNF_re4712519889275205905nt_int
      @ ^ [Y: int,Z: int] : ( Y = Z )
      @ ^ [Y: int,Z: int] : ( Y = Z ) )
    @ bit_se4203085406695923979it_int
    @ bit_se4203085406695923979it_int ) ).

% unset_bit_integer.rsp
thf(fact_8610_xor__integer_Orsp,axiom,
    ( bNF_re711492959462206631nt_int
    @ ^ [Y: int,Z: int] : ( Y = Z )
    @ ( bNF_re4712519889275205905nt_int
      @ ^ [Y: int,Z: int] : ( Y = Z )
      @ ^ [Y: int,Z: int] : ( Y = Z ) )
    @ bit_se6526347334894502574or_int
    @ bit_se6526347334894502574or_int ) ).

% xor_integer.rsp
thf(fact_8611_xor__natural_Orsp,axiom,
    ( bNF_re1345281282404953727at_nat
    @ ^ [Y: nat,Z: nat] : ( Y = Z )
    @ ( bNF_re5653821019739307937at_nat
      @ ^ [Y: nat,Z: nat] : ( Y = Z )
      @ ^ [Y: nat,Z: nat] : ( Y = Z ) )
    @ bit_se6528837805403552850or_nat
    @ bit_se6528837805403552850or_nat ) ).

% xor_natural.rsp
thf(fact_8612_set__bit__integer_Orsp,axiom,
    ( bNF_re4785983289428654063nt_int
    @ ^ [Y: nat,Z: nat] : ( Y = Z )
    @ ( bNF_re4712519889275205905nt_int
      @ ^ [Y: int,Z: int] : ( Y = Z )
      @ ^ [Y: int,Z: int] : ( Y = Z ) )
    @ bit_se7879613467334960850it_int
    @ bit_se7879613467334960850it_int ) ).

% set_bit_integer.rsp
thf(fact_8613_set__bit__natural_Orsp,axiom,
    ( bNF_re1345281282404953727at_nat
    @ ^ [Y: nat,Z: nat] : ( Y = Z )
    @ ( bNF_re5653821019739307937at_nat
      @ ^ [Y: nat,Z: nat] : ( Y = Z )
      @ ^ [Y: nat,Z: nat] : ( Y = Z ) )
    @ bit_se7882103937844011126it_nat
    @ bit_se7882103937844011126it_nat ) ).

% set_bit_natural.rsp
thf(fact_8614_drop__bit__natural_Orsp,axiom,
    ( bNF_re1345281282404953727at_nat
    @ ^ [Y: nat,Z: nat] : ( Y = Z )
    @ ( bNF_re5653821019739307937at_nat
      @ ^ [Y: nat,Z: nat] : ( Y = Z )
      @ ^ [Y: nat,Z: nat] : ( Y = Z ) )
    @ bit_se8570568707652914677it_nat
    @ bit_se8570568707652914677it_nat ) ).

% drop_bit_natural.rsp
thf(fact_8615_drop__bit__integer_Orsp,axiom,
    ( bNF_re4785983289428654063nt_int
    @ ^ [Y: nat,Z: nat] : ( Y = Z )
    @ ( bNF_re4712519889275205905nt_int
      @ ^ [Y: int,Z: int] : ( Y = Z )
      @ ^ [Y: int,Z: int] : ( Y = Z ) )
    @ bit_se8568078237143864401it_int
    @ bit_se8568078237143864401it_int ) ).

% drop_bit_integer.rsp
thf(fact_8616_or__natural_Orsp,axiom,
    ( bNF_re1345281282404953727at_nat
    @ ^ [Y: nat,Z: nat] : ( Y = Z )
    @ ( bNF_re5653821019739307937at_nat
      @ ^ [Y: nat,Z: nat] : ( Y = Z )
      @ ^ [Y: nat,Z: nat] : ( Y = Z ) )
    @ bit_se1412395901928357646or_nat
    @ bit_se1412395901928357646or_nat ) ).

% or_natural.rsp
thf(fact_8617_or__integer_Orsp,axiom,
    ( bNF_re711492959462206631nt_int
    @ ^ [Y: int,Z: int] : ( Y = Z )
    @ ( bNF_re4712519889275205905nt_int
      @ ^ [Y: int,Z: int] : ( Y = Z )
      @ ^ [Y: int,Z: int] : ( Y = Z ) )
    @ bit_se1409905431419307370or_int
    @ bit_se1409905431419307370or_int ) ).

% or_integer.rsp
thf(fact_8618_bit__integer_Orsp,axiom,
    ( bNF_re3376528473927230327_nat_o
    @ ^ [Y: int,Z: int] : ( Y = Z )
    @ ^ [Y: nat > $o,Z: nat > $o] : ( Y = Z )
    @ bit_se1146084159140164899it_int
    @ bit_se1146084159140164899it_int ) ).

% bit_integer.rsp
thf(fact_8619_not__integer_Orsp,axiom,
    ( bNF_re4712519889275205905nt_int
    @ ^ [Y: int,Z: int] : ( Y = Z )
    @ ^ [Y: int,Z: int] : ( Y = Z )
    @ bit_ri7919022796975470100ot_int
    @ bit_ri7919022796975470100ot_int ) ).

% not_integer.rsp
thf(fact_8620_bit__natural_Orsp,axiom,
    ( bNF_re578469030762574527_nat_o
    @ ^ [Y: nat,Z: nat] : ( Y = Z )
    @ ^ [Y: nat > $o,Z: nat > $o] : ( Y = Z )
    @ bit_se1148574629649215175it_nat
    @ bit_se1148574629649215175it_nat ) ).

% bit_natural.rsp
thf(fact_8621_mask__integer_Orsp,axiom,
    ( bNF_re6650684261131312217nt_int
    @ ^ [Y: nat,Z: nat] : ( Y = Z )
    @ ^ [Y: int,Z: int] : ( Y = Z )
    @ bit_se2000444600071755411sk_int
    @ bit_se2000444600071755411sk_int ) ).

% mask_integer.rsp
thf(fact_8622_mask__natural_Orsp,axiom,
    ( bNF_re5653821019739307937at_nat
    @ ^ [Y: nat,Z: nat] : ( Y = Z )
    @ ^ [Y: nat,Z: nat] : ( Y = Z )
    @ bit_se2002935070580805687sk_nat
    @ bit_se2002935070580805687sk_nat ) ).

% mask_natural.rsp
thf(fact_8623_division__segment__integer_Orsp,axiom,
    ( bNF_re4712519889275205905nt_int
    @ ^ [Y: int,Z: int] : ( Y = Z )
    @ ^ [Y: int,Z: int] : ( Y = Z )
    @ euclid3395696857347342551nt_int
    @ euclid3395696857347342551nt_int ) ).

% division_segment_integer.rsp
thf(fact_8624_division__segment__natural_Orsp,axiom,
    ( bNF_re5653821019739307937at_nat
    @ ^ [Y: nat,Z: nat] : ( Y = Z )
    @ ^ [Y: nat,Z: nat] : ( Y = Z )
    @ euclid3398187327856392827nt_nat
    @ euclid3398187327856392827nt_nat ) ).

% division_segment_natural.rsp
thf(fact_8625_Fract_Orsp,axiom,
    ( bNF_re157797125943740599nt_int
    @ ^ [Y: int,Z: int] : ( Y = Z )
    @ ( bNF_re6250860962936578807nt_int
      @ ^ [Y: int,Z: int] : ( Y = Z )
      @ ratrel )
    @ ^ [A2: int,B2: int] : ( if_Pro3027730157355071871nt_int @ ( B2 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ A2 @ B2 ) )
    @ ^ [A2: int,B2: int] : ( if_Pro3027730157355071871nt_int @ ( B2 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ A2 @ B2 ) ) ) ).

% Fract.rsp
thf(fact_8626_num__of__integer_Orsp,axiom,
    ( bNF_re7626690874201225453um_num
    @ ^ [Y: int,Z: int] : ( Y = Z )
    @ ^ [Y: num,Z: num] : ( Y = Z )
    @ ( comp_nat_num_int @ num_of_nat @ nat2 )
    @ ( comp_nat_num_int @ num_of_nat @ nat2 ) ) ).

% num_of_integer.rsp
thf(fact_8627_times__rat_Orsp,axiom,
    ( bNF_re5228765855967844073nt_int @ ratrel @ ( bNF_re7145576690424134365nt_int @ ratrel @ ratrel )
    @ ^ [X2: product_prod_int_int,Y6: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_fst_int_int @ Y6 ) ) @ ( times_times_int @ ( product_snd_int_int @ X2 ) @ ( product_snd_int_int @ Y6 ) ) )
    @ ^ [X2: product_prod_int_int,Y6: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_fst_int_int @ Y6 ) ) @ ( times_times_int @ ( product_snd_int_int @ X2 ) @ ( product_snd_int_int @ Y6 ) ) ) ) ).

% times_rat.rsp
thf(fact_8628_plus__rat_Orsp,axiom,
    ( bNF_re5228765855967844073nt_int @ ratrel @ ( bNF_re7145576690424134365nt_int @ ratrel @ ratrel )
    @ ^ [X2: product_prod_int_int,Y6: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_snd_int_int @ Y6 ) ) @ ( times_times_int @ ( product_fst_int_int @ Y6 ) @ ( product_snd_int_int @ X2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X2 ) @ ( product_snd_int_int @ Y6 ) ) )
    @ ^ [X2: product_prod_int_int,Y6: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_snd_int_int @ Y6 ) ) @ ( times_times_int @ ( product_fst_int_int @ Y6 ) @ ( product_snd_int_int @ X2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X2 ) @ ( product_snd_int_int @ Y6 ) ) ) ) ).

% plus_rat.rsp
thf(fact_8629_integer__of__nat__eq__of__nat,axiom,
    code_integer_of_nat = semiri4939895301339042750nteger ).

% integer_of_nat_eq_of_nat
thf(fact_8630_integer__of__nat__0,axiom,
    ( ( code_integer_of_nat @ zero_zero_nat )
    = zero_z3403309356797280102nteger ) ).

% integer_of_nat_0
thf(fact_8631_integer__of__nat_Oabs__eq,axiom,
    ( code_integer_of_nat
    = ( ^ [X2: nat] : ( code_integer_of_int @ ( semiri1314217659103216013at_int @ X2 ) ) ) ) ).

% integer_of_nat.abs_eq
thf(fact_8632_integer__of__nat__numeral,axiom,
    ! [N: num] :
      ( ( code_integer_of_nat @ ( numeral_numeral_nat @ N ) )
      = ( numera6620942414471956472nteger @ N ) ) ).

% integer_of_nat_numeral
thf(fact_8633_integer__of__nat__1,axiom,
    ( ( code_integer_of_nat @ one_one_nat )
    = one_one_Code_integer ) ).

% integer_of_nat_1
thf(fact_8634_uminus__rat_Orsp,axiom,
    ( bNF_re7145576690424134365nt_int @ ratrel @ ratrel
    @ ^ [X2: product_prod_int_int] : ( product_Pair_int_int @ ( uminus_uminus_int @ ( product_fst_int_int @ X2 ) ) @ ( product_snd_int_int @ X2 ) )
    @ ^ [X2: product_prod_int_int] : ( product_Pair_int_int @ ( uminus_uminus_int @ ( product_fst_int_int @ X2 ) ) @ ( product_snd_int_int @ X2 ) ) ) ).

% uminus_rat.rsp
thf(fact_8635_inverse__rat_Orsp,axiom,
    ( bNF_re7145576690424134365nt_int @ ratrel @ ratrel
    @ ^ [X2: product_prod_int_int] :
        ( if_Pro3027730157355071871nt_int
        @ ( ( product_fst_int_int @ X2 )
          = zero_zero_int )
        @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
        @ ( product_Pair_int_int @ ( product_snd_int_int @ X2 ) @ ( product_fst_int_int @ X2 ) ) )
    @ ^ [X2: product_prod_int_int] :
        ( if_Pro3027730157355071871nt_int
        @ ( ( product_fst_int_int @ X2 )
          = zero_zero_int )
        @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
        @ ( product_Pair_int_int @ ( product_snd_int_int @ X2 ) @ ( product_fst_int_int @ X2 ) ) ) ) ).

% inverse_rat.rsp
thf(fact_8636_plus__rat_Otransfer,axiom,
    ( bNF_re7627151682743391978at_rat @ pcr_rat @ ( bNF_re8279943556446156061nt_rat @ pcr_rat @ pcr_rat )
    @ ^ [X2: product_prod_int_int,Y6: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_snd_int_int @ Y6 ) ) @ ( times_times_int @ ( product_fst_int_int @ Y6 ) @ ( product_snd_int_int @ X2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X2 ) @ ( product_snd_int_int @ Y6 ) ) )
    @ plus_plus_rat ) ).

% plus_rat.transfer
thf(fact_8637_inverse__rat_Otransfer,axiom,
    ( bNF_re8279943556446156061nt_rat @ pcr_rat @ pcr_rat
    @ ^ [X2: product_prod_int_int] :
        ( if_Pro3027730157355071871nt_int
        @ ( ( product_fst_int_int @ X2 )
          = zero_zero_int )
        @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
        @ ( product_Pair_int_int @ ( product_snd_int_int @ X2 ) @ ( product_fst_int_int @ X2 ) ) )
    @ inverse_inverse_rat ) ).

% inverse_rat.transfer
thf(fact_8638_positive_Otransfer,axiom,
    ( bNF_re1494630372529172596at_o_o @ pcr_rat
    @ ^ [Y: $o,Z: $o] : ( Y = Z )
    @ ^ [X2: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_snd_int_int @ X2 ) ) )
    @ positive ) ).

% positive.transfer
thf(fact_8639_one__rat_Otransfer,axiom,
    pcr_rat @ ( product_Pair_int_int @ one_one_int @ one_one_int ) @ one_one_rat ).

% one_rat.transfer
thf(fact_8640_zero__rat_Otransfer,axiom,
    pcr_rat @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ zero_zero_rat ).

% zero_rat.transfer
thf(fact_8641_uminus__rat_Otransfer,axiom,
    ( bNF_re8279943556446156061nt_rat @ pcr_rat @ pcr_rat
    @ ^ [X2: product_prod_int_int] : ( product_Pair_int_int @ ( uminus_uminus_int @ ( product_fst_int_int @ X2 ) ) @ ( product_snd_int_int @ X2 ) )
    @ uminus_uminus_rat ) ).

% uminus_rat.transfer
thf(fact_8642_times__rat_Otransfer,axiom,
    ( bNF_re7627151682743391978at_rat @ pcr_rat @ ( bNF_re8279943556446156061nt_rat @ pcr_rat @ pcr_rat )
    @ ^ [X2: product_prod_int_int,Y6: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_fst_int_int @ Y6 ) ) @ ( times_times_int @ ( product_snd_int_int @ X2 ) @ ( product_snd_int_int @ Y6 ) ) )
    @ times_times_rat ) ).

% times_rat.transfer
thf(fact_8643_times__int_Otransfer,axiom,
    ( bNF_re7408651293131936558nt_int @ pcr_int @ ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int )
    @ ( produc27273713700761075at_nat
      @ ^ [X2: nat,Y6: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X2 @ U3 ) @ ( times_times_nat @ Y6 @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X2 @ V4 ) @ ( times_times_nat @ Y6 @ U3 ) ) ) ) )
    @ times_times_int ) ).

% times_int.transfer
thf(fact_8644_num__of__integer__def,axiom,
    ( code_num_of_integer
    = ( map_fu1227494855608507351um_num @ code_int_of_integer @ id_num @ ( comp_nat_num_int @ num_of_nat @ nat2 ) ) ) ).

% num_of_integer_def
thf(fact_8645_zero__int_Otransfer,axiom,
    pcr_int @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) @ zero_zero_int ).

% zero_int.transfer
thf(fact_8646_int__transfer,axiom,
    ( bNF_re6830278522597306478at_int
    @ ^ [Y: nat,Z: nat] : ( Y = Z )
    @ pcr_int
    @ ^ [N3: nat] : ( product_Pair_nat_nat @ N3 @ zero_zero_nat )
    @ semiri1314217659103216013at_int ) ).

% int_transfer
thf(fact_8647_uminus__int_Otransfer,axiom,
    ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int
    @ ( produc2626176000494625587at_nat
      @ ^ [X2: nat,Y6: nat] : ( product_Pair_nat_nat @ Y6 @ X2 ) )
    @ uminus_uminus_int ) ).

% uminus_int.transfer
thf(fact_8648_nat_Otransfer,axiom,
    ( bNF_re4555766996558763186at_nat @ pcr_int
    @ ^ [Y: nat,Z: nat] : ( Y = Z )
    @ ( produc6842872674320459806at_nat @ minus_minus_nat )
    @ nat2 ) ).

% nat.transfer
thf(fact_8649_one__int_Otransfer,axiom,
    pcr_int @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) @ one_one_int ).

% one_int.transfer
thf(fact_8650_less__int_Otransfer,axiom,
    ( bNF_re717283939379294677_int_o @ pcr_int
    @ ( bNF_re6644619430987730960nt_o_o @ pcr_int
      @ ^ [Y: $o,Z: $o] : ( Y = Z ) )
    @ ( produc8739625826339149834_nat_o
      @ ^ [X2: nat,Y6: nat] :
          ( produc6081775807080527818_nat_o
          @ ^ [U3: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X2 @ V4 ) @ ( plus_plus_nat @ U3 @ Y6 ) ) ) )
    @ ord_less_int ) ).

% less_int.transfer
thf(fact_8651_less__eq__int_Otransfer,axiom,
    ( bNF_re717283939379294677_int_o @ pcr_int
    @ ( bNF_re6644619430987730960nt_o_o @ pcr_int
      @ ^ [Y: $o,Z: $o] : ( Y = Z ) )
    @ ( produc8739625826339149834_nat_o
      @ ^ [X2: nat,Y6: nat] :
          ( produc6081775807080527818_nat_o
          @ ^ [U3: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X2 @ V4 ) @ ( plus_plus_nat @ U3 @ Y6 ) ) ) )
    @ ord_less_eq_int ) ).

% less_eq_int.transfer
thf(fact_8652_plus__int_Otransfer,axiom,
    ( bNF_re7408651293131936558nt_int @ pcr_int @ ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int )
    @ ( produc27273713700761075at_nat
      @ ^ [X2: nat,Y6: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X2 @ U3 ) @ ( plus_plus_nat @ Y6 @ V4 ) ) ) )
    @ plus_plus_int ) ).

% plus_int.transfer
thf(fact_8653_minus__int_Otransfer,axiom,
    ( bNF_re7408651293131936558nt_int @ pcr_int @ ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int )
    @ ( produc27273713700761075at_nat
      @ ^ [X2: nat,Y6: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X2 @ V4 ) @ ( plus_plus_nat @ Y6 @ U3 ) ) ) )
    @ minus_minus_int ) ).

% minus_int.transfer
thf(fact_8654_times__int_Orsp,axiom,
    ( bNF_re3099431351363272937at_nat @ intrel @ ( bNF_re2241393799969408733at_nat @ intrel @ intrel )
    @ ( produc27273713700761075at_nat
      @ ^ [X2: nat,Y6: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X2 @ U3 ) @ ( times_times_nat @ Y6 @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X2 @ V4 ) @ ( times_times_nat @ Y6 @ U3 ) ) ) ) )
    @ ( produc27273713700761075at_nat
      @ ^ [X2: nat,Y6: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X2 @ U3 ) @ ( times_times_nat @ Y6 @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X2 @ V4 ) @ ( times_times_nat @ Y6 @ U3 ) ) ) ) ) ) ).

% times_int.rsp
thf(fact_8655_intrel__iff,axiom,
    ! [X: nat,Y3: nat,U: nat,V: nat] :
      ( ( intrel @ ( product_Pair_nat_nat @ X @ Y3 ) @ ( product_Pair_nat_nat @ U @ V ) )
      = ( ( plus_plus_nat @ X @ V )
        = ( plus_plus_nat @ U @ Y3 ) ) ) ).

% intrel_iff
thf(fact_8656_int_Orel__eq__transfer,axiom,
    ( bNF_re717283939379294677_int_o @ pcr_int
    @ ( bNF_re6644619430987730960nt_o_o @ pcr_int
      @ ^ [Y: $o,Z: $o] : ( Y = Z ) )
    @ intrel
    @ ^ [Y: int,Z: int] : ( Y = Z ) ) ).

% int.rel_eq_transfer
thf(fact_8657_int_Oabs__eq__iff,axiom,
    ! [X: product_prod_nat_nat,Y3: product_prod_nat_nat] :
      ( ( ( abs_Integ @ X )
        = ( abs_Integ @ Y3 ) )
      = ( intrel @ X @ Y3 ) ) ).

% int.abs_eq_iff
thf(fact_8658_zero__int_Orsp,axiom,
    intrel @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) ).

% zero_int.rsp
thf(fact_8659_uminus__int_Orsp,axiom,
    ( bNF_re2241393799969408733at_nat @ intrel @ intrel
    @ ( produc2626176000494625587at_nat
      @ ^ [X2: nat,Y6: nat] : ( product_Pair_nat_nat @ Y6 @ X2 ) )
    @ ( produc2626176000494625587at_nat
      @ ^ [X2: nat,Y6: nat] : ( product_Pair_nat_nat @ Y6 @ X2 ) ) ) ).

% uminus_int.rsp
thf(fact_8660_nat_Orsp,axiom,
    ( bNF_re8246922863344978751at_nat @ intrel
    @ ^ [Y: nat,Z: nat] : ( Y = Z )
    @ ( produc6842872674320459806at_nat @ minus_minus_nat )
    @ ( produc6842872674320459806at_nat @ minus_minus_nat ) ) ).

% nat.rsp
thf(fact_8661_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_8662_intrel__def,axiom,
    ( intrel
    = ( produc8739625826339149834_nat_o
      @ ^ [X2: nat,Y6: nat] :
          ( produc6081775807080527818_nat_o
          @ ^ [U3: nat,V4: nat] :
              ( ( plus_plus_nat @ X2 @ V4 )
              = ( plus_plus_nat @ U3 @ Y6 ) ) ) ) ) ).

% intrel_def
thf(fact_8663_less__int_Orsp,axiom,
    ( bNF_re4202695980764964119_nat_o @ intrel
    @ ( bNF_re3666534408544137501at_o_o @ intrel
      @ ^ [Y: $o,Z: $o] : ( Y = Z ) )
    @ ( produc8739625826339149834_nat_o
      @ ^ [X2: nat,Y6: nat] :
          ( produc6081775807080527818_nat_o
          @ ^ [U3: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X2 @ V4 ) @ ( plus_plus_nat @ U3 @ Y6 ) ) ) )
    @ ( produc8739625826339149834_nat_o
      @ ^ [X2: nat,Y6: nat] :
          ( produc6081775807080527818_nat_o
          @ ^ [U3: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X2 @ V4 ) @ ( plus_plus_nat @ U3 @ Y6 ) ) ) ) ) ).

% less_int.rsp
thf(fact_8664_less__eq__int_Orsp,axiom,
    ( bNF_re4202695980764964119_nat_o @ intrel
    @ ( bNF_re3666534408544137501at_o_o @ intrel
      @ ^ [Y: $o,Z: $o] : ( Y = Z ) )
    @ ( produc8739625826339149834_nat_o
      @ ^ [X2: nat,Y6: nat] :
          ( produc6081775807080527818_nat_o
          @ ^ [U3: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X2 @ V4 ) @ ( plus_plus_nat @ U3 @ Y6 ) ) ) )
    @ ( produc8739625826339149834_nat_o
      @ ^ [X2: nat,Y6: nat] :
          ( produc6081775807080527818_nat_o
          @ ^ [U3: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X2 @ V4 ) @ ( plus_plus_nat @ U3 @ Y6 ) ) ) ) ) ).

% less_eq_int.rsp
thf(fact_8665_plus__int_Orsp,axiom,
    ( bNF_re3099431351363272937at_nat @ intrel @ ( bNF_re2241393799969408733at_nat @ intrel @ intrel )
    @ ( produc27273713700761075at_nat
      @ ^ [X2: nat,Y6: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X2 @ U3 ) @ ( plus_plus_nat @ Y6 @ V4 ) ) ) )
    @ ( produc27273713700761075at_nat
      @ ^ [X2: nat,Y6: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X2 @ U3 ) @ ( plus_plus_nat @ Y6 @ V4 ) ) ) ) ) ).

% plus_int.rsp
thf(fact_8666_minus__int_Orsp,axiom,
    ( bNF_re3099431351363272937at_nat @ intrel @ ( bNF_re2241393799969408733at_nat @ intrel @ intrel )
    @ ( produc27273713700761075at_nat
      @ ^ [X2: nat,Y6: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X2 @ V4 ) @ ( plus_plus_nat @ Y6 @ U3 ) ) ) )
    @ ( produc27273713700761075at_nat
      @ ^ [X2: nat,Y6: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X2 @ V4 ) @ ( plus_plus_nat @ Y6 @ U3 ) ) ) ) ) ).

% minus_int.rsp
thf(fact_8667_tl__upt,axiom,
    ! [M: nat,N: nat] :
      ( ( tl_nat @ ( upt @ M @ N ) )
      = ( upt @ ( suc @ M ) @ N ) ) ).

% tl_upt
thf(fact_8668_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_8669_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_8670_take__bit__num__def,axiom,
    ( bit_take_bit_num
    = ( ^ [N3: nat,M2: num] :
          ( if_option_num
          @ ( ( bit_se2925701944663578781it_nat @ N3 @ ( numeral_numeral_nat @ M2 ) )
            = zero_zero_nat )
          @ none_num
          @ ( some_num @ ( num_of_nat @ ( bit_se2925701944663578781it_nat @ N3 @ ( numeral_numeral_nat @ M2 ) ) ) ) ) ) ) ).

% take_bit_num_def
thf(fact_8671_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_8672_take__bit__num__simps_I3_J,axiom,
    ! [N: nat,M: num] :
      ( ( bit_take_bit_num @ ( suc @ N ) @ ( bit0 @ M ) )
      = ( case_o6005452278849405969um_num @ none_num
        @ ^ [Q3: num] : ( some_num @ ( bit0 @ Q3 ) )
        @ ( bit_take_bit_num @ N @ M ) ) ) ).

% take_bit_num_simps(3)
thf(fact_8673_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_8674_pred__nat__def,axiom,
    ( pred_nat
    = ( collec3392354462482085612at_nat
      @ ( produc6081775807080527818_nat_o
        @ ^ [M2: nat,N3: nat] :
            ( N3
            = ( suc @ M2 ) ) ) ) ) ).

% pred_nat_def
thf(fact_8675_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_8676_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_8677_inj__list__encode,axiom,
    ! [A3: set_list_nat] : ( inj_on_list_nat_nat @ nat_list_encode @ A3 ) ).

% inj_list_encode
thf(fact_8678_inj__prod__encode,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] : ( inj_on2178005380612969504at_nat @ nat_prod_encode @ A3 ) ).

% inj_prod_encode
thf(fact_8679_inj__Suc,axiom,
    ! [N5: set_nat] : ( inj_on_nat_nat @ suc @ N5 ) ).

% inj_Suc
thf(fact_8680_inj__int__encode,axiom,
    ! [A3: set_int] : ( inj_on_int_nat @ nat_int_encode @ A3 ) ).

% inj_int_encode
thf(fact_8681_inj__int__decode,axiom,
    ! [A3: set_nat] : ( inj_on_nat_int @ nat_int_decode @ A3 ) ).

% inj_int_decode
thf(fact_8682_inj__on__diff__nat,axiom,
    ! [N5: set_nat,K: nat] :
      ( ! [N4: nat] :
          ( ( member_nat @ N4 @ N5 )
         => ( ord_less_eq_nat @ K @ N4 ) )
     => ( inj_on_nat_nat
        @ ^ [N3: nat] : ( minus_minus_nat @ N3 @ K )
        @ N5 ) ) ).

% inj_on_diff_nat
thf(fact_8683_inj__on__set__encode,axiom,
    inj_on_set_nat_nat @ nat_set_encode @ ( collect_set_nat @ finite_finite_nat ) ).

% inj_on_set_encode
thf(fact_8684_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_8685_last__upt,axiom,
    ! [I2: nat,J: nat] :
      ( ( ord_less_nat @ I2 @ J )
     => ( ( last_nat @ ( upt @ I2 @ J ) )
        = ( minus_minus_nat @ J @ one_one_nat ) ) ) ).

% last_upt
thf(fact_8686_greaterThan__0,axiom,
    ( ( set_or1210151606488870762an_nat @ zero_zero_nat )
    = ( image_nat_nat @ suc @ top_top_set_nat ) ) ).

% greaterThan_0
thf(fact_8687_atLeast__0,axiom,
    ( ( set_ord_atLeast_nat @ zero_zero_nat )
    = top_top_set_nat ) ).

% atLeast_0
thf(fact_8688_atLeast__Suc__greaterThan,axiom,
    ! [K: nat] :
      ( ( set_ord_atLeast_nat @ ( suc @ K ) )
      = ( set_or1210151606488870762an_nat @ K ) ) ).

% atLeast_Suc_greaterThan
thf(fact_8689_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_8690_coprime__Suc__0__left,axiom,
    ! [N: nat] : ( algebr934650988132801477me_nat @ ( suc @ zero_zero_nat ) @ N ) ).

% coprime_Suc_0_left
thf(fact_8691_coprime__Suc__0__right,axiom,
    ! [N: nat] : ( algebr934650988132801477me_nat @ N @ ( suc @ zero_zero_nat ) ) ).

% coprime_Suc_0_right
thf(fact_8692_coprime__Suc__left__nat,axiom,
    ! [N: nat] : ( algebr934650988132801477me_nat @ ( suc @ N ) @ N ) ).

% coprime_Suc_left_nat
thf(fact_8693_coprime__Suc__right__nat,axiom,
    ! [N: nat] : ( algebr934650988132801477me_nat @ N @ ( suc @ N ) ) ).

% coprime_Suc_right_nat
thf(fact_8694_coprime__crossproduct__nat,axiom,
    ! [A: nat,D: nat,B: nat,C: nat] :
      ( ( algebr934650988132801477me_nat @ A @ D )
     => ( ( algebr934650988132801477me_nat @ B @ C )
       => ( ( ( times_times_nat @ A @ C )
            = ( times_times_nat @ B @ D ) )
          = ( ( A = B )
            & ( C = D ) ) ) ) ) ).

% coprime_crossproduct_nat
thf(fact_8695_coprime__common__divisor__nat,axiom,
    ! [A: nat,B: nat,X: nat] :
      ( ( algebr934650988132801477me_nat @ A @ B )
     => ( ( dvd_dvd_nat @ X @ A )
       => ( ( dvd_dvd_nat @ X @ B )
         => ( X = one_one_nat ) ) ) ) ).

% coprime_common_divisor_nat
thf(fact_8696_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_8697_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_8698_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_8699_floor__rat__def,axiom,
    ( archim3151403230148437115or_rat
    = ( ^ [X2: rat] :
          ( the_int
          @ ^ [Z6: int] :
              ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z6 ) @ X2 )
              & ( ord_less_rat @ X2 @ ( ring_1_of_int_rat @ ( plus_plus_int @ Z6 @ one_one_int ) ) ) ) ) ) ) ).

% floor_rat_def
thf(fact_8700_coprime__abs__left__iff,axiom,
    ! [K: int,L: int] :
      ( ( algebr932160517623751201me_int @ ( abs_abs_int @ K ) @ L )
      = ( algebr932160517623751201me_int @ K @ L ) ) ).

% coprime_abs_left_iff
thf(fact_8701_coprime__abs__right__iff,axiom,
    ! [K: int,L: int] :
      ( ( algebr932160517623751201me_int @ K @ ( abs_abs_int @ L ) )
      = ( algebr932160517623751201me_int @ K @ L ) ) ).

% coprime_abs_right_iff
thf(fact_8702_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_8703_normalize__stable,axiom,
    ! [Q4: int,P5: int] :
      ( ( ord_less_int @ zero_zero_int @ Q4 )
     => ( ( algebr932160517623751201me_int @ P5 @ Q4 )
       => ( ( normalize @ ( product_Pair_int_int @ P5 @ Q4 ) )
          = ( product_Pair_int_int @ P5 @ Q4 ) ) ) ) ).

% normalize_stable
thf(fact_8704_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_8705_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_8706_coprime__crossproduct__int,axiom,
    ! [A: int,D: int,B: int,C: int] :
      ( ( algebr932160517623751201me_int @ A @ D )
     => ( ( algebr932160517623751201me_int @ B @ C )
       => ( ( ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ C ) )
            = ( times_times_int @ ( abs_abs_int @ B ) @ ( abs_abs_int @ D ) ) )
          = ( ( ( abs_abs_int @ A )
              = ( abs_abs_int @ B ) )
            & ( ( abs_abs_int @ C )
              = ( abs_abs_int @ D ) ) ) ) ) ) ).

% coprime_crossproduct_int
thf(fact_8707_finite__vimage__Suc__iff,axiom,
    ! [F3: set_nat] :
      ( ( finite_finite_nat @ ( vimage_nat_nat @ suc @ F3 ) )
      = ( finite_finite_nat @ F3 ) ) ).

% finite_vimage_Suc_iff
thf(fact_8708_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_8709_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_8710_coprime__common__divisor__int,axiom,
    ! [A: int,B: int,X: int] :
      ( ( algebr932160517623751201me_int @ A @ B )
     => ( ( dvd_dvd_int @ X @ A )
       => ( ( dvd_dvd_int @ X @ B )
         => ( ( abs_abs_int @ X )
            = one_one_int ) ) ) ) ).

% coprime_common_divisor_int
thf(fact_8711_set__decode__div__2,axiom,
    ! [X: nat] :
      ( ( nat_set_decode @ ( divide_divide_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( vimage_nat_nat @ suc @ ( nat_set_decode @ X ) ) ) ).

% set_decode_div_2
thf(fact_8712_rat__floor__lemma,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( divide_divide_int @ A @ B ) ) @ ( fract @ A @ B ) )
      & ( ord_less_rat @ ( fract @ A @ B ) @ ( ring_1_of_int_rat @ ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ) ).

% rat_floor_lemma
thf(fact_8713_sgn__rat,axiom,
    ! [A: int,B: int] :
      ( ( sgn_sgn_rat @ ( fract @ A @ B ) )
      = ( ring_1_of_int_rat @ ( times_times_int @ ( sgn_sgn_int @ A ) @ ( sgn_sgn_int @ B ) ) ) ) ).

% sgn_rat
thf(fact_8714_minus__rat__cancel,axiom,
    ! [A: int,B: int] :
      ( ( fract @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
      = ( fract @ A @ B ) ) ).

% minus_rat_cancel
thf(fact_8715_minus__rat,axiom,
    ! [A: int,B: int] :
      ( ( uminus_uminus_rat @ ( fract @ A @ B ) )
      = ( fract @ ( uminus_uminus_int @ A ) @ B ) ) ).

% minus_rat
thf(fact_8716_divide__rat,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( divide_divide_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
      = ( fract @ ( times_times_int @ A @ D ) @ ( times_times_int @ B @ C ) ) ) ).

% divide_rat
thf(fact_8717_mult__rat,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( times_times_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
      = ( fract @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ).

% mult_rat
thf(fact_8718_less__rat,axiom,
    ! [B: int,D: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( ( ord_less_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
          = ( ord_less_int @ ( times_times_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ B @ D ) ) @ ( times_times_int @ ( times_times_int @ C @ B ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).

% less_rat
thf(fact_8719_le__rat,axiom,
    ! [B: int,D: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( ( ord_less_eq_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
          = ( ord_less_eq_int @ ( times_times_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ B @ D ) ) @ ( times_times_int @ ( times_times_int @ C @ B ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).

% le_rat
thf(fact_8720_add__rat,axiom,
    ! [B: int,D: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( ( plus_plus_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
          = ( fract @ ( plus_plus_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ C @ B ) ) @ ( times_times_int @ B @ D ) ) ) ) ) ).

% add_rat
thf(fact_8721_diff__rat,axiom,
    ! [B: int,D: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( ( minus_minus_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
          = ( fract @ ( minus_minus_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ C @ B ) ) @ ( times_times_int @ B @ D ) ) ) ) ) ).

% diff_rat
thf(fact_8722_Rat__cases,axiom,
    ! [Q4: rat] :
      ~ ! [A5: int,B4: int] :
          ( ( Q4
            = ( fract @ A5 @ B4 ) )
         => ( ( ord_less_int @ zero_zero_int @ B4 )
           => ~ ( algebr932160517623751201me_int @ A5 @ B4 ) ) ) ).

% Rat_cases
thf(fact_8723_Rat__induct,axiom,
    ! [P: rat > $o,Q4: rat] :
      ( ! [A5: int,B4: int] :
          ( ( ord_less_int @ zero_zero_int @ B4 )
         => ( ( algebr932160517623751201me_int @ A5 @ B4 )
           => ( P @ ( fract @ A5 @ B4 ) ) ) )
     => ( P @ Q4 ) ) ).

% Rat_induct
thf(fact_8724_Fract__of__nat__eq,axiom,
    ! [K: nat] :
      ( ( fract @ ( semiri1314217659103216013at_int @ K ) @ one_one_int )
      = ( semiri681578069525770553at_rat @ K ) ) ).

% Fract_of_nat_eq
thf(fact_8725_eq__rat_I3_J,axiom,
    ! [A: int,C: int] :
      ( ( fract @ zero_zero_int @ A )
      = ( fract @ zero_zero_int @ C ) ) ).

% eq_rat(3)
thf(fact_8726_eq__rat_I1_J,axiom,
    ! [B: int,D: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( ( ( fract @ A @ B )
            = ( fract @ C @ D ) )
          = ( ( times_times_int @ A @ D )
            = ( times_times_int @ C @ B ) ) ) ) ) ).

% eq_rat(1)
thf(fact_8727_mult__rat__cancel,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( ( fract @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
        = ( fract @ A @ B ) ) ) ).

% mult_rat_cancel
thf(fact_8728_eq__rat_I2_J,axiom,
    ! [A: int] :
      ( ( fract @ A @ zero_zero_int )
      = ( fract @ zero_zero_int @ one_one_int ) ) ).

% eq_rat(2)
thf(fact_8729_Rat__induct__pos,axiom,
    ! [P: rat > $o,Q4: rat] :
      ( ! [A5: int,B4: int] :
          ( ( ord_less_int @ zero_zero_int @ B4 )
         => ( P @ ( fract @ A5 @ B4 ) ) )
     => ( P @ Q4 ) ) ).

% Rat_induct_pos
thf(fact_8730_rat__number__collapse_I6_J,axiom,
    ! [K: int] :
      ( ( fract @ K @ zero_zero_int )
      = zero_zero_rat ) ).

% rat_number_collapse(6)
thf(fact_8731_rat__number__collapse_I1_J,axiom,
    ! [K: int] :
      ( ( fract @ zero_zero_int @ K )
      = zero_zero_rat ) ).

% rat_number_collapse(1)
thf(fact_8732_One__rat__def,axiom,
    ( one_one_rat
    = ( fract @ one_one_int @ one_one_int ) ) ).

% One_rat_def
thf(fact_8733_Fract__of__int__eq,axiom,
    ! [K: int] :
      ( ( fract @ K @ one_one_int )
      = ( ring_1_of_int_rat @ K ) ) ).

% Fract_of_int_eq
thf(fact_8734_quotient__of__def,axiom,
    ( quotient_of
    = ( ^ [X2: rat] :
          ( the_Pr4378521158711661632nt_int
          @ ^ [Pair: product_prod_int_int] :
              ( ( X2
                = ( 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_8735_Rat__cases__nonzero,axiom,
    ! [Q4: rat] :
      ( ! [A5: int,B4: int] :
          ( ( Q4
            = ( fract @ A5 @ B4 ) )
         => ( ( ord_less_int @ zero_zero_int @ B4 )
           => ( ( A5 != zero_zero_int )
             => ~ ( algebr932160517623751201me_int @ A5 @ B4 ) ) ) )
     => ( Q4 = zero_zero_rat ) ) ).

% Rat_cases_nonzero
thf(fact_8736_Zero__rat__def,axiom,
    ( zero_zero_rat
    = ( fract @ zero_zero_int @ one_one_int ) ) ).

% Zero_rat_def
thf(fact_8737_rat__number__expand_I3_J,axiom,
    ( numeral_numeral_rat
    = ( ^ [K3: num] : ( fract @ ( numeral_numeral_int @ K3 ) @ one_one_int ) ) ) ).

% rat_number_expand(3)
thf(fact_8738_rat__number__collapse_I3_J,axiom,
    ! [W2: num] :
      ( ( fract @ ( numeral_numeral_int @ W2 ) @ one_one_int )
      = ( numeral_numeral_rat @ W2 ) ) ).

% rat_number_collapse(3)
thf(fact_8739_quotient__of__unique,axiom,
    ! [R: rat] :
    ? [X3: product_prod_int_int] :
      ( ( R
        = ( 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] :
          ( ( ( R
              = ( 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_8740_Fract_Oabs__eq,axiom,
    ( fract
    = ( ^ [Xa4: int,X2: int] : ( abs_Rat @ ( if_Pro3027730157355071871nt_int @ ( X2 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ Xa4 @ X2 ) ) ) ) ) ).

% Fract.abs_eq
thf(fact_8741_positive__rat,axiom,
    ! [A: int,B: int] :
      ( ( positive @ ( fract @ A @ B ) )
      = ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ).

% positive_rat
thf(fact_8742_Fract__less__zero__iff,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_rat @ ( fract @ A @ B ) @ zero_zero_rat )
        = ( ord_less_int @ A @ zero_zero_int ) ) ) ).

% Fract_less_zero_iff
thf(fact_8743_zero__less__Fract__iff,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_rat @ zero_zero_rat @ ( fract @ A @ B ) )
        = ( ord_less_int @ zero_zero_int @ A ) ) ) ).

% zero_less_Fract_iff
thf(fact_8744_one__less__Fract__iff,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_rat @ one_one_rat @ ( fract @ A @ B ) )
        = ( ord_less_int @ B @ A ) ) ) ).

% one_less_Fract_iff
thf(fact_8745_Fract__less__one__iff,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_rat @ ( fract @ A @ B ) @ one_one_rat )
        = ( ord_less_int @ A @ B ) ) ) ).

% Fract_less_one_iff
thf(fact_8746_rat__number__collapse_I5_J,axiom,
    ( ( fract @ ( uminus_uminus_int @ one_one_int ) @ one_one_int )
    = ( uminus_uminus_rat @ one_one_rat ) ) ).

% rat_number_collapse(5)
thf(fact_8747_Fract_Otransfer,axiom,
    ( bNF_re3461391660133120880nt_rat
    @ ^ [Y: int,Z: int] : ( Y = Z )
    @ ( bNF_re2214769303045360666nt_rat
      @ ^ [Y: int,Z: int] : ( Y = Z )
      @ pcr_rat )
    @ ^ [A2: int,B2: int] : ( if_Pro3027730157355071871nt_int @ ( B2 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ A2 @ B2 ) )
    @ fract ) ).

% Fract.transfer
thf(fact_8748_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_8749_Fract__le__zero__iff,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_eq_rat @ ( fract @ A @ B ) @ zero_zero_rat )
        = ( ord_less_eq_int @ A @ zero_zero_int ) ) ) ).

% Fract_le_zero_iff
thf(fact_8750_zero__le__Fract__iff,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ ( fract @ A @ B ) )
        = ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).

% zero_le_Fract_iff
thf(fact_8751_one__le__Fract__iff,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_eq_rat @ one_one_rat @ ( fract @ A @ B ) )
        = ( ord_less_eq_int @ B @ A ) ) ) ).

% one_le_Fract_iff
thf(fact_8752_Fract__le__one__iff,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_eq_rat @ ( fract @ A @ B ) @ one_one_rat )
        = ( ord_less_eq_int @ A @ B ) ) ) ).

% Fract_le_one_iff
thf(fact_8753_rat__number__expand_I5_J,axiom,
    ! [K: num] :
      ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) )
      = ( fract @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) @ one_one_int ) ) ).

% rat_number_expand(5)
thf(fact_8754_rat__number__collapse_I4_J,axiom,
    ! [W2: num] :
      ( ( fract @ ( uminus_uminus_int @ ( numeral_numeral_int @ W2 ) ) @ one_one_int )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) ).

% rat_number_collapse(4)
thf(fact_8755_gcd__nat_Oordering__top__axioms,axiom,
    ( ordering_top_nat @ dvd_dvd_nat
    @ ^ [M2: nat,N3: nat] :
        ( ( dvd_dvd_nat @ M2 @ N3 )
        & ( M2 != N3 ) )
    @ zero_zero_nat ) ).

% gcd_nat.ordering_top_axioms
thf(fact_8756_bot__nat__0_Oordering__top__axioms,axiom,
    ( ordering_top_nat
    @ ^ [X2: nat,Y6: nat] : ( ord_less_eq_nat @ Y6 @ X2 )
    @ ^ [X2: nat,Y6: nat] : ( ord_less_nat @ Y6 @ X2 )
    @ zero_zero_nat ) ).

% bot_nat_0.ordering_top_axioms
thf(fact_8757_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_8758_prod__decode__eq,axiom,
    ! [X: nat,Y3: nat] :
      ( ( ( nat_prod_decode @ X )
        = ( nat_prod_decode @ Y3 ) )
      = ( X = Y3 ) ) ).

% prod_decode_eq
thf(fact_8759_prod__decode__inverse,axiom,
    ! [N: nat] :
      ( ( nat_prod_encode @ ( nat_prod_decode @ N ) )
      = N ) ).

% prod_decode_inverse
thf(fact_8760_prod__encode__inverse,axiom,
    ! [X: product_prod_nat_nat] :
      ( ( nat_prod_decode @ ( nat_prod_encode @ X ) )
      = X ) ).

% prod_encode_inverse
thf(fact_8761_inj__prod__decode,axiom,
    ! [A3: set_nat] : ( inj_on5538052773655684606at_nat @ nat_prod_decode @ A3 ) ).

% inj_prod_decode
thf(fact_8762_prod__decode__def,axiom,
    ( nat_prod_decode
    = ( nat_prod_decode_aux @ zero_zero_nat ) ) ).

% prod_decode_def
thf(fact_8763_surj__prod__decode,axiom,
    ( ( image_5846123807819985514at_nat @ nat_prod_decode @ top_top_set_nat )
    = top_to4669805908274784177at_nat ) ).

% surj_prod_decode
thf(fact_8764_list__decode_Oelims,axiom,
    ! [X: nat,Y3: list_nat] :
      ( ( ( nat_list_decode @ X )
        = Y3 )
     => ( ( ( X = zero_zero_nat )
         => ( Y3 != nil_nat ) )
       => ~ ! [N4: nat] :
              ( ( X
                = ( suc @ N4 ) )
             => ( Y3
               != ( produc2761476792215241774st_nat
                  @ ^ [X2: nat,Y6: nat] : ( cons_nat @ X2 @ ( nat_list_decode @ Y6 ) )
                  @ ( nat_prod_decode @ N4 ) ) ) ) ) ) ).

% list_decode.elims
thf(fact_8765_list__decode_Osimps_I2_J,axiom,
    ! [N: nat] :
      ( ( nat_list_decode @ ( suc @ N ) )
      = ( produc2761476792215241774st_nat
        @ ^ [X2: nat,Y6: nat] : ( cons_nat @ X2 @ ( nat_list_decode @ Y6 ) )
        @ ( nat_prod_decode @ N ) ) ) ).

% list_decode.simps(2)
thf(fact_8766_list__encode__inverse,axiom,
    ! [X: list_nat] :
      ( ( nat_list_decode @ ( nat_list_encode @ X ) )
      = X ) ).

% list_encode_inverse
thf(fact_8767_list__decode__inverse,axiom,
    ! [N: nat] :
      ( ( nat_list_encode @ ( nat_list_decode @ N ) )
      = N ) ).

% list_decode_inverse
thf(fact_8768_list__decode__eq,axiom,
    ! [X: nat,Y3: nat] :
      ( ( ( nat_list_decode @ X )
        = ( nat_list_decode @ Y3 ) )
      = ( X = Y3 ) ) ).

% list_decode_eq
thf(fact_8769_inj__list__decode,axiom,
    ! [A3: set_nat] : ( inj_on_nat_list_nat @ nat_list_decode @ A3 ) ).

% inj_list_decode
thf(fact_8770_list__decode_Osimps_I1_J,axiom,
    ( ( nat_list_decode @ zero_zero_nat )
    = nil_nat ) ).

% list_decode.simps(1)
thf(fact_8771_surj__list__decode,axiom,
    ( ( image_nat_list_nat @ nat_list_decode @ top_top_set_nat )
    = top_top_set_list_nat ) ).

% surj_list_decode
thf(fact_8772_list__decode_Opelims,axiom,
    ! [X: nat,Y3: list_nat] :
      ( ( ( nat_list_decode @ X )
        = Y3 )
     => ( ( accp_nat @ nat_list_decode_rel @ X )
       => ( ( ( X = zero_zero_nat )
           => ( ( Y3 = nil_nat )
             => ~ ( accp_nat @ nat_list_decode_rel @ zero_zero_nat ) ) )
         => ~ ! [N4: nat] :
                ( ( X
                  = ( suc @ N4 ) )
               => ( ( Y3
                    = ( produc2761476792215241774st_nat
                      @ ^ [X2: nat,Y6: nat] : ( cons_nat @ X2 @ ( nat_list_decode @ Y6 ) )
                      @ ( nat_prod_decode @ N4 ) ) )
                 => ~ ( accp_nat @ nat_list_decode_rel @ ( suc @ N4 ) ) ) ) ) ) ) ).

% list_decode.pelims
thf(fact_8773_list__decode_Opsimps_I1_J,axiom,
    ( ( accp_nat @ nat_list_decode_rel @ zero_zero_nat )
   => ( ( nat_list_decode @ zero_zero_nat )
      = nil_nat ) ) ).

% list_decode.psimps(1)
thf(fact_8774_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 ) )
             => ( ! [X4: nat,Y5: nat] :
                    ( ( ( product_Pair_nat_nat @ X4 @ Y5 )
                      = ( nat_prod_decode @ N4 ) )
                   => ( P @ Y5 ) )
               => ( P @ ( suc @ N4 ) ) ) )
         => ( P @ A0 ) ) ) ) ).

% list_decode.pinduct
thf(fact_8775_list__decode_Opsimps_I2_J,axiom,
    ! [N: nat] :
      ( ( accp_nat @ nat_list_decode_rel @ ( suc @ N ) )
     => ( ( nat_list_decode @ ( suc @ N ) )
        = ( produc2761476792215241774st_nat
          @ ^ [X2: nat,Y6: nat] : ( cons_nat @ X2 @ ( nat_list_decode @ Y6 ) )
          @ ( nat_prod_decode @ N ) ) ) ) ).

% list_decode.psimps(2)
thf(fact_8776_division__segment__integer__def,axiom,
    ( euclid6289375185220004616nteger
    = ( map_fu2599414010547811884nteger @ code_int_of_integer @ code_integer_of_int @ euclid3395696857347342551nt_int ) ) ).

% division_segment_integer_def
thf(fact_8777_uminus__integer__def,axiom,
    ( uminus1351360451143612070nteger
    = ( map_fu2599414010547811884nteger @ code_int_of_integer @ code_integer_of_int @ uminus_uminus_int ) ) ).

% uminus_integer_def
thf(fact_8778_abs__integer__def,axiom,
    ( abs_abs_Code_integer
    = ( map_fu2599414010547811884nteger @ code_int_of_integer @ code_integer_of_int @ abs_abs_int ) ) ).

% abs_integer_def
thf(fact_8779_sgn__integer__def,axiom,
    ( sgn_sgn_Code_integer
    = ( map_fu2599414010547811884nteger @ code_int_of_integer @ code_integer_of_int @ sgn_sgn_int ) ) ).

% sgn_integer_def
thf(fact_8780_not__integer__def,axiom,
    ( bit_ri7632146776885996613nteger
    = ( map_fu2599414010547811884nteger @ code_int_of_integer @ code_integer_of_int @ bit_ri7919022796975470100ot_int ) ) ).

% not_integer_def
thf(fact_8781_Code__Numeral_Odup__def,axiom,
    ( code_dup
    = ( map_fu2599414010547811884nteger @ code_int_of_integer @ code_integer_of_int
      @ ^ [K3: int] : ( plus_plus_int @ K3 @ K3 ) ) ) ).

% Code_Numeral.dup_def
thf(fact_8782_flip__bit__integer__def,axiom,
    ( bit_se1345352211410354436nteger
    = ( map_fu8224359078269604234nteger @ id_nat @ ( map_fu2599414010547811884nteger @ code_int_of_integer @ code_integer_of_int ) @ bit_se2159334234014336723it_int ) ) ).

% flip_bit_integer_def
thf(fact_8783_set__bit__integer__def,axiom,
    ( bit_se2793503036327961859nteger
    = ( map_fu8224359078269604234nteger @ id_nat @ ( map_fu2599414010547811884nteger @ code_int_of_integer @ code_integer_of_int ) @ bit_se7879613467334960850it_int ) ) ).

% set_bit_integer_def
thf(fact_8784_Code__Numeral_Odup__code_I1_J,axiom,
    ( ( code_dup @ zero_z3403309356797280102nteger )
    = zero_z3403309356797280102nteger ) ).

% Code_Numeral.dup_code(1)
thf(fact_8785_dup_Orep__eq,axiom,
    ! [X: code_integer] :
      ( ( code_int_of_integer @ ( code_dup @ X ) )
      = ( plus_plus_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ X ) ) ) ).

% dup.rep_eq
thf(fact_8786_dup_Oabs__eq,axiom,
    ! [X: int] :
      ( ( code_dup @ ( code_integer_of_int @ X ) )
      = ( code_integer_of_int @ ( plus_plus_int @ X @ X ) ) ) ).

% dup.abs_eq
thf(fact_8787_take__bit__integer__def,axiom,
    ( bit_se1745604003318907178nteger
    = ( map_fu8224359078269604234nteger @ id_nat @ ( map_fu2599414010547811884nteger @ code_int_of_integer @ code_integer_of_int ) @ bit_se2923211474154528505it_int ) ) ).

% take_bit_integer_def
thf(fact_8788_drop__bit__integer__def,axiom,
    ( bit_se3928097537394005634nteger
    = ( map_fu8224359078269604234nteger @ id_nat @ ( map_fu2599414010547811884nteger @ code_int_of_integer @ code_integer_of_int ) @ bit_se8568078237143864401it_int ) ) ).

% drop_bit_integer_def
thf(fact_8789_push__bit__integer__def,axiom,
    ( bit_se7788150548672797655nteger
    = ( map_fu8224359078269604234nteger @ id_nat @ ( map_fu2599414010547811884nteger @ code_int_of_integer @ code_integer_of_int ) @ bit_se545348938243370406it_int ) ) ).

% push_bit_integer_def
thf(fact_8790_unset__bit__integer__def,axiom,
    ( bit_se8260200283734997820nteger
    = ( map_fu8224359078269604234nteger @ id_nat @ ( map_fu2599414010547811884nteger @ code_int_of_integer @ code_integer_of_int ) @ bit_se4203085406695923979it_int ) ) ).

% unset_bit_integer_def
thf(fact_8791_Code__Numeral_Osub__code_I9_J,axiom,
    ! [M: num,N: num] :
      ( ( code_sub @ ( bit0 @ M ) @ ( bit1 @ N ) )
      = ( minus_8373710615458151222nteger @ ( code_dup @ ( code_sub @ M @ N ) ) @ one_one_Code_integer ) ) ).

% Code_Numeral.sub_code(9)
thf(fact_8792_Code__Numeral_Osub__code_I8_J,axiom,
    ! [M: num,N: num] :
      ( ( code_sub @ ( bit1 @ M ) @ ( bit0 @ N ) )
      = ( plus_p5714425477246183910nteger @ ( code_dup @ ( code_sub @ M @ N ) ) @ one_one_Code_integer ) ) ).

% Code_Numeral.sub_code(8)
thf(fact_8793_xor__integer__def,axiom,
    ( bit_se3222712562003087583nteger
    = ( map_fu8272188784021352819nteger @ code_int_of_integer @ ( map_fu2599414010547811884nteger @ code_int_of_integer @ code_integer_of_int ) @ bit_se6526347334894502574or_int ) ) ).

% xor_integer_def
thf(fact_8794_Code__Numeral_Osub__code_I1_J,axiom,
    ( ( code_sub @ one @ one )
    = zero_z3403309356797280102nteger ) ).

% Code_Numeral.sub_code(1)
thf(fact_8795_Code__Numeral_Osub__code_I6_J,axiom,
    ! [M: num,N: num] :
      ( ( code_sub @ ( bit0 @ M ) @ ( bit0 @ N ) )
      = ( code_dup @ ( code_sub @ M @ N ) ) ) ).

% Code_Numeral.sub_code(6)
thf(fact_8796_Code__Numeral_Osub__code_I7_J,axiom,
    ! [M: num,N: num] :
      ( ( code_sub @ ( bit1 @ M ) @ ( bit1 @ N ) )
      = ( code_dup @ ( code_sub @ M @ N ) ) ) ).

% Code_Numeral.sub_code(7)
thf(fact_8797_sub_Orep__eq,axiom,
    ! [X: num,Xa3: num] :
      ( ( code_int_of_integer @ ( code_sub @ X @ Xa3 ) )
      = ( minus_minus_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Xa3 ) ) ) ).

% sub.rep_eq
thf(fact_8798_sub_Oabs__eq,axiom,
    ( code_sub
    = ( ^ [Xa4: num,X2: num] : ( code_integer_of_int @ ( minus_minus_int @ ( numeral_numeral_int @ Xa4 ) @ ( numeral_numeral_int @ X2 ) ) ) ) ) ).

% sub.abs_eq
thf(fact_8799_plus__integer__def,axiom,
    ( plus_p5714425477246183910nteger
    = ( map_fu8272188784021352819nteger @ code_int_of_integer @ ( map_fu2599414010547811884nteger @ code_int_of_integer @ code_integer_of_int ) @ plus_plus_int ) ) ).

% plus_integer_def
thf(fact_8800_times__integer__def,axiom,
    ( times_3573771949741848930nteger
    = ( map_fu8272188784021352819nteger @ code_int_of_integer @ ( map_fu2599414010547811884nteger @ code_int_of_integer @ code_integer_of_int ) @ times_times_int ) ) ).

% times_integer_def
thf(fact_8801_minus__integer__def,axiom,
    ( minus_8373710615458151222nteger
    = ( map_fu8272188784021352819nteger @ code_int_of_integer @ ( map_fu2599414010547811884nteger @ code_int_of_integer @ code_integer_of_int ) @ minus_minus_int ) ) ).

% minus_integer_def
thf(fact_8802_divide__integer__def,axiom,
    ( divide6298287555418463151nteger
    = ( map_fu8272188784021352819nteger @ code_int_of_integer @ ( map_fu2599414010547811884nteger @ code_int_of_integer @ code_integer_of_int ) @ divide_divide_int ) ) ).

% divide_integer_def
thf(fact_8803_modulo__integer__def,axiom,
    ( modulo364778990260209775nteger
    = ( map_fu8272188784021352819nteger @ code_int_of_integer @ ( map_fu2599414010547811884nteger @ code_int_of_integer @ code_integer_of_int ) @ modulo_modulo_int ) ) ).

% modulo_integer_def
thf(fact_8804_and__integer__def,axiom,
    ( bit_se3949692690581998587nteger
    = ( map_fu8272188784021352819nteger @ code_int_of_integer @ ( map_fu2599414010547811884nteger @ code_int_of_integer @ code_integer_of_int ) @ bit_se725231765392027082nd_int ) ) ).

% and_integer_def
thf(fact_8805_or__integer__def,axiom,
    ( bit_se1080825931792720795nteger
    = ( map_fu8272188784021352819nteger @ code_int_of_integer @ ( map_fu2599414010547811884nteger @ code_int_of_integer @ code_integer_of_int ) @ bit_se1409905431419307370or_int ) ) ).

% or_integer_def
thf(fact_8806_Sup__int__def,axiom,
    ( complete_Sup_Sup_int
    = ( ^ [X7: set_int] :
          ( the_int
          @ ^ [X2: int] :
              ( ( member_int @ X2 @ X7 )
              & ! [Y6: int] :
                  ( ( member_int @ Y6 @ X7 )
                 => ( ord_less_eq_int @ Y6 @ X2 ) ) ) ) ) ) ).

% Sup_int_def
thf(fact_8807_num__of__integer_Otransfer,axiom,
    ( bNF_re6718328864250387230um_num @ code_pcr_integer
    @ ^ [Y: num,Z: num] : ( Y = Z )
    @ ( comp_nat_num_int @ num_of_nat @ nat2 )
    @ code_num_of_integer ) ).

% num_of_integer.transfer
thf(fact_8808_integer_Oid__abs__transfer,axiom,
    ( bNF_re982302072995117890nteger
    @ ^ [Y: int,Z: int] : ( Y = Z )
    @ code_pcr_integer
    @ ^ [X2: int] : X2
    @ code_integer_of_int ) ).

% integer.id_abs_transfer
thf(fact_8809_integer_Orep__transfer,axiom,
    ( bNF_re3804157879324367682nt_int @ code_pcr_integer
    @ ^ [Y: int,Z: int] : ( Y = Z )
    @ ^ [X2: int] : X2
    @ code_int_of_integer ) ).

% integer.rep_transfer
thf(fact_8810_zero__integer_Otransfer,axiom,
    code_pcr_integer @ zero_zero_int @ zero_z3403309356797280102nteger ).

% zero_integer.transfer
thf(fact_8811_less__integer_Otransfer,axiom,
    ( bNF_re6321650412969554871eger_o @ code_pcr_integer
    @ ( bNF_re6574881592172037608er_o_o @ code_pcr_integer
      @ ^ [Y: $o,Z: $o] : ( Y = Z ) )
    @ ord_less_int
    @ ord_le6747313008572928689nteger ) ).

% less_integer.transfer
thf(fact_8812_plus__integer_Otransfer,axiom,
    bNF_re398004352372739002nteger @ code_pcr_integer @ ( bNF_re3379532845092657523nteger @ code_pcr_integer @ code_pcr_integer ) @ plus_plus_int @ plus_p5714425477246183910nteger ).

% plus_integer.transfer
thf(fact_8813_uminus__integer_Otransfer,axiom,
    bNF_re3379532845092657523nteger @ code_pcr_integer @ code_pcr_integer @ uminus_uminus_int @ uminus1351360451143612070nteger ).

% uminus_integer.transfer
thf(fact_8814_times__integer_Otransfer,axiom,
    bNF_re398004352372739002nteger @ code_pcr_integer @ ( bNF_re3379532845092657523nteger @ code_pcr_integer @ code_pcr_integer ) @ times_times_int @ times_3573771949741848930nteger ).

% times_integer.transfer
thf(fact_8815_one__integer_Otransfer,axiom,
    code_pcr_integer @ one_one_int @ one_one_Code_integer ).

% one_integer.transfer
thf(fact_8816_less__eq__integer_Otransfer,axiom,
    ( bNF_re6321650412969554871eger_o @ code_pcr_integer
    @ ( bNF_re6574881592172037608er_o_o @ code_pcr_integer
      @ ^ [Y: $o,Z: $o] : ( Y = Z ) )
    @ ord_less_eq_int
    @ ord_le3102999989581377725nteger ) ).

% less_eq_integer.transfer
thf(fact_8817_minus__integer_Otransfer,axiom,
    bNF_re398004352372739002nteger @ code_pcr_integer @ ( bNF_re3379532845092657523nteger @ code_pcr_integer @ code_pcr_integer ) @ minus_minus_int @ minus_8373710615458151222nteger ).

% minus_integer.transfer
thf(fact_8818_abs__integer_Otransfer,axiom,
    bNF_re3379532845092657523nteger @ code_pcr_integer @ code_pcr_integer @ abs_abs_int @ abs_abs_Code_integer ).

% abs_integer.transfer
thf(fact_8819_integer__of__nat_Otransfer,axiom,
    ( bNF_re4153400068438556298nteger
    @ ^ [Y: nat,Z: nat] : ( Y = Z )
    @ code_pcr_integer
    @ semiri1314217659103216013at_int
    @ code_integer_of_nat ) ).

% integer_of_nat.transfer
thf(fact_8820_divide__integer_Otransfer,axiom,
    bNF_re398004352372739002nteger @ code_pcr_integer @ ( bNF_re3379532845092657523nteger @ code_pcr_integer @ code_pcr_integer ) @ divide_divide_int @ divide6298287555418463151nteger ).

% divide_integer.transfer
thf(fact_8821_nat__of__integer_Otransfer,axiom,
    ( bNF_re2807294637932363402at_nat @ code_pcr_integer
    @ ^ [Y: nat,Z: nat] : ( Y = Z )
    @ nat2
    @ code_nat_of_integer ) ).

% nat_of_integer.transfer
thf(fact_8822_modulo__integer_Otransfer,axiom,
    bNF_re398004352372739002nteger @ code_pcr_integer @ ( bNF_re3379532845092657523nteger @ code_pcr_integer @ code_pcr_integer ) @ modulo_modulo_int @ modulo364778990260209775nteger ).

% modulo_integer.transfer
thf(fact_8823_sgn__integer_Otransfer,axiom,
    bNF_re3379532845092657523nteger @ code_pcr_integer @ code_pcr_integer @ sgn_sgn_int @ sgn_sgn_Code_integer ).

% sgn_integer.transfer
thf(fact_8824_dup_Otransfer,axiom,
    ( bNF_re3379532845092657523nteger @ code_pcr_integer @ code_pcr_integer
    @ ^ [K3: int] : ( plus_plus_int @ K3 @ K3 )
    @ code_dup ) ).

% dup.transfer
thf(fact_8825_take__bit__integer_Otransfer,axiom,
    ( bNF_re4935368626670024657nteger
    @ ^ [Y: nat,Z: nat] : ( Y = Z )
    @ ( bNF_re3379532845092657523nteger @ code_pcr_integer @ code_pcr_integer )
    @ bit_se2923211474154528505it_int
    @ bit_se1745604003318907178nteger ) ).

% take_bit_integer.transfer
thf(fact_8826_not__integer_Otransfer,axiom,
    bNF_re3379532845092657523nteger @ code_pcr_integer @ code_pcr_integer @ bit_ri7919022796975470100ot_int @ bit_ri7632146776885996613nteger ).

% not_integer.transfer
thf(fact_8827_and__integer_Otransfer,axiom,
    bNF_re398004352372739002nteger @ code_pcr_integer @ ( bNF_re3379532845092657523nteger @ code_pcr_integer @ code_pcr_integer ) @ bit_se725231765392027082nd_int @ bit_se3949692690581998587nteger ).

% and_integer.transfer
thf(fact_8828_bit__integer_Otransfer,axiom,
    ( bNF_re4711666741709854504_nat_o @ code_pcr_integer
    @ ^ [Y: nat > $o,Z: nat > $o] : ( Y = Z )
    @ bit_se1146084159140164899it_int
    @ bit_se9216721137139052372nteger ) ).

% bit_integer.transfer
thf(fact_8829_or__integer_Otransfer,axiom,
    bNF_re398004352372739002nteger @ code_pcr_integer @ ( bNF_re3379532845092657523nteger @ code_pcr_integer @ code_pcr_integer ) @ bit_se1409905431419307370or_int @ bit_se1080825931792720795nteger ).

% or_integer.transfer
thf(fact_8830_xor__integer_Otransfer,axiom,
    bNF_re398004352372739002nteger @ code_pcr_integer @ ( bNF_re3379532845092657523nteger @ code_pcr_integer @ code_pcr_integer ) @ bit_se6526347334894502574or_int @ bit_se3222712562003087583nteger ).

% xor_integer.transfer
thf(fact_8831_push__bit__integer_Otransfer,axiom,
    ( bNF_re4935368626670024657nteger
    @ ^ [Y: nat,Z: nat] : ( Y = Z )
    @ ( bNF_re3379532845092657523nteger @ code_pcr_integer @ code_pcr_integer )
    @ bit_se545348938243370406it_int
    @ bit_se7788150548672797655nteger ) ).

% push_bit_integer.transfer
thf(fact_8832_drop__bit__integer_Otransfer,axiom,
    ( bNF_re4935368626670024657nteger
    @ ^ [Y: nat,Z: nat] : ( Y = Z )
    @ ( bNF_re3379532845092657523nteger @ code_pcr_integer @ code_pcr_integer )
    @ bit_se8568078237143864401it_int
    @ bit_se3928097537394005634nteger ) ).

% drop_bit_integer.transfer
thf(fact_8833_mask__integer_Otransfer,axiom,
    ( bNF_re4153400068438556298nteger
    @ ^ [Y: nat,Z: nat] : ( Y = Z )
    @ code_pcr_integer
    @ bit_se2000444600071755411sk_int
    @ bit_se2119862282449309892nteger ) ).

% mask_integer.transfer
thf(fact_8834_unset__bit__integer_Otransfer,axiom,
    ( bNF_re4935368626670024657nteger
    @ ^ [Y: nat,Z: nat] : ( Y = Z )
    @ ( bNF_re3379532845092657523nteger @ code_pcr_integer @ code_pcr_integer )
    @ bit_se4203085406695923979it_int
    @ bit_se8260200283734997820nteger ) ).

% unset_bit_integer.transfer
thf(fact_8835_set__bit__integer_Otransfer,axiom,
    ( bNF_re4935368626670024657nteger
    @ ^ [Y: nat,Z: nat] : ( Y = Z )
    @ ( bNF_re3379532845092657523nteger @ code_pcr_integer @ code_pcr_integer )
    @ bit_se7879613467334960850it_int
    @ bit_se2793503036327961859nteger ) ).

% set_bit_integer.transfer
thf(fact_8836_flip__bit__integer_Otransfer,axiom,
    ( bNF_re4935368626670024657nteger
    @ ^ [Y: nat,Z: nat] : ( Y = Z )
    @ ( bNF_re3379532845092657523nteger @ code_pcr_integer @ code_pcr_integer )
    @ bit_se2159334234014336723it_int
    @ bit_se1345352211410354436nteger ) ).

% flip_bit_integer.transfer
thf(fact_8837_division__segment__integer_Otransfer,axiom,
    bNF_re3379532845092657523nteger @ code_pcr_integer @ code_pcr_integer @ euclid3395696857347342551nt_int @ euclid6289375185220004616nteger ).

% division_segment_integer.transfer
thf(fact_8838_euclidean__size__integer_Otransfer,axiom,
    ( bNF_re2807294637932363402at_nat @ code_pcr_integer
    @ ^ [Y: nat,Z: nat] : ( Y = Z )
    @ euclid4774559944035922753ze_int
    @ euclid6377331345833325938nteger ) ).

% euclidean_size_integer.transfer
thf(fact_8839_sub_Otransfer,axiom,
    ( bNF_re7876454716742015248nteger
    @ ^ [Y: num,Z: num] : ( Y = Z )
    @ ( bNF_re6501075790457514782nteger
      @ ^ [Y: num,Z: num] : ( Y = Z )
      @ code_pcr_integer )
    @ ^ [M2: num,N3: num] : ( minus_minus_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N3 ) )
    @ code_sub ) ).

% sub.transfer
thf(fact_8840_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_8841_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_8842_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_8843_char_Osize_I2_J,axiom,
    ! [X1: $o,X22: $o,X32: $o,X42: $o,X52: $o,X62: $o,X72: $o,X82: $o] :
      ( ( size_size_char @ ( char2 @ X1 @ X22 @ X32 @ X42 @ X52 @ X62 @ X72 @ X82 ) )
      = zero_zero_nat ) ).

% char.size(2)
thf(fact_8844_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_8845_char_Osize__gen,axiom,
    ! [X1: $o,X22: $o,X32: $o,X42: $o,X52: $o,X62: $o,X72: $o,X82: $o] :
      ( ( size_char @ ( char2 @ X1 @ X22 @ X32 @ X42 @ X52 @ X62 @ X72 @ X82 ) )
      = zero_zero_nat ) ).

% char.size_gen
thf(fact_8846_integer__of__natural__of__nat,axiom,
    ! [N: nat] :
      ( ( code_i5400310926305786745atural @ ( semiri3763490453095760265atural @ N ) )
      = ( semiri4939895301339042750nteger @ N ) ) ).

% integer_of_natural_of_nat
thf(fact_8847_integer__of__natural__of__integer,axiom,
    ! [K: code_integer] :
      ( ( code_i5400310926305786745atural @ ( code_n4118661773612635043nteger @ K ) )
      = ( ord_max_Code_integer @ zero_z3403309356797280102nteger @ K ) ) ).

% integer_of_natural_of_integer
thf(fact_8848_natural__of__integer__of__natural,axiom,
    ! [N: code_natural] :
      ( ( code_n4118661773612635043nteger @ ( code_i5400310926305786745atural @ N ) )
      = N ) ).

% natural_of_integer_of_natural
thf(fact_8849_less__than__iff,axiom,
    ! [X: nat,Y3: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ less_than )
      = ( ord_less_nat @ X @ Y3 ) ) ).

% less_than_iff
thf(fact_8850_Code__Numeral_Osub__code_I4_J,axiom,
    ! [N: num] :
      ( ( code_sub @ one @ ( bit0 @ N ) )
      = ( code_Neg @ ( bitM @ N ) ) ) ).

% Code_Numeral.sub_code(4)
thf(fact_8851_Code__Numeral_Odup__code_I3_J,axiom,
    ! [N: num] :
      ( ( code_dup @ ( code_Neg @ N ) )
      = ( code_Neg @ ( bit0 @ N ) ) ) ).

% Code_Numeral.dup_code(3)
thf(fact_8852_less__integer__code_I7_J,axiom,
    ! [K: num] : ( ord_le6747313008572928689nteger @ ( code_Neg @ K ) @ zero_z3403309356797280102nteger ) ).

% less_integer_code(7)
thf(fact_8853_less__integer__code_I3_J,axiom,
    ! [L: num] :
      ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( code_Neg @ L ) ) ).

% less_integer_code(3)
thf(fact_8854_plus__integer__code_I6_J,axiom,
    ! [M: num,N: num] :
      ( ( plus_p5714425477246183910nteger @ ( code_Neg @ M ) @ ( code_Neg @ N ) )
      = ( code_Neg @ ( plus_plus_num @ M @ N ) ) ) ).

% plus_integer_code(6)
thf(fact_8855_less__eq__integer__code_I7_J,axiom,
    ! [K: num] : ( ord_le3102999989581377725nteger @ ( code_Neg @ K ) @ zero_z3403309356797280102nteger ) ).

% less_eq_integer_code(7)
thf(fact_8856_less__eq__integer__code_I3_J,axiom,
    ! [L: num] :
      ~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( code_Neg @ L ) ) ).

% less_eq_integer_code(3)
thf(fact_8857_less__integer__code_I9_J,axiom,
    ! [K: num,L: num] :
      ( ( ord_le6747313008572928689nteger @ ( code_Neg @ K ) @ ( code_Neg @ L ) )
      = ( ord_less_num @ L @ K ) ) ).

% less_integer_code(9)
thf(fact_8858_less__eq__integer__code_I9_J,axiom,
    ! [K: num,L: num] :
      ( ( ord_le3102999989581377725nteger @ ( code_Neg @ K ) @ ( code_Neg @ L ) )
      = ( ord_less_eq_num @ L @ K ) ) ).

% less_eq_integer_code(9)
thf(fact_8859_minus__integer__code_I6_J,axiom,
    ! [M: num,N: num] :
      ( ( minus_8373710615458151222nteger @ ( code_Neg @ M ) @ ( code_Neg @ N ) )
      = ( code_sub @ N @ M ) ) ).

% minus_integer_code(6)
thf(fact_8860_divmod__abs__code_I2_J,axiom,
    ! [K: num,L: num] :
      ( ( code_divmod_abs @ ( code_Neg @ K ) @ ( code_Neg @ L ) )
      = ( unique3479559517661332726nteger @ K @ L ) ) ).

% divmod_abs_code(2)
thf(fact_8861_Code__Numeral_Osub__code_I5_J,axiom,
    ! [N: num] :
      ( ( code_sub @ one @ ( bit1 @ N ) )
      = ( code_Neg @ ( bit0 @ N ) ) ) ).

% Code_Numeral.sub_code(5)
thf(fact_8862_pair__less__def,axiom,
    ( fun_pair_less
    = ( lex_prod_nat_nat @ less_than @ less_than ) ) ).

% pair_less_def
thf(fact_8863_Gcd__int__set__eq__fold,axiom,
    ! [Xs: list_int] :
      ( ( gcd_Gcd_int @ ( set_int2 @ Xs ) )
      = ( fold_int_int @ gcd_gcd_int @ Xs @ zero_zero_int ) ) ).

% Gcd_int_set_eq_fold
thf(fact_8864_binomial__def,axiom,
    ( binomial
    = ( ^ [N3: nat,K3: nat] :
          ( finite_card_set_nat
          @ ( collect_set_nat
            @ ^ [K6: set_nat] :
                ( ( member_set_nat @ K6 @ ( pow_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N3 ) ) )
                & ( ( finite_card_nat @ K6 )
                  = K3 ) ) ) ) ) ) ).

% binomial_def
thf(fact_8865_int_Obi__total,axiom,
    bi_tot896582865486249351at_int @ pcr_int ).

% int.bi_total
thf(fact_8866_integer_Obi__total,axiom,
    bi_tot1331153423839324337nteger @ code_pcr_integer ).

% integer.bi_total
thf(fact_8867_Gcd__nat__set__eq__fold,axiom,
    ! [Xs: list_nat] :
      ( ( gcd_Gcd_nat @ ( set_nat2 @ Xs ) )
      = ( fold_nat_nat @ gcd_gcd_nat @ Xs @ zero_zero_nat ) ) ).

% Gcd_nat_set_eq_fold
thf(fact_8868_Code__Numeral_Osub__code_I2_J,axiom,
    ! [M: num] :
      ( ( code_sub @ ( bit0 @ M ) @ one )
      = ( code_Pos @ ( bitM @ M ) ) ) ).

% Code_Numeral.sub_code(2)
thf(fact_8869_Code__Numeral_OPos__def,axiom,
    code_Pos = numera6620942414471956472nteger ).

% Code_Numeral.Pos_def
thf(fact_8870_Pos__fold_I2_J,axiom,
    ! [K: num] :
      ( ( numera6620942414471956472nteger @ ( bit0 @ K ) )
      = ( code_Pos @ ( bit0 @ K ) ) ) ).

% Pos_fold(2)
thf(fact_8871_Pos__fold_I1_J,axiom,
    ( ( numera6620942414471956472nteger @ one )
    = ( code_Pos @ one ) ) ).

% Pos_fold(1)
thf(fact_8872_Pos__fold_I3_J,axiom,
    ! [K: num] :
      ( ( numera6620942414471956472nteger @ ( bit1 @ K ) )
      = ( code_Pos @ ( bit1 @ K ) ) ) ).

% Pos_fold(3)
thf(fact_8873_one__integer__code,axiom,
    ( one_one_Code_integer
    = ( code_Pos @ one ) ) ).

% one_integer_code
thf(fact_8874_Code__Numeral_Odup__code_I2_J,axiom,
    ! [N: num] :
      ( ( code_dup @ ( code_Pos @ N ) )
      = ( code_Pos @ ( bit0 @ N ) ) ) ).

% Code_Numeral.dup_code(2)
thf(fact_8875_less__integer__code_I4_J,axiom,
    ! [K: num] :
      ~ ( ord_le6747313008572928689nteger @ ( code_Pos @ K ) @ zero_z3403309356797280102nteger ) ).

% less_integer_code(4)
thf(fact_8876_less__integer__code_I2_J,axiom,
    ! [L: num] : ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( code_Pos @ L ) ) ).

% less_integer_code(2)
thf(fact_8877_plus__integer__code_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( plus_p5714425477246183910nteger @ ( code_Pos @ M ) @ ( code_Pos @ N ) )
      = ( code_Pos @ ( plus_plus_num @ M @ N ) ) ) ).

% plus_integer_code(3)
thf(fact_8878_less__eq__integer__code_I4_J,axiom,
    ! [K: num] :
      ~ ( ord_le3102999989581377725nteger @ ( code_Pos @ K ) @ zero_z3403309356797280102nteger ) ).

% less_eq_integer_code(4)
thf(fact_8879_less__eq__integer__code_I2_J,axiom,
    ! [L: num] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( code_Pos @ L ) ) ).

% less_eq_integer_code(2)
thf(fact_8880_Code__Numeral_ONeg__def,axiom,
    ( code_Neg
    = ( ^ [N3: num] : ( uminus1351360451143612070nteger @ ( code_Pos @ N3 ) ) ) ) ).

% Code_Numeral.Neg_def
thf(fact_8881_uminus__integer__code_I2_J,axiom,
    ! [M: num] :
      ( ( uminus1351360451143612070nteger @ ( code_Pos @ M ) )
      = ( code_Neg @ M ) ) ).

% uminus_integer_code(2)
thf(fact_8882_uminus__integer__code_I3_J,axiom,
    ! [M: num] :
      ( ( uminus1351360451143612070nteger @ ( code_Neg @ M ) )
      = ( code_Pos @ M ) ) ).

% uminus_integer_code(3)
thf(fact_8883_less__integer__code_I8_J,axiom,
    ! [K: num,L: num] : ( ord_le6747313008572928689nteger @ ( code_Neg @ K ) @ ( code_Pos @ L ) ) ).

% less_integer_code(8)
thf(fact_8884_less__integer__code_I6_J,axiom,
    ! [K: num,L: num] :
      ~ ( ord_le6747313008572928689nteger @ ( code_Pos @ K ) @ ( code_Neg @ L ) ) ).

% less_integer_code(6)
thf(fact_8885_times__integer__code_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( times_3573771949741848930nteger @ ( code_Pos @ M ) @ ( code_Pos @ N ) )
      = ( code_Pos @ ( times_times_num @ M @ N ) ) ) ).

% times_integer_code(3)
thf(fact_8886_less__eq__integer__code_I8_J,axiom,
    ! [K: num,L: num] : ( ord_le3102999989581377725nteger @ ( code_Neg @ K ) @ ( code_Pos @ L ) ) ).

% less_eq_integer_code(8)
thf(fact_8887_less__eq__integer__code_I6_J,axiom,
    ! [K: num,L: num] :
      ~ ( ord_le3102999989581377725nteger @ ( code_Pos @ K ) @ ( code_Neg @ L ) ) ).

% less_eq_integer_code(6)
thf(fact_8888_less__integer__code_I5_J,axiom,
    ! [K: num,L: num] :
      ( ( ord_le6747313008572928689nteger @ ( code_Pos @ K ) @ ( code_Pos @ L ) )
      = ( ord_less_num @ K @ L ) ) ).

% less_integer_code(5)
thf(fact_8889_less__eq__integer__code_I5_J,axiom,
    ! [K: num,L: num] :
      ( ( ord_le3102999989581377725nteger @ ( code_Pos @ K ) @ ( code_Pos @ L ) )
      = ( ord_less_eq_num @ K @ L ) ) ).

% less_eq_integer_code(5)
thf(fact_8890_minus__integer__code_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( minus_8373710615458151222nteger @ ( code_Pos @ M ) @ ( code_Pos @ N ) )
      = ( code_sub @ M @ N ) ) ).

% minus_integer_code(3)
thf(fact_8891_divmod__abs__code_I1_J,axiom,
    ! [K: num,L: num] :
      ( ( code_divmod_abs @ ( code_Pos @ K ) @ ( code_Pos @ L ) )
      = ( unique3479559517661332726nteger @ K @ L ) ) ).

% divmod_abs_code(1)
thf(fact_8892_minus__integer__code_I5_J,axiom,
    ! [M: num,N: num] :
      ( ( minus_8373710615458151222nteger @ ( code_Neg @ M ) @ ( code_Pos @ N ) )
      = ( code_Neg @ ( plus_plus_num @ M @ N ) ) ) ).

% minus_integer_code(5)
thf(fact_8893_minus__integer__code_I4_J,axiom,
    ! [M: num,N: num] :
      ( ( minus_8373710615458151222nteger @ ( code_Pos @ M ) @ ( code_Neg @ N ) )
      = ( code_Pos @ ( plus_plus_num @ M @ N ) ) ) ).

% minus_integer_code(4)
thf(fact_8894_times__integer__code_I6_J,axiom,
    ! [M: num,N: num] :
      ( ( times_3573771949741848930nteger @ ( code_Neg @ M ) @ ( code_Neg @ N ) )
      = ( code_Pos @ ( times_times_num @ M @ N ) ) ) ).

% times_integer_code(6)
thf(fact_8895_times__integer__code_I5_J,axiom,
    ! [M: num,N: num] :
      ( ( times_3573771949741848930nteger @ ( code_Neg @ M ) @ ( code_Pos @ N ) )
      = ( code_Neg @ ( times_times_num @ M @ N ) ) ) ).

% times_integer_code(5)
thf(fact_8896_times__integer__code_I4_J,axiom,
    ! [M: num,N: num] :
      ( ( times_3573771949741848930nteger @ ( code_Pos @ M ) @ ( code_Neg @ N ) )
      = ( code_Neg @ ( times_times_num @ M @ N ) ) ) ).

% times_integer_code(4)
thf(fact_8897_plus__integer__code_I4_J,axiom,
    ! [M: num,N: num] :
      ( ( plus_p5714425477246183910nteger @ ( code_Pos @ M ) @ ( code_Neg @ N ) )
      = ( code_sub @ M @ N ) ) ).

% plus_integer_code(4)
thf(fact_8898_plus__integer__code_I5_J,axiom,
    ! [M: num,N: num] :
      ( ( plus_p5714425477246183910nteger @ ( code_Neg @ M ) @ ( code_Pos @ N ) )
      = ( code_sub @ N @ M ) ) ).

% plus_integer_code(5)
thf(fact_8899_divmod__abs__code_I3_J,axiom,
    ! [K: num,L: num] :
      ( ( code_divmod_abs @ ( code_Neg @ K ) @ ( code_Pos @ L ) )
      = ( unique3479559517661332726nteger @ K @ L ) ) ).

% divmod_abs_code(3)
thf(fact_8900_divmod__abs__code_I4_J,axiom,
    ! [K: num,L: num] :
      ( ( code_divmod_abs @ ( code_Pos @ K ) @ ( code_Neg @ L ) )
      = ( unique3479559517661332726nteger @ K @ L ) ) ).

% divmod_abs_code(4)
thf(fact_8901_Code__Numeral_Osub__code_I3_J,axiom,
    ! [M: num] :
      ( ( code_sub @ ( bit1 @ M ) @ one )
      = ( code_Pos @ ( bit0 @ M ) ) ) ).

% Code_Numeral.sub_code(3)
thf(fact_8902_cr__int__def,axiom,
    ( cr_int
    = ( ^ [X2: product_prod_nat_nat] :
          ( ^ [Y: int,Z: int] : ( Y = Z )
          @ ( abs_Integ @ X2 ) ) ) ) ).

% cr_int_def
thf(fact_8903_Least__eq__0,axiom,
    ! [P: nat > $o] :
      ( ( P @ zero_zero_nat )
     => ( ( ord_Least_nat @ P )
        = zero_zero_nat ) ) ).

% Least_eq_0
thf(fact_8904_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_8905_Least__Suc,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ N )
     => ( ~ ( P @ zero_zero_nat )
       => ( ( ord_Least_nat @ P )
          = ( suc
            @ ( ord_Least_nat
              @ ^ [M2: nat] : ( P @ ( suc @ M2 ) ) ) ) ) ) ) ).

% Least_Suc
thf(fact_8906_int_Opcr__cr__eq,axiom,
    pcr_int = cr_int ).

% int.pcr_cr_eq
thf(fact_8907_Quotient__int,axiom,
    quotie1194848508323700631at_int @ intrel @ abs_Integ @ rep_Integ @ cr_int ).

% Quotient_int
thf(fact_8908_gcd__nat_Osemilattice__neutr__axioms,axiom,
    semila9081495762789891438tr_nat @ gcd_gcd_nat @ zero_zero_nat ).

% gcd_nat.semilattice_neutr_axioms
thf(fact_8909_max__nat_Osemilattice__neutr__axioms,axiom,
    semila9081495762789891438tr_nat @ ord_max_nat @ zero_zero_nat ).

% max_nat.semilattice_neutr_axioms
thf(fact_8910_natLess__def,axiom,
    ( bNF_Ca8459412986667044542atLess
    = ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ord_less_nat ) ) ) ).

% natLess_def
thf(fact_8911_Restr__natLeq,axiom,
    ! [N: nat] :
      ( ( inf_in2572325071724192079at_nat @ bNF_Ca8665028551170535155natLeq
        @ ( produc457027306803732586at_nat
          @ ( collect_nat
            @ ^ [X2: nat] : ( ord_less_nat @ X2 @ N ) )
          @ ^ [Uu: nat] :
              ( collect_nat
              @ ^ [X2: nat] : ( ord_less_nat @ X2 @ N ) ) ) )
      = ( collec3392354462482085612at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [X2: nat,Y6: nat] :
              ( ( ord_less_nat @ X2 @ N )
              & ( ord_less_nat @ Y6 @ N )
              & ( ord_less_eq_nat @ X2 @ Y6 ) ) ) ) ) ).

% Restr_natLeq
thf(fact_8912_natLeq__def,axiom,
    ( bNF_Ca8665028551170535155natLeq
    = ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ord_less_eq_nat ) ) ) ).

% natLeq_def
thf(fact_8913_Restr__natLeq2,axiom,
    ! [N: nat] :
      ( ( inf_in2572325071724192079at_nat @ bNF_Ca8665028551170535155natLeq
        @ ( produc457027306803732586at_nat @ ( order_underS_nat @ bNF_Ca8665028551170535155natLeq @ N )
          @ ^ [Uu: nat] : ( order_underS_nat @ bNF_Ca8665028551170535155natLeq @ N ) ) )
      = ( collec3392354462482085612at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [X2: nat,Y6: nat] :
              ( ( ord_less_nat @ X2 @ N )
              & ( ord_less_nat @ Y6 @ N )
              & ( ord_less_eq_nat @ X2 @ Y6 ) ) ) ) ) ).

% Restr_natLeq2
thf(fact_8914_natLeq__underS__less,axiom,
    ! [N: nat] :
      ( ( order_underS_nat @ bNF_Ca8665028551170535155natLeq @ N )
      = ( collect_nat
        @ ^ [X2: nat] : ( ord_less_nat @ X2 @ N ) ) ) ).

% natLeq_underS_less
thf(fact_8915_wf__int__ge__less__than2,axiom,
    ! [D: int] : ( wf_int @ ( int_ge_less_than2 @ D ) ) ).

% wf_int_ge_less_than2
thf(fact_8916_wf__int__ge__less__than,axiom,
    ! [D: int] : ( wf_int @ ( int_ge_less_than @ D ) ) ).

% wf_int_ge_less_than
thf(fact_8917_wf__pair__less,axiom,
    wf_Pro7803398752247294826at_nat @ fun_pair_less ).

% wf_pair_less
thf(fact_8918_wf__less,axiom,
    wf_nat @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ord_less_nat ) ) ).

% wf_less
thf(fact_8919_gcd__nat_Omonoid__axioms,axiom,
    monoid_nat @ gcd_gcd_nat @ zero_zero_nat ).

% gcd_nat.monoid_axioms
thf(fact_8920_max__nat_Omonoid__axioms,axiom,
    monoid_nat @ ord_max_nat @ zero_zero_nat ).

% max_nat.monoid_axioms
thf(fact_8921_pair__leq__def,axiom,
    ( fun_pair_leq
    = ( sup_su718114333110466843at_nat @ fun_pair_less @ id_Pro2258643101195443293at_nat ) ) ).

% pair_leq_def
thf(fact_8922_cr__integer__def,axiom,
    ( code_cr_integer
    = ( ^ [X2: int,Y6: code_integer] :
          ( X2
          = ( code_int_of_integer @ Y6 ) ) ) ) ).

% cr_integer_def
thf(fact_8923_bit__integer__def,axiom,
    ( bit_se9216721137139052372nteger
    = ( map_fu186220595405829473_nat_o @ code_int_of_integer @ id_nat_o @ bit_se1146084159140164899it_int ) ) ).

% bit_integer_def
thf(fact_8924_integer_Opcr__cr__eq,axiom,
    code_pcr_integer = code_cr_integer ).

% integer.pcr_cr_eq
thf(fact_8925_lcm__altdef__int,axiom,
    ( gcd_lcm_int
    = ( ^ [A2: int,B2: int] : ( divide_divide_int @ ( times_times_int @ ( abs_abs_int @ A2 ) @ ( abs_abs_int @ B2 ) ) @ ( gcd_gcd_int @ A2 @ B2 ) ) ) ) ).

% lcm_altdef_int
thf(fact_8926_lcm__0__iff__int,axiom,
    ! [M: int,N: int] :
      ( ( ( gcd_lcm_int @ M @ N )
        = zero_zero_int )
      = ( ( M = zero_zero_int )
        | ( N = zero_zero_int ) ) ) ).

% lcm_0_iff_int
thf(fact_8927_lcm__1__iff__int,axiom,
    ! [M: int,N: int] :
      ( ( ( gcd_lcm_int @ M @ N )
        = one_one_int )
      = ( ( ( M = one_one_int )
          | ( M
            = ( uminus_uminus_int @ one_one_int ) ) )
        & ( ( N = one_one_int )
          | ( N
            = ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).

% lcm_1_iff_int
thf(fact_8928_lcm__ge__0__int,axiom,
    ! [X: int,Y3: int] : ( ord_less_eq_int @ zero_zero_int @ ( gcd_lcm_int @ X @ Y3 ) ) ).

% lcm_ge_0_int
thf(fact_8929_lcm__neg1__int,axiom,
    ! [X: int,Y3: int] :
      ( ( gcd_lcm_int @ ( uminus_uminus_int @ X ) @ Y3 )
      = ( gcd_lcm_int @ X @ Y3 ) ) ).

% lcm_neg1_int
thf(fact_8930_lcm__neg2__int,axiom,
    ! [X: int,Y3: int] :
      ( ( gcd_lcm_int @ X @ ( uminus_uminus_int @ Y3 ) )
      = ( gcd_lcm_int @ X @ Y3 ) ) ).

% lcm_neg2_int
thf(fact_8931_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_8932_gcd__nat_Ocomm__monoid__axioms,axiom,
    comm_monoid_nat @ gcd_gcd_nat @ zero_zero_nat ).

% gcd_nat.comm_monoid_axioms
thf(fact_8933_lcm__cases__int,axiom,
    ! [X: int,Y3: int,P: int > $o] :
      ( ( ( ord_less_eq_int @ zero_zero_int @ X )
       => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
         => ( P @ ( gcd_lcm_int @ X @ Y3 ) ) ) )
     => ( ( ( ord_less_eq_int @ zero_zero_int @ X )
         => ( ( ord_less_eq_int @ Y3 @ zero_zero_int )
           => ( P @ ( gcd_lcm_int @ X @ ( uminus_uminus_int @ Y3 ) ) ) ) )
       => ( ( ( ord_less_eq_int @ X @ zero_zero_int )
           => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
             => ( P @ ( gcd_lcm_int @ ( uminus_uminus_int @ X ) @ Y3 ) ) ) )
         => ( ( ( ord_less_eq_int @ X @ zero_zero_int )
             => ( ( ord_less_eq_int @ Y3 @ zero_zero_int )
               => ( P @ ( gcd_lcm_int @ ( uminus_uminus_int @ X ) @ ( uminus_uminus_int @ Y3 ) ) ) ) )
           => ( P @ ( gcd_lcm_int @ X @ Y3 ) ) ) ) ) ) ).

% lcm_cases_int
thf(fact_8934_max__nat_Ocomm__monoid__axioms,axiom,
    comm_monoid_nat @ ord_max_nat @ zero_zero_nat ).

% max_nat.comm_monoid_axioms
thf(fact_8935_lcm__unique__int,axiom,
    ! [D: int,A: int,B: int] :
      ( ( ( ord_less_eq_int @ zero_zero_int @ D )
        & ( dvd_dvd_int @ A @ D )
        & ( dvd_dvd_int @ B @ D )
        & ! [E3: int] :
            ( ( ( dvd_dvd_int @ A @ E3 )
              & ( dvd_dvd_int @ B @ E3 ) )
           => ( dvd_dvd_int @ D @ E3 ) ) )
      = ( D
        = ( gcd_lcm_int @ A @ B ) ) ) ).

% lcm_unique_int
thf(fact_8936_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_8937_Lcm__int__set__eq__fold,axiom,
    ! [Xs: list_int] :
      ( ( gcd_Lcm_int @ ( set_int2 @ Xs ) )
      = ( fold_int_int @ gcd_lcm_int @ Xs @ one_one_int ) ) ).

% Lcm_int_set_eq_fold
thf(fact_8938_lcm__0__iff__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( ( gcd_lcm_nat @ M @ N )
        = zero_zero_nat )
      = ( ( M = zero_zero_nat )
        | ( N = zero_zero_nat ) ) ) ).

% lcm_0_iff_nat
thf(fact_8939_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_8940_lcm__int__int__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( gcd_lcm_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
      = ( semiri1314217659103216013at_int @ ( gcd_lcm_nat @ M @ N ) ) ) ).

% lcm_int_int_eq
thf(fact_8941_lcm__nat__abs__right__eq,axiom,
    ! [N: nat,K: int] :
      ( ( gcd_lcm_nat @ N @ ( nat2 @ ( abs_abs_int @ K ) ) )
      = ( nat2 @ ( gcd_lcm_int @ ( semiri1314217659103216013at_int @ N ) @ K ) ) ) ).

% lcm_nat_abs_right_eq
thf(fact_8942_lcm__nat__abs__left__eq,axiom,
    ! [K: int,N: nat] :
      ( ( gcd_lcm_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ N )
      = ( nat2 @ ( gcd_lcm_int @ K @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).

% lcm_nat_abs_left_eq
thf(fact_8943_prod__gcd__lcm__nat,axiom,
    ( times_times_nat
    = ( ^ [M2: nat,N3: nat] : ( times_times_nat @ ( gcd_gcd_nat @ M2 @ N3 ) @ ( gcd_lcm_nat @ M2 @ N3 ) ) ) ) ).

% prod_gcd_lcm_nat
thf(fact_8944_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_8945_Lcm__int__greater__eq__0,axiom,
    ! [K5: set_int] : ( ord_less_eq_int @ zero_zero_int @ ( gcd_Lcm_int @ K5 ) ) ).

% Lcm_int_greater_eq_0
thf(fact_8946_lcm__nat__def,axiom,
    ( gcd_lcm_nat
    = ( ^ [X2: nat,Y6: nat] : ( divide_divide_nat @ ( times_times_nat @ X2 @ Y6 ) @ ( gcd_gcd_nat @ X2 @ Y6 ) ) ) ) ).

% lcm_nat_def
thf(fact_8947_lcm__int__def,axiom,
    ( gcd_lcm_int
    = ( ^ [X2: int,Y6: int] : ( semiri1314217659103216013at_int @ ( gcd_lcm_nat @ ( nat2 @ ( abs_abs_int @ X2 ) ) @ ( nat2 @ ( abs_abs_int @ Y6 ) ) ) ) ) ) ).

% lcm_int_def
thf(fact_8948_Lcm__int__def,axiom,
    ( gcd_Lcm_int
    = ( ^ [K6: set_int] : ( semiri1314217659103216013at_int @ ( gcd_Lcm_nat @ ( image_int_nat @ ( comp_int_nat_int @ nat2 @ abs_abs_int ) @ K6 ) ) ) ) ) ).

% Lcm_int_def
thf(fact_8949_Lcm__int__eq,axiom,
    ! [N5: set_nat] :
      ( ( gcd_Lcm_int @ ( image_nat_int @ semiri1314217659103216013at_int @ N5 ) )
      = ( semiri1314217659103216013at_int @ ( gcd_Lcm_nat @ N5 ) ) ) ).

% Lcm_int_eq
thf(fact_8950_Lcm__eq__0__I__nat,axiom,
    ! [A3: set_nat] :
      ( ( member_nat @ zero_zero_nat @ A3 )
     => ( ( gcd_Lcm_nat @ A3 )
        = zero_zero_nat ) ) ).

% Lcm_eq_0_I_nat
thf(fact_8951_Lcm__0__iff__nat,axiom,
    ! [A3: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( ( gcd_Lcm_nat @ A3 )
          = zero_zero_nat )
        = ( member_nat @ zero_zero_nat @ A3 ) ) ) ).

% Lcm_0_iff_nat
thf(fact_8952_Lcm__nat__infinite,axiom,
    ! [M7: set_nat] :
      ( ~ ( finite_finite_nat @ M7 )
     => ( ( gcd_Lcm_nat @ M7 )
        = zero_zero_nat ) ) ).

% Lcm_nat_infinite
thf(fact_8953_Lcm__nat__empty,axiom,
    ( ( gcd_Lcm_nat @ bot_bot_set_nat )
    = one_one_nat ) ).

% Lcm_nat_empty
thf(fact_8954_Lcm__nat__set__eq__fold,axiom,
    ! [Xs: list_nat] :
      ( ( gcd_Lcm_nat @ ( set_nat2 @ Xs ) )
      = ( fold_nat_nat @ gcd_lcm_nat @ Xs @ one_one_nat ) ) ).

% Lcm_nat_set_eq_fold
thf(fact_8955_Lcm__eq__Max__nat,axiom,
    ! [M7: set_nat] :
      ( ( finite_finite_nat @ M7 )
     => ( ( M7 != bot_bot_set_nat )
       => ( ~ ( member_nat @ zero_zero_nat @ M7 )
         => ( ! [M6: nat,N4: nat] :
                ( ( member_nat @ M6 @ M7 )
               => ( ( member_nat @ N4 @ M7 )
                 => ( member_nat @ ( gcd_lcm_nat @ M6 @ N4 ) @ M7 ) ) )
           => ( ( gcd_Lcm_nat @ M7 )
              = ( lattic8265883725875713057ax_nat @ M7 ) ) ) ) ) ) ).

% Lcm_eq_Max_nat
thf(fact_8956_Lcm__nat__def,axiom,
    ( gcd_Lcm_nat
    = ( ^ [M8: set_nat] : ( if_nat @ ( finite_finite_nat @ M8 ) @ ( lattic7826324295020591184_F_nat @ gcd_lcm_nat @ one_one_nat @ M8 ) @ zero_zero_nat ) ) ) ).

% Lcm_nat_def
thf(fact_8957_normalize__nat__def,axiom,
    normal574172969982618308ze_nat = id_nat ).

% normalize_nat_def
thf(fact_8958_normalize__int__def,axiom,
    normal571682499473568032ze_int = abs_abs_int ).

% normalize_int_def
thf(fact_8959_unit__factor__simps_I1_J,axiom,
    ( ( unit_f2748546683901255202or_nat @ zero_zero_nat )
    = zero_zero_nat ) ).

% unit_factor_simps(1)
thf(fact_8960_unit__factor__simps_I2_J,axiom,
    ! [N: nat] :
      ( ( unit_f2748546683901255202or_nat @ ( suc @ N ) )
      = one_one_nat ) ).

% unit_factor_simps(2)
thf(fact_8961_unit__factor__nat__def,axiom,
    ( unit_f2748546683901255202or_nat
    = ( ^ [N3: nat] : ( if_nat @ ( N3 = zero_zero_nat ) @ zero_zero_nat @ one_one_nat ) ) ) ).

% unit_factor_nat_def
thf(fact_8962_unit__factor__int__def,axiom,
    unit_f2746056213392204926or_int = sgn_sgn_int ).

% unit_factor_int_def
thf(fact_8963_times__num__def,axiom,
    ( times_times_num
    = ( ^ [M2: num,N3: num] : ( num_of_nat @ ( times_times_nat @ ( nat_of_num @ M2 ) @ ( nat_of_num @ N3 ) ) ) ) ) ).

% times_num_def
thf(fact_8964_nat__of__num__pos,axiom,
    ! [X: num] : ( ord_less_nat @ zero_zero_nat @ ( nat_of_num @ X ) ) ).

% nat_of_num_pos
thf(fact_8965_nat__of__num_Osimps_I2_J,axiom,
    ! [X: num] :
      ( ( nat_of_num @ ( bit0 @ X ) )
      = ( plus_plus_nat @ ( nat_of_num @ X ) @ ( nat_of_num @ X ) ) ) ).

% nat_of_num.simps(2)
thf(fact_8966_nat__of__num__neq__0,axiom,
    ! [X: num] :
      ( ( nat_of_num @ X )
     != zero_zero_nat ) ).

% nat_of_num_neq_0
thf(fact_8967_num__eq__iff,axiom,
    ( ( ^ [Y: num,Z: num] : ( Y = Z ) )
    = ( ^ [X2: num,Y6: num] :
          ( ( nat_of_num @ X2 )
          = ( nat_of_num @ Y6 ) ) ) ) ).

% num_eq_iff
thf(fact_8968_nat__of__num__numeral,axiom,
    nat_of_num = numeral_numeral_nat ).

% nat_of_num_numeral
thf(fact_8969_nat__of__num__inverse,axiom,
    ! [X: num] :
      ( ( num_of_nat @ ( nat_of_num @ X ) )
      = X ) ).

% nat_of_num_inverse
thf(fact_8970_nat__of__num__code_I1_J,axiom,
    ( ( nat_of_num @ one )
    = one_one_nat ) ).

% nat_of_num_code(1)
thf(fact_8971_nat__of__num__add,axiom,
    ! [X: num,Y3: num] :
      ( ( nat_of_num @ ( plus_plus_num @ X @ Y3 ) )
      = ( plus_plus_nat @ ( nat_of_num @ X ) @ ( nat_of_num @ Y3 ) ) ) ).

% nat_of_num_add
thf(fact_8972_nat__of__num__mult,axiom,
    ! [X: num,Y3: num] :
      ( ( nat_of_num @ ( times_times_num @ X @ Y3 ) )
      = ( times_times_nat @ ( nat_of_num @ X ) @ ( nat_of_num @ Y3 ) ) ) ).

% nat_of_num_mult
thf(fact_8973_nat__of__num__inc,axiom,
    ! [X: num] :
      ( ( nat_of_num @ ( inc @ X ) )
      = ( suc @ ( nat_of_num @ X ) ) ) ).

% nat_of_num_inc
thf(fact_8974_less__num__def,axiom,
    ( ord_less_num
    = ( ^ [M2: num,N3: num] : ( ord_less_nat @ ( nat_of_num @ M2 ) @ ( nat_of_num @ N3 ) ) ) ) ).

% less_num_def
thf(fact_8975_less__eq__num__def,axiom,
    ( ord_less_eq_num
    = ( ^ [M2: num,N3: num] : ( ord_less_eq_nat @ ( nat_of_num @ M2 ) @ ( nat_of_num @ N3 ) ) ) ) ).

% less_eq_num_def
thf(fact_8976_nat__of__num__code_I2_J,axiom,
    ! [N: num] :
      ( ( nat_of_num @ ( bit0 @ N ) )
      = ( plus_plus_nat @ ( nat_of_num @ N ) @ ( nat_of_num @ N ) ) ) ).

% nat_of_num_code(2)
thf(fact_8977_nat__of__num__sqr,axiom,
    ! [X: num] :
      ( ( nat_of_num @ ( sqr @ X ) )
      = ( times_times_nat @ ( nat_of_num @ X ) @ ( nat_of_num @ X ) ) ) ).

% nat_of_num_sqr
thf(fact_8978_nat__of__num_Osimps_I1_J,axiom,
    ( ( nat_of_num @ one )
    = ( suc @ zero_zero_nat ) ) ).

% nat_of_num.simps(1)
thf(fact_8979_nat__of__num_Osimps_I3_J,axiom,
    ! [X: num] :
      ( ( nat_of_num @ ( bit1 @ X ) )
      = ( suc @ ( plus_plus_nat @ ( nat_of_num @ X ) @ ( nat_of_num @ X ) ) ) ) ).

% nat_of_num.simps(3)
thf(fact_8980_num__of__nat__inverse,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( nat_of_num @ ( num_of_nat @ N ) )
        = N ) ) ).

% num_of_nat_inverse
thf(fact_8981_plus__num__def,axiom,
    ( plus_plus_num
    = ( ^ [M2: num,N3: num] : ( num_of_nat @ ( plus_plus_nat @ ( nat_of_num @ M2 ) @ ( nat_of_num @ N3 ) ) ) ) ) ).

% plus_num_def
thf(fact_8982_nat__of__num__code_I3_J,axiom,
    ! [N: num] :
      ( ( nat_of_num @ ( bit1 @ N ) )
      = ( suc @ ( plus_plus_nat @ ( nat_of_num @ N ) @ ( nat_of_num @ N ) ) ) ) ).

% nat_of_num_code(3)
thf(fact_8983_int__of__integer__of__natural,axiom,
    ! [N: code_natural] :
      ( ( code_int_of_integer @ ( code_i5400310926305786745atural @ N ) )
      = ( semiri1314217659103216013at_int @ ( code_nat_of_natural @ N ) ) ) ).

% int_of_integer_of_natural
thf(fact_8984_integer__of__natural_Orep__eq,axiom,
    ! [X: code_natural] :
      ( ( code_int_of_integer @ ( code_i5400310926305786745atural @ X ) )
      = ( semiri1314217659103216013at_int @ ( code_nat_of_natural @ X ) ) ) ).

% integer_of_natural.rep_eq
thf(fact_8985_zero__natural_Orep__eq,axiom,
    ( ( code_nat_of_natural @ zero_z2226904508553997617atural )
    = zero_zero_nat ) ).

% zero_natural.rep_eq
thf(fact_8986_plus__natural_Orep__eq,axiom,
    ! [X: code_natural,Xa3: code_natural] :
      ( ( code_nat_of_natural @ ( plus_p4538020629002901425atural @ X @ Xa3 ) )
      = ( plus_plus_nat @ ( code_nat_of_natural @ X ) @ ( code_nat_of_natural @ Xa3 ) ) ) ).

% plus_natural.rep_eq
thf(fact_8987_nat__of__natural__numeral,axiom,
    ! [K: num] :
      ( ( code_nat_of_natural @ ( numera5444537566228673987atural @ K ) )
      = ( numeral_numeral_nat @ K ) ) ).

% nat_of_natural_numeral
thf(fact_8988_minus__natural_Orep__eq,axiom,
    ! [X: code_natural,Xa3: code_natural] :
      ( ( code_nat_of_natural @ ( minus_7197305767214868737atural @ X @ Xa3 ) )
      = ( minus_minus_nat @ ( code_nat_of_natural @ X ) @ ( code_nat_of_natural @ Xa3 ) ) ) ).

% minus_natural.rep_eq
thf(fact_8989_times__natural_Orep__eq,axiom,
    ! [X: code_natural,Xa3: code_natural] :
      ( ( code_nat_of_natural @ ( times_2397367101498566445atural @ X @ Xa3 ) )
      = ( times_times_nat @ ( code_nat_of_natural @ X ) @ ( code_nat_of_natural @ Xa3 ) ) ) ).

% times_natural.rep_eq
thf(fact_8990_one__natural_Orep__eq,axiom,
    ( ( code_nat_of_natural @ one_one_Code_natural )
    = one_one_nat ) ).

% one_natural.rep_eq
thf(fact_8991_modulo__natural_Orep__eq,axiom,
    ! [X: code_natural,Xa3: code_natural] :
      ( ( code_nat_of_natural @ ( modulo8411746178871703098atural @ X @ Xa3 ) )
      = ( modulo_modulo_nat @ ( code_nat_of_natural @ X ) @ ( code_nat_of_natural @ Xa3 ) ) ) ).

% modulo_natural.rep_eq
thf(fact_8992_divide__natural_Orep__eq,axiom,
    ! [X: code_natural,Xa3: code_natural] :
      ( ( code_nat_of_natural @ ( divide5121882707175180666atural @ X @ Xa3 ) )
      = ( divide_divide_nat @ ( code_nat_of_natural @ X ) @ ( code_nat_of_natural @ Xa3 ) ) ) ).

% divide_natural.rep_eq
thf(fact_8993_nat__of__natural__min,axiom,
    ! [K: code_natural,L: code_natural] :
      ( ( code_nat_of_natural @ ( ord_min_Code_natural @ K @ L ) )
      = ( ord_min_nat @ ( code_nat_of_natural @ K ) @ ( code_nat_of_natural @ L ) ) ) ).

% nat_of_natural_min
thf(fact_8994_nat__of__natural__max,axiom,
    ! [K: code_natural,L: code_natural] :
      ( ( code_nat_of_natural @ ( ord_max_Code_natural @ K @ L ) )
      = ( ord_max_nat @ ( code_nat_of_natural @ K ) @ ( code_nat_of_natural @ L ) ) ) ).

% nat_of_natural_max
thf(fact_8995_division__segment__natural_Orep__eq,axiom,
    ! [X: code_natural] :
      ( ( code_nat_of_natural @ ( euclid5112970336976722131atural @ X ) )
      = ( euclid3398187327856392827nt_nat @ ( code_nat_of_natural @ X ) ) ) ).

% division_segment_natural.rep_eq
thf(fact_8996_euclidean__size__natural_Orep__eq,axiom,
    ( euclid5200926497590043453atural
    = ( ^ [X2: code_natural] : ( euclid4777050414544973029ze_nat @ ( code_nat_of_natural @ X2 ) ) ) ) ).

% euclidean_size_natural.rep_eq
thf(fact_8997_of__nat__of__natural,axiom,
    ! [N: code_natural] :
      ( ( semiri3763490453095760265atural @ ( code_nat_of_natural @ N ) )
      = N ) ).

% of_nat_of_natural
thf(fact_8998_nat__of__natural__of__nat,axiom,
    ! [N: nat] :
      ( ( code_nat_of_natural @ ( semiri3763490453095760265atural @ N ) )
      = N ) ).

% nat_of_natural_of_nat
thf(fact_8999_nat__of__natural__induct,axiom,
    ! [Y3: nat,P: nat > $o] :
      ( ( member_nat @ Y3 @ top_top_set_nat )
     => ( ! [X3: code_natural] : ( P @ ( code_nat_of_natural @ X3 ) )
       => ( P @ Y3 ) ) ) ).

% nat_of_natural_induct
thf(fact_9000_nat__of__natural__cases,axiom,
    ! [Y3: nat] :
      ( ( member_nat @ Y3 @ top_top_set_nat )
     => ~ ! [X3: code_natural] :
            ( Y3
           != ( code_nat_of_natural @ X3 ) ) ) ).

% nat_of_natural_cases
thf(fact_9001_nat__of__natural,axiom,
    ! [X: code_natural] : ( member_nat @ ( code_nat_of_natural @ X ) @ top_top_set_nat ) ).

% nat_of_natural
thf(fact_9002_set__bit__natural_Orep__eq,axiom,
    ! [X: nat,Xa3: code_natural] :
      ( ( code_nat_of_natural @ ( bit_se1617098188084679374atural @ X @ Xa3 ) )
      = ( bit_se7882103937844011126it_nat @ X @ ( code_nat_of_natural @ Xa3 ) ) ) ).

% set_bit_natural.rep_eq
thf(fact_9003_unset__bit__natural_Orep__eq,axiom,
    ! [X: nat,Xa3: code_natural] :
      ( ( code_nat_of_natural @ ( bit_se7083795435491715335atural @ X @ Xa3 ) )
      = ( bit_se4205575877204974255it_nat @ X @ ( code_nat_of_natural @ Xa3 ) ) ) ).

% unset_bit_natural.rep_eq
thf(fact_9004_flip__bit__natural_Orep__eq,axiom,
    ! [X: nat,Xa3: code_natural] :
      ( ( code_nat_of_natural @ ( bit_se168947363167071951atural @ X @ Xa3 ) )
      = ( bit_se2161824704523386999it_nat @ X @ ( code_nat_of_natural @ Xa3 ) ) ) ).

% flip_bit_natural.rep_eq
thf(fact_9005_drop__bit__natural_Orep__eq,axiom,
    ! [X: nat,Xa3: code_natural] :
      ( ( code_nat_of_natural @ ( bit_se2751692689150723149atural @ X @ Xa3 ) )
      = ( bit_se8570568707652914677it_nat @ X @ ( code_nat_of_natural @ Xa3 ) ) ) ).

% drop_bit_natural.rep_eq
thf(fact_9006_bit__natural_Orep__eq,axiom,
    ( bit_se8040316288895769887atural
    = ( ^ [X2: code_natural] : ( bit_se1148574629649215175it_nat @ ( code_nat_of_natural @ X2 ) ) ) ) ).

% bit_natural.rep_eq
thf(fact_9007_take__bit__natural_Orep__eq,axiom,
    ! [X: nat,Xa3: code_natural] :
      ( ( code_nat_of_natural @ ( bit_se569199155075624693atural @ X @ Xa3 ) )
      = ( bit_se2925701944663578781it_nat @ X @ ( code_nat_of_natural @ Xa3 ) ) ) ).

% take_bit_natural.rep_eq
thf(fact_9008_or__natural_Orep__eq,axiom,
    ! [X: code_natural,Xa3: code_natural] :
      ( ( code_nat_of_natural @ ( bit_se9127793120404214118atural @ X @ Xa3 ) )
      = ( bit_se1412395901928357646or_nat @ ( code_nat_of_natural @ X ) @ ( code_nat_of_natural @ Xa3 ) ) ) ).

% or_natural.rep_eq
thf(fact_9009_xor__natural_Orep__eq,axiom,
    ! [X: code_natural,Xa3: code_natural] :
      ( ( code_nat_of_natural @ ( bit_se2046307713759805098atural @ X @ Xa3 ) )
      = ( bit_se6528837805403552850or_nat @ ( code_nat_of_natural @ X ) @ ( code_nat_of_natural @ Xa3 ) ) ) ).

% xor_natural.rep_eq
thf(fact_9010_and__natural_Orep__eq,axiom,
    ! [X: code_natural,Xa3: code_natural] :
      ( ( code_nat_of_natural @ ( bit_se2773287842338716102atural @ X @ Xa3 ) )
      = ( bit_se727722235901077358nd_nat @ ( code_nat_of_natural @ X ) @ ( code_nat_of_natural @ Xa3 ) ) ) ).

% and_natural.rep_eq
thf(fact_9011_mask__natural_Orep__eq,axiom,
    ! [X: nat] :
      ( ( code_nat_of_natural @ ( bit_se943457434206027407atural @ X ) )
      = ( bit_se2002935070580805687sk_nat @ X ) ) ).

% mask_natural.rep_eq
thf(fact_9012_push__bit__natural_Orep__eq,axiom,
    ! [X: nat,Xa3: code_natural] :
      ( ( code_nat_of_natural @ ( bit_se6611745700429515170atural @ X @ Xa3 ) )
      = ( bit_se547839408752420682it_nat @ X @ ( code_nat_of_natural @ Xa3 ) ) ) ).

% push_bit_natural.rep_eq
thf(fact_9013_natural__eq__iff,axiom,
    ( ( ^ [Y: code_natural,Z: code_natural] : ( Y = Z ) )
    = ( ^ [M2: code_natural,N3: code_natural] :
          ( ( code_nat_of_natural @ M2 )
          = ( code_nat_of_natural @ N3 ) ) ) ) ).

% natural_eq_iff
thf(fact_9014_natural__eqI,axiom,
    ! [M: code_natural,N: code_natural] :
      ( ( ( code_nat_of_natural @ M )
        = ( code_nat_of_natural @ N ) )
     => ( M = N ) ) ).

% natural_eqI
thf(fact_9015_nat__of__natural__inject,axiom,
    ! [X: code_natural,Y3: code_natural] :
      ( ( ( code_nat_of_natural @ X )
        = ( code_nat_of_natural @ Y3 ) )
      = ( X = Y3 ) ) ).

% nat_of_natural_inject
thf(fact_9016_less__natural_Orep__eq,axiom,
    ( ord_le5570908160329646204atural
    = ( ^ [X2: code_natural,Xa4: code_natural] : ( ord_less_nat @ ( code_nat_of_natural @ X2 ) @ ( code_nat_of_natural @ Xa4 ) ) ) ) ).

% less_natural.rep_eq
thf(fact_9017_less__eq__natural_Orep__eq,axiom,
    ( ord_le1926595141338095240atural
    = ( ^ [X2: code_natural,Xa4: code_natural] : ( ord_less_eq_nat @ ( code_nat_of_natural @ X2 ) @ ( code_nat_of_natural @ Xa4 ) ) ) ) ).

% less_eq_natural.rep_eq
thf(fact_9018_natural__of__integer_Orep__eq,axiom,
    ! [X: code_integer] :
      ( ( code_nat_of_natural @ ( code_n4118661773612635043nteger @ X ) )
      = ( nat2 @ ( code_int_of_integer @ X ) ) ) ).

% natural_of_integer.rep_eq
thf(fact_9019_natural__decr,axiom,
    ! [N: code_natural] :
      ( ( N != zero_z2226904508553997617atural )
     => ( ord_less_nat @ ( minus_minus_nat @ ( code_nat_of_natural @ N ) @ ( suc @ zero_zero_nat ) ) @ ( code_nat_of_natural @ N ) ) ) ).

% natural_decr
thf(fact_9020_Nat_Orep__eq,axiom,
    ! [X: code_integer] :
      ( ( code_nat_of_natural @ ( code_Nat @ X ) )
      = ( nat2 @ ( code_int_of_integer @ X ) ) ) ).

% Nat.rep_eq
thf(fact_9021_Code__Numeral_Osize__nat__def,axiom,
    code_size_nat = code_nat_of_natural ).

% Code_Numeral.size_nat_def
thf(fact_9022_natural__zero__minus__one,axiom,
    ( ( minus_7197305767214868737atural @ zero_z2226904508553997617atural @ one_one_Code_natural )
    = zero_z2226904508553997617atural ) ).

% natural_zero_minus_one
thf(fact_9023_log_Osimps,axiom,
    ( log
    = ( ^ [B2: code_natural,I4: code_natural] :
          ( if_Code_natural
          @ ( ( ord_le1926595141338095240atural @ B2 @ one_one_Code_natural )
            | ( ord_le5570908160329646204atural @ I4 @ B2 ) )
          @ one_one_Code_natural
          @ ( plus_p4538020629002901425atural @ one_one_Code_natural @ ( log @ B2 @ ( divide5121882707175180666atural @ I4 @ B2 ) ) ) ) ) ) ).

% log.simps
thf(fact_9024_log_Oelims,axiom,
    ! [X: code_natural,Xa3: code_natural,Y3: code_natural] :
      ( ( ( log @ X @ Xa3 )
        = Y3 )
     => ( ( ( ( ord_le1926595141338095240atural @ X @ one_one_Code_natural )
            | ( ord_le5570908160329646204atural @ Xa3 @ X ) )
         => ( Y3 = one_one_Code_natural ) )
        & ( ~ ( ( ord_le1926595141338095240atural @ X @ one_one_Code_natural )
              | ( ord_le5570908160329646204atural @ Xa3 @ X ) )
         => ( Y3
            = ( plus_p4538020629002901425atural @ one_one_Code_natural @ ( log @ X @ ( divide5121882707175180666atural @ Xa3 @ X ) ) ) ) ) ) ) ).

% log.elims
thf(fact_9025_minus__shift__def,axiom,
    ( minus_shift
    = ( ^ [R5: code_natural,K3: code_natural,L3: code_natural] : ( if_Code_natural @ ( ord_le5570908160329646204atural @ K3 @ L3 ) @ ( minus_7197305767214868737atural @ ( plus_p4538020629002901425atural @ R5 @ K3 ) @ L3 ) @ ( minus_7197305767214868737atural @ K3 @ L3 ) ) ) ) ).

% minus_shift_def
thf(fact_9026_range,axiom,
    ! [K: code_natural,S: produc7822875418678951345atural] :
      ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ K )
     => ( ord_le5570908160329646204atural @ ( produc497848011034438852atural @ ( range @ K @ S ) ) @ K ) ) ).

% range
thf(fact_9027_log_Opelims,axiom,
    ! [X: code_natural,Xa3: code_natural,Y3: code_natural] :
      ( ( ( log @ X @ Xa3 )
        = Y3 )
     => ( ( accp_P8126237942716283194atural @ log_rel @ ( produc3574140220909816553atural @ X @ Xa3 ) )
       => ~ ( ( ( ( ( ord_le1926595141338095240atural @ X @ one_one_Code_natural )
                  | ( ord_le5570908160329646204atural @ Xa3 @ X ) )
               => ( Y3 = one_one_Code_natural ) )
              & ( ~ ( ( ord_le1926595141338095240atural @ X @ one_one_Code_natural )
                    | ( ord_le5570908160329646204atural @ Xa3 @ X ) )
               => ( Y3
                  = ( plus_p4538020629002901425atural @ one_one_Code_natural @ ( log @ X @ ( divide5121882707175180666atural @ Xa3 @ X ) ) ) ) ) )
           => ~ ( accp_P8126237942716283194atural @ log_rel @ ( produc3574140220909816553atural @ X @ Xa3 ) ) ) ) ) ).

% log.pelims
thf(fact_9028_bit__natural__def,axiom,
    ( bit_se8040316288895769887atural
    = ( map_fu6244533035418998608_nat_o @ code_nat_of_natural @ id_nat_o @ bit_se1148574629649215175it_nat ) ) ).

% bit_natural_def
thf(fact_9029_euclidean__size__natural__def,axiom,
    ( euclid5200926497590043453atural
    = ( map_fu226235371393106610at_nat @ code_nat_of_natural @ id_nat @ euclid4777050414544973029ze_nat ) ) ).

% euclidean_size_natural_def
thf(fact_9030_natural__of__nat__of__natural__inverse,axiom,
    ! [N: code_natural] :
      ( ( code_natural_of_nat @ ( code_nat_of_natural @ N ) )
      = N ) ).

% natural_of_nat_of_natural_inverse
thf(fact_9031_nat__of__natural__of__nat__inverse,axiom,
    ! [N: nat] :
      ( ( code_nat_of_natural @ ( code_natural_of_nat @ N ) )
      = N ) ).

% nat_of_natural_of_nat_inverse
thf(fact_9032_nat__of__natural__inverse,axiom,
    ! [X: code_natural] :
      ( ( code_natural_of_nat @ ( code_nat_of_natural @ X ) )
      = X ) ).

% nat_of_natural_inverse
thf(fact_9033_natural__of__nat__of__nat,axiom,
    code_natural_of_nat = semiri3763490453095760265atural ).

% natural_of_nat_of_nat
thf(fact_9034_natural__of__nat__inverse,axiom,
    ! [Y3: nat] :
      ( ( member_nat @ Y3 @ top_top_set_nat )
     => ( ( code_nat_of_natural @ ( code_natural_of_nat @ Y3 ) )
        = Y3 ) ) ).

% natural_of_nat_inverse
thf(fact_9035_natural__of__nat__cases,axiom,
    ! [X: code_natural] :
      ~ ! [Y4: nat] :
          ( ( X
            = ( code_natural_of_nat @ Y4 ) )
         => ~ ( member_nat @ Y4 @ top_top_set_nat ) ) ).

% natural_of_nat_cases
thf(fact_9036_natural__of__nat__induct,axiom,
    ! [P: code_natural > $o,X: code_natural] :
      ( ! [Y4: nat] :
          ( ( member_nat @ Y4 @ top_top_set_nat )
         => ( P @ ( code_natural_of_nat @ Y4 ) ) )
     => ( P @ X ) ) ).

% natural_of_nat_induct
thf(fact_9037_natural__of__nat__inject,axiom,
    ! [X: nat,Y3: nat] :
      ( ( member_nat @ X @ top_top_set_nat )
     => ( ( member_nat @ Y3 @ top_top_set_nat )
       => ( ( ( code_natural_of_nat @ X )
            = ( code_natural_of_nat @ Y3 ) )
          = ( X = Y3 ) ) ) ) ).

% natural_of_nat_inject
thf(fact_9038_minus__natural_Oabs__eq,axiom,
    ! [Xa3: nat,X: nat] :
      ( ( minus_7197305767214868737atural @ ( code_natural_of_nat @ Xa3 ) @ ( code_natural_of_nat @ X ) )
      = ( code_natural_of_nat @ ( minus_minus_nat @ Xa3 @ X ) ) ) ).

% minus_natural.abs_eq
thf(fact_9039_one__natural__def,axiom,
    ( one_one_Code_natural
    = ( code_natural_of_nat @ one_one_nat ) ) ).

% one_natural_def
thf(fact_9040_divide__natural_Oabs__eq,axiom,
    ! [Xa3: nat,X: nat] :
      ( ( divide5121882707175180666atural @ ( code_natural_of_nat @ Xa3 ) @ ( code_natural_of_nat @ X ) )
      = ( code_natural_of_nat @ ( divide_divide_nat @ Xa3 @ X ) ) ) ).

% divide_natural.abs_eq
thf(fact_9041_plus__natural_Oabs__eq,axiom,
    ! [Xa3: nat,X: nat] :
      ( ( plus_p4538020629002901425atural @ ( code_natural_of_nat @ Xa3 ) @ ( code_natural_of_nat @ X ) )
      = ( code_natural_of_nat @ ( plus_plus_nat @ Xa3 @ X ) ) ) ).

% plus_natural.abs_eq
thf(fact_9042_zero__natural__def,axiom,
    ( zero_z2226904508553997617atural
    = ( code_natural_of_nat @ zero_zero_nat ) ) ).

% zero_natural_def
thf(fact_9043_less__natural_Oabs__eq,axiom,
    ! [Xa3: nat,X: nat] :
      ( ( ord_le5570908160329646204atural @ ( code_natural_of_nat @ Xa3 ) @ ( code_natural_of_nat @ X ) )
      = ( ord_less_nat @ Xa3 @ X ) ) ).

% less_natural.abs_eq
thf(fact_9044_less__eq__natural_Oabs__eq,axiom,
    ! [Xa3: nat,X: nat] :
      ( ( ord_le1926595141338095240atural @ ( code_natural_of_nat @ Xa3 ) @ ( code_natural_of_nat @ X ) )
      = ( ord_less_eq_nat @ Xa3 @ X ) ) ).

% less_eq_natural.abs_eq
thf(fact_9045_times__natural_Oabs__eq,axiom,
    ! [Xa3: nat,X: nat] :
      ( ( times_2397367101498566445atural @ ( code_natural_of_nat @ Xa3 ) @ ( code_natural_of_nat @ X ) )
      = ( code_natural_of_nat @ ( times_times_nat @ Xa3 @ X ) ) ) ).

% times_natural.abs_eq
thf(fact_9046_modulo__natural_Oabs__eq,axiom,
    ! [Xa3: nat,X: nat] :
      ( ( modulo8411746178871703098atural @ ( code_natural_of_nat @ Xa3 ) @ ( code_natural_of_nat @ X ) )
      = ( code_natural_of_nat @ ( modulo_modulo_nat @ Xa3 @ X ) ) ) ).

% modulo_natural.abs_eq
thf(fact_9047_take__bit__natural_Oabs__eq,axiom,
    ! [Xa3: nat,X: nat] :
      ( ( bit_se569199155075624693atural @ Xa3 @ ( code_natural_of_nat @ X ) )
      = ( code_natural_of_nat @ ( bit_se2925701944663578781it_nat @ Xa3 @ X ) ) ) ).

% take_bit_natural.abs_eq
thf(fact_9048_xor__natural_Oabs__eq,axiom,
    ! [Xa3: nat,X: nat] :
      ( ( bit_se2046307713759805098atural @ ( code_natural_of_nat @ Xa3 ) @ ( code_natural_of_nat @ X ) )
      = ( code_natural_of_nat @ ( bit_se6528837805403552850or_nat @ Xa3 @ X ) ) ) ).

% xor_natural.abs_eq
thf(fact_9049_or__natural_Oabs__eq,axiom,
    ! [Xa3: nat,X: nat] :
      ( ( bit_se9127793120404214118atural @ ( code_natural_of_nat @ Xa3 ) @ ( code_natural_of_nat @ X ) )
      = ( code_natural_of_nat @ ( bit_se1412395901928357646or_nat @ Xa3 @ X ) ) ) ).

% or_natural.abs_eq
thf(fact_9050_and__natural_Oabs__eq,axiom,
    ! [Xa3: nat,X: nat] :
      ( ( bit_se2773287842338716102atural @ ( code_natural_of_nat @ Xa3 ) @ ( code_natural_of_nat @ X ) )
      = ( code_natural_of_nat @ ( bit_se727722235901077358nd_nat @ Xa3 @ X ) ) ) ).

% and_natural.abs_eq
thf(fact_9051_bit__natural_Oabs__eq,axiom,
    ! [X: nat] :
      ( ( bit_se8040316288895769887atural @ ( code_natural_of_nat @ X ) )
      = ( bit_se1148574629649215175it_nat @ X ) ) ).

% bit_natural.abs_eq
thf(fact_9052_push__bit__natural_Oabs__eq,axiom,
    ! [Xa3: nat,X: nat] :
      ( ( bit_se6611745700429515170atural @ Xa3 @ ( code_natural_of_nat @ X ) )
      = ( code_natural_of_nat @ ( bit_se547839408752420682it_nat @ Xa3 @ X ) ) ) ).

% push_bit_natural.abs_eq
thf(fact_9053_mask__natural_Oabs__eq,axiom,
    ( bit_se943457434206027407atural
    = ( ^ [X2: nat] : ( code_natural_of_nat @ ( bit_se2002935070580805687sk_nat @ X2 ) ) ) ) ).

% mask_natural.abs_eq
thf(fact_9054_drop__bit__natural_Oabs__eq,axiom,
    ! [Xa3: nat,X: nat] :
      ( ( bit_se2751692689150723149atural @ Xa3 @ ( code_natural_of_nat @ X ) )
      = ( code_natural_of_nat @ ( bit_se8570568707652914677it_nat @ Xa3 @ X ) ) ) ).

% drop_bit_natural.abs_eq
thf(fact_9055_euclidean__size__natural_Oabs__eq,axiom,
    ! [X: nat] :
      ( ( euclid5200926497590043453atural @ ( code_natural_of_nat @ X ) )
      = ( euclid4777050414544973029ze_nat @ X ) ) ).

% euclidean_size_natural.abs_eq
thf(fact_9056_flip__bit__natural_Oabs__eq,axiom,
    ! [Xa3: nat,X: nat] :
      ( ( bit_se168947363167071951atural @ Xa3 @ ( code_natural_of_nat @ X ) )
      = ( code_natural_of_nat @ ( bit_se2161824704523386999it_nat @ Xa3 @ X ) ) ) ).

% flip_bit_natural.abs_eq
thf(fact_9057_unset__bit__natural_Oabs__eq,axiom,
    ! [Xa3: nat,X: nat] :
      ( ( bit_se7083795435491715335atural @ Xa3 @ ( code_natural_of_nat @ X ) )
      = ( code_natural_of_nat @ ( bit_se4205575877204974255it_nat @ Xa3 @ X ) ) ) ).

% unset_bit_natural.abs_eq
thf(fact_9058_division__segment__natural_Oabs__eq,axiom,
    ! [X: nat] :
      ( ( euclid5112970336976722131atural @ ( code_natural_of_nat @ X ) )
      = ( code_natural_of_nat @ ( euclid3398187327856392827nt_nat @ X ) ) ) ).

% division_segment_natural.abs_eq
thf(fact_9059_set__bit__natural_Oabs__eq,axiom,
    ! [Xa3: nat,X: nat] :
      ( ( bit_se1617098188084679374atural @ Xa3 @ ( code_natural_of_nat @ X ) )
      = ( code_natural_of_nat @ ( bit_se7882103937844011126it_nat @ Xa3 @ X ) ) ) ).

% set_bit_natural.abs_eq
thf(fact_9060_integer__of__natural_Oabs__eq,axiom,
    ! [X: nat] :
      ( ( code_i5400310926305786745atural @ ( code_natural_of_nat @ X ) )
      = ( code_integer_of_int @ ( semiri1314217659103216013at_int @ X ) ) ) ).

% integer_of_natural.abs_eq
thf(fact_9061_natural__of__integer_Oabs__eq,axiom,
    ! [X: int] :
      ( ( code_n4118661773612635043nteger @ ( code_integer_of_int @ X ) )
      = ( code_natural_of_nat @ ( nat2 @ X ) ) ) ).

% natural_of_integer.abs_eq
thf(fact_9062_Nat_Oabs__eq,axiom,
    ! [X: int] :
      ( ( code_Nat @ ( code_integer_of_int @ X ) )
      = ( code_natural_of_nat @ ( nat2 @ X ) ) ) ).

% Nat.abs_eq
thf(fact_9063_mask__natural__def,axiom,
    ( bit_se943457434206027407atural
    = ( map_fu4742413587575543474atural @ id_nat @ code_natural_of_nat @ bit_se2002935070580805687sk_nat ) ) ).

% mask_natural_def
thf(fact_9064_Nat__def,axiom,
    ( code_Nat
    = ( map_fu1051355602067684763atural @ code_int_of_integer @ code_natural_of_nat @ nat2 ) ) ).

% Nat_def
thf(fact_9065_natural__of__integer__def,axiom,
    ( code_n4118661773612635043nteger
    = ( map_fu1051355602067684763atural @ code_int_of_integer @ code_natural_of_nat @ nat2 ) ) ).

% natural_of_integer_def
thf(fact_9066_division__segment__natural__def,axiom,
    ( euclid5112970336976722131atural
    = ( map_fu1239815594074539274atural @ code_nat_of_natural @ code_natural_of_nat @ euclid3398187327856392827nt_nat ) ) ).

% division_segment_natural_def
thf(fact_9067_integer__of__natural__def,axiom,
    ( code_i5400310926305786745atural
    = ( map_fu2787874002554666395nteger @ code_nat_of_natural @ code_integer_of_int @ semiri1314217659103216013at_int ) ) ).

% integer_of_natural_def
thf(fact_9068_set__bit__natural__def,axiom,
    ( bit_se1617098188084679374atural
    = ( map_fu6981160461238011112atural @ id_nat @ ( map_fu1239815594074539274atural @ code_nat_of_natural @ code_natural_of_nat ) @ bit_se7882103937844011126it_nat ) ) ).

% set_bit_natural_def
thf(fact_9069_unset__bit__natural__def,axiom,
    ( bit_se7083795435491715335atural
    = ( map_fu6981160461238011112atural @ id_nat @ ( map_fu1239815594074539274atural @ code_nat_of_natural @ code_natural_of_nat ) @ bit_se4205575877204974255it_nat ) ) ).

% unset_bit_natural_def
thf(fact_9070_take__bit__natural__def,axiom,
    ( bit_se569199155075624693atural
    = ( map_fu6981160461238011112atural @ id_nat @ ( map_fu1239815594074539274atural @ code_nat_of_natural @ code_natural_of_nat ) @ bit_se2925701944663578781it_nat ) ) ).

% take_bit_natural_def
thf(fact_9071_push__bit__natural__def,axiom,
    ( bit_se6611745700429515170atural
    = ( map_fu6981160461238011112atural @ id_nat @ ( map_fu1239815594074539274atural @ code_nat_of_natural @ code_natural_of_nat ) @ bit_se547839408752420682it_nat ) ) ).

% push_bit_natural_def
thf(fact_9072_drop__bit__natural__def,axiom,
    ( bit_se2751692689150723149atural
    = ( map_fu6981160461238011112atural @ id_nat @ ( map_fu1239815594074539274atural @ code_nat_of_natural @ code_natural_of_nat ) @ bit_se8570568707652914677it_nat ) ) ).

% drop_bit_natural_def
thf(fact_9073_flip__bit__natural__def,axiom,
    ( bit_se168947363167071951atural
    = ( map_fu6981160461238011112atural @ id_nat @ ( map_fu1239815594074539274atural @ code_nat_of_natural @ code_natural_of_nat ) @ bit_se2161824704523386999it_nat ) ) ).

% flip_bit_natural_def
thf(fact_9074_and__natural__def,axiom,
    ( bit_se2773287842338716102atural
    = ( map_fu6549440983881763648atural @ code_nat_of_natural @ ( map_fu1239815594074539274atural @ code_nat_of_natural @ code_natural_of_nat ) @ bit_se727722235901077358nd_nat ) ) ).

% and_natural_def
thf(fact_9075_xor__natural__def,axiom,
    ( bit_se2046307713759805098atural
    = ( map_fu6549440983881763648atural @ code_nat_of_natural @ ( map_fu1239815594074539274atural @ code_nat_of_natural @ code_natural_of_nat ) @ bit_se6528837805403552850or_nat ) ) ).

% xor_natural_def
thf(fact_9076_plus__natural__def,axiom,
    ( plus_p4538020629002901425atural
    = ( map_fu6549440983881763648atural @ code_nat_of_natural @ ( map_fu1239815594074539274atural @ code_nat_of_natural @ code_natural_of_nat ) @ plus_plus_nat ) ) ).

% plus_natural_def
thf(fact_9077_minus__natural__def,axiom,
    ( minus_7197305767214868737atural
    = ( map_fu6549440983881763648atural @ code_nat_of_natural @ ( map_fu1239815594074539274atural @ code_nat_of_natural @ code_natural_of_nat ) @ minus_minus_nat ) ) ).

% minus_natural_def
thf(fact_9078_divide__natural__def,axiom,
    ( divide5121882707175180666atural
    = ( map_fu6549440983881763648atural @ code_nat_of_natural @ ( map_fu1239815594074539274atural @ code_nat_of_natural @ code_natural_of_nat ) @ divide_divide_nat ) ) ).

% divide_natural_def
thf(fact_9079_times__natural__def,axiom,
    ( times_2397367101498566445atural
    = ( map_fu6549440983881763648atural @ code_nat_of_natural @ ( map_fu1239815594074539274atural @ code_nat_of_natural @ code_natural_of_nat ) @ times_times_nat ) ) ).

% times_natural_def
thf(fact_9080_modulo__natural__def,axiom,
    ( modulo8411746178871703098atural
    = ( map_fu6549440983881763648atural @ code_nat_of_natural @ ( map_fu1239815594074539274atural @ code_nat_of_natural @ code_natural_of_nat ) @ modulo_modulo_nat ) ) ).

% modulo_natural_def
thf(fact_9081_or__natural__def,axiom,
    ( bit_se9127793120404214118atural
    = ( map_fu6549440983881763648atural @ code_nat_of_natural @ ( map_fu1239815594074539274atural @ code_nat_of_natural @ code_natural_of_nat ) @ bit_se1412395901928357646or_nat ) ) ).

% or_natural_def
thf(fact_9082_Code__Numeral_OSuc__def,axiom,
    ( code_Suc
    = ( map_fu1239815594074539274atural @ code_nat_of_natural @ code_natural_of_nat @ suc ) ) ).

% Code_Numeral.Suc_def
thf(fact_9083_cr__natural__def,axiom,
    ( code_cr_natural
    = ( ^ [X2: nat,Y6: code_natural] :
          ( X2
          = ( code_nat_of_natural @ Y6 ) ) ) ) ).

% cr_natural_def
thf(fact_9084_natural_Oinject,axiom,
    ! [Natural: code_natural,Natural2: code_natural] :
      ( ( ( code_Suc @ Natural )
        = ( code_Suc @ Natural2 ) )
      = ( Natural = Natural2 ) ) ).

% natural.inject
thf(fact_9085_Suc_Orep__eq,axiom,
    ! [X: code_natural] :
      ( ( code_nat_of_natural @ ( code_Suc @ X ) )
      = ( suc @ ( code_nat_of_natural @ X ) ) ) ).

% Suc.rep_eq
thf(fact_9086_natural_Oexhaust,axiom,
    ! [Y3: code_natural] :
      ( ( Y3 != zero_z2226904508553997617atural )
     => ~ ! [Natural3: code_natural] :
            ( Y3
           != ( code_Suc @ Natural3 ) ) ) ).

% natural.exhaust
thf(fact_9087_natural_Odistinct_I1_J,axiom,
    ! [Natural2: code_natural] :
      ( zero_z2226904508553997617atural
     != ( code_Suc @ Natural2 ) ) ).

% natural.distinct(1)
thf(fact_9088_natural_Odistinct_I2_J,axiom,
    ! [Natural2: code_natural] :
      ( ( code_Suc @ Natural2 )
     != zero_z2226904508553997617atural ) ).

% natural.distinct(2)
thf(fact_9089_Suc_Oabs__eq,axiom,
    ! [X: nat] :
      ( ( code_Suc @ ( code_natural_of_nat @ X ) )
      = ( code_natural_of_nat @ ( suc @ X ) ) ) ).

% Suc.abs_eq
thf(fact_9090_Suc__natural__minus__one,axiom,
    ! [N: code_natural] :
      ( ( minus_7197305767214868737atural @ ( code_Suc @ N ) @ one_one_Code_natural )
      = N ) ).

% Suc_natural_minus_one
thf(fact_9091_Nat_Otransfer,axiom,
    bNF_re1831474436612530402atural @ code_pcr_integer @ code_pcr_natural @ nat2 @ code_Nat ).

% Nat.transfer
thf(fact_9092_Suc_Otransfer,axiom,
    bNF_re3704215830270325841atural @ code_pcr_natural @ code_pcr_natural @ suc @ code_Suc ).

% Suc.transfer
thf(fact_9093_natural_Opcr__cr__eq,axiom,
    code_pcr_natural = code_cr_natural ).

% natural.pcr_cr_eq
thf(fact_9094_natural_Oid__abs__transfer,axiom,
    ( bNF_re2605341659958429177atural
    @ ^ [Y: nat,Z: nat] : ( Y = Z )
    @ code_pcr_natural
    @ ^ [X2: nat] : X2
    @ code_natural_of_nat ) ).

% natural.id_abs_transfer
thf(fact_9095_less__natural_Otransfer,axiom,
    ( bNF_re1639080489988575423ural_o @ code_pcr_natural
    @ ( bNF_re2785088596696291543al_o_o @ code_pcr_natural
      @ ^ [Y: $o,Z: $o] : ( Y = Z ) )
    @ ord_less_nat
    @ ord_le5570908160329646204atural ) ).

% less_natural.transfer
thf(fact_9096_plus__natural_Otransfer,axiom,
    bNF_re88643428490162567atural @ code_pcr_natural @ ( bNF_re3704215830270325841atural @ code_pcr_natural @ code_pcr_natural ) @ plus_plus_nat @ plus_p4538020629002901425atural ).

% plus_natural.transfer
thf(fact_9097_minus__natural_Otransfer,axiom,
    bNF_re88643428490162567atural @ code_pcr_natural @ ( bNF_re3704215830270325841atural @ code_pcr_natural @ code_pcr_natural ) @ minus_minus_nat @ minus_7197305767214868737atural ).

% minus_natural.transfer
thf(fact_9098_zero__natural_Otransfer,axiom,
    code_pcr_natural @ zero_zero_nat @ zero_z2226904508553997617atural ).

% zero_natural.transfer
thf(fact_9099_one__natural_Otransfer,axiom,
    code_pcr_natural @ one_one_nat @ one_one_Code_natural ).

% one_natural.transfer
thf(fact_9100_natural_Obi__total,axiom,
    bi_tot9006467052213973024atural @ code_pcr_natural ).

% natural.bi_total
thf(fact_9101_natural_Orep__transfer,axiom,
    ( bNF_re7333046803109187065at_nat @ code_pcr_natural
    @ ^ [Y: nat,Z: nat] : ( Y = Z )
    @ ^ [X2: nat] : X2
    @ code_nat_of_natural ) ).

% natural.rep_transfer
thf(fact_9102_less__eq__natural_Otransfer,axiom,
    ( bNF_re1639080489988575423ural_o @ code_pcr_natural
    @ ( bNF_re2785088596696291543al_o_o @ code_pcr_natural
      @ ^ [Y: $o,Z: $o] : ( Y = Z ) )
    @ ord_less_eq_nat
    @ ord_le1926595141338095240atural ) ).

% less_eq_natural.transfer
thf(fact_9103_divide__natural_Otransfer,axiom,
    bNF_re88643428490162567atural @ code_pcr_natural @ ( bNF_re3704215830270325841atural @ code_pcr_natural @ code_pcr_natural ) @ divide_divide_nat @ divide5121882707175180666atural ).

% divide_natural.transfer
thf(fact_9104_times__natural_Otransfer,axiom,
    bNF_re88643428490162567atural @ code_pcr_natural @ ( bNF_re3704215830270325841atural @ code_pcr_natural @ code_pcr_natural ) @ times_times_nat @ times_2397367101498566445atural ).

% times_natural.transfer
thf(fact_9105_modulo__natural_Otransfer,axiom,
    bNF_re88643428490162567atural @ code_pcr_natural @ ( bNF_re3704215830270325841atural @ code_pcr_natural @ code_pcr_natural ) @ modulo_modulo_nat @ modulo8411746178871703098atural ).

% modulo_natural.transfer
thf(fact_9106_take__bit__natural_Otransfer,axiom,
    ( bNF_re3692170009638431535atural
    @ ^ [Y: nat,Z: nat] : ( Y = Z )
    @ ( bNF_re3704215830270325841atural @ code_pcr_natural @ code_pcr_natural )
    @ bit_se2925701944663578781it_nat
    @ bit_se569199155075624693atural ) ).

% take_bit_natural.transfer
thf(fact_9107_or__natural_Otransfer,axiom,
    bNF_re88643428490162567atural @ code_pcr_natural @ ( bNF_re3704215830270325841atural @ code_pcr_natural @ code_pcr_natural ) @ bit_se1412395901928357646or_nat @ bit_se9127793120404214118atural ).

% or_natural.transfer
thf(fact_9108_xor__natural_Otransfer,axiom,
    bNF_re88643428490162567atural @ code_pcr_natural @ ( bNF_re3704215830270325841atural @ code_pcr_natural @ code_pcr_natural ) @ bit_se6528837805403552850or_nat @ bit_se2046307713759805098atural ).

% xor_natural.transfer
thf(fact_9109_and__natural_Otransfer,axiom,
    bNF_re88643428490162567atural @ code_pcr_natural @ ( bNF_re3704215830270325841atural @ code_pcr_natural @ code_pcr_natural ) @ bit_se727722235901077358nd_nat @ bit_se2773287842338716102atural ).

% and_natural.transfer
thf(fact_9110_bit__natural_Otransfer,axiom,
    ( bNF_re2895713572246619415_nat_o @ code_pcr_natural
    @ ^ [Y: nat > $o,Z: nat > $o] : ( Y = Z )
    @ bit_se1148574629649215175it_nat
    @ bit_se8040316288895769887atural ) ).

% bit_natural.transfer
thf(fact_9111_mask__natural_Otransfer,axiom,
    ( bNF_re2605341659958429177atural
    @ ^ [Y: nat,Z: nat] : ( Y = Z )
    @ code_pcr_natural
    @ bit_se2002935070580805687sk_nat
    @ bit_se943457434206027407atural ) ).

% mask_natural.transfer
thf(fact_9112_push__bit__natural_Otransfer,axiom,
    ( bNF_re3692170009638431535atural
    @ ^ [Y: nat,Z: nat] : ( Y = Z )
    @ ( bNF_re3704215830270325841atural @ code_pcr_natural @ code_pcr_natural )
    @ bit_se547839408752420682it_nat
    @ bit_se6611745700429515170atural ) ).

% push_bit_natural.transfer
thf(fact_9113_drop__bit__natural_Otransfer,axiom,
    ( bNF_re3692170009638431535atural
    @ ^ [Y: nat,Z: nat] : ( Y = Z )
    @ ( bNF_re3704215830270325841atural @ code_pcr_natural @ code_pcr_natural )
    @ bit_se8570568707652914677it_nat
    @ bit_se2751692689150723149atural ) ).

% drop_bit_natural.transfer
thf(fact_9114_set__bit__natural_Otransfer,axiom,
    ( bNF_re3692170009638431535atural
    @ ^ [Y: nat,Z: nat] : ( Y = Z )
    @ ( bNF_re3704215830270325841atural @ code_pcr_natural @ code_pcr_natural )
    @ bit_se7882103937844011126it_nat
    @ bit_se1617098188084679374atural ) ).

% set_bit_natural.transfer
thf(fact_9115_division__segment__natural_Otransfer,axiom,
    bNF_re3704215830270325841atural @ code_pcr_natural @ code_pcr_natural @ euclid3398187327856392827nt_nat @ euclid5112970336976722131atural ).

% division_segment_natural.transfer
thf(fact_9116_unset__bit__natural_Otransfer,axiom,
    ( bNF_re3692170009638431535atural
    @ ^ [Y: nat,Z: nat] : ( Y = Z )
    @ ( bNF_re3704215830270325841atural @ code_pcr_natural @ code_pcr_natural )
    @ bit_se4205575877204974255it_nat
    @ bit_se7083795435491715335atural ) ).

% unset_bit_natural.transfer
thf(fact_9117_flip__bit__natural_Otransfer,axiom,
    ( bNF_re3692170009638431535atural
    @ ^ [Y: nat,Z: nat] : ( Y = Z )
    @ ( bNF_re3704215830270325841atural @ code_pcr_natural @ code_pcr_natural )
    @ bit_se2161824704523386999it_nat
    @ bit_se168947363167071951atural ) ).

% flip_bit_natural.transfer
thf(fact_9118_euclidean__size__natural_Otransfer,axiom,
    ( bNF_re7333046803109187065at_nat @ code_pcr_natural
    @ ^ [Y: nat,Z: nat] : ( Y = Z )
    @ euclid4777050414544973029ze_nat
    @ euclid5200926497590043453atural ) ).

% euclidean_size_natural.transfer
thf(fact_9119_integer__of__natural_Otransfer,axiom,
    bNF_re5252274238750452962nteger @ code_pcr_natural @ code_pcr_integer @ semiri1314217659103216013at_int @ code_i5400310926305786745atural ).

% integer_of_natural.transfer
thf(fact_9120_natural__of__integer_Otransfer,axiom,
    bNF_re1831474436612530402atural @ code_pcr_integer @ code_pcr_natural @ nat2 @ code_n4118661773612635043nteger ).

% natural_of_integer.transfer
thf(fact_9121_bij__betw__Suc,axiom,
    ! [M7: set_nat,N5: set_nat] :
      ( ( bij_betw_nat_nat @ suc @ M7 @ N5 )
      = ( ( image_nat_nat @ suc @ M7 )
        = N5 ) ) ).

% bij_betw_Suc
thf(fact_9122_bij__int__decode,axiom,
    bij_betw_nat_int @ nat_int_decode @ top_top_set_nat @ top_top_set_int ).

% bij_int_decode
thf(fact_9123_bij__int__encode,axiom,
    bij_betw_int_nat @ nat_int_encode @ top_top_set_int @ top_top_set_nat ).

% bij_int_encode
thf(fact_9124_bij__prod__decode,axiom,
    bij_be8693218025023041337at_nat @ nat_prod_decode @ top_top_set_nat @ top_to4669805908274784177at_nat ).

% bij_prod_decode
thf(fact_9125_bij__list__decode,axiom,
    bij_be6293887246118711976st_nat @ nat_list_decode @ top_top_set_nat @ top_top_set_list_nat ).

% bij_list_decode
thf(fact_9126_bij__list__encode,axiom,
    bij_be8532844293280997160at_nat @ nat_list_encode @ top_top_set_list_nat @ top_top_set_nat ).

% bij_list_encode
thf(fact_9127_bij__prod__encode,axiom,
    bij_be5333170631980326235at_nat @ nat_prod_encode @ top_to4669805908274784177at_nat @ top_top_set_nat ).

% bij_prod_encode
thf(fact_9128_natLeq__on__wo__rel,axiom,
    ! [N: nat] :
      ( bNF_We3818239936649020644el_nat
      @ ( collec3392354462482085612at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [X2: nat,Y6: nat] :
              ( ( ord_less_nat @ X2 @ N )
              & ( ord_less_nat @ Y6 @ N )
              & ( ord_less_eq_nat @ X2 @ Y6 ) ) ) ) ) ).

% natLeq_on_wo_rel
thf(fact_9129_Quotient__natural,axiom,
    ( quotie743640454666494480atural
    @ ^ [Y: nat,Z: nat] : ( Y = Z )
    @ code_natural_of_nat
    @ code_nat_of_natural
    @ code_cr_natural ) ).

% Quotient_natural
thf(fact_9130_Quotient__integer,axiom,
    ( quotie2291698863146621601nteger
    @ ^ [Y: int,Z: int] : ( Y = Z )
    @ code_integer_of_int
    @ code_int_of_integer
    @ code_cr_integer ) ).

% Quotient_integer
thf(fact_9131_type__definition__integer,axiom,
    type_d8366093980585677751er_int @ code_int_of_integer @ code_integer_of_int @ top_top_set_int ).

% type_definition_integer
thf(fact_9132_wait__def,axiom,
    ( heap_Time_wait
    = ( ^ [N3: nat] :
          ( heap_T6183433275982383450t_unit
          @ ^ [H4: heap_e7401611519738050253t_unit] : ( some_P1914260805536162275it_nat @ ( produc7133225469290080770it_nat @ product_Unity @ ( produc584006145561248582it_nat @ H4 @ N3 ) ) ) ) ) ) ).

% wait_def
thf(fact_9133_uminus__unit__def,axiom,
    ( uminus2952777764628376836t_unit
    = ( ^ [Uu2: product_unit] : product_Unity ) ) ).

% uminus_unit_def
thf(fact_9134_Code__Numeral_Osub__def,axiom,
    ( code_sub
    = ( map_fu6891787308814931657nteger @ id_num @ ( map_fu8638147718074629079nteger @ id_num @ code_integer_of_int )
      @ ^ [M2: num,N3: num] : ( minus_minus_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N3 ) ) ) ) ).

% Code_Numeral.sub_def
thf(fact_9135_Heap_Oempty__def,axiom,
    ( empty
    = ( heap_e2437179713268326763t_unit
      @ ^ [Uu: typerep,Uv: nat] : nil_nat
      @ ^ [Uu: typerep,Uv: nat] : zero_zero_nat
      @ zero_zero_nat
      @ product_Unity ) ) ).

% Heap.empty_def
thf(fact_9136_Field__natLeq__on,axiom,
    ! [N: nat] :
      ( ( field_nat
        @ ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [X2: nat,Y6: nat] :
                ( ( ord_less_nat @ X2 @ N )
                & ( ord_less_nat @ Y6 @ N )
                & ( ord_less_eq_nat @ X2 @ Y6 ) ) ) ) )
      = ( collect_nat
        @ ^ [X2: nat] : ( ord_less_nat @ X2 @ N ) ) ) ).

% Field_natLeq_on
thf(fact_9137_less__nat__rel,axiom,
    ( ord_less_nat
    = ( transi2163837189807498211lp_nat
      @ ^ [M2: nat,N3: nat] :
          ( N3
          = ( suc @ M2 ) ) ) ) ).

% less_nat_rel
thf(fact_9138_natLeq__on__Well__order,axiom,
    ! [N: nat] :
      ( order_2888998067076097458on_nat
      @ ( field_nat
        @ ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [X2: nat,Y6: nat] :
                ( ( ord_less_nat @ X2 @ N )
                & ( ord_less_nat @ Y6 @ N )
                & ( ord_less_eq_nat @ X2 @ Y6 ) ) ) ) )
      @ ( collec3392354462482085612at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [X2: nat,Y6: nat] :
              ( ( ord_less_nat @ X2 @ N )
              & ( ord_less_nat @ Y6 @ N )
              & ( ord_less_eq_nat @ X2 @ Y6 ) ) ) ) ) ).

% natLeq_on_Well_order
thf(fact_9139_Quotient3__int,axiom,
    quotie6776551016481293500at_int @ intrel @ abs_Integ @ rep_Integ ).

% Quotient3_int
thf(fact_9140_natLeq__on__well__order__on,axiom,
    ! [N: nat] :
      ( order_2888998067076097458on_nat
      @ ( collect_nat
        @ ^ [X2: nat] : ( ord_less_nat @ X2 @ N ) )
      @ ( collec3392354462482085612at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [X2: nat,Y6: nat] :
              ( ( ord_less_nat @ X2 @ N )
              & ( ord_less_nat @ Y6 @ N )
              & ( ord_less_eq_nat @ X2 @ Y6 ) ) ) ) ) ).

% natLeq_on_well_order_on
thf(fact_9141_type__definition__natural,axiom,
    type_d4410041424927559462al_nat @ code_nat_of_natural @ code_natural_of_nat @ top_top_set_nat ).

% type_definition_natural
thf(fact_9142_less__int__code_I3_J,axiom,
    ! [L: num] :
      ~ ( ord_less_int @ zero_zero_int @ ( neg @ L ) ) ).

% less_int_code(3)
thf(fact_9143_less__int__code_I7_J,axiom,
    ! [K: num] : ( ord_less_int @ ( neg @ K ) @ zero_zero_int ) ).

% less_int_code(7)
thf(fact_9144_less__eq__int__code_I7_J,axiom,
    ! [K: num] : ( ord_less_eq_int @ ( neg @ K ) @ zero_zero_int ) ).

% less_eq_int_code(7)
thf(fact_9145_less__eq__int__code_I3_J,axiom,
    ! [L: num] :
      ~ ( ord_less_eq_int @ zero_zero_int @ ( neg @ L ) ) ).

% less_eq_int_code(3)
thf(fact_9146_nat__code_I1_J,axiom,
    ! [K: num] :
      ( ( nat2 @ ( neg @ K ) )
      = zero_zero_nat ) ).

% nat_code(1)
thf(fact_9147_plus__int__code_I6_J,axiom,
    ! [M: num,N: num] :
      ( ( plus_plus_int @ ( neg @ M ) @ ( neg @ N ) )
      = ( neg @ ( plus_plus_num @ M @ N ) ) ) ).

% plus_int_code(6)
thf(fact_9148_less__eq__int__code_I9_J,axiom,
    ! [K: num,L: num] :
      ( ( ord_less_eq_int @ ( neg @ K ) @ ( neg @ L ) )
      = ( ord_less_eq_num @ L @ K ) ) ).

% less_eq_int_code(9)
thf(fact_9149_less__int__code_I9_J,axiom,
    ! [K: num,L: num] :
      ( ( ord_less_int @ ( neg @ K ) @ ( neg @ L ) )
      = ( ord_less_num @ L @ K ) ) ).

% less_int_code(9)
thf(fact_9150_Int_Osub__code_I4_J,axiom,
    ! [N: num] :
      ( ( sub @ one @ ( bit0 @ N ) )
      = ( neg @ ( bitM @ N ) ) ) ).

% Int.sub_code(4)
thf(fact_9151_Int_Osub__code_I5_J,axiom,
    ! [N: num] :
      ( ( sub @ one @ ( bit1 @ N ) )
      = ( neg @ ( bit0 @ N ) ) ) ).

% Int.sub_code(5)
thf(fact_9152_Int_Osub__code_I1_J,axiom,
    ( ( sub @ one @ one )
    = zero_zero_int ) ).

% Int.sub_code(1)
thf(fact_9153_Int_Osub__def,axiom,
    ( sub
    = ( ^ [M2: num,N3: num] : ( minus_minus_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N3 ) ) ) ) ).

% Int.sub_def
thf(fact_9154_minus__int__code_I6_J,axiom,
    ! [M: num,N: num] :
      ( ( minus_minus_int @ ( neg @ M ) @ ( neg @ N ) )
      = ( sub @ N @ M ) ) ).

% minus_int_code(6)
thf(fact_9155_Int_Osub__code_I9_J,axiom,
    ! [M: num,N: num] :
      ( ( sub @ ( bit0 @ M ) @ ( bit1 @ N ) )
      = ( minus_minus_int @ ( dup @ ( sub @ M @ N ) ) @ one_one_int ) ) ).

% Int.sub_code(9)
thf(fact_9156_Int_Osub__code_I8_J,axiom,
    ! [M: num,N: num] :
      ( ( sub @ ( bit1 @ M ) @ ( bit0 @ N ) )
      = ( plus_plus_int @ ( dup @ ( sub @ M @ N ) ) @ one_one_int ) ) ).

% Int.sub_code(8)
thf(fact_9157_Int_Odup__code_I1_J,axiom,
    ( ( dup @ zero_zero_int )
    = zero_zero_int ) ).

% Int.dup_code(1)
thf(fact_9158_Int_Odup__def,axiom,
    ( dup
    = ( ^ [K3: int] : ( plus_plus_int @ K3 @ K3 ) ) ) ).

% Int.dup_def
thf(fact_9159_Int_Odup__code_I3_J,axiom,
    ! [N: num] :
      ( ( dup @ ( neg @ N ) )
      = ( neg @ ( bit0 @ N ) ) ) ).

% Int.dup_code(3)
thf(fact_9160_Int_Osub__code_I6_J,axiom,
    ! [M: num,N: num] :
      ( ( sub @ ( bit0 @ M ) @ ( bit0 @ N ) )
      = ( dup @ ( sub @ M @ N ) ) ) ).

% Int.sub_code(6)
thf(fact_9161_Int_Osub__code_I7_J,axiom,
    ! [M: num,N: num] :
      ( ( sub @ ( bit1 @ M ) @ ( bit1 @ N ) )
      = ( dup @ ( sub @ M @ N ) ) ) ).

% Int.sub_code(7)
thf(fact_9162_Int_Osub__code_I2_J,axiom,
    ! [M: num] :
      ( ( sub @ ( bit0 @ M ) @ one )
      = ( pos @ ( bitM @ M ) ) ) ).

% Int.sub_code(2)
thf(fact_9163_Int_Osub__code_I3_J,axiom,
    ! [M: num] :
      ( ( sub @ ( bit1 @ M ) @ one )
      = ( pos @ ( bit0 @ M ) ) ) ).

% Int.sub_code(3)
thf(fact_9164_less__int__code_I8_J,axiom,
    ! [K: num,L: num] : ( ord_less_int @ ( neg @ K ) @ ( pos @ L ) ) ).

% less_int_code(8)
thf(fact_9165_less__int__code_I6_J,axiom,
    ! [K: num,L: num] :
      ~ ( ord_less_int @ ( pos @ K ) @ ( neg @ L ) ) ).

% less_int_code(6)
thf(fact_9166_less__eq__int__code_I6_J,axiom,
    ! [K: num,L: num] :
      ~ ( ord_less_eq_int @ ( pos @ K ) @ ( neg @ L ) ) ).

% less_eq_int_code(6)
thf(fact_9167_less__eq__int__code_I8_J,axiom,
    ! [K: num,L: num] : ( ord_less_eq_int @ ( neg @ K ) @ ( pos @ L ) ) ).

% less_eq_int_code(8)
thf(fact_9168_Int_ONeg__def,axiom,
    ( neg
    = ( ^ [N3: num] : ( uminus_uminus_int @ ( pos @ N3 ) ) ) ) ).

% Int.Neg_def
thf(fact_9169_uminus__int__code_I2_J,axiom,
    ! [M: num] :
      ( ( uminus_uminus_int @ ( pos @ M ) )
      = ( neg @ M ) ) ).

% uminus_int_code(2)
thf(fact_9170_uminus__int__code_I3_J,axiom,
    ! [M: num] :
      ( ( uminus_uminus_int @ ( neg @ M ) )
      = ( pos @ M ) ) ).

% uminus_int_code(3)
thf(fact_9171_nat__code_I3_J,axiom,
    ! [K: num] :
      ( ( nat2 @ ( pos @ K ) )
      = ( nat_of_num @ K ) ) ).

% nat_code(3)
thf(fact_9172_times__int__code_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( times_times_int @ ( pos @ M ) @ ( pos @ N ) )
      = ( pos @ ( times_times_num @ M @ N ) ) ) ).

% times_int_code(3)
thf(fact_9173_Int_OPos__def,axiom,
    pos = numeral_numeral_int ).

% Int.Pos_def
thf(fact_9174_less__eq__int__code_I2_J,axiom,
    ! [L: num] : ( ord_less_eq_int @ zero_zero_int @ ( pos @ L ) ) ).

% less_eq_int_code(2)
thf(fact_9175_less__eq__int__code_I4_J,axiom,
    ! [K: num] :
      ~ ( ord_less_eq_int @ ( pos @ K ) @ zero_zero_int ) ).

% less_eq_int_code(4)
thf(fact_9176_one__int__code,axiom,
    ( one_one_int
    = ( pos @ one ) ) ).

% one_int_code
thf(fact_9177_plus__int__code_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( plus_plus_int @ ( pos @ M ) @ ( pos @ N ) )
      = ( pos @ ( plus_plus_num @ M @ N ) ) ) ).

% plus_int_code(3)
thf(fact_9178_less__int__code_I4_J,axiom,
    ! [K: num] :
      ~ ( ord_less_int @ ( pos @ K ) @ zero_zero_int ) ).

% less_int_code(4)
thf(fact_9179_less__int__code_I2_J,axiom,
    ! [L: num] : ( ord_less_int @ zero_zero_int @ ( pos @ L ) ) ).

% less_int_code(2)
thf(fact_9180_less__eq__int__code_I5_J,axiom,
    ! [K: num,L: num] :
      ( ( ord_less_eq_int @ ( pos @ K ) @ ( pos @ L ) )
      = ( ord_less_eq_num @ K @ L ) ) ).

% less_eq_int_code(5)
thf(fact_9181_less__int__code_I5_J,axiom,
    ! [K: num,L: num] :
      ( ( ord_less_int @ ( pos @ K ) @ ( pos @ L ) )
      = ( ord_less_num @ K @ L ) ) ).

% less_int_code(5)
thf(fact_9182_minus__int__code_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( minus_minus_int @ ( pos @ M ) @ ( pos @ N ) )
      = ( sub @ M @ N ) ) ).

% minus_int_code(3)
thf(fact_9183_Int_Odup__code_I2_J,axiom,
    ! [N: num] :
      ( ( dup @ ( pos @ N ) )
      = ( pos @ ( bit0 @ N ) ) ) ).

% Int.dup_code(2)
thf(fact_9184_minus__int__code_I4_J,axiom,
    ! [M: num,N: num] :
      ( ( minus_minus_int @ ( pos @ M ) @ ( neg @ N ) )
      = ( pos @ ( plus_plus_num @ M @ N ) ) ) ).

% minus_int_code(4)
thf(fact_9185_minus__int__code_I5_J,axiom,
    ! [M: num,N: num] :
      ( ( minus_minus_int @ ( neg @ M ) @ ( pos @ N ) )
      = ( neg @ ( plus_plus_num @ M @ N ) ) ) ).

% minus_int_code(5)
thf(fact_9186_times__int__code_I4_J,axiom,
    ! [M: num,N: num] :
      ( ( times_times_int @ ( pos @ M ) @ ( neg @ N ) )
      = ( neg @ ( times_times_num @ M @ N ) ) ) ).

% times_int_code(4)
thf(fact_9187_times__int__code_I5_J,axiom,
    ! [M: num,N: num] :
      ( ( times_times_int @ ( neg @ M ) @ ( pos @ N ) )
      = ( neg @ ( times_times_num @ M @ N ) ) ) ).

% times_int_code(5)
thf(fact_9188_times__int__code_I6_J,axiom,
    ! [M: num,N: num] :
      ( ( times_times_int @ ( neg @ M ) @ ( neg @ N ) )
      = ( pos @ ( times_times_num @ M @ N ) ) ) ).

% times_int_code(6)
thf(fact_9189_plus__int__code_I5_J,axiom,
    ! [M: num,N: num] :
      ( ( plus_plus_int @ ( neg @ M ) @ ( pos @ N ) )
      = ( sub @ N @ M ) ) ).

% plus_int_code(5)
thf(fact_9190_plus__int__code_I4_J,axiom,
    ! [M: num,N: num] :
      ( ( plus_plus_int @ ( pos @ M ) @ ( neg @ N ) )
      = ( sub @ M @ N ) ) ).

% plus_int_code(4)
thf(fact_9191_GreatestI__nat,axiom,
    ! [P: nat > $o,K: nat,B: nat] :
      ( ( P @ K )
     => ( ! [Y4: nat] :
            ( ( P @ Y4 )
           => ( ord_less_eq_nat @ Y4 @ B ) )
       => ( P @ ( order_Greatest_nat @ P ) ) ) ) ).

% GreatestI_nat
thf(fact_9192_Greatest__le__nat,axiom,
    ! [P: nat > $o,K: nat,B: nat] :
      ( ( P @ K )
     => ( ! [Y4: nat] :
            ( ( P @ Y4 )
           => ( ord_less_eq_nat @ Y4 @ B ) )
       => ( ord_less_eq_nat @ K @ ( order_Greatest_nat @ P ) ) ) ) ).

% Greatest_le_nat
thf(fact_9193_GreatestI__ex__nat,axiom,
    ! [P: nat > $o,B: nat] :
      ( ? [X_1: nat] : ( P @ X_1 )
     => ( ! [Y4: nat] :
            ( ( P @ Y4 )
           => ( ord_less_eq_nat @ Y4 @ B ) )
       => ( P @ ( order_Greatest_nat @ P ) ) ) ) ).

% GreatestI_ex_nat
thf(fact_9194_sum__encode__def,axiom,
    ( nat_sum_encode
    = ( sum_ca6763686470577984908at_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      @ ^ [B2: nat] : ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) ) ) ).

% sum_encode_def
thf(fact_9195_inj__sum__encode,axiom,
    ! [A3: set_Sum_sum_nat_nat] : ( inj_on6343450744447823682at_nat @ nat_sum_encode @ A3 ) ).

% inj_sum_encode
thf(fact_9196_sum__encode__eq,axiom,
    ! [X: sum_sum_nat_nat,Y3: sum_sum_nat_nat] :
      ( ( ( nat_sum_encode @ X )
        = ( nat_sum_encode @ Y3 ) )
      = ( X = Y3 ) ) ).

% sum_encode_eq
thf(fact_9197_bij__sum__encode,axiom,
    bij_be5432664580149595207at_nat @ nat_sum_encode @ top_to6661820994512907621at_nat @ top_top_set_nat ).

% bij_sum_encode
thf(fact_9198_surj__sum__encode,axiom,
    ( ( image_1320371278474632150at_nat @ nat_sum_encode @ top_to6661820994512907621at_nat )
    = top_top_set_nat ) ).

% surj_sum_encode
thf(fact_9199_int__encode__def,axiom,
    ( nat_int_encode
    = ( ^ [I4: int] : ( nat_sum_encode @ ( if_Sum_sum_nat_nat @ ( ord_less_eq_int @ zero_zero_int @ I4 ) @ ( sum_Inl_nat_nat @ ( nat2 @ I4 ) ) @ ( sum_Inr_nat_nat @ ( nat2 @ ( minus_minus_int @ ( uminus_uminus_int @ I4 ) @ one_one_int ) ) ) ) ) ) ) ).

% int_encode_def
thf(fact_9200_le__sum__encode__Inl,axiom,
    ! [X: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ X @ Y3 )
     => ( ord_less_eq_nat @ X @ ( nat_sum_encode @ ( sum_Inl_nat_nat @ Y3 ) ) ) ) ).

% le_sum_encode_Inl
thf(fact_9201_le__sum__encode__Inr,axiom,
    ! [X: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ X @ Y3 )
     => ( ord_less_eq_nat @ X @ ( nat_sum_encode @ ( sum_Inr_nat_nat @ Y3 ) ) ) ) ).

% le_sum_encode_Inr
thf(fact_9202_sum__decode__def,axiom,
    ( nat_sum_decode
    = ( ^ [N3: nat] : ( if_Sum_sum_nat_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ ( sum_Inl_nat_nat @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( sum_Inr_nat_nat @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% sum_decode_def
thf(fact_9203_sum__decode__inverse,axiom,
    ! [N: nat] :
      ( ( nat_sum_encode @ ( nat_sum_decode @ N ) )
      = N ) ).

% sum_decode_inverse
thf(fact_9204_sum__encode__inverse,axiom,
    ! [X: sum_sum_nat_nat] :
      ( ( nat_sum_decode @ ( nat_sum_encode @ X ) )
      = X ) ).

% sum_encode_inverse
thf(fact_9205_sum__decode__eq,axiom,
    ! [X: nat,Y3: nat] :
      ( ( ( nat_sum_decode @ X )
        = ( nat_sum_decode @ Y3 ) )
      = ( X = Y3 ) ) ).

% sum_decode_eq
thf(fact_9206_inj__sum__decode,axiom,
    ! [A3: set_nat] : ( inj_on5701776251185195458at_nat @ nat_sum_decode @ A3 ) ).

% inj_sum_decode
thf(fact_9207_surj__sum__decode,axiom,
    ( ( image_678696785212003926at_nat @ nat_sum_decode @ top_top_set_nat )
    = top_to6661820994512907621at_nat ) ).

% surj_sum_decode
thf(fact_9208_bij__sum__decode,axiom,
    bij_be4790990086886966983at_nat @ nat_sum_decode @ top_top_set_nat @ top_to6661820994512907621at_nat ).

% bij_sum_decode
thf(fact_9209_int__decode__def,axiom,
    ( nat_int_decode
    = ( ^ [N3: nat] :
          ( sum_ca7763040182479039464nt_nat @ semiri1314217659103216013at_int
          @ ^ [B2: nat] : ( minus_minus_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ B2 ) ) @ one_one_int )
          @ ( nat_sum_decode @ N3 ) ) ) ) ).

% int_decode_def
thf(fact_9210_Abs__int__cases,axiom,
    ! [X: int] :
      ~ ! [Y4: set_Pr1261947904930325089at_nat] :
          ( ( X
            = ( abs_int @ Y4 ) )
         => ~ ( member2643936169264416010at_nat @ Y4
              @ ( collec5514110066124741708at_nat
                @ ^ [C3: set_Pr1261947904930325089at_nat] :
                  ? [X2: product_prod_nat_nat] :
                    ( ( intrel @ X2 @ X2 )
                    & ( C3
                      = ( collec3392354462482085612at_nat @ ( intrel @ X2 ) ) ) ) ) ) ) ).

% Abs_int_cases
thf(fact_9211_Abs__int__induct,axiom,
    ! [P: int > $o,X: int] :
      ( ! [Y4: set_Pr1261947904930325089at_nat] :
          ( ( member2643936169264416010at_nat @ Y4
            @ ( collec5514110066124741708at_nat
              @ ^ [C3: set_Pr1261947904930325089at_nat] :
                ? [X2: product_prod_nat_nat] :
                  ( ( intrel @ X2 @ X2 )
                  & ( C3
                    = ( collec3392354462482085612at_nat @ ( intrel @ X2 ) ) ) ) ) )
         => ( P @ ( abs_int @ Y4 ) ) )
     => ( P @ X ) ) ).

% Abs_int_induct
thf(fact_9212_Abs__int__inject,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
      ( ( member2643936169264416010at_nat @ X
        @ ( collec5514110066124741708at_nat
          @ ^ [C3: set_Pr1261947904930325089at_nat] :
            ? [X2: product_prod_nat_nat] :
              ( ( intrel @ X2 @ X2 )
              & ( C3
                = ( collec3392354462482085612at_nat @ ( intrel @ X2 ) ) ) ) ) )
     => ( ( member2643936169264416010at_nat @ Y3
          @ ( collec5514110066124741708at_nat
            @ ^ [C3: set_Pr1261947904930325089at_nat] :
              ? [X2: product_prod_nat_nat] :
                ( ( intrel @ X2 @ X2 )
                & ( C3
                  = ( collec3392354462482085612at_nat @ ( intrel @ X2 ) ) ) ) ) )
       => ( ( ( abs_int @ X )
            = ( abs_int @ Y3 ) )
          = ( X = Y3 ) ) ) ) ).

% Abs_int_inject
thf(fact_9213_Abs__Integ__def,axiom,
    ( abs_Integ
    = ( quot_a8334514980957697477at_int @ intrel @ abs_int ) ) ).

% Abs_Integ_def
thf(fact_9214_Abs__int__inverse,axiom,
    ! [Y3: set_Pr1261947904930325089at_nat] :
      ( ( member2643936169264416010at_nat @ Y3
        @ ( collec5514110066124741708at_nat
          @ ^ [C3: set_Pr1261947904930325089at_nat] :
            ? [X2: product_prod_nat_nat] :
              ( ( intrel @ X2 @ X2 )
              & ( C3
                = ( collec3392354462482085612at_nat @ ( intrel @ X2 ) ) ) ) ) )
     => ( ( rep_int @ ( abs_int @ Y3 ) )
        = Y3 ) ) ).

% Abs_int_inverse
thf(fact_9215_Rep__int__inject,axiom,
    ! [X: int,Y3: int] :
      ( ( ( rep_int @ X )
        = ( rep_int @ Y3 ) )
      = ( X = Y3 ) ) ).

% Rep_int_inject
thf(fact_9216_Rep__int__inverse,axiom,
    ! [X: int] :
      ( ( abs_int @ ( rep_int @ X ) )
      = X ) ).

% Rep_int_inverse
thf(fact_9217_Rep__int,axiom,
    ! [X: int] :
      ( member2643936169264416010at_nat @ ( rep_int @ X )
      @ ( collec5514110066124741708at_nat
        @ ^ [C3: set_Pr1261947904930325089at_nat] :
          ? [X2: product_prod_nat_nat] :
            ( ( intrel @ X2 @ X2 )
            & ( C3
              = ( collec3392354462482085612at_nat @ ( intrel @ X2 ) ) ) ) ) ) ).

% Rep_int
thf(fact_9218_Rep__int__cases,axiom,
    ! [Y3: set_Pr1261947904930325089at_nat] :
      ( ( member2643936169264416010at_nat @ Y3
        @ ( collec5514110066124741708at_nat
          @ ^ [C3: set_Pr1261947904930325089at_nat] :
            ? [X2: product_prod_nat_nat] :
              ( ( intrel @ X2 @ X2 )
              & ( C3
                = ( collec3392354462482085612at_nat @ ( intrel @ X2 ) ) ) ) ) )
     => ~ ! [X3: int] :
            ( Y3
           != ( rep_int @ X3 ) ) ) ).

% Rep_int_cases
thf(fact_9219_Rep__int__induct,axiom,
    ! [Y3: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
      ( ( member2643936169264416010at_nat @ Y3
        @ ( collec5514110066124741708at_nat
          @ ^ [C3: set_Pr1261947904930325089at_nat] :
            ? [X2: product_prod_nat_nat] :
              ( ( intrel @ X2 @ X2 )
              & ( C3
                = ( collec3392354462482085612at_nat @ ( intrel @ X2 ) ) ) ) ) )
     => ( ! [X3: int] : ( P @ ( rep_int @ X3 ) )
       => ( P @ Y3 ) ) ) ).

% Rep_int_induct
thf(fact_9220_Rep__Integ__def,axiom,
    ( rep_Integ
    = ( quot_r1512469584000298464at_nat @ rep_int ) ) ).

% Rep_Integ_def
thf(fact_9221_type__definition__int,axiom,
    ( type_d8752208705193531015at_nat @ rep_int @ abs_int
    @ ( collec5514110066124741708at_nat
      @ ^ [C3: set_Pr1261947904930325089at_nat] :
        ? [X2: product_prod_nat_nat] :
          ( ( intrel @ X2 @ X2 )
          & ( C3
            = ( collec3392354462482085612at_nat @ ( intrel @ X2 ) ) ) ) ) ) ).

% type_definition_int
thf(fact_9222_Domainp__cr__rat,axiom,
    ( ( domain9213661015745956397nt_rat @ pcr_rat )
    = ( ^ [X2: product_prod_int_int] :
          ( ( product_snd_int_int @ X2 )
         != zero_zero_int ) ) ) ).

% Domainp_cr_rat
thf(fact_9223_typerep_Osize__neq,axiom,
    ! [X: typerep] :
      ( ( size_size_typerep @ X )
     != zero_zero_nat ) ).

% typerep.size_neq
thf(fact_9224_typerep_Osize_I2_J,axiom,
    ! [X1: literal,X22: list_typerep] :
      ( ( size_size_typerep @ ( typerep2 @ X1 @ X22 ) )
      = ( plus_plus_nat @ ( size_list_typerep @ size_size_typerep @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).

% typerep.size(2)
thf(fact_9225_typerep_Osize__gen,axiom,
    ! [X1: literal,X22: list_typerep] :
      ( ( size_typerep @ ( typerep2 @ X1 @ X22 ) )
      = ( plus_plus_nat @ ( size_list_typerep @ size_typerep @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).

% typerep.size_gen
thf(fact_9226_Fract__def,axiom,
    ( fract
    = ( map_fu7831380289885515383nt_rat @ id_int @ ( map_fu3424225382358772769nt_rat @ id_int @ abs_Rat )
      @ ^ [A2: int,B2: int] : ( if_Pro3027730157355071871nt_int @ ( B2 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ A2 @ B2 ) ) ) ) ).

% Fract_def
thf(fact_9227_of__int__eq__id,axiom,
    ring_1_of_int_int = id_int ).

% of_int_eq_id
thf(fact_9228_natural_Odomain,axiom,
    ! [P: nat > $o] :
      ( ( ( domainp_nat_nat
          @ ^ [Y: nat,Z: nat] : ( Y = Z ) )
        = P )
     => ( ( domain8128281535334753374atural @ code_pcr_natural )
        = P ) ) ).

% natural.domain
thf(fact_9229_integer_Odomain,axiom,
    ! [P: int > $o] :
      ( ( ( domainp_int_int
          @ ^ [Y: int,Z: int] : ( Y = Z ) )
        = P )
     => ( ( domain452967906960104687nteger @ code_pcr_integer )
        = P ) ) ).

% integer.domain
thf(fact_9230_int_Odomain,axiom,
    ! [DR1: nat > $o,DR2: nat > $o] :
      ( ( ( domainp_nat_nat
          @ ^ [Y: nat,Z: nat] : ( Y = Z ) )
        = DR1 )
     => ( ( ( domainp_nat_nat
            @ ^ [Y: nat,Z: nat] : ( Y = Z ) )
          = DR2 )
       => ( ( domain7287725470393535433at_int @ pcr_int )
          = ( basic_570253317919918037at_nat @ DR1 @ DR2 ) ) ) ) ).

% int.domain

% Helper facts (39)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
    ! [X: int,Y3: int] :
      ( ( if_int @ $false @ X @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
    ! [X: int,Y3: int] :
      ( ( if_int @ $true @ X @ Y3 )
      = X ) ).

thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y3: nat] :
      ( ( if_nat @ $false @ X @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y3: nat] :
      ( ( if_nat @ $true @ X @ Y3 )
      = X ) ).

thf(help_If_2_1_If_001t__Num__Onum_T,axiom,
    ! [X: num,Y3: num] :
      ( ( if_num @ $false @ X @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Num__Onum_T,axiom,
    ! [X: num,Y3: num] :
      ( ( if_num @ $true @ X @ Y3 )
      = X ) ).

thf(help_If_2_1_If_001t__Rat__Orat_T,axiom,
    ! [X: rat,Y3: rat] :
      ( ( if_rat @ $false @ X @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Rat__Orat_T,axiom,
    ! [X: rat,Y3: rat] :
      ( ( if_rat @ $true @ X @ Y3 )
      = X ) ).

thf(help_If_2_1_If_001t__Code____Numeral__Ointeger_T,axiom,
    ! [X: code_integer,Y3: code_integer] :
      ( ( if_Code_integer @ $false @ X @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Code____Numeral__Ointeger_T,axiom,
    ! [X: code_integer,Y3: code_integer] :
      ( ( if_Code_integer @ $true @ X @ Y3 )
      = X ) ).

thf(help_If_2_1_If_001t__Code____Numeral__Onatural_T,axiom,
    ! [X: code_natural,Y3: code_natural] :
      ( ( if_Code_natural @ $false @ X @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Code____Numeral__Onatural_T,axiom,
    ! [X: code_natural,Y3: code_natural] :
      ( ( if_Code_natural @ $true @ X @ Y3 )
      = X ) ).

thf(help_If_2_1_If_001t__Set__Oset_It__Int__Oint_J_T,axiom,
    ! [X: set_int,Y3: set_int] :
      ( ( if_set_int @ $false @ X @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Set__Oset_It__Int__Oint_J_T,axiom,
    ! [X: set_int,Y3: set_int] :
      ( ( if_set_int @ $true @ X @ Y3 )
      = X ) ).

thf(help_If_2_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
    ! [X: set_nat,Y3: set_nat] :
      ( ( if_set_nat @ $false @ X @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
    ! [X: set_nat,Y3: set_nat] :
      ( ( if_set_nat @ $true @ X @ Y3 )
      = X ) ).

thf(help_If_2_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
    ! [X: list_int,Y3: list_int] :
      ( ( if_list_int @ $false @ X @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
    ! [X: list_int,Y3: list_int] :
      ( ( if_list_int @ $true @ X @ Y3 )
      = X ) ).

thf(help_If_2_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [X: list_nat,Y3: list_nat] :
      ( ( if_list_nat @ $false @ X @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [X: list_nat,Y3: list_nat] :
      ( ( if_list_nat @ $true @ X @ Y3 )
      = X ) ).

thf(help_If_2_1_If_001_062_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X: int > int,Y3: int > int] :
      ( ( if_int_int @ $false @ X @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001_062_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X: int > int,Y3: int > int] :
      ( ( if_int_int @ $true @ X @ Y3 )
      = X ) ).

thf(help_If_2_1_If_001t__Option__Ooption_It__Num__Onum_J_T,axiom,
    ! [X: option_num,Y3: option_num] :
      ( ( if_option_num @ $false @ X @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Option__Ooption_It__Num__Onum_J_T,axiom,
    ! [X: option_num,Y3: option_num] :
      ( ( if_option_num @ $true @ X @ Y3 )
      = X ) ).

thf(help_If_2_1_If_001t__Heap____Time____Monad__OHeap_It__Nat__Onat_J_T,axiom,
    ! [X: heap_Time_Heap_nat,Y3: heap_Time_Heap_nat] :
      ( ( if_Hea2662716070787841314ap_nat @ $false @ X @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Heap____Time____Monad__OHeap_It__Nat__Onat_J_T,axiom,
    ! [X: heap_Time_Heap_nat,Y3: heap_Time_Heap_nat] :
      ( ( if_Hea2662716070787841314ap_nat @ $true @ X @ Y3 )
      = X ) ).

thf(help_If_2_1_If_001t__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X: sum_sum_nat_nat,Y3: sum_sum_nat_nat] :
      ( ( if_Sum_sum_nat_nat @ $false @ X @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X: sum_sum_nat_nat,Y3: sum_sum_nat_nat] :
      ( ( if_Sum_sum_nat_nat @ $true @ X @ Y3 )
      = X ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X: product_prod_int_int,Y3: product_prod_int_int] :
      ( ( if_Pro3027730157355071871nt_int @ $false @ X @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X: product_prod_int_int,Y3: product_prod_int_int] :
      ( ( if_Pro3027730157355071871nt_int @ $true @ X @ Y3 )
      = X ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X: product_prod_nat_nat,Y3: product_prod_nat_nat] :
      ( ( if_Pro6206227464963214023at_nat @ $false @ X @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X: product_prod_nat_nat,Y3: product_prod_nat_nat] :
      ( ( if_Pro6206227464963214023at_nat @ $true @ X @ Y3 )
      = X ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_T,axiom,
    ! [X: produc6271795597528267376eger_o,Y3: produc6271795597528267376eger_o] :
      ( ( if_Pro5737122678794959658eger_o @ $false @ X @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_T,axiom,
    ! [X: produc6271795597528267376eger_o,Y3: produc6271795597528267376eger_o] :
      ( ( if_Pro5737122678794959658eger_o @ $true @ X @ Y3 )
      = X ) ).

thf(help_If_2_1_If_001_062_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
    ! [X: code_integer > code_integer,Y3: code_integer > code_integer] :
      ( ( if_Cod4779417660136461971nteger @ $false @ X @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001_062_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
    ! [X: code_integer > code_integer,Y3: code_integer > code_integer] :
      ( ( if_Cod4779417660136461971nteger @ $true @ X @ Y3 )
      = X ) ).

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,
    ! [X: produc8923325533196201883nteger,Y3: produc8923325533196201883nteger] :
      ( ( if_Pro6119634080678213985nteger @ $false @ X @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
    ! [X: produc8923325533196201883nteger,Y3: produc8923325533196201883nteger] :
      ( ( if_Pro6119634080678213985nteger @ $true @ X @ Y3 )
      = X ) ).

% Conjectures (2)
thf(conj_0,hypothesis,
    $true ).

thf(conj_1,conjecture,
    thesis ).

%------------------------------------------------------------------------------